A MULTI-PRODUCT, MULTI-FACTOR THERMAL INACTIVATION MODEL FOR
SALMONELLA IN MEAT AND POULTRY PRODUCTS

By

María Isabel Tenorio Bernal

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

MASTER OF SCIENCE

Biosystems Engineering

2012

ABSTRACT
A MULTI-PRODUCT, MULTI-FACTOR THERMAL INACTIVATION MODEL FOR
SALMONELLA IN TURKEY, BEEF, PORK, AND CHICKEN
By
María Isabel Tenorio Bernal
No current modeling tool accounts for all the necessary factors to validate a thermal
process for ready-to-eat (RTE) meat and poultry; some do not address Salmonella and/or
important product/process attributes, and most have not been validated against industry-relevant
data. Therefore, the objective of this project was to develop a multi-product, multi-factor thermal
inactivation model for Salmonella in meat and poultry products. First, the effect of sublethal
thermal injury on subsequent bacterial heat resistance was quantified by expanding the
capabilities of a previously published path-dependent Salmonella inactivation model. Salmonella
inoculated ground turkey, beef, and pork samples were subjected to multiple non-isothermal
treatments. The resulting path-dependent model was validated against equivalent data, showing
error reductions of 63 to 82%, relative to the state-dependent model, thus confirming the
importance of accounting for sublethal injury in inactivation models. In the second part, thermal
inactivation data for Salmonella in turkey, beef, and pork were selected from published sources
(nobs=411, 764, and 446 for each, respectively) and used to parameterize various versions of a
multi-product, multi-factor model, using ordinary least squares and mixed-effects statistical
methods. Validated against industry-relevant data, most models performed favorably when
considering fat content, sublethal injury, and muscle structure. Overall, this project illustrated the
current difficulties and positive outcomes of pooling thermal inactivation data from different
sources, parameterizing models with them, and validating them against industry-relevant data.

ACKNOWLEDGEMENTS
This project could not have been realized without the extensive support and expertise
from my major advisor, Dr. Bradley Marks. I thank him for pushing me to grow and to improve
my problem-solving and writing skills. Special thanks to my committee members: Dr. Kirk
Dolan for always taking the time to assist with my programming and statistical problems; and
Dr. Elliot Ryser and Dr. Alden Booren for their specialized knowledge relevant to my studies.
A special recognition to all my professors for sharing their time, knowledge, and
experience which positively contributed to my academic growth and understanding. Thank you
to CSTAT, members of Dr. Marks’ research team, especially those who contributed to
generating the vast amount of data used for validation of my work, and graduate student
colleagues for their contributions to this project.
To my Michigan sisters and US family, who have always provided support and the homeaway-from-home environment that allowed me to complete this undertaking.
And finally, to my father, mother, and sister, whose support and love was always heartfelt
regardless of the physical distance between us.

Thank you all.

This material is based upon work supported by the Cooperative State Research,
Education, and Extension Service (now the National Institute of Food and Agriculture), U.S.
Department of Agriculture, under Award No. 2007-51110-18247, by the American Meat
Institute Foundation, and by the National Pork Board.
iii

TABLE OF CONTENTS

LIST OF TABLES……………………………………………………………………….........VII
LIST OF FIGURES ..................................................................................................................... X
1.

INTRODUCTION ................................................................................................................. 1
1.1
Background/problem statement........................................................................................ 1
1.2
Goals and objectives......................................................................................................... 2

2.

LITERATURE REVIEW ..................................................................................................... 4
2.1
Salmonella and foodborne illness .................................................................................... 4
2.2
Federal regulations regarding heat processing in RTE products ..................................... 4
2.2.1
Safe Harbors ............................................................................................................. 5
2.2.2
End-point lethality .................................................................................................... 5
2.3
Factors affecting bacterial inactivation ............................................................................ 6
2.3.1
Temperature .............................................................................................................. 8
2.3.2
Fat ............................................................................................................................. 8
2.3.3
Muscle structure........................................................................................................ 9
2.3.4
Media moisture content............................................................................................. 9
2.3.5
Sublethal injury ....................................................................................................... 10
2.3.6
Inactivation media (liquid vs. meat and species) .................................................... 12
2.4
Quantifying bacterial inactivation .................................................................................. 13
2.4.1
Mathematical models .............................................................................................. 13
2.4.1.1
Primary models ................................................................................................ 15
2.4.1.2
Secondary models ............................................................................................ 18
2.4.1.3
Tertiary models ................................................................................................ 21
2.4.2
Challenge studies .................................................................................................... 25

3.

MODELING SUBLETHAL THERMAL INJURY ......................................................... 26
3.1
Introduction .................................................................................................................... 26
3.2
Materials and Methods ................................................................................................... 28
3.2.1
Model development ................................................................................................. 29
3.2.2
Inoculum ................................................................................................................. 31
3.2.3
Meat preparation .................................................................................................... 31
3.2.4
Inoculation .............................................................................................................. 32
3.2.5
Heat treatments ....................................................................................................... 32
3.2.6
Recovery of samples ................................................................................................ 33
3.2.7
Model parameterization and validation.................................................................. 33
3.3
Results and discussion.................................................................................................... 36
3.3.1
Meat sterility and composition................................................................................ 36
3.3.2
Calibration: state-dependent model parameterization ........................................... 37
3.3.3
Validation: application of the state-dependent model to non-isothermal data ...... 40
iv

3.3.4
Calibration: path-dependent model parameterization ........................................... 43
3.3.5
Validation: application of the path-dependent model to isothermal and nonisothermal data ...................................................................................................................... 43
3.4
Conclusions .................................................................................................................... 46
4.

MULTI-PRODUCT, MULTI-FACTOR MODEL........................................................... 47
4.1
Introduction .................................................................................................................... 47
4.2
Materials and Methods ................................................................................................... 49
4.2.1
Data compilation .................................................................................................... 49
4.2.1.1
MSU laboratory data ....................................................................................... 52
4.2.1.2
Scientific journals ............................................................................................ 52
4.2.1.3
ComBase Browser ........................................................................................... 53
4.2.2
Model development ................................................................................................. 54
4.2.3
Non-linear parameter estimation............................................................................ 57
4.2.3.1
Ordinary least squares (OLS) method ............................................................. 58
4.2.3.2
Non-linear mixed-effects (NLME) method ..................................................... 64
4.2.4
Model validation against pilot-scale data .............................................................. 69
4.2.4.1
Steaks/fillets and patties in impingement oven ............................................... 70
4.2.4.2
Whole-muscle roasts........................................................................................ 71
4.2.4.3
Hot dogs ........................................................................................................... 71
4.3
Results and discussion.................................................................................................... 72
4.3.1
Data compilation results ......................................................................................... 72
4.3.2
OLS model parameters, statistics, and PIs ............................................................. 76
4.3.3
Mixed-effects model parameters and statistics ....................................................... 80
4.3.4
Validation against pilot-scale data ......................................................................... 84
4.3.4.1
Steaks/fillets and patties .................................................................................. 88
4.3.4.2
Whole-muscle roasts...................................................................................... 108
4.3.4.3
Hot dogs ......................................................................................................... 115
4.3.4.4
Overall model performance on all pilot-scale data ........................................ 122
4.4
Conclusions .................................................................................................................. 124

5.

OVERALL CONCLUSIONS ........................................................................................... 125
5.1
Sublethal injury ............................................................................................................ 125
5.2
Model validation .......................................................................................................... 125
5.3
Use of models in industry............................................................................................. 125
5.4
The need for standardized testing methods .................................................................. 126
5.5
Future work .................................................................................................................. 127
5.5.1
Enhancement of the AMI lethality spreadsheet .................................................... 127
5.5.2
Methods comparison with other research groups ................................................ 128
5.5.3
The need for standardized testing methods ........................................................... 128
5.5.4
Statistical improvements and model modifications............................................... 128
v

6.

APPENDICES .................................................................................................................... 130
6.1
MATLAB code for sublethal injury model parameter estimation (section 3) ............. 131
6.2
Correlation coefficients, and standard and relative errors for path-dependent model
parameters (section 3) ............................................................................................................. 134
6.3
MATLAB and R programming codes for multi-product multi-factor model parameter
estimation (section 4) .............................................................................................................. 135
6.3.1
MATLAB code example (OLS method) ................................................................. 135
6.3.2
R code example (mixed-effects method)................................................................ 137
6.4
Hot dog methods and data (section 4) .......................................................................... 140
6.4.1
Preparation of Salmonella .................................................................................... 140
6.4.2
Preparation of inoculated frankfurters ................................................................. 140
6.4.3
Frankfurter cooking and survivor recovery .......................................................... 141
6.4.4
Results ................................................................................................................... 142
6.5
Standard and relative errors for the multi-product, multi-factor model parameters
(section 4) ................................................................................................................................ 144
6.5.1
OLS method ........................................................................................................... 144
6.5.2
Mixed-effects method ............................................................................................ 145
6.6
Correlation coefficients for multi-product multi-factor model parameters (section 4) 147
6.6.1
OLS models ........................................................................................................... 147
6.6.2
Mixed-effects models ............................................................................................. 148
6.7
OLS method prediction interval calculations (section 4) ............................................. 150
6.7.1
Ellipse, bootstrapping, and Monte Carlo (PI methods) contours for each
calibration product .............................................................................................................. 150
6.7.2
Prediction intervals for each PI method and each calibration product. .............. 153
6.8
Random effects variance for the mixed-effects models (section 4) ............................. 158
6.9
Validation against pilot-scale data (section 4) ............................................................. 159
6.9.1
Minimum and maximum errors for each model .................................................... 159
6.9.1.1
Impingement oven data ................................................................................. 159
6.9.1.2
Big roasts ....................................................................................................... 169
6.9.1.3
Hot dogs ......................................................................................................... 172
6.9.1.4
Overall model performance on pilot-scale data ............................................. 174
6.9.2
Plots showing model predictions and PIs for representative data sets and models
175
6.9.2.1
OLS models ................................................................................................... 175
6.9.2.2
Mixed-effects models .................................................................................... 178

7.

REFERENCES .................................................................................................................. 185

vi

LIST OF TABLES
Table 1. Model parameters and results against calibration and validation sets. ........................... 39
Table 2. Description of parameters and variables for secondary model. ...................................... 57
Table 3. Versions of the "full" model parameterized with OLS method. ..................................... 61
Table 4. Versions of the "full" model parameterized with mixed-effects method (continued next
page). ............................................................................................................................................. 67
Table 5. MSU data characteristics by source. ............................................................................... 72
Table 6. Scientific journals and ComBase Browser data characteristics by source. .................... 73
Table 7. Final compiled data characteristics and sources (continued next page). ........................ 74
Table 8. Reference temperatures by data set. ............................................................................... 76
Table 9. Parameters and statistics for OLS models (calibrated with only MSU data). ................ 77
Table 10. Parameters and statistics for mixed-effects model (continued next page).................... 81
Table 11. OLS models validated against chicken fillets and turkey patties (impingement cooked).
....................................................................................................................................................... 90
Table 12. OLS models validated against beef steaks and patties (impingement cooked). ........... 91
Table 13. OLS models validated against pork steaks and patties (impingement cooked). ........... 92
Table 14. OLS models validated against ALL impingement oven data. ...................................... 96
Table 15. Mixed-effects models validated against chicken steaks and turkey patties
(impingement cooked) (continued next page). ............................................................................. 98
Table 16. Mixed-effects models validated against beef steaks and patties (impingement cooked)
(continued next page). ................................................................................................................. 101
Table 17. Mixed-effects models validated against pork steaks and patties (impingement cooked)
(continued next page). ................................................................................................................. 104
Table 18. Mixed-effects models validated against ALL impingement oven data. ..................... 107
Table 19. OLS models validated against roasts. ......................................................................... 109
Table 20. OLS models validation against ALL roast data. ......................................................... 111
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Table 21. Mixed-effects models validated against roasts (continued next page). ...................... 112
Table 22. Mixed-effects models validated against ALL roast data. ........................................... 115
Table 23. OLS models validated against hot dogs. ..................................................................... 116
Table 24. OLS models validated against ALL hot dog data. ...................................................... 118
Table 25. Mixed-effects models validated against hot dogs. ...................................................... 119
Table 26. Mixed-effects models validated against ALL hot dog data. ....................................... 121
Table 27. OLS models performance against ALL pilot-scale data. ............................................ 122
Table 28. Mixed-effects models performance against ALL pilot-scale data. ............................. 123
Table 29. Correlation coefficients for path-dependent model parameters. ................................. 134
Table 30. Standard and relative errors for path-dependent model parameters. .......................... 134
Table 31. Commercial cooking schedule for frankfurters. ......................................................... 141
Table 32. D- and z- values from previous research used to predict real-time lethality. ............ 141
Table 33. Predicted and experimental log reductions for turkey hot dogs. ................................ 142
Table 34. Predicted and experimental log reductions for beef hot dogs. .................................... 143
Table 35. Parameter standard and relative errors for β3 estimation. ........................................... 144
Table 36. Parameter standard and relative errors for OLS models. ............................................ 144
Table 36. Parameter standard and relative errors for mixed-effects models .............................. 145
Table 38. Parameter correlation coefficients for β3 estimation. ................................................. 147
Table 37. Correlation coefficients for OLS models parameters. ................................................ 148
Table 40. Correlation coefficients for mixed-effects models parameters. .................................. 148
Table 40. Random effects variance for mixed-effects models. .................................................. 158
Table 41. OLS models prediction errors chicken fillets and turkey patties (impingement cooked).
..................................................................................................................................................... 159
Table 42. OLS models prediction errors on beef steaks and patties (impingement cooked)...... 160
Table 43. OLS models prediction errors on pork steaks and patties (impingement cooked). .... 161
viii

Table 44. OLS models prediction errors on ALL impingement oven data. ............................... 162
Table 45. Mixed-effects models prediction errors on chicken steaks and turkey patties
(impingement cooked) (continued next page). ........................................................................... 162
Table 46. Mixed-effects models prediction errors on beef steaks and patties (impingement
cooked) (continued next page). ................................................................................................... 164
Table 47. Mixed-effects models prediction errors on pork steaks and patties (impingement
cooked) (continued next page). ................................................................................................... 166
Table 48. Mixed-effects models prediction errors on ALL impingement oven data.................. 168
Table 49. OLS models prediction errors on roasts. .................................................................... 169
Table 50. Mixed-effects models prediction errors on roasts (continued next page). .................. 170
Table 51. OLS models prediction errors on ALL roast data....................................................... 171
Table 52. Mixed-effects models prediction errors on ALL roast data. ....................................... 172
Table 53. OLS models prediction errors on hot dogs. ................................................................ 172
Table 54. OLS models predictions errors on ALL hot dog data. ................................................ 173
Table 55. Mixed-effects models prediction errors on hot dogs. ................................................. 173
Table 56. Mixed-effects models prediction errors on ALL hot dog data. .................................. 174
Table 57. OLS models prediction errors on ALL pilot-scale data. ............................................. 174
Table 58. Mixed-effects models prediction errors on ALL pilot-scale data. .............................. 175

ix

LIST OF FIGURES
Figure 1. Representative heating profiles, where τ is the integral of the time-temperature profile
within a prescribed heat shock region. .......................................................................................... 30
Figure 2. Log reduction errors for the state-dependent model and path-dependent model for the
turkey calibration set. .................................................................................................................... 41
Figure 3. Log reduction errors for the state-dependent model and path-dependent model for the
beef calibration set. ....................................................................................................................... 42
Figure 4. Log reduction errors for the state-dependent model and path-dependent model for the
pork calibration set. ....................................................................................................................... 42
Figure 5. Log reduction errors for the state-dependent model and path-dependent model for the
turkey validation set. ..................................................................................................................... 44
Figure 6. Log reduction errors for the state-dependent model and path-dependent model for the
beef validation set. ........................................................................................................................ 45
Figure 7. Log reduction errors for the state-dependent model and path-dependent model for the
pork validation set. ........................................................................................................................ 45
Figure 8. Definition of "data set" and "data point" for raw data. .................................................. 51
Figure 9. Graphical representations of mixed-effects models. ..................................................... 65
Figure 10. PI methods parameter contours for ground turkey calibration set. ............................. 78
Figure 11. PIs with all methods for ground turkey calibration data set. ....................................... 79
Figure 12. Zoom-in section from Figure 11.................................................................................. 79
Figure 13. Example performance of an ideal model. .................................................................... 85
Figure 14. Example performance of a practically possible ideal model. ...................................... 86
Figure 15. Example performance of a conservative, but possibly useful model. ......................... 86
Figure 16. Example performance of a potentially useful model; all data points fall above the faildangerous PI, but PI width would need to be evaluated by the user to determine usefulness,
especially as several data points are quite close to the lower PI band. ......................................... 87
Figure 17. Example performance of a possibly useful model; all data points fall above the faildangerous PI, but bias is highly negative, and PI width would need to be evaluated by the user to
determine usefulness. .................................................................................................................... 87
x

Figure 18. Example performance of a useless model. In addition to not capturing all data points
above the fail-dangerous PI band, notice the width of the PI. ...................................................... 88
Figure 19. OLS T (g) model validated against ground pork data. Compare with Figure 20. ....... 93
Figure 20. OLS T F τ (g) model validated against ground pork data. Notice that by considering
fat content (F) and sublethal history (τ), model performance is improved. .................................. 93
Figure 21. OLS T F τ (g) model validated against ground turkey data. Compare with Figure 22.
....................................................................................................................................................... 95
Figure 22. OLS T F τ (w) model validated against whole-muscle chicken breast data. .............. 95
Figure 23. Mixed-effects T F (w) model validated against whole-muscle chicken fillets. .......... 97
Figure 24. Mixed-effects T F M τ S model validated against whole-muscle beef steaks. ......... 100
Figure 25. Mixed-effects T F M τ S model validated against ground-muscle pork patties. ....... 103
Figure 26. OLS T F τ (g) model validated with turkey roast data. Compare with Figure 27. .... 110
Figure 27. OLS T F τ (w) model validated against turkey roast data. Notice the significantly
better performance than its (g) counterpart (Figure 26).............................................................. 110
Figure 28. Mixed-effects T F τ S (w) model validated against pork roasts. Notice the favorable
fitting statistics. ........................................................................................................................... 114
Figure 29. OLS T (g) model validated against beef hot dogs. Compare with Figure 30. ........... 117
Figure 30. OLS T F τ (g) model validated against beef hot dog data. Notice the significantly
better performance than the simpler T (g) model (Figure 29). ................................................... 117
Figure 31. Mixed-effects T F M S τ model validated against beef hot dogs. ............................. 120
Figure 32. PI methods parameter contours for whole turkey calibration set. ............................. 150
Figure 33. PI methods parameter contours for ground beef calibration set. ............................... 151
Figure 34. PI methods parameter contours for whole beef calibration set. ................................ 151
Figure 35. PI methods parameter contours for ground pork calibration set. .............................. 152
Figure 36. PI methods parameter contours for whole pork calibration set. ................................ 152
Figure 37. PIs with all methods for whole turkey calibration data set. ...................................... 153
Figure 38. Zoom-in section from Figure 37................................................................................ 153
xi

Figure 39. PIs with all methods for ground beef calibration data set. ........................................ 154
Figure 40. Zoom-in section from Figure 39................................................................................ 154
Figure 41. PIs with all methods for whole beef calibration data set. .......................................... 155
Figure 42. Zoom-in section from Figure 41................................................................................ 155
Figure 43. PIs with all methods for ground pork calibration data set. ........................................ 156
Figure 44. Zoom-in section from Figure 43................................................................................ 156
Figure 45. PIs with all methods for whole pork calibration data set. ......................................... 157
Figure 46. Zoom-in section from Figure 45................................................................................ 157
Figure 47. OLS T (g) model validated with whole-muscle pork steaks. .................................... 176
Figure 48. OLS T F (g) model validated against whole-muscle pork steaks. ............................. 176
Figure 49. OLS T τ (g) model validated against whole-muscle pork steaks. ............................. 176
Figure 50. OLS T F τ (g) model validated against whole-muscle pork steaks. .......................... 177
Figure 51. OLS T (w) model validated against whole-muscle pork steaks. ............................... 177
Figure 52. OLS T F (w) model validated against whole-muscle pork steaks. ............................ 177
Figure 53. OLS T τ (w) model validated against whole-muscle pork steaks. ............................ 178
Figure 54. OLS T F t (w) model validated against whole-muscle pork steaks. .......................... 178
Figure 55. Mixed-effects T (g) model validated against whole-muscle chicken breasts............ 179
Figure 56. Mixed-effects T F (g) model validated against whole-muscle chicken breasts. ....... 179
Figure 57. Mixed-effects T F τ (g) model validated against whole-muscle chicken steaks. ...... 179
Figure 58. Mixed-effects T F S (g) model validated against whole-muscle chicken breasts. .... 180
Figure 59. Mixed-effects T F S τ (g) model validated against whole-muscle chicken breasts. . 180
Figure 60. Mixed-effects T (w) model validated against whole-muscle chicken breasts. .......... 180
Figure 61. Mixed-effects T F (w) model validated against whole-muscle chicken breasts. ...... 181
Figure 62. Mixed-effects T F τ (w) model validated against whole-muscle chicken breasts. .... 181
xii

Figure 63. Mixed-effects T F S (w) model validated against whole-muscle chicken breasts. ... 181
Figure 64. Mixed-effects T F S τ (w) model validated against whole-muscle chicken breasts. 182
Figure 65. Mixed-effects T F M model validated against whole-muscle chicken breasts. ........ 182
Figure 66. Mixed-effects T F M τ model validated against whole-muscle chicken breasts. ...... 182
Figure 67. Mixed-effects T F M S model validated against whole-muscle chicken breasts. ..... 183
Figure 68. Mixed-effects T F M S τ model validated against whole-muscle chicken breasts. .. 183

xiii

1. INTRODUCTION

1.1

Background/problem statement
Ready-to-eat (RTE) food products have become an important part of the American diet,

due to their convenience, nutritional value, and palatability. However, because the production
chain to make these goods is highly sophisticated, there is a high risk for physical, chemical, and
biological contamination to occur. While it is easier to monitor physical and chemical impurities,
it is harder to do so for biological elements, such as pathogenic microorganisms, because they
are naturally occurring in the food materials.
Regarding meat and poultry products, the pathogens of concern are Escherichia coli,
Salmonella, Campylobacter, and Listeria monocytogenes. Together with four other pathogens,
these microbial agents are considered to cause most of the foodborne illnesses in the United
States (CDC 2011).

Therefore, to protect consumers, the United States Department of

Agriculture (USDA) bases thermal processing regulations for these products on worst-case
scenarios for Salmonella contamination, due to the pathogen’s higher resistance to heat
inactivation treatments (FSIS-USDA 1999b). The regulation requires that the products be cooked
to reach a specific lethality of 6.5 or 7.0 log10 reduction in Salmonella population for beef or
poultry, respectively (FSIS-USDA 1999b). To fulfill these requirements, processors have two
options: to follow pre-established cooking conditions, also known as “safe harbors”, or to choose
their own cooking conditions. The safe harbors prescribe how long to hold a product at a certain
temperature to comply with the regulation (e.g., 12 minutes at 60°C). The problem with this
option is that it can cause over-processing, which in turn leads to higher energy costs and a
decrease in product yield and quality. On the other hand, if processors use their own cooking
1

schedules, they have to prove “based on scientific rationale [and] experimental data” that they
comply with the lethality performance standard (FSIS-USDA 1999b). The drawback in this case
is that most establishments do not have the financial or scientific resources to provide the
evidence specific to their process, and the scientific information available is related almost
exclusively to laboratory studies, and likely have not been validated for industrial processes.
Essentially, the regulation delegates the problem of validating process lethality to
industry, which in turn does not have the necessary tools to achieve this. Nonetheless, there are
microbial inactivation computer programs available to help processors validate their own
cooking conditions. These are the product of the combined effort of several food safety agencies
and concerned food industry groups, including the USDA, the American Meat Institute (AMI),
the Food Standards Agency (FSA) and the Institute of Food Research (IFR) in the United
Kingdom, and the Food Safety Centre (FSC) in Australia. The federal regulation only mentions
the program developed by the USDA’s Agricultural Research Service (ARS), the Pathogen
Modeling Program, and states that “[o]ther programs may be available commercially”. However,
as explained in more depth in the next chapter, they present several drawbacks and fall short
from the tool industry needs to provide the scientific evidence required in the federal regulation.

1.2

Goals and objectives
Given the importance of assuring food safety for consumers, it is imperative that industry

obtain the necessary tools to achieve this. A model applicable to any meat or poultry cooking
process, that could provide processors a documented scientific means to comply with federal
regulations, would be an important step forward. Therefore, the main goal of this project was to
develop a multi-product multi-factor model that could be used by industry for thermal
2

inactivation of Salmonella in different meat products. Specific objectives were: (i) to test an
improved secondary model for thermal inactivation, accounting for enhanced thermal resistance
resulting from sublethal injury of Salmonella in multiple meat products; (ii) to determine the
most relevant parameters that should be included in a multi-product multi-factor thermal
inactivation model for Salmonella; (iii) to propose and parameterize multiple options for a multiproduct multi-factor thermal inactivation model; and (iv) to validate model performance when
applied to independent pilot-scale data.

3

2. LITERATURE REVIEW

2.1

Salmonella and foodborne illness
According to the Center for Disease Control (CDC), there are approximately 48 million

cases of foodborne illness cases each year in the United States (CDC 2011). These in turn cause
an estimated 128,000 hospitalizations, and 3,000 deaths (CDC 2011). Among the most common
bacterial pathogens in food, the CDC lists Campylobacter, Salmonella, and E. coli O157:H7
(CDC 2011). Salmonella, the leading bacterial cause of foodborne illness, and the pathogen of
concern for this project, is naturally found in the intestinal tract of birds, reptiles, and mammals
(USDA-FSIS 2010), and reaches food for human consumption via fecal contamination during
processing, especially in the case of meat and poultry (Adams and Moss 2008). When infected, a
person can develop non-bloody diarrhea, abdominal cramps, and fever, which characterize
salmonellosis (USDA-FSIS 2010). The illness is not considered life-threatening if treated
promptly. However, small children, the elderly, and individuals with weak immune systems can
be more susceptible to the infection and develop additional long term complications, such as
Reiter’s syndrome and chronic arthritis (USDA-FSIS 2010).

2.2

Federal regulations regarding heat processing in RTE products
Effective March 8, 1999, there is one regulation governing the performance standards for

the manufacture of certain ready-to-eat (RTE) meat and poultry products: USDA-FSIS, 9 CFR
Parts 301, 317, 318, 320, and 381 (FSIS-USDA 1999b). The regulation encompasses three parts:
lethality, stabilization, and handling. Relevant to this project are the lethality performance
standards. According to the regulation, the kill can be accomplished with several antimicrobial
4

methods, but inclusion of a cooking step is mandatory, hence the importance of the heating
process. In addition, there are no documented means of assessing the cumulative effect of several
antimicrobial methods. Therefore, if this path is chosen, processors must scientifically prove that
the lethality levels are reached (FSIS-USDA 1999b).

2.2.1

Safe Harbors
One of the options available to producers to comply with federal regulations regarding

RTE meat and poultry products is to follow the pre-established time/temperature cooking
schedules developed by the FSIS, also known as “safe harbors”. These indicate how long to cook
a certain product once a minimum specified temperature is reached to achieve the regulatory
standard (FSIS-USDA 1999a). For example, a beef roast would be deemed safe if cooked for 12
min once 60°C is reached throughout the product. The full table can be found in “Compliance
Guidelines For Meeting Lethality Performance Standards For Certain Meat And Poultry
Products” (FSIS-USDA 1999a). The safe harbors are the easiest way to comply with the
regulation, and while not required, are given as an option. However, the use of these cooking
schedules may limit a processor from manipulating product attributes only acquirable during the
cooking step, such as moisture retention or crust-layered roasts.

2.2.2

End-point lethality
When concerned about the final quality of the product and total energy expenditure in

cooking, processors may prefer to customize their own cooking schedules. In this case, they must
prove “based on scientific rationale [and] experimental data” that these special cooking
conditions comply with the regulatory standards (FSIS-USDA 1999b). Currently, processes need
to accomplish a 6.5 log10 reduction and 7.0 log10 reduction in Salmonella population for RTE
5

meat and poultry products, respectively (FSIS-USDA 1999b). Even though this choice clearly
gives processors more flexibility in their cooking options, the challenge is proving that their
time/temperature choices reach the required lethality levels. This is because there currently there
is no scientifically proven, generalized tool available to present this evidence, and producing
proof for each specific process is economically burdensome.
A new regulation involving 9 CFR Parts 301, 303, 317, 318, 319, 320, 325, 331, 381,
417, and 430 was proposed by the FSIS on February, 2001 (USDA-FSIS 2001). If finalized, the
new regulation would require that the processing method for all RTE and partially-cooked meat
and poultry products comply with the 6.5 and 7.0 log reduction currently applicable to only
certain products.

2.3

Factors affecting bacterial inactivation
There are numerous methods to inhibit bacterial growth in foods; these include addition

of chemical agents (preservatives), freezing, drying, controlled atmospheres, high pressure
systems, among others. On the other hand, current methods for complete pathogen inactivation
via processing are mostly limited to heating and irradiation. Even with the application of these
two technologies, food processors commonly use the “Hurdle concept”, which means several of
these methods are used to conjointly prevent microbial growth in foods (Jay and others 2005;
Adams and Moss 2008). For example, in a broad perspective, the ingredients for a RTE chicken
dinner would come from a high quality source, the meal would be prepared with the addition of
preservatives, and then cooked to kill any possible incident pathogens, be packaged under
sanitary conditions, and finally frozen and kept at freezing temperatures until consumed. The
combination of controls is designed to help ensure product safety. In the case of certain RTE
6

meat and poultry products, because the use of an inactivation method is imperative for the
destruction of pathogens, a heating step is always included. While irradiation is approved for
fresh meat and poultry products, regulations regarding these RTE products require the inclusion
of a heating step (FSIS-USDA 1999b). For that reason, heating is used as the preferred method
for bacterial destruction in manufacturing RTE meat and poultry products.
Bacterial behavior under heating conditions is well documented for a variety of extrinsic
and intrinsic factors. Extrinsic factors refer to environmental conditions, which include relative
humidity, temperature, and gaseous atmosphere – the most important of these for thermal
inactivation being temperature (Adams and Moss 2008). Intrinsic factors denote properties of the
food that affect bacterial response, such as nutrients, pH and buffering capacity, oxidationreduction potential, moisture or water activity, antimicrobial constituents, food structures, and fat
in the product (Adams and Moss 2008). The most relevant for heat processing would be
moisture, food structure, and fat (Juneja and Eblen 2000; Juneja and others 2001; Juneja and
others 2000a; Tuntivanich and others 2008; Mogollon and others 2009; Velasquez and others
2010; Orta-Ramirez and others 2005; Carlson and others 2005).
Other critical factors cover the characteristics of the microorganisms, which refer to
specific growth rate, physiological state of the cells, mutualism, and antagonism (Adams and
Moss 2008). For thermal processing, the factor of main interest among these would be the
physiological state of the bacterial cell, specifically referring to sublethal injury. Factors most
significant for heat inactivation are described in detail in the following sections.

7

2.3.1

Temperature
Thermal processing inherently involves an increase of temperature in the processing

environment and consequently in the food product. Subjecting bacterial cells to high
temperatures for a certain period will injure them and eventually kill them. There is no cut-off
temperature at which bacterial cells will instantly deactivate; rather, it is a gradual process. This
is because naturally occurring bacterial populations contain cells in different stages of growth,
and log-phase cells are more susceptible to heat than their stationary-phase counterparts (Adams
and Moss 2008). Nonetheless, higher temperature causes faster pathogen inactivation. For
example, cooked beef is deemed equally safe if held for 71 mins at 55°C or 54 s at 66.1°C (FSISUSDA 1999a). While longer cook times also have an influence on bacterial inactivation,
temperature is the most determining factor (Jay et al. 2005).

2.3.2

Fat
Studies conducted by Juneja et al. (Juneja and Eblen 2000; Juneja et al. 2000a; Juneja et

al. 2001) showed that fat percentage (%) in different meat and poultry products significantly
increased Salmonella thermal resistance. The same conclusions were reached by Ahmed et al.
(1995) with regard to E. coli O157:H7. However, the goals of these studies did not include
investigating the reason behind this. While it is presumed that fat globules present in food can act
as a shield for bacterial cells against heat (thus increasing their heat resistance) (Adams and
Moss 2008), it can be argued that from a biological standpoint, the increased resistance could be
due to biochemical interactions between the pathogens’ cell membrane (a lipid bilayer) and the
fats in the product. Regardless of the real cause, this means that a high-fat product would need to

8

be cooked for longer times and/or higher temperatures to achieve the same log reduction than a
low-fat product.

2.3.3

Muscle structure
Food structures have been proven to affect bacterial growth and inactivation (Adams and

Moss 2008). For the case of meat and poultry products, this is relevant to ground vs. wholemuscle products. Although the exact mechanism by which cells increase their thermal resistance
in these two different environments is not completely understood, it may be due to the different
internal structures in the meat and/or available water (Tuntivanich et al. 2008). Orta-Ramirez et
al. (2005), Tuntivanich et al. (2008), and Velasquez et al. (2010) reported that Salmonella had
significantly higher thermal resistance (~double) in whole muscle meat and poultry products
when compared to their ground muscle counterparts. Mogollon et al. (2009) tested whole muscle,
coarsely ground, finely ground, and pureed beef. Salmonella was significantly more resistant in
whole-muscle beef than in the other products, but there was no effect of the degree of grinding.

2.3.4

Media moisture content
Studies from Carlson et al. (2005), McCann et al. (2009), Goepfert et al. (1970), and

Reichart (1994) found that pathogens present in a dry environment portray a higher thermal
resistance than those residing in a moist medium. For example, Carlson et al. reported that the
thermal inactivation for an 8-servorar Salmonella cocktail decreased 64% (p<0.01) when water
activity (aw, a measure of available water in a product) in ground turkey was decreased from
0.99 to 0.95 (2005). In Goepfert et al.’s study, media with different aw were prepared by using
sucrose, fructose, glycerol, and sorbitol; results showed that the cells in environments with the
9

lower aw yielded D-values 25 to 75% lower than those tested in the media with higher aw
(1970).
Moisture as a percentage of product composition is also commonly related to fat
percentage, meaning that a high-fat product will more likely have lower moisture content than a
low-fat product. Consequently, the same principle for inactivating pathogens with longer
cooking times for high-fat products applies to low-moisture products. In addition to the
biological effect, a secondary consequence of a dry system is that with less water present, the
heat transfer process is less efficient, making the cooking less lethal to the bacterial cells.

2.3.5

Sublethal injury
Cell injury results when a process affects the bacterial cell in a negative way, but fails to

kill it. Pathogens are inevitably injured when exposed to heating, freezing, and starvation
environments (Wesche and others 2005). In any of these cases, cells can either become more
susceptible to further inactivation procedures, or react to the changing environment by adapting,
thus becoming more resistant to the processes (Mackey and Derrick 1987b; Wesche et al. 2005).
Under sublethal heating conditions, which occurs for example when slowly cooking a
beef roast, a portion of the bacterial population adapts to the gradual increase in temperature by
developing heat-shock proteins (Xavier and Ingham 1997a; Jorgensen and others 1996), or by
other unknown mechanism (Mackey and Derrick 1990). These physiological changes allow cells
to resist heat inactivation at higher temperatures, thereby affecting cooking time needed to
inactivate the microorganisms (Jorgensen et al. 1996; Xavier and Ingham 1997a).

10

Many studies agree that heat-shocking bacterial cells increases the population resistance
to heat. For example, Knabel et al. (1990) reported that the thermotolerance of Listeria
monocytogenes increased when cells were heat shocked at 43°C for 5, 30, and 60 min. Farber et
al. (1990) subjected the same pathogen to four different heat shock temperatures (40, 44, 48, and
52°C), and reported the same conclusions. Pagan et al.’s (1997) results agree with these findings
with respect to time, but their study also analyzed the effect of heat shock temperature on L.
monocytogenes. They found that just before lethal temperatures were reached, the higher the
temperature, the more thermotolerant the cells became. Therefore, in processing conditions
where sublethal heating is expected, microbial adaptation should be taken into account when
predicting subsequent lethality.
A preliminary model addressing this issue was developed by Stasiewicz et al. (2008) for
Salmonella inactivation in turkey thigh. The new “path-dependent” model incorporated a term
that accounted for sublethal injury, which considered the amount of time the cells remained in
the “heat shock region”, which was determined to be between 38°C and 52°C (Stasiewicz et al.
2008). The study fitted data from an 8-serovar Salmonella cocktail at different residence times
and hold temperatures within the heat shock region (thus yielding different degrees of sublethal
injury) to the path-dependent model. Results showed that the use of the latter reduced prediction
error by 56% when compared against a traditional “state-dependent” model, which did not
account for sublethal injury (Stasiewicz et al. 2008).
Cold shock of cells can also lead to a change of bacterial thermal tolerance. This can
happen in the processing environment when raw food materials are stored in chilled settings
prior to the heat treatment. The effect of cold shock on cells is not fully documented, and
different studies have reached contradicting conclusions. For example, Leenanon et al. (2001)
11

found in a first study that heat tolerance for E. coli O157:H7 was decreased with cold shock
(TSB for 1 week at 5°C), while a second study (Elhanafi and others 2004) revealed that heat
tolerance increased, although the cold shock treatment was different (tryptic soy broth (TSB) for
4 weeks at 4°C). Also, Bang et al. (2002) observed minimal heat tolerance increase in Vibrio
vulnificus after a cold-shock treatment. On the other hand, Wesche et al. (2005) did not find a
significant difference between the thermal tolerance of cold-shocked and control Salmonella
cells in turkey. These conclusions, in addition to limited inactivation data on this area, make it
difficult to account for this type of bacterial injury in heat inactivation models. Results are
similar for the case of starvation stress. For example, Bang et al. (2002) reported a slight increase
in thermotolerance for one of the three strains of Vibrio vulnificus tested, but Wesche et al.
(2005) found no significant difference between control and starved-cell heat tolerance of
Salmonella. In addition, because different pathogens experience starvation under many different
conditions (e.g., phosphate buffer, peptone water, nitrogen or carbon starvation), the lack of data
availability makes this phenomenon extremely difficult to characterize and introduce in heat
inactivation models.

2.3.6

Inactivation media (liquid vs. meat and species)
Most of the initial laboratory studies measuring bacterial thermal inactivation were

carried out in laboratory media, such as tryptic soy broth (TSB). However, it has been
demonstrated that bacterial inactivation rates vary with the medium in which the cells develop
(Shah and others 1991; Sergelidis and Abrahim 2009; Smith and others 2001; Murphy and others
2000). Therefore, if the results of any thermal inactivation study are to be applied to industry
processes, it is imperative to consider the media in which the tests were carried out. This would
12

mean, for example, using data from ground beef when evaluating ground beef, but not for other
products.

2.4

Quantifying bacterial inactivation
Given the impossibility of taking samples from every single RTE product leaving the

processing line to test for pathogen presence, food processors and academics alike utilize
mathematical models that describe the inactivation kinetics of these microorganisms. Another
option considered in industry is “challenge studies” specific to the manufacturing plant and
cooking process. These will be explained in detail in the following sections.

2.4.1

Mathematical models
Generally, thermal inactivation models are developed and tested on laboratory-scale data.

This fact poses a difficulty when the model is applied to an industrial setting. This is because
industrial conditions (i.e., product characteristics, process conditions, etc) rarely resemble a
laboratory setting. As a result, due to the aforementioned reasons, the bacterial response may be
different. Nonetheless, a recent study (Breslin 2009) tested these models on pilot plant data,
which have a closer resemblance to an industry plant, thus producing better estimates of model
performance.
A significant difficulty that arises when trying to apply laboratory data to industrial
settings is that not all scientific studies use the same experimental methods or the same model
(except for maybe the traditional log-linear model); every researcher has preferences, and
specific reasons are typically not reported for choosing a certain model (Murphy et al. 2000);

13

(Quintavalla and others 2001; Smith et al. 2001). As a result, it is difficult for processors to
decide which data and/or model best applies to their own processes.
Another obstacle processors face when choosing models is that multiple studies focus
primarily on developing mathematically improved models (Vaidya and Corvalan 2009; Corradini
and Peleg 2009; Corradini and others 2010) . Although they may describe certain inactivation
data sets extremely well, or are able to account for multiple environmental factors, their
usefulness in industry can be hindered because of the extensive experimental data needed to
characterize them, or due to the lack of sufficient technical expertise or resources. Furthermore,
numerous reports simply develop the models mathematically and only show theoretical results;
there is no parameterization with real data, and the crucial validation against independent data is
not carried out (Vaidya and Corvalan 2009; Corradini and Peleg 2009; Corradini and others
2009). As a result, the scientific literature contain numerous models, but very few reach industry
or have significant implications on the food safety system.
The following section attempts to describe the different types of thermal inactivation
models along with the characteristics that affect their utility for diverse applications. Notice that,
as stated by McKellar et al. (2004), “the most appropriate model would be the simplest model
possible for a given purpose and the given data quality, provided that it is validated and precise”.
Although there are multiple studies dedicated to the use of these models, few books have been
devoted to their categorization and description, and most of these contain in turn more
information and detail about bacterial growth than about inactivation methods (McKellar and Lu
2004; Brul and others 2007; McMeekin and others 1993).

14

2.4.1.1 Primary models
Primary models define a relationship between bacterial behavior and time at a single
environmental condition (Whiting and Buchanan 1993) . The most used primary inactivation
model follows log-linear kinetics. However, multiple studies have proven that shoulders and tails
in deactivation curves (log N/N0 vs. time) are clearly not well described by this linear model
(Peleg 2006; Juneja and Marks 2003). Therefore, several other models have been studied. These
include those following log-normal, Weibull, or log-logistic distributions, and those based on
probabilistic models, sigmoidal, or semi-logarithmic survival curves, among others.

2.4.1.1.1 Log-linear kinetics (D-value)
The log-linear model describes the most basic form of relationship between pathogen
cells, time, and thermal inactivation. It assumes that a bacterial population present in food will
decrease exponentially with time at a constant temperature. This is normally referred to as the
“log-linear model” (or first-order kinetics), and can be expressed as D-values to characterize
pathogen thermal kinetics. The D-value is the time it takes at a specified temperature to reach a
decimal reduction in a bacterial population. A variety of D-values for different products,
pathogens, and applications can be easily found in literature. Although this model is widely
accepted and used, in part because of its simplicity, in part because it is the most widely studied,
it presents one, but very important weakness. The drawback when trying to fit the model to
inactivation data is that bacteria will often not show a log-linear decay. This is because the loglinearity assumption does not account for any type of bacterial adaptation, heating and cooling
lag times, cell growth stage, among other natural factors. Van Boekel (van Boekel 2002)
collected inactivation data from 55 different studies and found that the bacteria presented the
15

behavior characteristic of the log-linear model in only 2 of them, proving that this traditional
model represents “the exception rather than the rule”. As a result, researchers have in recent
years increasingly sought and studied other models with better prediction abilities.

2.4.1.1.2 Weibull distribution
The Weibull model is commonly used to describe failure phenomena. In the case of
inactivation kinetics, it can be interpreted as the failure of microorganisms to survive lethal
environmental conditions after a certain time (van Boekel 2002). The solved form of the
equation that is relevant for inactivation microbiology is (Peleg 2006):

log S = log

N
= −bt n
N0

(1)

where S is the survival ratio N/N0, N is the current bacterial population (CFU/g), N0 is the initial
population (CFU/g), b is a inactivation rate parameter, and n is the shape of the survival curve
(Peleg 2006). Depending on the bacterial survival curve, the shape will be either described as
concave upward or concave downward, yielding values of n<1 and n>1 respectively (Peleg
2006; van Boekel 2002). Common speculations as to why the curvature is upward or downward
refer to the bacterial population characteristics. When the curve shows an initial “shoulder”, it
might mean that the cells are hardy and are adapting or strongly resisting thermal kill; in this
case the curve is said to be concave downward and n>1. On the other hand, there might be a
swift decrease in microorganism numbers with “tailing” as time increases, which may mean that
most of the population was weak enough to be deactivated fast (Peleg 2006; van Boekel 2002).
Note that when n=1, the model becomes log-linear, a special case of the Weibull model.

16

This model has become increasingly popular among the research community for its
simplicity, and the versatility to describe different inactivation data sets. Van Boekel (van Boekel
2002) applied the model to 55 different data sets and obtained acceptable 95% confidence
intervals for each case.

2.4.1.1.3 Logistic distribution function
The survival curve described by this model always has a prominent “shoulder”, a section
in the inactivation curve where there was heating, but negligible pathogen inactivation. Short
shoulders are often the result of thermal lag times, which is the time delay it takes for the cold
spot of a food product to reach lethal temperatures. This is usually due to the size and mass of the
food, and the processing equipment. In such a case, the shoulder is an experimental artifact, not a
biological phenomenon. However, when such an effect is observed, and with longer times than
commonly expected, it might be due to characteristics of the bacterial population. This effect can
also be described by the Weibull distribution with n>1 or any unimodal distribution. However,
the main difference with the Logistic distribution function is that it will portray a log-linear
behavior once the shoulder disappears. In the case of the other models, the inactivation lines are
significantly more curved (Peleg 2006; McKellar and Lu 2004).

2.4.1.1.4 Sigmoidal survival curves
In a way, sigmoidal curves present both aspects of the Weibull distribution, because they
contain both the concave upward and the concave downward behaviors. In terms of cell
characteristics, this means that populations are a mixture of highly-resistant and highly-sensitive
individuals. There are two types of these curves; the first ones initially show the concave
downward behavior, and then, as the pathogens are deactivated, the behavior switches to concave
17

upward. The second group of curves is the exact opposite as the first: they initially show concave
upward trends and then concave downward (Peleg 2006). Although less used than the Weibull
distribution, several studies (Miller and others 2009; Feng and others 2011) cite these curves and
show experimental data that are adequately described by this model.
There is no unique equation that describes sigmodial survival curves; what groups them
are the observed characteristics described above. Rather, these curves are represented by
different empirical models (Peleg 2006).

2.4.1.2 Secondary models
Secondary models describe the relationship between primary model parameters and the
conditions in which the bacteria reside, such as temperature, pH, salt concentration, among
others (Whiting and Buchanan 1993). Just like industry has a preference for the D-value model,
the secondary model most commonly used and studied is the z-value, which is generally used in
conjunction with the D-value. However, just as with the log-linear model, the z-value presents
several drawbacks, which has led scientists, to search for a better model.

2.4.1.2.1 Z-value
The z-value model follows the same log-linearity assumption the D-value model does,
and describes the temperature dependence of the D-value (Van Boekel 2008). Basically, it
assumes that the D-value decreases exponentially with temperature, and so it can be defined as
the change in temperature it takes for 10-fold change in D-value. Most studies in which D-values
have been determined also include a z-value, and so it is common for the scientific community
and industry to associate one parameter with the other, making together the traditionally used
18

complete Bigelow-type model (Bigelow 1921). However, as stated by van Boekel, for this
widely accepted model to be valid, both the rate of bacterial change with time and the rate of Dvalue change with temperature have to be semilogarithmic- something that rarely happens,
because when either the D-value or the z-value deviate, the predictions become questionable
(Van Boekel 2008). For these reasons, research is continuously being carried out to develop
better inactivation models.

2.4.1.2.2 Arrhenius relationship
The Arrhenius model found its way to microbiology from its common and useful
application in chemistry and other sciences, in which chemical reactions are continuously
studied. The Arrhenius equation can be presented in several ways, a common one being
(McKellar and Lu 2004):
 Ea 
−

 RT 
k = Ae

(2)

where k is the reaction rate constant, A is a constant related to the reaction, Ea the reaction
activation energy, R the Universal gas constant (8.314 J/molK), and T the absolute temperature.
Because it is common that the Arrhenius equation yields high deviations at extreme
temperatures, for microbiology it is generally depicted with the use of a reference temperature
(Tref) (Nunes and others 1993):

k = k ref

 Ea  1
1
−  −
 R  T Tref

e

19






(3)

where kref is the corresponding reaction rate at Tref, T is the temperature of interest, and k the
reaction rate at T. Notice that with the inclusion of a reference temperature, A is replaced by kref,
making this form of the model to behave less erratically in the region around Tref, and giving a
physically relevant meaning to this parameter, rather than using A. For that reason, in thermal
inactivation kinetics for Salmonella in meat and poultry products, Tref is commonly chosen as
60°C (333 K) or similar.
Peleg (2006) discredits the use of the Arrhenius equation by mentioning that its
predictions include high errors in low or high temperatures. However, he fails to mention that
choosing an adequate Tref significantly lowers the risk for erroneous predictions in the
temperature range of interest. Similarly, he states that the Arrhenius equation is a second
logarithmic transformation of the inactivation data (the first being log N/N0), which presumes
the primary model parameters to be linear. According to him, this is not a common case in
inactivation kinetics, and so the Arrhenius relationship should be invalid. However, he does not
consider the fact that the parameters for this equation can be estimated via global, non-linear
regression. If this is the case, the model parameters are free to be linear or non-linear as they fit
best (vanBoekel 1996; Van Boekel 2008) . In addition, Dolan et al. (Dolan and others 2007) have
demonstrated that when estimating model parameters for this equation, it is possible to choose an
optimum Tref so that the correlation coefficient between the equation’s parameters is minimized
and so all parameter estimates from the regression are the best possible achievable. As stated
above (McKellar and Lu 2004), a model is useful if it has been appropriately validated and
20

fittingly describes a given set of data. For the case of the Arrhenius model, it has proven to be the
best fit for several heat inactivation studies (Stasiewicz et al. 2008; McQuestin and others 2009;
Murphy et al. 2000).

2.4.1.2.3 Log-logistic model
The log-logistic secondary model describes the relationship of primary model parameters
to environmental factors by marking a specific environmental “set point” where the primary
model parameter will change, so that it significantly affects bacterial inactivation (Peleg 2006).
For example, if using the Weibull distribution as primary model, temperature as the
environmental factor, and b as our parameter of interest, the log-logistic model will have the
following form (Peleg 2006):
b (T ) = exp{ + exp[k (T − Tc )]}
1

(4)

Where Tc is the “set point” that marks when b(T) will drastically change, and k is the rate
of change for b with respect to temperature after Tc is reached (Peleg 2006). This means that
once the system reaches a high enough temperature (Tc), b(T) will make bacterial inactivation
significant, otherwise, it is insignificant.

2.4.1.3 Tertiary models
Tertiary models bring together primary and secondary models and make them available
to users through a computer interface (Whiting and Buchanan 1993). Not until a tertiary model is
established do primary and secondary models become useful to industry. Currently, the only
available tertiary models, which are relatively useful for heat inactivation processes, are the
21

Pathogen Modeling Program (PMP) (USDA-ARS 2007), the American Meat Institute (AMI)
Lethality Spreadsheet (American Meat Institute Foundation 2010), and the ComBase Predictor
(ComBase 2012); all described in the following sections. Apart from their respective advantages
and disadvantages, the one main flaw present in all tools is that they do not provide the user with
confidence intervals and prediction intervals for the estimated lethality. Therefore, the user is
unable to know the prediction reliability. As a result, even if the process predicted lethality is
over the regulatory target, it might be that the actual outcome is outside of the confidence and/or
prediction intervals, still yielding an inadequately processed product.

2.4.1.3.1 Pathogen Modeling Program (PMP)
The Pathogen Modeling Program (USDA-ARS 2007) is a tool provided by the USDA’s
Agricultural Research Service (ARS). This is the only program mentioned in the federal
regulation as an option to provide “scientific evidence” (FSIS-USDA 1999a). Although the tool
is designed primarily for pathogen growth, it includes a few heat inactivation models. However,
there is none available to predict Salmonella lethality, which is the target pathogen in the
regulation. In addition, those available do not include any secondary model relationships, so that
the tool cannot account for different product or processing conditions that affect the thermal
treatment. Furthermore, the program is not customized to work under non-isothermal conditions,
an inherent feature of industrial processes. It is evident that although widely known, this tool
cannot be used at all to help processors comply with the specific regulations for meat cooking.

2.4.1.3.2 AMI Lethality Spreadsheet
The American Meat Institute (AMI) is an association that represents most of the red meat
and turkey processors and suppliers in the US. The Institute revises and distributes to its
22

members up-to-date information relevant to the meat and poultry industries. Through the AMI
Foundation, research is carried out to improve processing methods and products (AMI 2010).
One of the most important outcomes of the research related to food safety is the AMI Lethality
Spreadsheet (American Meat Institute Foundation 2010). This program can calculate total
process lethality for any pathogen using the log-linear model, as long as the adequate D- and zvalues are given. Values for Salmonella, E. coli O157:H7, and Listeria monocytogenes are listed
for certain products. However, they are only examples from specific laboratory studies, which
cannot adequately describe an industrial process (Breslin 2009). According to the spreadsheet
instructions, the users must obtain those figures from their company’s “challenge study data,
from scientific literature, or other reliable sources” (American Meat Institute Foundation 2010).
For most producers this is as problematic as proving a process meets the required lethality. Also,
just like the PMP, this spreadsheet is not adapted to include secondary models that account for
other product characteristics that can alter bacterial behavior in food. Therefore, this is tool is
only useful to a certain extent if processors have their own D- and z-values.

2.4.1.3.3 ComBase Predictor
ComBase is the result of the combined efforts of several food safety agencies across the
globe: the USDA’s Agricultural Research Service (ARS) in the United States, the Food
Standards Agency (FSA) and the Institute of Food Research (IFR) in the United Kingdom, and
the Food Safety Centre (FSC) in Australia (Baranyi and Tamplin 2004). Initially, it was a
compilation of microbiological data published in the UK, collected data from scientific literature
at IFR, data from European research institutions, and data from members of the USDA-ARS
Center of Excellence in Microbiology Modeling Informatics (CEMMI) (ComBase 2012). Today,
23

researchers are encouraged to submit their data to the database, and so the ComBase Browser has
grown to be the biggest source for food microbiological data, containing 50,474 data records
from all over the world (ComBase 2012).
Equivalent to the US’s PMP, the UK’s agencies developed the Food MicroModel, which
eventually evolved to become the ComBase Predictor, by incorporating all the available data in
ComBase (ComBase 2012). Although highly oriented to predict microbial growth with 23
models, the tool also has 6 thermal inactivation models. A variety of foodborne pathogens are
targeted on both types of models, while spoilage microorganisms are also included in the growth
models (ComBase 2012). Differing from the PMP and the AMI Lethality Spreadsheet, the
ComBase Predictor does include secondary models, allowing the user to adjust the prediction to
several environmental factors such as salt concentration, changing temperature, pH, CO2
concentration, etc (ComBase 2012). Also, predictions for up to four microorganisms can be
carried out simultaneously (ComBase 2012). However, because the data used for the models
comes from those compiled by ComBase, and most such data were developed in non-food
media, the Predictor warns the user that the “growth models represent fail-safe, conservative
predictions” (ComBase 2012). This statement is consistent with the results of the study by
Tamplin et al. (2005), where it was determined that “the absence of an appreciable lag period at
6, 8, and 10°C suggest that more occurrences of growth at refrigeration temperatures should be
expected than are typically assumed in risk assessment models”. In addition, the specific models
used for each prediction are not documented, and the exact sources of the data used to develop
model parameters are not listed. Therefore, the quality of the tool’s predictions can easily be
questioned.
24

An additional drawback is that the Predictor asks for data within the temperature range of
54.5 and 65°C, and the “first time-point must be [zero]” (ComBase 2012). This represents a
problem for processors, because normally time/temperature data are logged once the products
enter the heating instrument and until the end of cooling, meaning that there would have to be
manual manipulations of the data to select the appropriate points that the model can use. In
addition, some processes may reach temperatures higher than 65°C, in which case the Predictor
does not accept the data to carry out the prediction.
While the ComBase Predictor may have several useful characteristics for growth
predictions, the reasons stated above are sufficient to deem this tool inadequate for thermal
processing validations in meat and poultry applications.

2.4.2

Challenge studies
Challenge studies involve monitoring a product process from beginning to end and

assessing the bacterial growth and inactivation throughout all its stages. This inevitably involves
actually inoculating a product with the pathogen of concern and running it through all the steps
in the production process, then analyzing the lethal effects the complete process had on the
bacterial population (Adams and Moss 2008). This option may seem appealing from the point of
view of obtaining results that are specific to the product and process in question and thus
unequivocally assessing the effectiveness of the procedure. However, deliberately introducing a
pathogen in an industrial setting is not feasible, due to obvious contamination concerns. For that
reason, this method is not highly considered by processors. However, pilot-plant challenge
studies, such as those carried out by Breslin (2009) and Wiegand (2012) come closer to
resembling an industrial setting and give good approximations of bacterial death kinetics.
25

3. MODELING SUBLETHAL THERMAL INJURY

3.1

Introduction
Although important to certain industrial processes, the potential effects of sublethal injury

on foodborne pathogens have not been quantified in a manner applicable to prediction models.
Currently, most secondary models (whether Arrhenius-type, Bigelow-type z-value, or other
empirical form) assume that the rate of inactivation is a function of the instantaneous state of the
system (e.g., temperature, fat content). Such state-dependent models may be ineffective when
pathogens are subjected to sublethal heating (Jorgensen et al. 1996; Stephens and others 1994;
Mackey and Derrick 1986), which can occur during slow-cooking processes, and which can
cause cells to increase their thermotolerance (Bunning and others 1990; Mackey and Derrick
1986; Shah et al. 1991; Xavier and Ingham 1997a). If this occurs, then a state-dependent model
might over-predict the process lethality, as bacterial inactivation rate does not depend solely on
the state of the system, but also on the thermal path preceding the lethal condition.
Several prior studies have reported the effect of heating rate on inactivation rate (Mackey
and Derrick 1987a; Quintavalla and Campanini 1991; Stephens et al. 1994). These studies
reported that slow heating rates (on the order of < 1°C/min) induce a higher heat resistance in
cells than did fast or instantaneous heating. This is consistent with the understanding that slow
heating rates inherently expose the bacteria to extended periods in the temperature range
previously described as inducing the heat shock response (i.e., approximately 40-50°C), thereby
allowing sufficient time for that response to be expressed.
From this evidence, other studies did modify secondary models and incorporate heating
rate as a variable to account for increasing thermotolerance due to heat adaptation (Corradini and
26

Peleg 2009; Valdramidis and others 2007; Stephens et al. 1994). However, the previously cited
heat shock literature indicates that heat shock is a direct function of time spent in a critical
temperature range, not of heating rate (Farber and Brown 1990; Pagan et al. 1997; Knabel et al.
1990; Diller 2006). In other words, the adaptive response of the bacteria is expressed over time
when exposed to certain temperatures (i.e., a function of time and temperature), so that the
cellular response is not a function specifically of the rate of temperature change. Therefore,
heating rate as a variable in an inactivation model is a surrogate for the underlying cellular
mechanisms of adaptation, and therefore may not be phenomenologically consistent with
outcomes that can occur during certain treatments, such as those that include rapid heating rates
but static holding periods at sublethal temperatures.
An alternative method to modeling heat inactivation with prior heat shock was developed
by Vaidya et al. (2009). They added a “memory kernel” to their model, which allowed the model
to account for “events at a temporal distance” that influenced the present state of a system.
However, because a specific value was not assigned to the “temporal distance”, the latter is only
an arbitrary measure. This means that for any point in time, it is unknown how far the memory
kernel goes back to account for heat effects. Also, using a constant “temporal distance” means
the memory kernel makes no distinction between the sublethal and lethal regions. This is again
inconsistent with heat shock literature, given that bacterial adaptation occurs only over a known
temperature range (Pagan et al. 1997), and once the lethal temperatures are reached, adaptation
essentially ceases, and only bacterial inactivation occurs.
Overall, the previously proposed models have potential for industry use, but they still
present significant weaknesses. For instance, with the exception of Valdramidis et al. (2007),
who reported accuracy and bias factors, sufficient statistical measures of model performance,
27

such as root mean squared errors (RMSEs), are rarely given to quantitatively test the predictive
ability of the models. Such quantitative measures of the predictive ability and robustness of
models are critical before they can be adapted to industrial applications.
A different approach was taken by Stasiewicz et al. (2008) (section 2.3.5). Their model
incorporated an integral of thermal history in the sublethal region as a variable influencing
subsequent inactivation rate. In comparing this path-dependent secondary model to a statedependent secondary model, both were incorporated into a Weibull primary model applied to
non-isothermal treatments with varying sublethal histories, the RMSE was lowered from 2.5 to
1.1 log CFU/g for the state- and path-dependent models, respectively (Stasiewicz et al. 2008).
However, the experiments were constrained to only one type of meat (ground turkey thigh meat),
and few data points accounted for very long exposures to sublethal temperatures that might
correspond to commercial cooking schedules of slow-roasted products. Hence, it is important to
test whether these results can be extended to longer exposures in the sublethal region, and to
what degree substrate affects the model parameters. Therefore, the goals of this section were: (i)
to extend the capabilities of the previously developed path-dependent model to account for
longer sublethal heating times and three different meat species, and (ii) to validate the model
against isothermal and non-isothermal independent data.

3.2

Materials and Methods
This study entailed non-isothermal heat treatment of Salmonella-inoculated meat samples

(turkey, beef, and pork), including treatments designed to impart sublethal injury. The resulting
data were used to estimate the parameters for a novel, path-dependent secondary inactivation

28

model that accounts for sublethal history. The model was then validated and compared to a
traditional, state-dependent model.

3.2.1

Model development
The traditional, state-dependent inactivation model follows a Weibull distribution form

(Peleg 2006):

log S = log

N
= −bt n
N0

(1)

where S is the survivor ratio, N is the number of microorganisms at time t, and N0 is the initial
microorganism population. In this work, the expression “log reduction” will be used, just as is in
the federal regulation; log reduction is equivalent to the negative of the survivor ratio (i.e., -log
S) . The parameters b and n are estimated via non-linear regression, where n describes the shape
of the survival curve. Although any suitable secondary model might be used to describe b as a
function of temperature, Stasiewicz et al. (2008) previously reported, for data similar to the
present study, that b can be described with an Arrhenius-type dependency:


 1
1
b(T ) = bref ⋅ exp− β1
−
 T (t ) T

ref








(5)

where β1 describes the effect of temperature (K) on b. The state-dependent model in this study
was equations (1) and (2). A previously reported, path-dependent inactivation model (Stasiewicz
et al. 2008) takes the following modified form:

 1
1
−
b(T , t ) = bref ⋅ exp − β1 
 T (t ) Tref






 − β τ (T , t )
2




29

(6)

τ (T , t ) =

and

∫

tT = HS

lower
T (t ) − HS lower dt

(7)

tT = HS upper

where the sublethal history (τ) is quantified as the integral of the temperature vs. time curve in
the heat shock region (i.e., from T=HSlower to T=HSupper), where Salmonella can increase its
thermal tolerance (Figure 1). In equation (3), β2 scales the impact of this phenomenon so that
increasing sublethal history (τ) causes a decrease in b. Based on prior research on heat shock
response, Stasiewicz et al. (2008) set the heat shock region to be 38 to 52°C (HSupper and
HSlower). The final, path-dependent model is obtained by combining equations 1, 3, and 4.

Type A

Type B

Temperature (°
C)

52
Sublethal
region

38

25

time (min)

Figure 1. Representative heating profiles, where τ is the integral of the time-temperature
profile within a prescribed heat shock region.

30

3.2.2

Inoculum
The inoculum was an 8-serovar Salmonella cocktail, previously obtained from Dr. V.K.

Juneja (Agricultural Research Service, Eastern Regional Research Center, USDA-ARS,
Wyndmoor, PA), which included: S. Thompson FSIS 120 (chicken isolate), S. Enteritidis H3527
and H3502 (clinical isolates phage types 13A and 4 respectively), S. Typhimurium DT 104
H3380 (human isolate), S. Hadar MF60404 (turkey isolate), S. Copenhagen 8457 (pork isolate),
S. Montevideo FSIS 051 (beef isolate), and S. Heidelberg F5038BGI (human isolate). Before
use, all serovars were kept separately at -80°C in vials containing tryptic soy broth (TSB; Difco
Laboratories, Sparks, MD) and 20% glycerol. Cultures were started by transferring one loop of
each frozen culture into separate tubes containing 9 ml of tryptic soy broth with 0.6% w/v yeast
extract (TSBYE; Difco Laboratories, Sparks, MD) and incubating at 37°C. All serovars were
separately grown for 24-36 h and transferred at least twice at ~24 h before use for inoculation.

3.2.3

Meat preparation
Whole-muscle skinless turkey breast, beef round, and pork loin were obtained from a

local supplier as close to the time of harvest as possible, and transferred to Michigan State
University’s meat laboratory at <4.4°C. The meat was ground (Hobart, model 4146, Troy, OH)
(twice through the 4.8 mm hole plate, then once through the 3.2 mm hole plate), vacuum
packaged in double plastic bags, frozen and kept at -23°C. The packaged and frozen samples
were irradiated to >10kGy (Food Technology Services, Incorporated, Mulberry, FL, 33860) to
eliminate background microflora. The effectiveness of the irradiation was confirmed by thawing
random samples, diluting them (1:5) in sterile 0.1% peptone water (Difco, Becton, Dickinson
and Company, Sparks, MD), and plating them on Petrifilm aerobic count plates (3M
31

Microbiology Products, St. Paul, MN). The irradiated samples were kept frozen until needed.
Prior to inoculation, meat packets were thawed overnight at 4°C. Moisture and fat percentage
measurements were determined on fresh samples using AOAC methods 950.46B and 960.39,
respectively.

3.2.4

Inoculation
On the day of each experiment, a mixture containing 9 ml of each serovar in TSBYE was

centrifuged (6000xg for 20 min at 4°C). The resulting pellet was then resuspended in 7.2 ml of
peptone water to an inoculum population of ~1010 CFU/ml (confirmed by serially diluting in
0.1% peptone water and plating on Petrifilm

TM

aerobic count plates). The inoculum (400 µl)

was manually and aseptically mixed into 40 g of meat for 5 min, targeting a homogeneous
population of ~108 CFU/g in the sample. Individual 1 g samples were then pulled from the 40 g
for the heat treatments.

3.2.5

Heat treatments
All treatments were carried out in a temperature-controlled programmable thermocycler

(ENE Mate, Model FPROGO2G, ISC Bioexpress, Kaysville, UT), with a manufacturer-stated
accuracy of ±0.1°C. For each test, the 1 g sample was divided into 0.2 g portions and inserted
into five 0.2 ml thin-walled PCR microtubes with attached caps (Dot Scientific Incorporated,
Burton, MI). For triplicate testing, fifteen microtubes were placed in the thermocycler and then
equilibrated to 25°C before being subjected to one of the 53 different heating profiles (described
below). Immediately after the heat treatment, all samples were cooled in ice and held at ~4°C for
recovery and plating on the same day.
32

Each heating profile consisted of a randomly selected combination of a linear heating rate
(1, 2, 3, 4, or 7 K/min), a variable-length sublethal holding period (at 40, 45, or 50°C), and a
final lethal temperature (55, 58, 61, or 64°C). The sublethal holding period was determined
depending on the randomly selected sublethal history (τ) target (15, 25, 34, 50, 100, 200, 370, or
500 K·min). The holding time at the lethal temperature was chosen to achieve a nominal target
lethality of ~3 or 5 log reductions. Total treatment times were between 8.17 and 251.92 minutes,
and the full sample set consisted of 159 data points for each species. There were two types of
heating profiles (Figure 1): type A, which included the sublethal holding period and used only a
heating rate of 7 K/min, and type B, which did not include a sublethal holding period, but did use
all of the stated heating rates.

3.2.6

Recovery of samples
For each of the triplicate tests, five 0.2 g cooked subsamples were recovered, recombined

into a 1 g sample, diluted (1:5), and serially diluted in 0.1% sterile peptone water for duplicate
plating on Petrifilm™ aerobic count plates, which were incubated at 37°C for ~48 h before
enumeration. A 5 g sample of uncooked, inoculated meat was diluted (1:5) and plated as a
positive control against which the heated samples were compared to determine the process
lethality (i.e., log reductions). In addition, a 5 g sample of non-inoculated meat was diluted (1:5)
and plated as a negative control to verify meat sterility.

3.2.7

Model parameterization and validation
Before carrying out the state-dependent model parameter estimation on the non-

isothermal data from this study, the Weibull model (equation (1)) and the corresponding
simplified log-linear version (n=1 in equation (1)) were fitted to raw isothermal inactivation data
33

(individual inactivation curves) previously obtained for the same Salmonella cocktail and turkey,
beef, and pork used in this study (Tuntivanich et al. 2008; Breslin 2009; Velasquez et al. 2010)
(nobs=90, 148, and 121 for each species, respectively, from a total of 13 different speciestemperature combinations). Temperatures tested were 55, 58, 60, 62, 62.5, and 63°C. Model
parameters were estimated via non-linear regression (Gauss-Newton with step halving, as used
by JMP, Version 7. SAS Institute Inc., Cary, NC, 1989-2007) for the Weibull model and linear
regression for the log-linear version. Regressions were performed for each temperature and for
each species separately. Following fitting, two tests were conducted to evaluate which model
better described the data: (i) Akaike’s Information Criterion corrected for sample sizes (AICc),
and (ii) a t-test. AICc compares models by creating a balance between goodness-of-fit and the
number of parameters; then it determines the likelihood that one model is better at describing a
set of data than the other (Motulsky and Christopoulos 2004). For example, model A might have
a better goodness-of-fit (represented by the sum of squared errors, SSE) than a simpler model B,
but when AICc is applied, model A is penalized for having more parameters than model B, and
the result might show that the latter is more likely to be correct in describing the data. The t-test
also can be used to test whether a certain parameter estimate is statistically different (α=0.05)
from a fixed value (Bardsley and others 1995), in this case, for n=1 in equation (1). A t-value is
calculated:

t=

fixed value − parameter estimate
estimated parameter standard error

34

(8)

and compared to the t distribution with nobs minus npara degrees of freedom, where nobs is the
total number of observations and npara the number of parameters (Bardsley et al. 1995), to test
the null hypothesis that n=1. Both AICc and the t-test calculations were completed in Excel
(Microsoft Excel. Microsoft, 2003. Redmond, WA). Parameter estimates for each individual
inactivation curve and their corresponding standard errors were obtained from the non-linear and
linear regressions.
To obtain the state-dependent model parameters for all the pooled data, the same
isothermal inactivation data sets were used (nobs=90, 148, and 121, for turkey, beef, and pork,
respectively). Global regressions on the respective data sets were done using completed by
minimizing the sum of squared errors (SSE) using Excel’s Solver function. Error was defined as
the difference between the experimental log reduction and the state-dependent model predicted
lethality (equations 1 and 2). Additionally, the non-isothermal calibration sets developed for this
study (section 3.2.5) were used to calculate new parameters for the state-dependent model.
Parameter estimation for these was done using MATLAB’s nlinfit function (MATLAB R2011a,
The MathWorks Inc., Natick, MA, 2011). Also using a global regression method, this function
estimates the coefficients of a nonlinear function using the least squares estimation via the
Gauss-Newton algorithm (with Levenberg-Marquardt modifications for global convergence). A
sample code can be found in section 6.1. This analysis was carried out to determine whether poor
predictions by the state-dependent model were due to sublethal heating rather than the use of an
isothermally-calibrated model to predict non-isothermal microbial inactivation.

35

For the path-dependent model, all data from the 53 triplicated non-isothermal tests
(nobs=159 for each species) were used. To obtain the model parameters (bref, β1, and β2), 36
randomly selected tests were used (calibration set, nobs=108 for each species). All parameters
were obtained using MATLAB’s nlinfit function, where the error was defined as the difference
between the experimental and the path-dependent model prediction of log reductions (equations
1, 3, and 4). A sample code for this can be found in section 6.1.
Validation of the calibration results for each meat species was carried out against the
remaining 17 non-isothermal tests (validation set, nobs=51 for each species). In addition, the nonisothermal calibrations for both models were validated against the isothermal data to test
whether: (i) the path-dependent model was reducible and applicable to the simpler isothermal
case, and (ii) it was possible to obtain better inactivation predictions by changing from a statedependent to a path-dependent model while using the same inactivation data.

3.3
3.3.1

Results and discussion
Meat sterility and composition
Tests determining irradiation effectiveness returned negative results (i.e., zero plate

counts) for all irradiated samples and negative controls. For the isothermal tests, the product
compositions were as follows: turkey breast was 72.5±0.2% water and 1.0±0.6% fat
(Tuntivanich et al. 2008), beef round was 72.5±1.2% water and 2.7±1.3% fat (Breslin 2009), and
pork loin was 73.6±2.7% water and 2.5±0.9% fat (Velasquez et al. 2010). For the non-isothermal
tests, turkey breast was 74.0±0.9% water and 1.1±0.2% fat, beef round was 73.8±0.3% water and
2.3±0.6% fat, and pork loin was 68.5±0.9% water and 10.0±3.3% fat. Product composition
36

between isothermal and non-isothermal tests was noticeably different only for pork. However,
based on prior results (Juneja et al. 2000a), for a fat difference of 2.5 to 10.0% and the timetemperature combinations used, lethality would be expected to vary at the most by 0.5 log
CFU/g. Therefore, it was assumed for this study, that the estimated variation would not
meaningfully affect the final conclusions.

3.3.2

Calibration: state-dependent model parameterization
AICc results showed that in 8 out of the 13 isothermal treatments (species-temperature

combinations), the log-linear model (n=1 in equation 2) was more likely to be correct than the
Weibull model (n≠1) in describing the isothermal data. Likeliness ranged from 52% to 100%,
with the average being 74%. For the remaining 5 tests where the Weibull model was more likely
to be correct, likeliness ranged from 62% to 99%, with the average being 86%. In average, for
the 13 tests, the log-linear model was 51% more likely to be correct in describing the data. In
addition, even with a relaxed significance level (α=0.1), the t-test did not reject the null
hypothesis (H0: n=1) in 10 out of the 13 tests. The results from both tests do not indicate that the
use of a log-linear model will always give a better outcome, but did give enough evidence that it
was marginally the better choice for most of the data used in this study. Although Stasiewicz et
al. (2008) reported successful use of the Weibull model (n≠1) with the proposed path-dependent
model (equations (1), (3), and (4)), the simpler log-linear version was used in the present study,
given the results of the statistical tests reported above. In either case, the modification and testing
of the path-dependent secondary model (equations (3) and (4)) was the primary objective of this
study, so that confirmation of the performance of equations (3) and (4) with multiple primary
models will further support the underlying construct of the secondary model form.
37

The calculated parameters and accuracy of the model, represented by RMSE and bias
(mean residual), for the state-dependent model with both the isothermal and non-isothermal
calibration sets are shown in Table 1. The reference temperature used was 60°C, as it is
approximately the average of the lethal temperatures used in the treatments. The correlation
coefficients, and standard and relative errors for the path-dependent model parameters (bref, β1,
β2) can be found in section 6.2.

38

Table 1. Model parameters and results against calibration and validation sets.
Validation
against
isothermal
data

Calibration

Parameters
Model

Data source

nobs

Statistics
β2

RMSE

(min )

β1
(K)

bias

(K min )

bref

-1

-1

-1

Validation against
non-isothermal
data (validation
set, nobs=51)

Statistics

Statistics

RMSE

bias

RMSE

bias*

(log CFU/g)

Isothermal turkey

49,315

NA

0.43

0.06

NA

NA

2.91

-1.50

148

1.02

46,829

NA

0.90

-0.03

NA

NA

2.22

-1.25

Isothermal pork

121

0.90

42,590

NA

0.99

-0.06

NA

NA

4.55

-2.58

Non-isothermal
turkey calibration set

108

0.57

48,491

NA

1.42

0.51

1.04

-0.77

1.75

0.80

Non-isothermal beef
calibration set

108

0.57

40,851

NA

1.53

0.56

1.45

-0.98

1.77

0.88

Non-isothermal pork
calibration set

108

0.45

52,382

NA

1.33

0.53

1.94

-1.45

1.49

0.82

Non-isothermal
turkey calibration set
Pathdependent
†

0.97

Isothermal beef
State dependent
♦

90

108

0.91

50,787

0.0017

0.66

0.07

0.46

-0.09

0.90

0.14

Non-isothermal beef
calibration set

108

0.94

44,710

0.0018

0.93

0.12

0.99

-0.19

0.81

0.24

0.18

1.52

-0.73

0.75

0.24

Non-isothermal pork
108
0.70
54,713
0.0016
0.87
calibration set
*bias: mean residual.
♦ state-dependent model was equations (1) and (2), with n=1 in equation (1).
†path-dependent model was equations (1), (3), and (4), with n=1 in equation (1).
39

3.3.3

Validation: application of the state-dependent model to non-isothermal data
When the isothermally-calibrated state-dependent model was applied to the non-

isothermal validation sets, substantial over-prediction errors were observed (Figure 2, Figure 3,
and Figure 4). The largest deviations were -10.1 log CFU/g, -7.5 log CFU/g, and -14.7 log
CFU/g for turkey, beef, and pork respectively ̶ all caused at the largest tested sublethal history
(τ=500 K⋅min). Analysis of variance (ANOVA) revealed a statistically significant relationship
between the sublethal history (τ) and the traditional state-dependent model error (p<0.0001 for
all species), indicating that the state-dependent model error increased as τ increased. This agrees
with previous work (Stasiewicz et al. 2008), and reaffirms the importance of a model that can
account for sublethal injury in a variety of heating profiles. In addition, this concurs with heat
shock literature (Farber and Brown 1990; Knabel et al. 1990; Pagan et al. 1997), in that a
combination of time and temperature in the sublethal region, described by τ, causes an increase
in bacterial thermotolerance. It can be seen (Table 1) that the RMSEs for the isothermallycalibrated, state-dependent model applied to the non-isothermal data are the largest of all model
fittings, and that the corresponding biases are located in the fail-dangerous zone (i.e., bias<0
indicates over-prediction of lethality). These results show that the combined use of isothermal
data and state-dependent models is ineffective for predicting microbial inactivation in cooking
conditions where significant sublethal injury can occur.
The state-dependent model also was fitted to the non-isothermal calibration sets to verify
that it was the model, rather than the isothermal calibration, that caused poor inactivation
predictions. When applied to the non-isothermal validation sets, the RMSEs for this case (Table
1) were reduced from the isothermal calibration model predictions, but still reached values of
40

more than 1.49 log CFU/g. Overall, the biases were ~0.8 log CFU/g, which could be considered
as being fail-safe. Even though application of such a model might mean a safe product, it would
also mean overprocessing, which leads to lower product yield and therefore monetary losses.
These findings confirm that the state-dependent model is unable to effectively describe
inactivation profiles where sublethal heating takes place, even if non-isothermal data are used to
estimate the model parameters.

Log reduction (obs – pred) (log CFU/g)

State-dependent
Path-dependent

Sublethal history (τ, K·min)
Figure 2. Log reduction errors (observed-predicted) for the state-dependent model and
path-dependent model for the turkey calibration set.

41

Log reduction (obs – pred) (log CFU/g)

State-dependent
Path-dependent

Sublethal history (τ, K·min)

Log reduction (obs – pred) (log CFU/g)

Figure 3. Log reduction errors (observed-predicted) for the state-dependent model and
path-dependent model for the beef calibration set.

State-dependent
Path-dependent

Sublethal history (τ, K·min)
Figure 4. Log reduction errors (observed-predicted) for the state-dependent model and
path-dependent model for the pork calibration set.
42

3.3.4

Calibration: path-dependent model parameterization
In comparison to the state-dependent model, the path-dependent model yielded better

RMSEs (Table 1) in the case of beef and pork, and slightly worse values for turkey (0.43 vs. 0.66
log CFU/g), while bias was better for all species.
Evaluation of the residuals from the application of both the state- and path-dependent
models to all calibration sets (Figure 2, Figure 3, and Figure 4) shows that the path-dependent
model performed better than the state-dependent model, with maximum fail-dangerous errors of
-3.3 and -14.7 log CFU/g, respectively. Prediction improvement by the path-dependent model is
especially evident as sublethal history increases, showing the positive effect of accounting for
this phenomenon in the thermal inactivation model.

3.3.5

Validation: application of the path-dependent model to isothermal and non-isothermal
data
The use of the path-dependent model on the non-isothermal validation tests showed a

substantial reduction in prediction error when compared to the state-dependent model (Table 1
and Figures 5-7). All RMSEs were below 0.9 log CFU/g, which translates into a 69%, 63%, and
82% reduction in RMSE from the state-dependent model for turkey, beef, and pork, respectively.
This shows that the path-dependent model provides much improved accuracy, compared to the
state-dependent model when describing data where sublethal heating has occurred. Additionally,
all bias values are in the slight underprediction range (0.14-0.24 log CFU/g), a great
improvement over the high overprediction biases from the state-dependent model (-1.5 to -2.6
CFU/g). Because the negative values are considered fail-dangerous, they show that a state-

43

dependent model cannot be relied upon to effectively predict microbial inactivation where nonisothermal conditions and significant sublethal heating has occurred.
When applied to the isothermal data, the state-dependent model also performed favorably
(Table 1), with RMSEs comparable to those yielded by the isothermally-calibrated statedependent model for turkey and beef, but slightly larger in the case of pork. The bias values
show the same trend, and results are also on the slight overprediction range. These findings
indicate that the path-dependent model is capable to also predict microbial inactivation when
applied to isothermal data. In contrast, when the non-isothermally-calibrated state-dependent
model was applied to the isothermal data, results were notably poorer (Table 1). This shows that
even when the same calibration sets were used, the path-dependent model had a better predictive

Log reduction (obs – pred) (log CFU/g)

ability than the state-dependent model.

State-dependent
Path-dependent

Sublethal history (τ, K·min)
Figure 5. Log reduction errors (observed-predicted) for the state-dependent model and
path-dependent model for the turkey validation set.

44

Log reduction (obs – pred) (log CFU/g)

State-dependent
Path-dependent

Sublethal history (τ, K·min)

Log reduction (obs – pred) (log CFU/g)

Figure 6. Log reduction errors (observed-predicted) for the state-dependent model and
path-dependent model for the beef validation set.

State-dependent
Path-dependent

Sublethal history (τ, K·min)
Figure 7. Log reduction errors (observed-predicted) for the state-dependent model and
path-dependent model for the pork validation set.
45

3.4

Conclusions
This study revealed that the use of a model that considers sublethal thermal history in

addition to the current state of the product can predict microbial lethality with significantly
improved accuracy when applied to cases with or without sublethal heating, and across multiple
meat products, and therefore can be useful to assure the safety of slow-cooked meat products.
All possible combinations between the use of isothermal and non-isothermal calibration
and validation data sets, and the use of a state-dependent or a path-dependent model were
analyzed for turkey, beef, and pork. Results demonstrated that isothermally-calibrated statedependent models, as are typically reported and used for meat products, are ineffective when
predicting lethality in processes where significant sublethal heating has occurred, and can
produce fail-dangerous results that could jeopardize the safety of the products in question. On the
other hand, the non-isothermally-calibrated, path-dependent model effectively described both
isothermal and non-isothermal data sets. Slight fail-safe values resulted when validating against
non-isothermal data, but slight fail-dangerous errors did result when the model was applied to
isothermal data. These systematic errors could indicate that further studies are needed to improve
the function that describes sublethal history (τ), so that it better reflects the complex cellular
processes of stress adaptation.

46

4. MULTI-PRODUCT, MULTI-FACTOR MODEL

4.1

Introduction
As described in the literature review (section 2.4.1.3), there are modeling tools available

to predict bacterial inactivation (i.e., ComBase, PMP, AMI Lethality Spreadsheet). However,
these lack the elements necessary to comply with the lethality performance standards set by
federal regulations, such as addressing Salmonella or considering product characteristics. On the
other hand, secondary models are presented extensively in scientific literature (ICMSF 1996;
FDA/CFSAN 2000), but each addresses a different factor affecting inactivation (in addition to
temperature), and, with few exceptions, none has been validated against industry-relevant data.
Therefore, the problem once again is delegated to processors to find data and/or models that
would be applicable.
Ideally, a universal Salmonella thermal inactivation model applicable to various meat and
poultry products would meet these requirements for the industry. Gathering inactivation data to
understand pathogen inactivation behavior and develop models across product and processes has
been attempted by few researchers, such as van Asselt and Zwietering (2006), Farakos and
Zwietering (2011), and Halder and others (2010). Van Asselt gathered 4066 D-values for
different pathogens at different temperatures in several food products and found that the
parameter variability between sources was greater than that expected to be caused by product
conditions (e.g., fat, pH). Farakos et al. collected DP-values (equivalent to DT-value for high
hydrostatic pressure processes, HPP) across different temperatures, pressures, and microbial
species in an attempt to develop global HPP inactivation models for each pathogen. They found
that the obtained data were highly variable across these factors, especially temperature and
47

species, resulting in model parameter estimations with high standard deviations (e.g., log DPref =
0.27±0.25 log min for Bacillus spp.). Halder defined food groups based on the USDA national
nutrient database food groups and set out to obtain growth and inactivation data for foodborne
pathogens from ComBase and scientific literature, with the goal of developing “comprehensive
food safety prediction software” (Halder et al. 2010) . The study avoided treating pathogenspecific effects as general, or being too specific on process and/or product characteristics.
However, in the case of Salmonella, the inactivation model obtained was not sufficiently precise
to validate industry processes in accordance to federal regulations (FSIS 2001). For example, one
model is specified for red meat, but this food group can be understood to encompass beef, pork,
lamb, and other meat species, not being specific enough to be used to validate a process; on the
other hand, poultry is grouped with baby foods, soups and sauces, vegetables, and seafood,
making the resulting model even less specific than in the red meat case.
Additionally, independent validation of a model is critically important if the model is to
be used for industrial applications. Few studies have considered this step, and most of those
validated against only laboratory-setting data, often with few quantitative measures. For
example, although Halder stated the importance of model validation, only an example using one
model from the study is shown, and no indication is given as to whether the other obtained
models were validated (Halder et al. 2010). Wiegand et al. (2012) did validate an inactivation
model against industry-scale data. Among other objectives, they evaluated the thermotolerance
of E. coli O157:H7 in beef roasts under industry-relevant cooking conditions and compared those
results to predictions by a model with parameters obtained under isothermal conditions. They
reported high variability from the roasts lethality data, and found that the model predictions
48

greatly overestimated lethality, shedding light onto some of the few issues encountered when
doing this kind of validation work. Breslin also conducted industry-relevant validations, by slowcooking turkey, beef, or pork roasts inoculated with Salmonella (Breslin 2009). The study
concluded that replication error significantly increases when scaling up from controlled
laboratory experiments to pilot-scale cooks, which is an important consideration when applying
lethality models to industry-scale processes. While insight from both projects is valuable in terms
of scaled-up data, they do not provide quantitative means to account for the high variability
possible in commercial applications.
Therefore, there is a need for a validated multi-product multi-factor Salmonella thermal
inactivation model applicable for meat and poultry processors to use as means of complying with
federal regulations. Therefore, the objectives of this study were to: (i) gather thermal inactivation
data for Salmonella in poultry, beef, and pork products, (ii) compare multiple statistical methods
to develop a multi-product multi-factor thermal inactivation model and its corresponding
prediction intervals, and (iii) validate candidate models against industry-relevant, pilot-scale
inactivation data.

4.2
4.2.1

Materials and Methods
Data compilation
To develop a multi-product multi-factor thermal inactivation model (i.e., a model suitable

for predicting Salmonella thermal inactivation in various types of meat and poultry products and
under differing conditions), all the relevant published raw data ideally should be compiled. Apart
from laboratory data developed within our research group at Michigan State University, other
data sets were sought in the electronic database ComBase (ComBase 2012) and in scientific
49

literature. To comply with the model goal described above, the data ultimately selected had to
fulfill the following requirements:
1. Data had to describe Salmonella thermal inactivation: log N vs. time or log N/N0 vs. time
at constant or variable temperature. Plots of (a) log N vs. time, or (b) log N/N0 vs. time
with multiple sets indistinguishable from each other on one graph, were not used. This
was because, for (a), it was impossible to determine which data points belonged to each
set and thus obtain the standardized log N/N0 value needed for parameter determination;
and in the case of (b) it would not be possible to distinguish between data sets (see Figure
8), a factor needed for the mixed-effects statistical method (explained in detail in section
4.2.3.2). D- and z-values are the result of fitting raw data to the traditional log-linear
model. Therefore, these values, without raw log N or log N/N0 data, were not sufficient
for the model development.
2. Data from Salmonella Senftenberg were not included, as this serovar is significantly
more heat resistant than the rest of the serovars and is not linked to meat or poultry
outbreaks (Goepfert et al. 1970; CDC 2011; Smith et al. 2001).
3. If experiments were done in replicates, the corresponding number of data points was
expected (i.e., triplicate: 3 log N/N0 points per time point); averaged values were already
manipulated data, so they were not included in the raw data pool.
4. Muscle type (ground or whole), and species (e.g., turkey, beef, or pork) had to be stated.

50

5. Degree of grinding was not considered, as this variable does not affect Salmonella
thermal inactivation (Mogollon et al. 2009).
6. Samples of any size/weight were considered. However, meat had to be inoculated raw
and then cooked to simulate industrial processes.
7. Meat fat percentage (%) had to be reported.
A raw “data set” was considered to be a complete series of log N/N0 vs. time observations at
constant or variable temperature in one experimental trial. Each observation was then considered
to be an individual raw data point (Figure 8).

Data at 58°
C
time (sec)

log N/N0
Data set

0

0

140

-0.7285

280

-1.2047

420

-2.4520

560

-3.3073

Data point

Figure 8. Definition of "data set" and "data point" for raw data.

51

The general approach was to include or exclude data sources following the above
guidelines. The specific search methods in each case are described in detail in the following
sections.

4.2.1.1 MSU laboratory data
The raw data collected from MSU and used for model development were all isothermal
and came from the following experimental studies: (Breslin 2009), (Carlson et al. 2005),
(Mogollon et al. 2009), (Orta-Ramirez et al. 2005), (Tuntivanich et al. 2008), (Velasquez et al.
2010), and (Wesche et al. 2005). For Carlson, only the data with the original product moisture
was considered, as the model will not consider moisture content explicitly. In the case of
Wesche, the data including pre-injured cells were ignored. For the rest, the isothermal data in
both ground and whole muscle were considered.

4.2.1.2 Scientific journals
Relevant data from other research groups were sought electronically via searches of the
Thomson Reuters’ Web of Knowledge (Reuters 2011). Multiple combinations for the following
search terms were used under the fields for Topic and Title:
•

Salmonella

•

Turkey

•

Ground

•

Heat(ing)

•

Beef

•

Whole

•

Thermal

•

Pork

•

Whole muscle

•

Inactivation

•

Poultry

•

Fat

•

Temperature

•

Meat

•

Effect of…

52

•

Resistance

•

Processing

•

Survival

•

Lethality

•

Cooking

•

Inoculated

•

Tolerance

Once all possible articles were listed, a complete copy of each paper was obtained either
electronically or, for those not available online, through the MSU Library, to verify whether met
criteria described in section 4.2.1. From this second evaluation, it was not possible to obtain raw
data from some of the articles, so they were set aside, and their data sought in ComBase (see
section 4.2.1.3). For the remainder, the available data sets were added to the pool to be used in
this study.

4.2.1.3 ComBase Browser
As mentioned in section 2.4.1.3.3, ComBase is an online resource where food
microbiology data are submitted from research groups on a voluntary basis. Although most of
the data describe microbial growth, several sets for thermal inactivation where successfully
obtained. The search parameters used to find relevant data were:
•

Food type: Beef, Pork, Poultry

•

Organisms: Salmonella spp

•

Atmosphere: all selected

•

Preparation: all selected

•

Additives: all selected

•

Other: all selected
53

•

Temperature: 54°C to 120°C

•

pH: 4.0 – 7.5 (default settings)

•

Water activity: 0.71 – 1.0 (default settings)
As with the scientific journals, a second review was done on the initially collected data to

ensure that the data complied with the criteria described in section 4.2.1. In addition, sources
providing only D- and/or z-values were sought in the Web of Knowledge index and/or through
other sources to see whether raw data could be obtained from them.

4.2.2

Model development
As described in section 3.2.1, the primary model can be described with a log-linear

relationship:

log S = log

N
= −bt
N0

(9)

where S is the survival ratio, N is the number of microorganisms at time t, and N0 is the initial
microorganism population. In this work, the expression “log reduction” will be used, just as in
the federal regulation; log reduction is equivalent to the negative of the survivor ratio (i.e., -log
S). For this project, log-linear bacterial inactivation was assumed because, as described in
section 3.3.2, AICc results indicated that this model was superior to the Weibull model (n≠1 in
equation (2)) when describing the source data. Indeed, the results from section 3 show that the
log-linear model performed well in describing both the isothermal and non-isothermal data
(Table 1). Also, there is no known usage of non-log-linear inactivation models in the meat and
54

poultry industry or associated regulations. Therefore, to maximize the likelihood that the results
of this study will be useful, it is reasonable to begin with the log-linear assumption. Additionally,
the purpose of this work is to shed light onto the consequences of gathering data from different
sources, not to address specific effects of tailing and similar non-log-linear phenomena that are
addressed with non-log-linear models. Overall, this study does not suggest that the log-linear
model is always the best choice, but only that it is the best for the data and goals of this study.
Parameter b, the rate of inactivation, can be described by a variety of secondary models,
including an Arrhenius-type secondary model dependency for temperature (Stasiewicz et al.
2008), which is the most important factor for thermal inactivation (Jay et al. 2005):


 1
1
b(T ) = bref ⋅ exp− β1
−
 T (t ) T

ref








(5)

Although other models could also be suitable, the Arrhenius model has been shown to
work in other studies. For example, a section of the study by McQuestin et al. (2009) assessed
the goodness-of-fit between the Arrhenius temperature dependency and “other empirical
equations” when fitted to inactivation data of Escherichia coli in fermented meat; the study
concluded that the Arrhenius version was more adequate for describing the data (RMSE of 1.01
vs. 1.19 log CFU/g for the empirical model). Xu et al. fitted Bigelow and Arrhenius models to
Bacillus anthracis thermal inactivation data, and found that the Arrhenius version described the
2

2

data better (R >0.99 in all strains, vs. R ~0.86-0.94 for the Bigelow model) (Xu and others
2006). Stasiewicz et al. (2008), among other objectives, used Salmonella inactivation data to fit
Arrhenius, log-logistic, and empirical models; results showed that the Arrhenius version gave the
best fit to the model parameters (RMSE of 0.16 log CFU/g vs. 0.45 and 0.23 log CFU/g for log55

logistic and empirical models, respectively). Based on these results and preliminary analyses of
the data included in the study, the Arrhenius temperature dependency was used.
From section 2.3, other factors that can be considered to include in the secondary model
include fat, muscle type, moisture, sublethal injury, and media. From these, moisture, cold shock
injury, and starvation injury were excluded from the model. In the case of moisture content, fat
percentage in the product was sufficient to account for these two factors, as they are inversely
proportional to each other (section 2.3). Cold and starvation shock were excluded because, apart
from the lack of data to obtain their describing parameters, it is not feasible, with current
technology, to quantify them during processes. For media, meat products from turkey, beef, and
pork were chosen due to their dominance in the available data pool. Therefore, the final version
for the secondary model was established as:

b(T , τ , F , M , K , B, P ) =


 1
1 
 − β τ − β F − β M − β Y − β B − β P
−
bref ⋅ exp − β1 
2
3
4
5
6
7
 T (t ) Tref 







(10)

The meaning of each parameter and variable is described in Table 2. Except for the case
of temperature, a simple relationship was chosen for all parameters, as their mathematical
association to lethality has not yet been determined. Additionally, this allowed the model to
remain simple, which was beneficial when individually determining the effects of each factor.

56

Table 2. Description of parameters and variables for secondary model.
Parameter or
Variable

Description

b

b, inactivation rate (min), dependent on T, τ, F, M, Y, B, and P (see definitions
below)
-1

bref

Inactivation rate (min ) at a reference temperature Tref

T

Product temperature (K), dependent on process time (t)

Tref

Reference temperature (K)

τ
F

Meat product fat content (%)

M

Muscle structure of the meat product. M = 1 for ground meat, M = 0 for whole
muscle

Y

Variable takes value of 1 if meat product is poultry. Otherwise, it is 0

B

Variable takes value of 1 if meat product is beef. Otherwise, it is 0

P

Variable takes value of 1 if meat product is pork. Otherwise, it is 0

β1

Temperature parameter (K). Describes the effect of temperature on b

β2

Sublethal history parameter (K·min) . Describes the effect of τ on b

β3

Fat content parameter (% ). Describes the effect of F on b

β4

Muscle type parameter (unitless). Describes the effect of M on b

β5

Species parameter (unitless). Describes the effect of Y on b

β6

Species parameter (unitless). Describes the effect of B on b

β7

4.2.3

Thermal sublethal (injury) history (K·min). Explained in detail in section 3.2.1,
(Stasiewicz et al. 2008)

Species parameter (unitless). Describes the effect of P on b

-1

-1

Non-linear parameter estimation

Two methods for model calibration were used on the different versions derived from
equation (10): the ordinary least squares method, and the mixed-effects method. Each one is
detailed in the following sections. For initial estimates, the parameter values obtained from
57

section 3 were used. In addition, model parameters were recalculated by varying these initial
estimates by at least 500%, and were compared to the original calibration values to test for
parameter sensitivity to initial estimates.
To estimate the sublethal history parameter (β2), the collected isothermal data were not
useful because of the parameter’s nature (equation (4)). Therefore, the parameter values used in
the models were based on those found from the studies in section 3 (Table 1).
The statistical measures of performance obtained from each model fitting were the RMSE
(a measure of the goodness-of-fit), the bias (mean residuals), and the AICc (Akaike’s
information criterion corrected for finite sample sizes, as described in section 3.2.7 ).

4.2.3.1 Ordinary least squares (OLS) method

4.2.3.1.1 Theory
The ordinary least squares method (OLS) tests different parameter values to minimize the
sum of squares of the errors. For these tests, the residuals were defined as the difference between
the experimental log reductions and the log reductions predicted by the model:
Error = log reduction experimental – log reduction predicted

4.2.3.1.2 Parameter estimation
As mentioned in section 4.2.3, the values used for the sublethal injury parameter (β2)
were those estimated from section 3. In addition, for the OLS method, the temperature effect was
accounted for using only the MSU data. This was because:

58

•

The inactivation rates resulting from the MSU data and the information from other sources
were highly different; for example, for ground pork at 60°C, D-value was 6.21 min from a
study by Juneja and others (2000b), while it was 0.99 min from a MSU study (Velasquez et
al. 2010). Using them both together with the OLS method would cause the prediction
intervals to be particularly large, which would not be useful for our purposes.

•

Data differences between sources could not be explained from simple analysis of the data and
acquisition methods, so it was not possible to scale all the data to a common baseline.

•

Most of the data obtained for model calibration and all the data for pilot-scale validation
came from MSU, so by choosing the MSU data to parameterize the temperature effect, better
consistency across data sets would be kept.
This also meant that it would not be possible to estimate the fat parameter (β3), as the

MSU data did not contain studies analyzing the effect of fat on thermal inactivation. Therefore,
the parameter used was obtained with model regression from data that did study this effect
(Juneja et al. 2001; Juneja and Eblen 2000; Juneja and others 2000b). First, an optimum Tref was
calculated (procedure described below) with these data, and then model parameters were
estimated using MATLAB’s nlinfit function (described in section 3.2.7). The β3 used in the final
OLS models was based on the parameter obtained from this procedure.
For parameter estimation, the first step was to find an optimum reference temperature
(Tref) to minimize errors in future calculations (Datta 1993). The process chosen minimizes the
correlation coefficient between the inactivation rate parameter (bref) and the temperature
59

dependency parameter (β1) (Dolan et al. 2007). To do this, the parameters related to Tref (bref
and β1, equation(2)) were estimated several times while varying Tref. For each case, the
correlation coefficient between bref and β1 was plotted against Tref. The optimum Tref is the one
-4

at which the correlation coefficient is a minimum (in this case, r~1x10 ).
Parameter estimation was done using MATLAB’s nlinfit function A sample code can be
found in section 6.3.1. For parameterization, different versions of the “full” model (equation
(10)) were independently calibrated. This is because, even though all factors described by the
model are relevant, the model with the most parameters is not necessarily the best (Zwietering
and den Besten 2011; Motulsky and Christopoulos 2004). Following a naming convention, the
model took the initials of the factors it included (i.e., T F (w) model includes temperature and fat
effects, and uses the whole-muscle data for obtaining parameters). The models tested are shown
in Table 3. Note that the fat and sublethal history parameters (β2 and β3) were obtained
separately as mentioned at the beginning of this section.

60

Table 3. Versions of the "full" model parameterized with OLS method.
Model
♦
name

T (g)

Factors
included

Temperature

Temperature
T F (g)
Fat

T τ (g)

TFτ
(g)

T(w)

TF
(w)

Tτ
(w)

TFτ
(w)

Temperature
Sublethal history

Data set(s) used for
parameterization*

Factors excluded

Fat
Muscle type
Sublethal history
Species

bref, β1

3

bref, β1, β3

3

bref, β1, β2

3

G turkey

Fat
Muscle type
Species

No. of
models

G turkey

Muscle type
Sublethal history
Secies

Parameters
in model

G turkey

G beef
G pork

G beef
G pork
G beef
G pork
G turkey

Temperature
Fat
Sublethal history

Muscle type
Species

W turkey

Temperature

Fat
Muscle type
Sublethal history
Species
Muscle type
Sublethal history
Species

W turkey

Fat
Muscle type
Species

W turkey

Temperature
Fat

Temperature
Sublethal history
Temperature
Fat
Sublethal history

G beef
G pork

W beef

bref,
β1, β2, β3

3

bref, β1

3

bref, β1, β3

3

bref, β1, β2

3

W pork

W beef
W pork
W beef
W pork
W turkey

Muscle type
Species

W beef
W pork

bref,
β1, β2, β3

3

*G: ground muscle, W: whole muscle
♦
(g): model calibrated with ground-muscle data, (w): model calibrated with whole-muscle data
61

4.2.3.1.3 Prediction intervals
A 95% prediction interval (PI) is the region around a predicted value where a new
individual observation is expected to fall with a confidence of 95%. In contrast, a 95%
confidence interval (CI) is the region around a predicted value where the mean of new
observations is expected to fall with a confidence of 95% (Motulsky and Christopoulos 2004).
Although CIs are more commonly used in model predictions, the application of our model in
food safety requires the use of PIs, because food safety goods should be based on the safety of
individual servings, not on the mean.
When using the OLS method, there are two ways to calculate a close approximation of
the asymmetric PIs. The first is to use the nlpredci function in MATLAB, which gives the
asymptotic PIs. Although these are good approximate PIs, their width can be further
approximated to the asymmetric value by further parameter estimation with the ellipse method
via QR decomposition (Bates and Watts 1988), bootstrapping (Mishra and others 2011), or
Monte Carlo (van Boekel 1996) simulations. These will be referred to as the PI methods from
this point forward. As our objective is to estimate PIs useful to industry (i.e., those that are both
reliable and small estimates), the PI methods were preferred over the nlpredci function in
MATLAB.
The end result of the PI methods is a region made of nsim different combinations of
possible parameter values with a 1-α confidence (when α = 0.05, confidence = 95%), where nsim
is the number of simulations executed. Carrying out these methods with two parameters gives a
contouring area; three parameters would mean the creation of a three-dimensional space; using
four parameters would add another dimension, and so on. In level of complexity, the ellipse
62

method is the simplest, followed by bootstrapping, and finally by the Monte Carlo simulations.
For each case of parameter estimation (Table 3), the three methods were carried out in
MATLAB, and compared against each other to determine the best approach for calculating PIs.
Although it is possible to use any number of parameters, studies where the PI methods
are applied on models with more than two parameters are extremely scarce, as the methodologies
for the use of more than two dimensions are not well developed (Dolan 2012). For that reason, a
different empirical approach had to be taken for the model cases with 3 or more parameters. The
two parameters chosen as main factors in the PI methods were bref and β1. This is because the
reference inactivation rate (bref) and temperature (β1) are the most determinant parameters for
thermal inactivation. The PIs calculated with these two parameters will be known as the main PIs
from this point forward. To calculate the PI when the remaining parameters (β2, β3, etc) were
used, it was determined that the main PI would be used and then it would be modified by adding
the remaining parameter confidence intervals to account for their variability. This empirical
approach would inevitably make the PIs wider, but the other option would have been to develop
three- and four- dimensional regions – work that is much beyond the scope of this project.
Once the best PI method was chosen, to obtain the PI for each validation data set, a
prediction was calculated with each of the nsim parameter combinations. This yielded nsim
predictions, which were arranged in increasing (or decreasing) order. The 2.5% and 97.5%
values represented the upper and lower PI for the data set in question.

63

4.2.3.2 Non-linear mixed-effects (NLME) method

4.2.3.2.1 Theory
All statistical models are composed of variables, which change value depending on the
conditions the model is using to estimate (e.g., temperature (T) in equation (6)), and fixed
effects, which are the model parameters – values related to an entire population or with
repeatable levels of experimental factors (e.g., bref in equation (6)) (Pinheiro 2000). In addition,
some models include a random effect, which are values linked with “individual experimental
units drawn at random from a population” (Pinheiro 2000). In the case of this project, an
“experimental unit” would be a data set (Figure 8). A mixed-effects model is a model that
incorporates both fixed and random effects, such as that shown by the simple linear equation
th

(11). Here, yi is the model prediction for the i “experimental unit”, x is the predictor, m and b
th

are the fixed effects (model parameters), and the random effect for the i “experimental unit” is
symbolized by ξi.
=

+

+

(11)
A simple graphical way to represent a mixed-effects model is shown in Figure 9. Here, we have
two different representations of a linear model (y=mx+b). The first shows a “random”
distribution in the y-axis intercept b, while the second shows the same for the lines’ slopes m.
Therefore, in mixed-effects modeling, a random effect would be associated with the intercept
parameter b in the first case, while it would be associated with the slope parameter m in the
second case. When parameterizing any of the model versions derived from equation (10), the
64

fitting software would evaluate the variability caused by the “experimental units” on each of the
model parameters (bref, b1, b2, etc), and determine which one would carry a random effect.

y

y

Random effect in slope parameter

Random effect in intercept parameter
x

x

Figure 9. Graphical representations of mixed-effects models.

For the purpose of developing a multi-product multi-factor model with Salmonella
inactivation data from different sources, the use of a mixed-effects model for repeated measures
is extremely helpful. This is because these types of models are designed to handle data
“generated by observing a number of individuals repeatedly under differing experimental
conditions where the individuals are assumed to constitute a random sample from a population of
interest” (Lindstrom and Bates 1990). Basically, if we assume the “population of interest” to be
all the data describing Salmonella thermal inactivation in meat and poultry products, and a
“random sample of individuals” treated under “differing experimental conditions” to be the data
obtained from different laboratory settings, then the mixed-effects model would be able to
characterize all the data sets accounting for the variability generated by different research groups.

65

4.2.3.2.2 Parameter estimation
Parameter estimation was done using the non-linear mixed-effects (nlme) function from
the R statistical package (R: A Language and Environment for Statistical Computing, Vienna,
Austria, 2011). This function fits a nonlinear mixed-effects model using the formulation and
computational methods described in Lindstrom and Bates (Lindstrom and Bates 1990). A sample
code for this method can be found in section 6.3.2.
To determine which parameters carried a random effect, several versions of the model
were fitted, associating a random effect with one parameter each time. The statistical measures to
determine the most appropriate model were AICc, RMSE, and variance of the random effect. For
AICc and RMSE, the model yielding the lowest values of both would be preferred. For the
variance of the random effect, a large value is desired, because its approximation to zero would
mean there is no unexplained variability between data sets, thus no random effect associated
(Pinheiro 2000). After the final model version was selected, the rest of the data were analyzed.
As with the least squares method (section 4.2.3.1.2), different versions for the “full”
model (equation (7)) were parameterized. The same naming convention was followed; the model
took the initials of the factors it described (e.g., T F (g) model describes temperature and fat
calibrated with ground-muscle data). The models tested are described in Table 4.

66

Table 4. Versions of the "full" model parameterized with mixed-effects method (continued
next page).
Model
♦
name

T (g)

T (w)

TF
(g)

TF
(w)

TFτ
(g)

TFτ
(w)

TFS
(g)

Factors
included

Temperature

Temperature

Temperature
Fat

Temperature
Fat

Data set(s) used for
parameterization*

Factors excluded

Fat
Muscle type
Sublethal history
Species

3

bref, β1

3

bref, β1, β3

3

bref, β1, β3

3

G turkey

Muscle type
Sublethal history
Species

bref, β1

W turkey

Muscle type
Sublethal history
Species

No. of
models

G turkey

Fat
Muscle type
Sublethal history
Species

Parameters
in model

W turkey

G beef
G pork

W beef
W pork

G beef
G pork
W beef
W pork
G turkey

Temperature
Fat
Sublethal history

Muscle type
Species

Temperature
Fat
Sublethal history

Muscle type
Species

Temperature
Fat
Species

Muscle type
Sublethal history

G beef

bref,
β1, β2, β3

3

G pork
W turkey
W beef
W pork

bref,
β1, β2, β3

3

bref,
G turkey, beef, pork

β1, β3, β5,

1

β6, β7

bref,
Temperature
Muscle type
W turkey, beef, pork
1
Fat
β1, β3, β5,
Sublethal history
Species
β6, β7
*G: ground muscle, W: whole muscle
♦
(g): model calibrated with ground-muscle data, (w): model calibrated with whole-muscle data.
Where not present, model was calibrated with both ground- and whole-muscle data.
TFS
(w)

67

Table 4 (cont’d). Versions of the "full" model parameterized with mixed-effects method
(continued next page).
Model
♦
name

Factors
considered

Temperature
T F S τ Fat
(g)
Species
Sublethal history
Temperature
T F S τ Fat
(w)
Species
Sublethal history
TFM

TFM
τ

TFM
S

Temperature
Fat
Muscle type
Temperature
Fat
Muscle type
Sublethal history
Temperature
Fat
Muscle type
Species

Data set(s) used for
parameterization*

Factors left out

Parameters
in model

No. of
models

bref,
Muscle type

G turkey, beef, pork

β1, β2, β3,

1

β5, β6, β7
bref,
Muscle type

W turkey, beef, pork

β1, β2, β3,

1

β5, β6, β7
G + W turkey
Sublethal history
Species

G + W beef
G + W pork
G + W turkey

bref,
β1, β3, β4
bref,

G + W beef

β1, β2, β3,

G + W pork

Species

3

β4

G+W
Turkey, beef, pork

Sublethal history

3

bref,
β1, β3, β4,

1

β5, β6, β7

Temperature
bref,
Fat
TFM
G+W
β1, β2, β3,
1
None
Muscle type
Turkey, beef, pork
τS
β4, β5, β6,
Sublethal history
β7
Species
*G: ground muscle, W: whole muscle
♦
(g): model calibrated with ground-muscle data, (w): model calibrated with whole-muscle data.
Where not present, model was calibrated with both ground- and whole-muscle data.

68

4.2.3.2.3 Prediction intervals
The importance of PIs for our application is described in section 4.2.3.1.3. For the nlme
method, the approach taken was that described by Gelman (Gelman and Hill 2007) (pp.272-275)
with the addition of Monte Carlo simulations. These calculations were also carried out in R. A
sample code for this method can be found in section 6.3.2. The code imported a validation data
set, and computed PIs doing nsim Monte Carlo simulations taking into account the random effect
and residual variances. As described in section 4.2.3.1.3, the simulations were carried out nsim
times, yielding nsim predictions that were arranged in increasing (or decreasing) order to obtain
the 2.5% and 97.5% percentiles, representing the upper and lower PI values for the
corresponding data set.

4.2.4

Model validation against pilot-scale data

Validation of a model against independent data is of utmost importance to determine
model performance. In addition, it provides insight into model suitability in cases where it is
tested outside its application range (Halder et al. 2010). Although manuscripts describing thermal
lethality abound in scientific journals, very few of them validate the resulting models. From these
few studies, most are validated against other laboratory-based data (Mattick and others 2001;
Peleg and others 2007), and the rationale for selecting a specific data set is rarely reported
(Corradini and Peleg 2009; Vaidya and Corvalan 2009; Aragao and others 2007; Peleg et al.
2007). In a few other instances, a validation is carried out, but few statistical parameters to
describe goodness-of-fit or quantify model performance are reported (Vaidya and Corvalan
2009; Porto-Fett and others 2008; Sallami and others 2006). While all these works provide
69

valuable information on their research topics, their results are not directly applicable for
validating industrial processes. If a thermal inactivation model is to be useful for lethality
predictions in independent tests, it is essential that the model be suitably validated. For our
application, it is imperative that the models be tested against data produced under pilot-scale,
industry-like conditions. For these reasons, each model was validated against the pilot-scale sets
of data described in the following sections, from both rapid (i.e., impingement) and slow
convection processes. For each validation data set, the following statistical values were
calculated: RMSE, bias (mean residual), maximum error (fail-safe), minimum error (faildangerous), and percentage (%) of data points that fell within the PIs and above the faildangerous PI.

4.2.4.1 Steaks/fillets and patties in impingement oven
7

8

These tests involved the cooking of Salmonella-inoculated (~10 – 10 CFU/g) wholemuscle cuts, and ground and formed patties in a pilot-scale, moist-air impingement oven, as
reported in detail by Hildebrandt (2012a). Overall, the samples (chicken breast fillets, beef steaks
from boneless round roasts, pork chops from boneless loin, and ground and formed patties of
turkey, beef, and pork) were ~120 g and ~11-12 mm thick, with fat contents of 0.33 to 10%.
There were 6 different cooking treatments, which combined conditions relevant to commercial
applications: oven air temperature (149 or 204°C), humidity (20 or 50% moisture by volume),
and target lethality (4 or 6 log reduction, in order to have reliable survivor counts). The full
factorial combination of the treatments (in triplicate for steaks/fillets, in duplicate for patties)
yielded a total of 144 data points, with total cooking times of 4 to 11 min. All raw data obtained

70

from these experiments complied with the criteria described in section 4.2.1, and were comprised
of log reduction results for the product cores.

4.2.4.2 Whole-muscle roasts

These experimental data, from Breslin (2009), resulted from pilot-scale, slow cooking of
6.5

Salmonella-inoculated (~10

CFU/g at the core) turkey breast, beef round, and pork loin roasts

in a commercial, moist-air convection oven. Roast sample size was ~680 g. There were 7
different cooking schedules representing industry processes, which combined the following
parameters: cook (in-bag or out-of-bag), time (total cook time 86-253 min), oven temperature
(constant at 93.3°C or ramp-up), and humidity (20, 50, or 78% RH). All raw data obtained from
these experiments complied with the parameters described in section 4.2.1. However, only the
data in which cooks were done to a specified lethality were taken into account, ignoring those
which reached 71.1°C, because those generally yielded no survivors.

4.2.4.3 Hot dogs

For these experiments, commercially formulated beef and turkey emulsions (batter) were
8

acquired from a local processor, inoculated with Salmonella (~10 CFU/g), vacuum-stuffed, and
cooked in a pilot-scale convection oven using a cook schedule for low-fat hot dogs similar to that
used in industry. Samples were ~60 g (~15.5 cm long, 2 cm diameter) and had a thermocouple
inserted for temperature logging. The cooking cycle increased temperature (60 to 82 °C) and
humidity (38 to 79%RH) over ~145 min. Cooking was stopped by quenching the samples in
liquid nitrogen when the data-logger signaled a predicted lethality of 4 or 6 log. All raw data
71

obtained from these experiments complied with the parameters described in section 4.2.1. A
more detailed description of the tests can be found in section 0.

4.3
4.3.1

Results and discussion
Data compilation results

Table 5 shows a description of the data obtained from the MSU studies. Although the
experiments did not include the effect of fat percentage (%) on Salmonella inactivation, the data
provided valuable information with respect to muscle type.

Table 5. MSU data characteristics by source.
Reference

Breslin (2009)
Carlson et al. (2005)
Mogollon et al. (2009)
Orta-Ramirez et al. (2005)
Tuntivanich et al. (2008)
Velasquez et al. (2010)

Species and
muscle type

Data
points*

Data
sets*

Test
temperatures

Fat %

Ground beef

156

21

Whole beef

164

25

55, 58, 60, 62,
63°C

2.7%

Ground turkey

22

4

60°C

1.8%

Ground beef

67

7

Whole beef

22

2

60°C

4.5%

Ground beef

80

10

Whole beef

27

6

55, 60, 62.5°C

5.6%

Ground turkey

90

9

Whole turkey

67

9

55, 60, 62.5°C

1%

Ground pork

121

16

Whole pork

120

15

55, 58, 60, 62,
63°C

2.5%

Wesche et al. (2005)
Ground turkey
48
6
60°C
*See Figure 8 for illustration and definition of data point and data set (section 4.2.1).

72

1.5%

Regarding data from other research groups, a total of 36 journal articles from 12 different
journals were considered possible sources for inactivation data. After applying the criteria set in
section 4.3.1, 4 journal articles were selected for our database, with all of them directed to
ComBase for raw-data search.
From the 43,153 total data sets available in ComBase at the time of the initial search, 419
matched the characteristics from section 4.2.1.3. From these, 106 data sets were selected
according to the criteria previously described for the study (section 4.2.1). The raw data missing
from scientific articles (section 4.2.1.2) were available to download from ComBase, allowing the
completion of the data pool.
Because these data were ultimately pooled with the combined searches from previous
literature and ComBase, it is fitting to present the results as one (Table 6). While muscle type
was not addressed in these studies, the data sets provide beneficial information regarding fat
percentage (%) in the meat products.

Table 6. Scientific journals and ComBase Browser data characteristics by source.
Reference

Species and
muscle type

Data
points

Data
sets

Test
temperatures

Fat %

Juneja et al. (2001)

Ground turkey

244

32

58, 60, 62.5, 65°C

1, 7, 10, 12%

Juneja (2003)

Ground beef

44

6

55, 57.5, 60°C

25%

Juneja and Eblen
(2000)

Ground beef

100

12

58, 60, 62.5, 65°C

7, 12, 18, 24%

Ground beef

202

24

58°C

7, 12, 18, 24%

Ground pork

268

32

58, 60, 62.5, 65°C

4, 10, 24, 28%

Juneja et al.
(2000b)

73

Table 7 shows the characteristics for all the data compiled. Beef comprised most of the
collected data, particularly the ground product, while the least common data were for wholemuscle turkey, with only 9 data sets with 67 data points. Both the MSU data and those from
other sources included the temperature effect on Salmonella inactivation with a useful range of
temperatures. Generally, the MSU data focused on assessing the effect of muscle type, while
those from other sources included varying fat content. This shows that, depending on the
application, merging results from different studies could be beneficial in trying to explain
different phenomena that have not been studied together. In this case, the merged data showed
great variability across source studies, due to different sample preparation, processing, and
recovery methods. Therefore it was decided that both methods of parameter estimation would be
used to obtain the multi-product multi-factor model: the standard non-linear estimation (OLS)
(section 4.2.3.1), and the mixed-effects method (section 4.2.3.2).

Table 7. Final compiled data characteristics and sources (continued next page).
Data characteristics

MSU data

Scientific journals and
ComBase Browser

All sources

All data
Data points

995

908

1903

Data sets

134

106

240

Temperature range

55 - 63°C

55 - 65°C

55 - 65°C

Fat % range

1 – 5.6%

1 – 28%

1 – 28%

Ground Turkey
Data points

160

244

448

Data sets

19

32

59

55 – 62.5°C

58 - 65°C

55 - 65°C

1 – 1.8%

1 – 12%

1 – 12%

Temperature range
Fat % range

74

Table 7 (cont’d). Final compiled data characteristics and sources.
Data characteristics

MSU data

Scientific journals and
ComBase Browser

All sources

Whole Turkey
Data points

67

-

67

Data sets

9

-

9

55 – 62.5°C

-

55 – 62.5°C

1%

-

1%

Temperature range
Fat % range

Ground Beef
Data points

303

352

655

Data sets

38

42

80

Temperature range

55 - 63°C

55 - 65°C

55 - 65°C

Fat % range

2.7 – 5.6%

7 – 25%

2.7 – 25%

Whole Beef
Data points

224

-

224

Data sets

29

-

29

Temperature range

55 - 63°C

-

55 - 63°C

Fat % range

2.7 – 5.6%

-

2.7 – 5.6%

Ground Pork
Data points

121

268

389

Data sets

16

32

48

55 - 63°C

58 – 65°C

55 - 65°C

2.5%

4 – 28%

2.5 – 28%

Temperature range
Fat % range

Whole Pork
Data points

120

-

120

Data sets

15

-

15

55 - 63°C

-

55 - 63°C

2.5%

-

2.5%

Temperature range
Fat % range

75

4.3.2

OLS model parameters, statistics, and PIs

The reference temperatures obtained for each data set are shown in Table 8. As expected,
the reference temperature falls approximately in the middle of the lethal temperature range data
in the data pool. The obtained parameters and corresponding statistics for the OLS model are
shown in Table 9. Note that the RMSE and bias for the models does not change by the addition
of the estimated β2, the sublethal injury parameter from section 3, as the tests are carried out in
temperature ranges above that of the sublethal injury region. On the other hand, β2 should prove
useful when validating the pilot-scale data, especially the longer-cook roasts. The standard and
relative errors for the model parameters of all model versions, including the fits made to obtain
β3, can be found in section 6.5.1. The correlation coefficients can be found in section 6.6.1.

Table 8. Reference temperatures by data set.
Data

Reference temperature (Tref, °C)*

Ground turkey

59.5943

Ground beef

59.7356

Ground pork

59.0460

Whole turkey

58.9800

Whole beef

59.4870

Whole pork

59.2068

*Temperature in models is in K.

76

Table 9. Parameters and statistics for OLS models (calibrated with only MSU data).
Parameters
Data
source for
calibration

Model

bref
min

-1

β1

β2*

K

K

-1

Statistics

β3**
fat %

-1

RMSE

Bias

log (CFU/g)

T (g)

1.11

52,269

-

-

1.21

-0.04

T F (g)

1.11

52,269

-

0.0300

1.24

-0.13

T τ (g)

1.11

52,269

0.0017

-

1.21

-0.04

T F τ (g)

1.11

52,269

0.0017

0.0300

1.24

-0.13

T (w)

0.37

48,589

-

-

0.42

0.003

T F (w)

0.37

48,589

-

0.0300

0.43

-0.03

T τ (w)

0.37

48,589

0.0017

-

0.42

0.003

T F τ (w)

0.37

48,589

0.0017

0.0300

0.43

-0.13

T (g)

0.83

44,242

-

-

0.77

-0.02

T F (g)

0.83

44,242

-

0.0227

0.80

-0.18

T τ (g)

0.83

44,242

0.0018

-

0.77

-0.02

T F τ (g)

0.83

44,242

0.0018

0.0277

0.80

-0.18

T (w)

0.44

44,799

-

-

0.87

-0.03

Whole beef T F (w)

0.44

44,799

-

0.0227

0.87

-0.16

T τ (w)

0.44

44,799

0.0018

-

0.87

-0.03

T F τ (w)

0.44

44,799

0.0018

0.0277

0.87

-0.16

T (g)

0.63

41,750

-

-

0.99

-0.05

T F (g)

0.63

41,750

-

0.0137

0.99

-0.13

T τ (g)

0.63

41,750

0.0016

-

0.99

-0.05

T F τ (g)

0.63

41,750

0.0016

0.0137

0.99

-0.13

T (w)

0.45

47,164

-

-

1.03

-0.04

Whole pork T F (w)

0.45

47,164

-

0.0137

1.03

-0.12

T τ (w)

0.45

47,164

0.0016

-

1.03

-0.04

1.03

-0.12

Ground
turkey
nobs = 160

Whole
turkey
nobs = 67

Ground
beef
nobs = 303

nobs = 224

Ground
pork
nobs= 121

nobs =120

0.45
47,164
0.0016
0.0137
T F τ (w)
*Parameters obtained from sublethal injury studies, Chapter 3, Table 1.
**Obtained from calibrating only data from Table 6.
77

Regarding the PIs, the contours produced by all methods (ellipse, bootstrap, Monte
Carlo) were essentially the same in all cases (ground turkey in Figure 10, the rest in section
6.7.1), and so the PIs generated were also extremely similar, with <0.05 log N/N0 difference
between them (ground turkey in Figure 11 and Figure 12), the rest in section 6.7.2). Therefore, to
avoid unneeded complexity, it was decided to use the results from the ellipse method to

-1

bref (min )

determine the final PIs for each model.

β1 x10

-4

(1/K)
Figure 10. PI methods parameter contours for ground turkey calibration set.

78

Asymptotic PI

log N/N0 (CFU/g)

Zoom-in area,
Figure 12

Prediction line

Ellipse,
bootstrapping,
Monte Carlo PIs

time (min)
Figure 11. PIs with all methods for ground turkey calibration data set.

Asymptotic PI
log N/N0 (CFU/g)

Ellipse PI
Bootstrap PI
Monte Carlo PI

time (min)
Figure 12. Zoom-in section from Figure 11.

79

4.3.3

Mixed-effects model parameters and statistics

Table 10 shows the parameters and statistics for the different model versions obtained
with the mixed-effects method. In the cases where the sublethal injury parameter (β2) was added,
the RMSE did not change, as mentioned in section 4.3.2, and the AICc could not be computed.
This is because the number of observations (nobs) changes, but the RMSE does not take this into
account; therefore, the AICc that would be obtained does not accurately represent the model in
question. Standard and relative errors for the all the parameters in the different model versions
can be found in section 6.5.2, while the correlation coefficients are in section 6.6.2, and the
random effects for all data sets (groups) in the same model versions are in section 6.8.

80

Table 10. Parameters and statistics for mixed-effects model (continued next page).
Parameters
Data
source for
calibration

nobs

G turkey

404

51

0.67

50,750

-

G beef

649

80

0.56

44,710

G pork

389

48

0.43

W turkey

67

9

W beef

224

W pork
G turkey

β5

β6

β7

RMSE

•

•

•

•

log
(CFU/g)

•

-

-

-

-

-

0.38

392

-

-

-

-

-

-

0.70

714

53,950

-

-

-

-

-

-

0.56

377

0.59

50,750

-

-

-

-

-

-

0.43

20

29

0.54

44,710

-

-

-

-

-

-

0.62

145

120

15

0.65

53,950

-

-

-

-

-

-

0.77

115

404

51

1.58

36,470

-

0.21

-

-

-

-

0.38

294

T F (g) G beef

649

80

0.95

36,320

-

0.06

-

-

-

-

0.70

640

G pork

389

48

0.74

35,940

-

0.06

-

-

-

-

0.55

313

W turkey

67

9

1.36

44,710

-

0.90

-

-

-

-

0.45

23

W beef

224

29

0.80

44,710

-

0.13

-

-

-

-

0.67

142

W pork

120

15

1.05

53,950

-

0.19

-

-

-

-

0.77

117

T (g)

T (w)

TF
(w)
♦

ngroups

bref
-1

min

K

β2

β3

♦

β4

Model

β1

Statistics

-1

K·min

fat %

-1

RMSE values for models containing τ are repeated from the same models without τ (see text).

81

AICc

Table 10 (cont’d). Parameters and statistics for mixed-effects model (continued next page).
Parameters
Model

Data
source for
calibration

nobs*

ngroups
*

bref
-1

min

β1

β2**

K

β3

K⋅min

-1

fat %

Statistics

β4
-1

β5

β6

β7

RMSE

AICc

•

•

•

•

log
(CFU/g)

•

51+36

1.58

36,470

0.0018

0.21

-

-

-

-

0.38

NA

G beef

649+108

80+36

0.95

36,320

0.0018

0.06

-

-

-

-

0.70

NA

389+108

48+36

0.74

35,940

0.0016

0.06

-

-

-

-

0.55

NA

W turkey

67+108

9+36

1.36

44,710

0.0018

0.90

-

-

-

-

0.45

NA

W beef

224+108

29+36

0.80

44,710

0.0018

0.13

-

-

-

-

0.67

NA

W pork

TFτ
(w)

404+108

G pork

TFτ
(g)

G turkey

120+108

15+36

1.05

53,950

0.0016

0.19

-

-

-

-

0.77

NA

TFS
(g)

G turkey,
beef, pork

1442

179

1.53

49,800

-

0.07

-

0.83

0.78

1.09

0.58

1409

TFS
(w)

W turkey,
beef, pork

358

53

5.20

45,690

-

0.13

-

2.15

1.88

1.79

0.70

267

T F S τ G turkey,
beef, pork
(g)

1442 +
324 ˟

179+108
˟

1.53

49,800

0.0017

†

0.07

-

0.83

0.78

1.09

0.58

NA

T F S τ W turkey,
beef, pork
(w)

358 +
324˟

53+108˟

5.20

45,690

0.0017

†

0.13

-

2.15

1.88

1.79

0.70

NA

*108 data points correspond to the data used to obtain β2 in section 3. These correspond in turn to 36 data groups.
** Parameters obtained from sublethal injury studies, Chapter 3, Table 1.
†
Average from the 3 species; individual values obtained from Table 1.
˟324 data points correspond to the data used to obtain β2 in section 3 (for the 3 species). These correspond in turn to 108 data groups.

NA: not applicable for models containing τ, as nobs is modified (see text).
82

Table 10 (cont’d). Parameters and statistics for mixed-effects model.
Parameters
Model

Data source
for
calibration

nobs*

ngroups
*

bref
-1

min

β1

β2**

K

β3

K⋅min

-1

fat %

Statistics

β4
-1

β5

β6

β7

RMSE

AICc

•

•

•

•

log
(CFU/g)

•

G+W
turkey

471

60

0.52

36,860

-

0.21

-1.14

-

-

-

0.39

321

G + W beef

873

109

0.61

37,490

-

0.06

-0.45

-

-

-

0.68

791

G + W pork

509

63

0.67

40,300

-

0.07

-0.14

-

-

-

0.61

428

G+W
turkey

471+108

60+36

0.52

36,860

0.0018

0.21

-1.14

-

-

-

0.39

NA

G + W beef

873+108

109+36

0.61

37,490

0.0018

0.06

-0.45

-

-

-

0.68

NA

G + W pork

509+108

63 + 36

0.67

40,300

0.0016

0.07

-0.14

-

-

-

0.61

NA

TFM
S

G+W
turkey, beef,
pork

1853

232

1.09

49,410

-

0.06

-0.51

0.65

0.65

0.86

0.59

1612

TFM
Sτ

G+W
turkey, beef,
pork

1853 +
324˟

232 +
108˟

1.09

49,410

0.0017

0.06

-0.51

0.65

0.65

0.86

0.59

NA

TFM

TFM
τ

†

*108 data points correspond to the data used to obtain β2 in section 3. These correspond in turn to 36 data groups.
** Parameters obtained from sublethal injury studies, Chapter 3, Table 1.
†
Average from the 3 species; individual values obtained from Table 1.
˟324 data points correspond to the data used to obtain β2 in section 3 (for the 3 species). These correspond in turn to 108 data groups.

NA: not applicable for models containing τ, as nobs is modified (see text).
83

4.3.4

Validation against pilot-scale data

Validation for all developed models against the independent pilot-scale data is shown in
the following sections. The comparisons to consider when determining model performance are
those within the same types of products, i.e., ground-muscle calibrated models against groundmuscle data and not against whole-muscle data. However, current limitations in practice might
entail use of ground-muscle calibrated models to validate whole-muscle products (because of the
lack of adequately-calibrated models). These comparisons were also carried out to showcase
possible outcomes. When validating turkey-calibrated models against the impingement oven
whole-muscle samples, the cooked poultry product was chicken instead of turkey. In all cases,
the maximum (+, most fail-safe) and minimum (-, most fail-dangerous) errors, and plots showing
model predictions for representative data sets can be found in section 6.9.
Given the various statistical measures of model performance, it was considered for this
study that an ideal (although impossible) model would predict lethality with RMSE and bias of
0.0 log CFU/g, i.e., a perfect fit, PI width would be near zero, and percentage of data points
inside it would be 100% (Figure 13). However, due to the inherent variability and experimental
error in real-world tests, a best practically possible and industry-useful model would present a
low prediction RMSE (say ~1.0 log CFU/g), positive bias to avoid fail-dangerous errors,
moderately narrow 95% PIs (say ~±1.5 log CFU/g), and 100% of the predicted data points
captured by the 95% PI (Figure 14). When a model like this is not obtained, one yielding
conservative predictions could also be highly functional (Figure 15). Also, a model with all data
points above the fail-dangerous PI band could still be useful, depending on the other statistical
parameters. For example, with a model capturing 100% of the data points above the fail84

dangerous PI band, but with a wide PI, negative bias and/or high RMSE, the model user would
need to determine whether the width of the PI is useful for a particular process or not (Figure 16
and Figure 17). In this case, bias and RMSE would be secondary parameters to consider, as there
can be a highly negative bias or a highly scattered data set (high RMSE); however, with all
points above the fail-dangerous PI band and an acceptable PI width, process safety can still be
assured. On the other hand, a model of no use would not necessarily be the complete opposite of
the ideal model; it suffices to have most of the data points below the fail-dangerous PI band for
the model to be unacceptable, regardless of PI width (Figure 18).

95% PI widthavg= ±0.21 log CFU/g
RMSE = 0.0 log CFU/g
bias = 0.0 log CFU/g
% above low PI = 100%

Figure 13. Example performance of an ideal model.

85

95% PI widthavg= ±1.63 log CFU/g
RMSE = 0.81 log CFU/g
bias = 0.24 log CFU/g
% above low PI = 100%

Figure 14. Example performance of a practically possible ideal model.

95% PI widthavg= ±1.76 log CFU/g
RMSE = 1.07 log CFU/g
bias = 0.97 log CFU/g
% above low PI = 100%

Figure 15. Example performance of a conservative, but possibly useful model.

86

RMSE = 1.40 log CFU/g
bias = 0.30 log CFU/g
95% PI widthavg= ±2.11 log CFU/g
% above low PI = 100%

Figure 16. Example performance of a potentially useful model; all data points fall above
the fail-dangerous PI, but PI width would need to be evaluated by the user to determine
usefulness, especially as several data points are quite close to the lower PI band.

RMSE = 1.72 log CFU/g
bias = -1.38 log CFU/g
95% PI widthavg= ±2.77 log CFU/g
% above low PI = 100%

Figure 17. Example performance of a possibly useful model; all data points fall above the
fail-dangerous PI, but bias is highly negative, and PI width would need to be evaluated by
the user to determine usefulness.

87

RMSE = 3.44 log CFU/g
bias = -3.25 log CFU/g
95% PI widthavg= ±2.49 log CFU/g
% above low PI = 22%

Figure 18. Example performance of a useless model. In addition to not capturing all data
points above the fail-dangerous PI band, notice the width of the PI.

In the next sections, model performance on the pilot-scale data is analyzed based on the
criteria described above.

4.3.4.1 Steaks/fillets and patties

Tables 11-13 show the OLS models’ performance when validated against the
impingement-cooked products (whole-muscle and ground-and-formed poultry, beef, and pork).
Overall, as expected, the whole-muscle-calibrated models fared better than their ground-musclecalibrated counterparts when predicting lethality in whole-muscle products. This can be
attributed to the fact that the whole-muscle model parameters reflect the significantly greater
thermal resistance of Salmonella in whole-muscle than in ground products (Tuntivanich et al.
2008; Orta-Ramirez et al. 2005; Velasquez et al. 2010). In both the ground-muscle calibrated and
the whole-muscle calibrated versions, and against most of the validation data sets, the addition of
88

the fat and sublethal injury parameters (β3 and β2, respectively) improved the models’
performance; decreased RMSE, “pulled” all points to the fail-safe side of prediction, which
improved the bias and the percentage of observations that fell above the lower PI. In some cases,
for example, the ground beef samples, the RMSE did not improve when β2 and/or β3 were
added, but the percentage of points falling above the lower PI did, which is more important for
the food safety application of this project. In other cases, for example the whole chicken samples,
this percentage did not change, but the bias was improved, meaning that the predictions were less
fail-dangerous. Finally, in all cases, the addition of the fat parameter (β3) was fairly more
beneficial to the model prediction than the addition of the sublethal injury parameter (β2). This is
especially noticeable in the ground pork samples (10% fat) where RMSE improved by 0.4 log
CFU/g and bias by 0.6 log CFU/g with the addition of the fat content term (Figure 19 and Figure
20). From these observations, it was concluded that the consideration of fat percentage and
sublethal injury by the models was beneficial to their predictions of the independent validation
results.

89

Table 11. OLS models validated against chicken fillets and turkey patties (impingement
cooked).
Average 95%
PI (+/- value)

RMSE

bias

% above
low PI

% that
fit in PI

T (g)

27

27

2.61

4.51

-3.94

T F (g)

27

27

2.60

4.36

-3.80

T τ (g)

27

27

2.60

4.41

-3.84

T F τ (g)

27

27

2.60

4.26

-3.70

T (g) + T (w)

52

34

1.83

3.59

-1.99

T F (g) + T F (w)

52

34

1.81

3.44

-1.86

T τ (g) + T τ (w)

52

34

1.82

3.52

-1.91

T F τ (g) +
T F τ (w)

55

34

1.81

3.37

-1.79

T (g)

26

26

2.62

4.79

-4.17

T F (g)

26

26

2.59

4.58

-3.96

nobs = 23
fat = 1.05%

T τ (g)

26

26

2.61

4.69

-4.07

τavg =
8.77 K·min

T F τ (g)

26

26

2.58

4.48

-3.86

T (g)

29

29

2.60

4.17

-3.68

T F (g)

29

29

2.60

4.11

-3.62

T τ (g)

29

29

2.59

4.08

-3.59

T F τ (g)

29

29

2.60

4.01

-3.52

T (w)

81

43

0.96

1.34

0.41

T F (w)

81

43

0.95

1.34

0.44

T τ (w)

81

43

0.95

1.35

0.45

Data source
for validation

Ground turkey
+
Whole chicken
nobs = 44

Ground turkey

Whole chicken
nobs = 21
fat = 0.33%
τavg =
7.92 K·min

Model*

log (CFU/g)

86
43
0.95
1.35
0.47
T F τ (w)
*where (g)+(w) models appear, ground-muscle calibrated models predicted lethality for groundmuscle data and whole-muscle calibrated models did for whole-muscle data.

90

Table 12. OLS models validated against beef steaks and patties (impingement cooked).
Average 95%
PI (+/- value)

RMSE

bias

% above
low PI

% that
fit in PI

T (g)

70

66

1.56

1.88

-0.93

T F (g)

77

70

1.55

1.72

-0.69

T τ (g)

75

70

1.56

1.84

-0.86

T F τ (g)

80

73

1.55

1.69

-0.62

T (g) + T (w)

91

80

1.65

1.35

0.20

T F (g) + T F (w)

95

75

1.64

1.38

0.38

T τ (g) + T τ (w)

95

82

1.65

1.36

0.25

T F τ (g) +
T F τ (w)

95

73

1.64

1.39

0.43

T (g)

84

74

1.54

1.45

-0.07

T F (g)

95

79

1.54

1.44

0.12

nobs = 19
fat = 2.32%

T τ (g)

95

84

1.54

1.45

-0.01

τavg =
9.28 K·min

T F τ (g)

95

79

1.54

1.45

0.18

T (g)

60

60

1.57

2.14

-1.58

T F (g)

64

64

1.56

1.91

-1.30

T τ (g)

60

60

1.57

2.09

-1.51

T F τ (g)

68

68

1.56

1.86

-1.23

T (w)

96

84

1.73

1.27

0.41

T F (w)

96

72

1.73

1.33

0.58

T τ (w)

96

80

1.73

1.29

0.45

Data source
for validation

Ground +
Whole beef
nobs = 44

Ground beef

Whole beef
nobs = 25
fat = 2.68%
τavg =
8.18 K·min

Model*

log (CFU/g)

96
68
1.73
1.35
0.62
T F τ (w)
*where (g)+(w) models appear, ground-muscle calibrated models predicted lethality for groundmuscle data and whole-muscle calibrated models did for whole-muscle data.

91

Table 13. OLS models validated against pork steaks and patties (impingement cooked).
Average 95%
PI (+/- value)

RMSE

bias

% above
low PI

% that
fit in PI

T (g)

77

75

2.06

1.97

-1.30

T F (g)

79

75

2.02

1.79

-0.88

T τ (g)

75

73

2.05

1.98

-1.13

T F τ (g)

79

75

2.02

1.76

-0.82

T (g) + T (w)

86

82

2.09

1.89

-0.76

T F (g) + T F (w)

91

84

2.06

1.66

-0.43

T τ (g) + T τ (w)

88

84

2.08

1.86

-0.71

T F τ (g) +
T F τ (w)

91

84

2.06

1.65

-0.40

T (g)

78

78

2.05

2.06

-1.53

T F (g)

89

85

2.00

1.64

-0.98

T τ (g)

81

81

2.05

2.01

-1.47

T F τ (g)

89

85

2.00

1.61

-0.93

T (g)

83

79

2.12

1.95

-0.82

T F (g)

69

66

2.04

1.92

-0.78

T τ (g)

69

66

2.05

1.94

-0.82

nobs = 29
fat = 1.53%

T F τ (g)

69

66

2.04

1.89

-0.72

T (w)

93

86

2.12

1.71

-0.04

τ avg =
8.21 K·min

T F (w)

93

83

2.11

1.69

0.08

T τ (w)

93

86

2.11

1.71

0.00

Data source
for validation

Ground +
Whole pork
nobs = 56

Ground pork
nobs = 27
fat = 10%
τavg =
9.34 K·min

Whole pork

Model*

log (CFU/g)

93
83
2.11
1.69
0.08
T F τ (w)
*where (g)+(w) models appear, ground-muscle calibrated models predicted lethality for groundmuscle data and whole-muscle calibrated models did for whole-muscle data.

92

RMSE = 2.06 log CFU/g
bias = -1.53 log CFU/g
95% PI widthavg= ±2.05 log CFU/g
% above low PI = 78%

95%

Figure 19. OLS T (g) model validated against ground pork data. Compare with Figure 20.

RMSE = 1.61 log CFU/g
bias = -0.93 log CFU/g
95% PI widthavg= ±2.0 log CFU/g
% above low PI = 89%

Figure 20. OLS T F τ (g) model validated against ground pork data. Notice that by
considering fat content (F) and sublethal history (τ), model performance is improved.

93

In the poultry products, the PIs for the whole-muscle validated models also performed
better than their ground-muscle counterparts, in terms of data points above the fail-dangerous PI
band (~80% vs. ~25%) (Figure 21 and Figure 22). However, for beef and pork, both model
versions showed high percentages for this measure (>80%) in their corresponding validation data
sets. Additionally, the prediction interval “widths” were different across species and whole- or
ground-muscle validated models. For example, for turkey, the ground-muscle-validated models
PIs were more than twice as wide than their whole-muscle-validated counterparts (~2.60 vs
~0.95 log CFU/g). In the case of beef, PI width for the ground-muscle-validated models was only
~0.4 log CFU/g narrower than their whole-muscle-validated partners’; and finally for pork, they
were essentially the same. While these interval widths evidently have an effect on the percentage
of data points falling above the fail-dangerous PI band, they are mostly a consequence of the
source data used to calibrate the models. Also, for our application, the choice of a model can be
balanced between percentage of data points above the fail-dangerous PI band and the PI width.

94

RMSE = 4.48 log CFU/g
bias = -3.86 log CFU/g
95% PI widthavg= ±2.59 log CFU/g
% above low PI = 26%

95%

Figure 21. OLS T F τ (g) model validated against ground turkey data. Compare with
Figure 22.

RMSE = 1.35 log CFU/g
bias = 0.47 log CFU/g
95% PI widthavg= ±0.95 log CFU/g
% above low PI = 86%

95%

Figure 22. OLS T F τ (w) model validated against whole-muscle chicken breast data.

95

Table 14 shows model predictions when all the impingement oven data were pooled
together. As with the individual species and muscle samples, the addition of the fat and sublethal
injury parameters (β2 and β3, represented by F and τ in model names) improved model
performance. However, substantial differences in the performance parameters were not evident,
meaning, as expected, that temperature is the most influential factor in determining process
lethality. On the other hand, the fat and sublethal injury factors may influence model predictions
significantly more in products where they appear to be more prominent, such as sublethal history
for slow-cooking roasts, and fat content for products such as hot dogs (as reported in the next
section).

Table 14. OLS models validated against ALL impingement oven data.

ALL
impingement
oven data

Average 95%
PI (+/- value)

RMSE

bias

% above
low PI

% that
fit in PI

T (g) + T (w)

77

67

1.87

2.42

-0.84

T F (g) + T F (w)

81

66

1.85

2.30

-0.62

T τ (g) + T τ (w)

79

68

1.87

2.38

-0.78

Data source
for validation

Model*

log (CFU/g)

T F τ (g) +
81
65
1.86
2.26
-0.57
T F τ (w)
*In all cases, ground-muscle calibrated models predicted lethality for ground-muscle data and
whole-muscle calibrated models did for whole-muscle data. In addition, models used were
species specific, that is, turkey models predicted for turkey data, and so on.
nobs = 144

Tables 15-17 show the mixed-effects models performance when validated against wholemuscle products and ground patties in the impingement oven. These are discussed in detail in the
following paragraphs.

96

For the poultry products (Table 15), the models fared better when predicting lethality in
the whole chicken breasts. For example, while the PIs for the more complex models
encompassed high percentages (~80-100%) of data points for both ground- and whole-muscle
products, the simpler models produced much lower numbers (~40%) against the ground turkey
patties than against the whole-muscle chicken samples (~70%). The exception to this would be
the low percentage (38%) obtained with the T F and T F τ models in the whole chicken breasts
(Figure 23); however, notice here that PI width is only 0.85 log CFU/g. On the other hand, the
RMSE and PI widths were slightly larger for the ground turkey patties, while the bias was also
more prominent towards the fail-dangerous side. Overall, when both poultry data sets were
pooled together (nobs=44), the most complex model (T F M τ S) performed, in a conservative
manner, the best.
RMSE = 1.86 log CFU/g
bias = -1.20 log CFU/g
95% PI widthavg= ±0.85 log CFU/g
% above low PI = 33%

Figure 23. Mixed-effects T F (w) model validated against whole-muscle chicken fillets.

97

Table 15. Mixed-effects models validated against chicken steaks and turkey patties
(impingement cooked) (continued next page).
Average
95% PI
(+/- value)

%
above
low PI

% that
fit in
PI

T (g)

100

100

3.66

1.54

-0.38

T F (g)

66

66

2.90

2.65

-2.14

T F τ (g)

66

66

2.89

2.58

-2.06

T F S (g)

100

100

4.75

1.46

-0.16

T F τ S (g)

100

100

4.67

1.45

-0.12

T (g) + T (w)

91

73

2.33

1.53

-0.20

T F (g) + T F (w)

50

48

1.85

2.27

-1.64

T F τ (g) + T F τ (w)

50

48

1.86

2.21

-1.57

T F S (g) + T F S (w)

98

91

3.24

1.48

0.09

T F τ S (g) +
T F τ S (w)

98

91

3.21

1.47

0.13

TFM

73

45

2.09

2.28

-0.54

TFMτ

73

45

2.04

2.25

-0.49

TFMS

100

98

3.15

1.90

-0.58

TFMτS

100

98

3.14

1.88

-0.53

T (g)

100

100

3.70

1.70

-0.59

T F (g)

61

61

2.76

2.58

-2.03

T F τ (g)

61

61

2.78

2.52

-1.96

T F S (g)

100

100

4.69

1.57

-0.28

nobs = 23
fat = 1.05%

T F τ S (g)

100

100

4.62

1.56

-0.24

TFM

48

48

2.76

2.69

-2.16

τavg =
8.77 K·min

TFMτ

48

48

2.68

2.62

-2.08

TFMS

100

100

3.69

2.29

-1.52

Data source
for
validation

Ground
Turkey
+
Whole
Chicken
nobs = 44

Ground
turkey

Model*

RMSE

bias

log (CFU/g)

TFMτS
100
100
3.72
2.24
-1.46
*where (g)+(w) models appear, ground-muscle calibrated models predicted lethality for groundmuscle data and whole-muscle calibrated models did for whole-muscle data.
98

Table 15 (cont’d). Mixed-effects models validated against chicken steaks and turkey patties
(impingement cooked).
Data source
for validation

Model*

% above
low PI

% that
fit in PI

Average
95% PI
(+/- value)

RMSE

bias

log (CFU/g)

T (g)

1.34

-0.16

71

71

3.05

2.72

-2.25

71

71

3.02

2.65

-2.18

T F S (g)

100

100

4.82

1.33

-0.04

T F τ S (g)

100

100

4.73

1.32

0.01

T (w)

81

43

0.83

1.31

0.23

T F (w)

38

33

0.85

1.86

-1.20

T F τ (w)

38

33

0.85

1.82

-1.14

T F S (w)

95

81

1.66

1.36

0.50

T F τ S (w)

95

81

1.66

1.38

0.53

TFM

100

43

1.35

1.73

1.22

TFMτ

100

43

1.34

1.75

1.25

TFMS

τavg =
7.92 K·min

3.30

T F τ (g)

nobs = 21
fat = 0.33%

100

T F (g)

Whole
chicken

100

100

95

2.56

1.36

0.45

TFMτS
100
95
2.51
1.37
0.48
*where (g)+(w) models appear, ground-muscle calibrated models predicted lethality for groundmuscle data and whole-muscle calibrated models did for whole-muscle data.

99

For the beef samples (Table 16), all models performed relatively well, with high
percentages of data points (>95%) above the lower fail-dangerous PI band (Figure 24), RMSEs
ranging from ~1.2-1.9 log CFU/g, and almost all bias values on the fail-safe side. Some models
against the whole beef samples produced negative bias values, but no larger than -0.34 log
CFU/g. Overall, the PIs were slightly, but consistently across models, wider than in the poultry
samples.

RMSE = 1.33 log CFU/g
bias = 0.41 log CFU/g
95% PI widthavg= ±2.76 log CFU/g
% above low PI = 100%

Figure 24. Mixed-effects T F M τ S model validated against whole-muscle beef steaks.

100

Table 16. Mixed-effects models validated against beef steaks and patties (impingement
cooked) (continued next page).
Data source
for validation

Average 95%
PI (+/- value)

RMSE

bias

93

2.56

1.58

0.76

100

95

2.83

1.33

0.14

T F τ (g)

100

95

2.80

1.34

0.19

T F S (g)

100

100

4.96

1.55

0.20

T F τ S (g)

100

100

4.83

1.56

0.25

T (g) + T (w)

98

82

2.11

1.59

0.81

T F (g) + T F (w)

98

86

2.01

1.37

0.43

T F τ (g) +
T F τ (w)

98

82

2.00

1.38

0.48

T F S (g) +
T F S (w)

98

95

2.78

1.47

0.50

T F τ S (g) +
T F τ S (w)

98

95

2.72

1.48

0.55

TFM

100

86

2.26

1.48

0.78

TFMτ

100

84

2.23

1.51

0.85

TFMS

100

95

2.90

1.41

0.22

TFMτS

100

98

2.90

1.41

0.27

T (g)

100

84

2.75

1.93

1.34

T F (g)
Ground beef

100

T F (g)

nobs = 44

% that
fit in PI

T (g)

Ground
+
Whole beef

% above
low PI

100

89

2.59

1.51

0.63

T F τ (g)

100

89

2.56

1.53

0.70

Model*

log (CFU/g)

T F S (g)

100

100

4.14

1.71

0.92

nobs = 19
fat = 2.32%

T F τ S (g)

100

100

4.00

1.73

0.96

τavg =
9.28 K·min

TFM

100

89

2.42

1.48

0.56

TFMτ

100

89

2.38

1.50

0.61

TFMS

100

95

3.13

1.51

0.03

TFMτS
100
100
3.08
1.51
0.09
*where (g)+(w) models appear, ground-muscle calibrated models predicted lethality for groundmuscle data and whole-muscle calibrated models did for whole-muscle data.
101

Table 16 (cont’d). Mixed-effects models validated against beef steaks and patties
(impingement cooked).
Data source
for validation

Average 95%
PI (+/- value)

RMSE

bias

% above
low PI

% that
fit in PI

T (g)

100

100

3.46

1.25

0.32

T F (g)

100

95

3.01

1.18

-0.26

T F τ (g)

100

100

2.98

1.17

-0.20

T F S (g)

100

100

5.59

1.42

-0.34

T F τ S (g)

100

100

5.46

1.40

-0.29

T (w)

96

80

1.62

1.27

0.41

nobs = 25
fat = 2.68%

T F (w)

96

84

1.57

1.24

0.26

τavg =
8.18 K·min

T F τ (w)

96

76

1.58

1.25

0.31

T F S (w)

96

92

1.75

1.25

0.19

T F τ S (w)

96

92

1.74

1.26

0.23

TFM

100

84

2.14

1.47

0.95

TFMτ

100

80

2.11

1.51

1.03

TFMS

100

96

2.72

1.32

0.37

Whole beef

Model*

log (CFU/g)

TFMτS
100
96
2.76
1.33
0.41
*where (g)+(w) models appear, ground-muscle calibrated models predicted lethality for groundmuscle data and whole-muscle calibrated models did for whole-muscle data.

102

For the pork products (Table 17), the models gave slightly better predictions for the
ground-muscle patties in terms of the percentage of data points above the fail-dangerous PI band.
In addition, the PI widths for the ground-muscle calibrated models (g) were slightly narrower
than those of the whole-muscle calibrated models (w). RMSE was consistent (~1.4-1.9 log
CFU/g) throughout all the data set, except for a few exceptions (~2.3-2.6 and ~3.1 log CFU/g).
On the other hand, bias values were more scattered, ranging from -2.2 to 1.2 log CFU/g.
However, when both ground- and whole-muscle samples were pooled together (nobs=56),
RMSE, bias, and PI width were consistent across models once more.

RMSE = 1.42 log CFU/g
bias = 0.36 log CFU/g
95% PI widthavg= ±2.42 log CFU/g
% above low PI = 100%

Figure 25. Mixed-effects T F M τ S model validated against ground-muscle pork patties.

103

Table 17. Mixed-effects models validated against pork steaks and patties (impingement
cooked) (continued next page).
Data source
for validation

Average 95%
PI (+/- value)

RMSE

bias

% above
low PI

% that
fit in PI

T (g)

93

89

3.54

1.55

0.12

T F (g)

82

100

1.87

1.60

0.80

T F τ (g)

100

82

1.87

1.62

0.82

T F S (g)

100

93

3.54

1.82

0.93

T F τ S (g)

100

93

3.66

1.71

0.84

T (g) + T (w)

82

80

2.79

1.98

-0.72

T F (g) + T F (w)

89

75

2.36

2.60

-0.60

T F τ (g) +
T F τ (w)

88

73

2.34

2.57

-0.56

T F S (g) +
T F S (w)

86

50

1.27

1.98

0.57

T F τ S (g) +
T F τ S (w)

89

80

2.30

1.86

0.46

TFM

100

88

1.98

1.54

0.70

TFMτ

100

84

1.98

1.56

0.73

TFMS

100

82

2.49

1.66

0.67

TFMτS

100

86

2.48

1.53

0.60

T (g)

85

85

2.87

1.47

-0.13

T F (g)

100

74

1.59

1.83

1.11

T F τ (g)

100

74

1.59

1.85

1.14

nobs = 27
fat = 10%

T F S (g)

100

89

2.43

1.85

1.16

T F τ S (g)

100

89

2.72

1.87

1.18

τavg =
9.34 K·min

TFM

100

85

1.73

1.67

0.92

TFMτ

100

81

1.75

1.70

0.95

TFMS

100

89

2.43

1.40

0.32

Ground
+
Whole pork
nobs = 56

Ground pork

Model*

log (CFU/g)

TFMτS
100
93
2.42
1.42
0.36
*where (g)+(w) models appear, ground-muscle calibrated models predicted lethality for groundmuscle data and whole-muscle calibrated models did for whole-muscle data.
104

Table 17 (cont’d). Mixed-effects models validated against pork steaks and patties
(impingement cooked).
Data source
for validation

Average 95%
PI (+/- value)

RMSE

bias

% above
low PI

% that
fit in PI

T (g)

100

93

4.17

1.61

0.35

T F (g)

100

90

2.12

1.35

0.50

T F τ (g)

100

90

2.13

1.36

0.53

T F S (g)

100

97

4.57

1.80

0.71

T F τ S (g)

100

97

4.54

1.56

0.52

T (w)

79

76

2.71

2.35

-1.26

nobs = 29
fat = 1.53 %

T F (w)

79

76

3.08

3.15

-2.20

τavg =
8.21 K·min

T F τ (w)

76

72

3.03

3.10

-2.13

T F S (w)

86

76

1.92

1.87

-0.25

T F τ S (w)

79

72

1.91

1.86

-0.21

TFM

100

90

2.20

1.40

0.49

TFMτ

100

86

2.19

1.41

0.52

TFMS

100

76

2.55

1.87

1.00

Whole pork

Model*

log (CFU/g)

TFMτS
100
79
2.54
1.64
0.83
*where (g)+(w) models appear, ground-muscle calibrated models predicted lethality for groundmuscle data and whole-muscle calibrated models did for whole-muscle data.

The fact that the mixed-effects models performed better with the beef and pork samples
(especially in terms of the percentage of data points on the safe side) is a measure of the
suitability of the calibration data for the chosen models. Just as with the OLS models, the
addition of the sublethal injury parameter (β2, represented by τ in model names), improved
overall, even if slightly, in some manner the models- be it by reducing RMSE, bias, PI width, or
a combination of the three. On the other hand, the combination of ground- and whole-muscle
data to obtain models with the M (muscle) parameter (Table 4) had mixed outcomes; it improved
105

predictions for the poultry samples, but had both positive and negative effects for the beef and
pork products. In addition, the separation of data only by muscle type (that is, lumping all ground
turkey, beef, and pork data in one calibration group, and their whole-muscle counterparts in
another (Table 4) to yield the S (species) parameter, had this same effect. On the other hand,
within the TFS models, those calibrated with ground-muscle data (T F S (g) and T F τ S (g)) had
wider PIs (~±2.50 vs. ~±1.0 log CFU/g), but consequently were able to capture higher
percentages of data points above the fail-dangerous PI band (~±100% vs. ~±90%). Although
these percentage differences could be considered small, on food safety applications it is desirable
to err on the safe side. In terms of RMSE, the ground-calibrated models fared slightly better
(~0.2 log CFU/g) for the beef and pork samples, while the bias values were evenly fail-safe and
fail-dangerous across all samples.
Table 18 shows the mixed-effects models performing against all impingement oven data.
Percentage of data points above the fail-dangerous PI band was overall satisfactory, with the two
most complicated models (T F M S and T F M τ S) capturing 100% of them, and all capturing at
least 79%. RMSEs were also acceptable, given the expected variability in this kind of data, and
bias values were mostly positive, except for the first three model versions. On the other hand, PI
width was large -bigger than what could possibly be useful for industrial applications, especially
in the case of the last two models.
When comparing the OLS models with the mixed-effects versions against all the
impingement oven data pooled together (Table 14 and Table 18), it is evident each has its
positive features and drawbacks. For example, while the OLS models placed, at most, 81% of the
data points above the fail-dangerous PI band, the mixed-effects models were able to capture
106

>80% of them in the same region. However, this could mostly be due to the PI widths (~2.0-2.8
log CFU/g for mixed-effects models vs. ~1.86 log CFU/g for the OLS versions). On the other
hand, RMSE and bias were more favorable to the mixed-effects models, with a maximum RMSE
= 2.18 log CFU/g for the mixed-effects models and minimum RMSE=2.26 log CFU/g for OLS
versions; and in the case of bias, all were fail-dangerous for the OLS models, while 67% of them
were fail-safe for the mixed-effects versions.

Table 18. Mixed-effects models validated against ALL impingement oven data.
Average 95%
PI (+/- value)

RMSE

bias

90

78

2.44

1.73

-0.09

80

70

2.10

2.18

-0.60

T F τ (g) +
T F τ (w)

79

68

2.09

2.16

-0.55

T F S (g) +
T F S (w)

93

76

2.34

1.69

0.40

T F τ S (g) +
T F τ S (w)

94

88

2.70

1.64

0.39

TFM

92

74

2.10

1.78

0.34

TFMτ

92

72

2.07

1.78

0.39

TFMS

nobs =144

% that
fit in PI

T F (g) + T F (w)
ALL
impingement
oven data

% above
low PI

T (g) + T (w)

Data source
for validation

100

91

2.82

1.67

0.15

Model*

log (CFU/g)

TFMτS
100
93
2.81
1.61
0.16
*In all cases, ground-muscle calibrated models predicted lethality for ground-muscle data and
whole-muscle calibrated models did for whole-muscle data. In addition, models used were
species specific, that is, turkey models predicted for turkey data, and so on.

107

4.3.4.2 Whole-muscle roasts

Table 19 shows the OLS models predictions against the pilot-scale roasts. Because these
are the biggest pilot-scale samples, high variability and potential fail-dangerous predictions were
expected (Breslin 2009). For the turkey roasts, the ground-calibrated models did not predict
satisfactorily; PIs were wide, RMSEs very big, and bias values extremely high on the faildangerous side, as would be expected (Figure 26). On the other hand, even though the percentage
of data points above the fail-dangerous PI band remained the same at 0%, when the wholemuscle validated models were applied to these whole-muscle roasts, the prediction statistics were
better: narrower PIs, lower RMSEs, and less dangerous bias values (Figure 27). This shows the
importance of using models calibrated with products that match the characteristics of those to be
involved in model predictions. In the case of beef and pork, the same trends were produced with
whole-muscle calibrated models predicting better than their ground-muscle counterparts.
However, the pork models performed significantly better, in terms of reaching 100% of data
points above the fail-dangerous PI band compared to 62% in the case of beef. This could be
attributed to the PI widths of pork being larger than those of beef (~2.0 CFU/g vs ~1.75 CFU/g).
On the other hand, the RMSEs for beef were significantly smaller than those of pork (~1.0
CFU/g vs. ~3.0 CFU/g), and bias values were also more prominent on the fail-safe side for the
same products (~-1.6 CFU/g vs. ~0.2 CFU/g for beef and pork, respectively).

108

Table 19. OLS models validated against roasts.
Data source
for
validation

Average 95%
PI (+/- value)

RMSE

bias

τrange =
100 – 600
K·min

0

0

2.80

12.27

-12.23

0

0

2.68

10.95

-10.89

T F τ (g)

0

0

2.66

10.83

-10.77

T (w)

0

0

1.11

3.44

-3.25

T F (w)

0

0

1.10

3.39

-3.19

T τ (w)

0

0

1.02

2.81

-2.58

0

0

1.01

2.76

-2.53

0

0

1.63

7.57

-6.56

T F (g)

0

0

1.59

6.94

-5.92

0

0

1.58

6.94

-5.90

T F τ (g)

8

8

1.55

6.34

-5.30

T (w)

46

46

1.75

2.72

-1.91

T F (w)

62

62

1.74

2.98

-1.71

T τ (w)

62

62

1.74

3.00

-1.69

62

62

1.72

2.73

-1.35

60

60

2.02

2.37

-1.98

T F (g)

65

65

2.01

2.26

-1.86

T τ (g)
nobs = 20
fat =1.53%

T F (g)

T (g)
Pork roasts

-12.36

T F τ (w)

τrange =
100 – 480
K·min

12.41

T τ (g)
nobs = 13
fat = 2.68%

2.82

T (g)
Beef roasts

0

T F τ (w)

τrange =
200 – 500
K·min

0

T τ (g)
nobs = 9
fat = 0.27%

% that
fit in PI

T (g)
Turkey roasts

% above
low PI

65

65

2.00

1.96

-1.63

T F τ (g)

70

70

1.99

1.86

-1.52

T (w)

100

100

2.06

0.94

0.10

T F (w)

100

100

2.06

0.96

0.22

T τ (w)

100

100

2.05

0.90

0.31

T F τ (w)

100

100

2.04

0.93

0.42

Model

109

log (CFU/g)

RMSE = 10.83 log CFU/g
bias = -10.77 log CFU/g
95% PI widthavg= ±2.66 log CFU/g
% above low PI = 0%

Figure 26. OLS T F τ (g) model validated with turkey roast data. Compare with Figure 27.

RMSE = 2.76 log CFU/g
bias = -2.53 log CFU/g
95% PI widthavg= ±1.01 log CFU/g
% above low PI = 0%

Figure 27. OLS T F τ (w) model validated against turkey roast data. Notice the significantly
better performance than its (g) counterpart (Figure 26).

110

Table 20 shows model predictions when the roast data for all three species were pooled
together. As with the species-specific roasts and the impingement oven data, the addition of the
fat and sublethal injury parameters (β3 and β2, respectively) enhanced model performance, but
not substantially. On the other hand, while β3 had a greater effect on the impingement oven data
predictions, β2 here has a larger impact, reducing PI width, RMSE, and bias values more
significantly than β3. This is due to the effect the high sublethal history values of these data
(ranging from 100 to 600 K·min) have on the model predictions (compared to ~8-9 K·min for the
impingement oven data).
Table 20. OLS models validation against ALL roast data.
Average 95%
PI (+/- value)

RMSE

bias

ALL roast
data
nobs = 42

% above
low PI

% that
fit in PI

T (w)

62

62

1.76

2.29

-1.24

T F (w)

67

67

1.75

2.38

-1.11

T τ (w)

Data source
for validation

67

67

1.73

2.20

-0.93

T F τ (w)

67

67

1.72

2.09

-0.76

Model

log (CFU/g)

Table 21 shows the mixed-effects models when validated against the roast products.
Performance is once more negatively affected when using ground-muscle calibrated models to
predict lethality on whole-muscle data, as expected. Just as with the impingement oven samples,
turkey roasts were once more the most affected by this, with highly dangerous bias (> -5.0 log
CFU/g), extremely wide PIs (> ±8.0 log CFU/g), large RMSEs (<5.50 log CFU/g), and low
percentage of data points above the fail-dangerous PI band. It can be argued that the T F τ S (g)
model on the turkey roasts encompassed 100% of the data points, but looking at the PI width
111

(±8.05 log CFU/g), this becomes obvious and not useful for our application. For the beef and
pork roasts, the ground-calibrated modes fared better in terms of the percentage values (>90%
above low PI), but most PI widths were still outside the scope of usefulness, as explained in
section 4.2.4 (>±5.5 log CFU/g for beef, and >2.2 log CFU/g for pork, one of the few exceptions
in Figure 28). Even though performance on pork was satisfactory in the case of the T F τ (g)
model, the whole-muscle validated counterparts provided better predictions. Overall, the wholemuscle validated models, those including sublethal history (τ), or those accounting for muscle
type (M) had better predictions across the products, with the more complicated model (T F M τ
S) performing the best conservatively.
Table 21. Mixed-effects models validated against roasts (continued next page).
Data source
for
validation

Average 95%
PI (+/- value)

RMSE

bias

% above
low PI

% that
fit in PI

100

100

9.00

4.34

-4.19

T F (g)

0

0

12.20

18.65

-18.54

T F τ (g)

11

0

7.70

9.77

-9.57

T F S (g)

100

100

10.64

11.53

-11.38

T F τ S (g)

100

100

7.83

1.59

-0.92

T (w)

0

0

0.83

3.27

-3.00

T F (w)

0

0

0.85

18.65

-18.54

T F τ (w)

0

0

0.85

4.83

-4.54

T F S (w)

33

33

2.79

3.77

-3.58

T F τ S (w)

89

89

2.10

1.37

-0.61

TFM

78

78

3.87

3.42

-3.15

TFMτ

89

89

2.52

1.18

-0.25

TFMS

100

100

5.92

5.44

-5.25

TFMτS

100

100

3.82

1.25

-0.18

Model

T (g)

Turkey roasts
nobs = 9
fat = 0.27%
τrange =
200 – 500
K·min

112

log (CFU/g)

Table 21 (cont’d). Mixed-effects models validated against roasts.
Data source
for
validation

Average 95%
PI (+/- value)

RMSE

bias

-2.30

85

85

7.44

6.58

-5.80

92

92

5.34

4.06

-3.00

100

100

10.08

3.92

-2.61

T F τ S (g)

100

92

7.44

2.68

-0.68

57

50

2.35

3.18

-2.10

T F (w)

54

54

2.24

3.58

-2.42

T F τ (w)

77

54

1.89

2.42

-0.52

T F S (w)

62

62

2.49

3.58

-2.38

T F τ S (w)

96

92

1.74

1.26

0.23

TFM

92

92

4.32

3.17

-2.19

TFMτ

100

92

3.21

2.10

-0.32

TFMS

100

100

4.70

2.85

-1.35

TFMτS

100

85

3.53

2.26

0.22

T (g)

100

100

3.14

1.71

1.50

T F (g)

90

90

3.12

1.91

-1.17

T F τ (g)

100

90

2.39

0.82

0.19

T F S (g)

100

100

4.04

1.46

1.15

T F τ S (g)

100

100

3.18

1.97

1.82

T (w)

τrange =
100 – 600
K·min

3.48

T (w)

nobs = 20
fat = 1.53%

6.55

T F S (g)

Pork roasts

92

T F τ (g)

τrange =
100 – 480
K·min

100

T F (g)

nobs = 13
fat = 2.68%

% that
fit in PI

T (g)

Beef roasts

% above
low PI

100

100

2.25

0.86

0.33

T F (w)

100

100

2.50

0.92

-0.35

T F τ (w)

100

90

2.12

0.97

0.69

T F S (w)

90

90

1.95

1.12

-0.50

T F τ S (w)

100

95

1.75

0.91

0.63

TFM

100

100

2.70

1.23

-0.28

TFMτ

100

100

2.19

1.11

0.82

TFMS

100

85

2.31

1.66

1.38

TFMτS

100

45

1.97

2.15

1.99

Model

113

log (CFU/g)

RMSE = 0.91 log CFU/g
bias = 0.63 log CFU/g
95% PI widthavg= ±1.75 log CFU/g
% above low PI = 100%

Figure 28. Mixed-effects T F τ S (w) model validated against pork roasts. Notice the
favorable fitting statistics.

Table 22 shows the whole-muscle-validated and those accounting for muscle type (M)
mixed-effects models against all roast data put together. The most complicated models (lower
down the table) fared better in terms of the percentage of data points captured above the faildangerous PI band, mostly because of the low percentage values from the simpler models’
predictions in the turkey roasts (Table 21). On the other hand, the simpler models had narrower
PIs, but also higher RMSE and more fail-dangerous bias values.

114

Table 22. Mixed-effects models validated against ALL roast data.
Data source
for validation

Average 95%
PI (+/- value)

RMSE

bias

67

64

1.98

4.00

-2.28

T F (w)

64

64

2.07

8.88

-4.89

T F τ (w)

71

60

1.83

2.67

-1.04

T F S (w)

69

69

2.30

2.76

-1.74

T F τ S (w)

90

83

1.92

1.63

0.02

TFM

93

93

3.45

2.52

-1.49

TFMτ

98

95

2.58

1.50

0.24

TFMS

100

93

3.82

3.19

-0.89

TFMτS

nobs = 42

% that
fit in PI

T (w)
ALL roast
data

% above
low PI

100

69

2.85

2.03

0.98

Model

log (CFU/g)

Comparing the OLS models (Table 20) vs. the mixed-effects models (Table 22)
performances in the all the roast data pooled together, once more the OLS models reach a plateau
in capturing data points above the fail-dangerous PI band (~67% maximum), while model
complexity in the mixed-effects versions allows the percentage to reach 100%. On the other
hand, PI widths on the OLS models are more industry-useful than those yielded by the mixedeffects models (~1.75 log CFU/g vs. 1.83-3.82 log CFU/g). Finally RMSE and bias values were
inconsistent across all models, but overall presenting more high numbers compared to the
impingement oven data.

4.3.4.3 Hot dogs

Table 23 shows the OLS models predicting lethality for the hot dog data. With these few
validations and data sets, it is easier to see the positive effect fat and sublethal history have when
included in the models. In the case of the turkey hot dogs, no improvement in the percentage of
115

data points above the fail-dangerous PI band was observed with the separate addition of the two
parameters in question. However, RMSE and bias did improve. Nevertheless, when applied
together, the percent captured increased from 33% to 58%, RMSE decreased from 3.31 to 2.19
log CFU/g and bias decreased from -2.60 to -1.40 log CFU/g. On the other hand, for the beef hot
dogs, fat content and sublethal injury had significant effects when considered individually by the
model; each increased the percentage of data points above the lower PI band from 33% to 92%.
Additionally, when acting together, the PI encompassed 100% of the data points. It can be noted
that the fat parameter performed particularly well in these ~15% fat beef products, decreasing the
RMSE and bias more than the sublethal injury parameter did.

Table 23. OLS models validated against hot dogs.
Data source for
validation

Average 95%
PI (+/- value)

RMSE

bias

% above
low PI

% that
fit in PI

T (g)

33

33

2.47

3.31

-2.60

T F (g)

33

33

2.43

2.74

-2.02

T τ (g)

33

33

2.43

2.63

-1.90

T F τ (g)

58

50

2.41

2.19

-1.40

Beef hot dogs

T (g)

33

33

1.55

2.03

-1.90

nobs = 12
fat =15.42%

T F (g)

92

92

1.52

0.81

-0.69

92

92

1.52

0.97

-0.82

100

100

1.52

0.40

0.07

Turkey hot dogs
nobs = 12
fat = 4.28%
τrange =
50 – 125 K·min

Model

T τ (g)
τrange =
100 – 275 K·min T F τ (g)

116

log (CFU/g)

RMSE = 2.03 log CFU/g
bias = -1.90 log CFU/g
95% PI widthavg= ±1.55 log CFU/g
% above low PI = 33%

Figure 29. OLS T (g) model validated against beef hot dogs. Compare with Figure 30.

RMSE = 0.40 log CFU/g
bias = 0.07 log CFU/g
95% PI widthavg= ±1.71 log CFU/g
% above low PI = 100%

Figure 30. OLS T F τ (g) model validated against beef hot dog data. Notice the significantly
better performance than the simpler T (g) model (Figure 29).

117

Table 24 shows the OLS models validated against both the turkey and beef hot dogs as a
single data set. Again, the trends shown in the individual data sets are evident; fat and sublethal
injury considerations meaningfully improved model predictions.

Table 24. OLS models validated against ALL hot dog data.
Data source
for validation

Average 95%
PI (+/- value)

RMSE

bias

nobs = 24

% that
fit in PI

T (g)

33

33

2.01

2.75

-2.25

T F (g)

63

63

1.97

2.02

-1.35

T τ (g)

ALL hot dog
data

% above
low PI

63

63

2.06

1.98

-1.36

T F τ (g)

75

75

2.03

1.69

-1.11

Model

log (CFU/g)

Table 25 shows the different versions of the mixed-effects models validated against the
hot dog data. Again, the models performed less satisfactorily on the turkey than on the beef
samples, with wider PIs, and higher RMSEs and fail-dangerous bias values. However, the most
complicated model (T F M τ S) fared sufficiently well for both cases (i.e., Figure 31), with 100%
of the data points above the lower PI band, even though the interval was relatively wide for the
turkey samples (±2.77 log CFU/g).

118

Table 25. Mixed-effects models validated against hot dogs.
Average 95%
PI (+/- value)

Beef hot dogs
nobs = 12
fat = 15.42%
τrange =
100 – 275 K·min

83

2.89

1.40

-0.29

T F (g)

67

58

2.29

2.18

-1.48

T F τ (g)

67

50

2.03

3.73

-0.94

T F S (g)

100

83

3.25

1.34

0.29

T F τ S (g)

100

83

2.83

2.41

0.57

TFM

67

50

2.19

2.82

-1.50

TFMτ

67

50

1.93

1.82

-0.96

100

83

2.67

3.49

-0.59

100

83

2.40

2.51

-0.18

T (g)

τrange =
50 – 125 K·min

100

TFMτS

nobs = 12
fat =4.28%

% that
fit in PI

TFMS

Turkey hot dogs

%
above
low PI

T (g)

Data source for
validation

100

100

2.99

0.43

-0.25

T F (g)

100

100

2.18

0.33

0.07

T F τ (g)

100

100

1.91

1.07

0.59

T F S (g)

100

100

1.98

1.36

1.23

T F τ S (g)

100

100

2.59

1.57

1.48

TFM

100

100

1.99

0.34

0.10

TFMτ

100

100

1.82

0.78

0.65

TFMS

100

100

1.75

1.38

0.82

TFMτS

100

92

1.60

1.65

1.18

Model

119

RMSE

bias

log (CFU/g)

RMSE = 1.65 log CFU/g
bias = 1.18 log CFU/g
95% PI widthavg= ±1.60 log CFU/g
% above low PI = 100%

Figure 31. Mixed-effects T F M S τ model validated against beef hot dogs.

Table 26 shows the performance of the mixed-effects models when the turkey and beef
hot dog data were pooled together. Here, it is more evident that the T F M τ S model gave the
best predictions among the model versions; 100% of the data points fell above the fail-dangerous
PI band, RMSE was 1.34 log CFU/g, bias was 0.50 log CFU/g, and the PI width was of ±2.0 log
CFU/g, which is a fairly acceptable value, compared to the other validated products. It can be
argued that models T F S (g) and T F τ S (g) also covered 100% of the data points in the safe
region. However, even though their RMSE and bias values were similar, their PI widths were
bigger (±2.62 log CFU/g and ±2.71 log CFU/g).

120

Table 26. Mixed-effects models validated against ALL hot dog data.
Data source
for validation

Average 95%
PI (± value)

RMSE

bias

100

4

2.94

1.04

-0.27

T F (g)

83

4

2.23

1.56

-0.71

T F τ (g)

83

4

1.97

1.38

-0.17

T F S (g)

100

4

2.62

1.30

0.76

T F τ S (g)

100

4

2.71

1.48

1.02

TFM

83

4

2.09

1.57

-0.70

TFMτ

83

4

1.87

1.40

-0.15

TFMS

100

4

2.21

1.27

0.12

TFMτS

nobs = 24

% that
fit in PI

T (g)

ALL hot dog
data

% above
low PI

100

4

2.00

1.34

0.50

Model

log (CFU/g)

Comparing the OLS and mixed-effects models against both the turkey and beef hot dog
data put together (Table 24 and Table 26) it is evident that the mixed-effects versions fared better
in almost all cases. For example, percentage of data points above the fail-dangerous PI band in
the mixed-effects models was at least 83%, compared to a maximum 75% from the OLS models.
Additionally, PI widths were practically the same across all models, giving the mixed-effects
versions the performance advantage. Finally, RMSEs and bias were also better for the mixedeffects versions; with a maximum RMSE of 1.57 log CFU/g vs. a minimum of 1.69 log CFU/g
from the OLS models, and all bias being fail-dangerous in the OLS versions vs. positive bias in
~50% of the cases in the mixed-effects models.

121

4.3.4.4 Overall model performance on all pilot-scale data

Table 27 shows OLS models performance against all pilot-scale data. It can be seen that
PI width remains practically constant across models, while the percentage of data points above
the fail-dangerous PI band increases. Although this change is not substantial (~10%), RMSE and
bias do improve more notably: by 0.27 log CFU/g and 0.41 log CFU/g, for each parameter,
respectively. Therefore, the statement from previous sections that a model accounting for fat
content and sublethal history has a better overall performance is reinforced.

Table 27. OLS models performance against ALL pilot-scale data.
Data
source for
validation

Model

T (g) + T (w)
ALL
pilot-scale T F (g) + T F (w)
data
T τ (g) + T τ (w)
nobs = 210
T F τ (g) + T F τ (w)

Average 95%
RMSE
PI (+/- value)

%
above
low PI

% that
fit in
PI

69

62

1.83

2.44

-1.08

76

66

1.85

2.28

-0.80

75

67

1.83

2.31

-0.88

78

67

1.83

2.17

-0.67

bias

log (CFU/g)

Table 28 shows the different versions of the mixed-effects models when validated against
all pilot-scale data put together. It can be seen that average PI widths remain relatively constant
(although it does reach its maximum values in the most complicated models), but the percentage
of data points above the fail-dangerous PI band increases with model complexity, reaching the
100%. This does not necessarily mean that the model with the most parameters will always give
the best predictions and/or is the most adequate to predict lethality in a certain process, but it
122

does show the possible positive implications of pooling data from products with differing
characteristics and obtaining a model that could possibly account for most of the variability
between them.

Table 28. Mixed-effects models performance against ALL pilot-scale data.
Data source
for validation

Average 95%
PI (+/- value)

RMSE

bias

86

77

2.40

2.32

-0.55

T F (g) + T F (w)

nobs =210

% that
fit in PI

T (g) + T (w)

ALL
pilot-scale
data

% above
low PI

77

70

2.11

4.40

-1.47

T F τ (g) +
T F τ (w)

78

67

2.02

2.20

-0.61

T F S (g) +
T F S (w)

89

77

2.36

1.91

0.02

T F τ S (g) +
T F τ S (w)

94

88

2.55

1.62

0.39

TFM

91

78

2.37

1.93

-0.14

TFMτ

92

77

2.15

1.69

0.30

TFMS

100

91

2.95

2.03

-0.06

TFMτS

100

88

2.73

1.68

0.36

Model*

123

log (CFU/g)

4.4

Conclusions

This study showed possible sources and methods that can be used to develop a multiproduct, multi-factor model for Salmonella thermal inactivation in meat and poultry products. It
provides insight on the challenges and difficulties of attempting to complete such a task with the
existing experimental data and without making use of extremely complex statistical methods.
Data were gathered, multiple models were developed with different techniques, and they were all
validated against pilot-scale data to test their usefulness to industrial applications. While there
were significant deviations between model predictions and experimental results, explaining them
was not the purpose of this project, but rather to demonstrate the degree of the expected
variability and the possible effects of applying the laboratory-developed models in industrial
settings.
There is no individual model that can be deemed the “best”, as each would be able to
perform differently under differing processing conditions, in which case the “best” model to use
would be the one that accounts for the parameters important to the user’s cooking process and
product characteristics. Therefore, while this project does not completely close the gap between
scientific work and real-life applications, it does provide new information that should be helpful
both in directing future research in this field and in improving utilization of inactivation models
in industry.

124

5. OVERALL CONCLUSIONS

5.1

Sublethal injury

•

Models not accounting for bacterial sublethal injury can overestimate kill in meat
products cooked under heat regimes where this phenomenon occurs (i.e., slow
cooking/roasting).

5.2

Model validation

•

The literature on thermal inactivation data and models developed in laboratory settings is
extensive. However, almost none of these models are validated. Those that are provide
few quantitative measures of model performance.

•

Very little work exists where thermal inactivation models are validated against pilot-scale
data or industry-relevant conditions, and they illustrate the difficulties associated with
this kind of work – largely scattered data and model predictions.

•

These results indicate that the existing information on thermal inactivation has limitations
in directly helping to improve “real time” food safety.

5.3

Use of models in industry

•

Research shows that multiple product and pathogen factors affect thermal inactivation:
temperature, fat content, muscle structure, meat species, pH, salt %, sublethal injury, etc.
Processors should identify which of these are truly important to their specific product and
choose a model accordingly.

125

•

When choosing a model, industry users should match, to the extent possible, the process
to be validated and the nature of the data under which the considered models were
obtained, especially when allowing for the use of models developed from laboratorybased experiments.

•

Although model complexity does not necessarily translate into better lethality
predictions, using an overly simplified model that neglects key factors can lead to faildangerous predictions, depending on product and process characteristics.

5.4

The need for standardized testing methods

•

Gathering data from different sources to develop more comprehensive models can be
done with mixed-effects or other statistical methods. However, due to generally
unquantified data variability, prediction intervals will inevitably be large, decreasing
model utility in industry.

•

The introduction of standardized microbiological and analytical methods for carrying out
thermal inactivation studies would allow qualitative comparison of data across studies
and the quantification of their differences.

•

While the ComBase database is a useful resource, the thermal inactivation data presented
cannot be pooled together to develop models or statistics without expecting high
variability due to the diverse methodologies in the original studies.

126

5.5
5.5.1

Future work
Enhancement of the AMI lethality spreadsheet

As mentioned in section 2.4.1.3.2, the AMI lethality spreadsheet is a tool currently used
by meat processors to aid in determining process lethality. The results from this project could
potentially help to enhance this tool, as several models were developed and validated against
industry-relevant data. From what was concluded in this study, it is recommended that the model
chosen should account for product fat content, sublethal history, muscle structure, and species, as
their incorporation always showed prediction improvements in some way or another (RMSE,
bias, etc.). The question then remains as to which model version, from those presented in this
study, should be incorporated into the tool, especially in the case of the mixed-effects varieties,
as even those considering the four aforementioned factors could have been calibrated with
different data sets. For example, the T F M S τ model and ground-muscle-beef-specific T F τ
model are both equally appropriate to predict lethality in ground-beef products, but the T F M S
τ model was parameterized with both ground- and whole-muscle data, and turkey, beef, and pork
data sets, while only ground beef data was used to calibrate the T F τ version. However each
model yielded different fitting statistics, factors that should also be considered when determining
which model to use. This can lead to three possible solutions: (i) allow the user to choose the
model based on validation statistics and a comparison of the process characteristics and the
conditions under which the calibration and/or validation data were generated, (ii) have the tool
predict lethality and PIs with all the validated models relevant to the product, and output the
most conservative values, or (iii) a combination of (i) and (ii), where the user can choose the
model, but the tool will additionally output predicted values from other relevant model versions
and allow the user to confirm or change the initial model selection. While these choices do not
127

entirely solve the issue of validating a process with 100% certainty, the tool’s outputs based on
these recommendations could certainly be an improvement from the currently available options.

5.5.2

Methods comparison with other research groups

As stated throughout this project, one of the main issues when attempting to pool and
manipulate data from different sources was that methodologies varied widely between them,
causing high unexplained variability in the final results. Further studies, such as that carried out
by Hildebrandt et al. (2012b) could potentially help quantify the effects of methodology
variability and gain insight into possible ways to solve this problem for future studies.

5.5.3

The need for standardized testing methods

Related to the work described in the previous section, this proposed study could be the
next step to possibly separately quantify the contributions of methodology and
human/experimental error of data variability. A project where the importance of standardized
testing methods is assessed can provide further insight into the problems and necessities for this,
and hopefully lead to potential improvements across this field of research.

5.5.4

Statistical improvements and model modifications

While several versions of the multi-product multi-factor model were tested, there is
definitely room for improvement to obtain a better mathematical relationship between all the
parameters and/or variables in the exponential term in equation (7) (i.e., β2, τ, β3, F, etc) and
process lethality. The current relationships are quite basic. For example, the way sublethal
history (τ) is determined (equation (4)) assumes that there is a linear relationship between the
sublethal region temperature and acquired bacterial resistance. However, taking into account the
128

biological aspect of bacterial adaptation, it is likely that the vegetative cells become more
thermotolerant up to a certain temperature within the sublethal range and then the effect tails at
the transition to the lethal temperature range (Diller 2006). A study analyzing this behavior could
lead to improvement of equation (4).
Another example is fat content; the current model version implies that the difficulty of
achieving a certain process lethality increases exponentially (and proportionally to β3) with the
product fat content. However, future research should test this rigorously, or whether a different
mathematical relationship is needed. All of these parameters could potentially be improved with
different mathematical relationships.

129

6. APPENDICES

130

6.1

MATLAB code for sublethal injury model parameter estimation (section 3)

%thank you to Dr. Dolan for providing the file template, modified by: Isabel Tenorio
% units:
% *time: minutes!!!
% *log reductions: CFU/g or CFU/ml, (-) in model function, and (-) in
% experimental observations (Excel file)
% *Temperature in Excel file in Celsius!!
%Column order in excel: T(C)--time(min)--Tau(sublethal injury)
clear all
nlinfitcheck = statset('nlinfit');
nlinfitcheck.FunValCheck='off';
global nsets nrows
nsets=36;
nrows=2401;
%read in data
%data format must be first column is temperature (C), second column is time (sec), 3rd column
is Tau (sublethal injury history), 3rd dimension is set number
%make it global so that modelTTau function can read it
xTTau1=zeros(nrows,4,nsets);
for k=1:nsets
xTTau1(:,:,k)=xlsread('G Turkey 2009 calibration.xlsx',k); % sheets are read in order!!!!
end
%now put them together to make replicates
global xTTau
xTTau=zeros(2401,4,nsets*3);
xTTau(:,:,1:36)=xTTau1;
xTTau(:,:,37:72)=xTTau1;
xTTau(:,:,73:108)=xTTau1;
%initial estimates
param0=xlsread('G Turkey 2009 calibration.xlsx','beta0');
%experimental log reductions observed
XYobs = xlsread('G Turkey 2009 calibration.xlsx','X and Yobs');
Yobs=XYobs(:,2);
%X is the data set number

131

X=XYobs(:,1);
%estimate the parameters
%---only-T model
[param,resids,J,covar,mse] = nlinfit(X,Yobs,'modelTTau',param0);
%parameter 95% asymptotic confidence intervals
ci = nlparci(param,resids,J);
%final params, RMSE, and bias of fit
finals(1:3)=param;
finals(4)=sqrt(mse);
finals(5)=mean(resids);
% %asymptotic simultaneous CONFIDENCE intervals for Y
% T-only model
[ypred,delta] = nlpredci('modelTTau',X,param,resids,J,0.05,'on','curve');
asyCIup=ypred+delta;
asyCIdo=ypred-delta;
%asymptotic simultaneous PREDICTION intervals for Y (number 2 in ypred2
%and such is just to differentiate from CI's parameter).
% T-only model
[ypred2, deltaob] = nlpredci('modelTTau',X,param,resids,J,0.05,'on','observation');
asyPIup=ypred+deltaob;
asyPIdo=ypred-deltaob;
%time=xgtT(:,2); %for plotting later
% Correlation between parameters
%R is the correlation matrix for the parameters, sigma is the standard error vector
[R,sigma]=corrcov(covar);
RTref=R(2,1);
%Relative standard error for parameters
%RSE=zeros(6,9);
RSE(1,1)=sigma(1)/param(1,1);
RSE(2,1)=sigma(2)/param(2,1);
RSE(3,1)=sigma(3)/param(3,1);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% generic function estimating logN/No for data with T and Tau, nonisothermal
%inputs are: parameters and data set number
function result = modelTTau(params,X)

132

%params=param0;
global xTTau nsets nrows
%columns:
%xTTau(1)= Temperature (C)
%xTTau(2)= time(min)
%xTTau(3)= Tau
%xTTau(4)= total cook time (min) (value repeated through rows for MatLab to accept matrix)
%xTTau(5) = Tref (C) (value repeated through rows also)
bref=params(1);
B1=params(2);
B2=params(3); %Tau
%Predict b first
Tref=60; %same for all data sets 60 C
b=zeros(nrows,1,nsets*3);
pred=zeros(nrows,1,nsets*3);
for i=1:nrows; %row is b at t=i;
for j=1:nsets*3; % 3rd dimension is set number (with replicates)
b(i,1,j)=bref*exp(-B1*((1/(xTTau(i,1,j)+273.15))-(1/(Tref+273.15)))-B2*xTTau(i,3,j)); %b at
t=i
end
end
% Get log reductions
pred(1,1,:)=0;
for i=2:nrows; %row is log N/No prediction at t=i;
for j=1:nsets*3; % 3rd dimension is set number (with replicates)
pred(i,1,j)=-((b(i,1,j)+b(i-1,1,j))/2).*(xTTau(i,2,j)-xTTau(i-1,2,j))+pred(i-1,1,j);
end
end
%get result with total cooking time index (total cook time was different
%for each set)
%result has to be nsets*3 long (rows, 1 column)
cooktimeind(X,1)=round(xTTau(1,4,X)*12)+1; %this is for first replicate, repeat to x3
cooktimeind=[cooktimeind;cooktimeind;cooktimeind];
result=zeros(nsets,1);
for i=1:nsets;
result(i)=-pred(cooktimeind(i),i);
end
133

6.2

Correlation coefficients, and standard and relative errors for path-dependent model
parameters (section 3)
Table 29. Correlation coefficients for path-dependent model parameters.

β1

bref
0.18

β1
-

β2

0.74

0.14

β1

0.26

-

β2

0.77

0.24

β1

0.14

-

β2

0.76

0.17

Species

Parameter*

Turkey
Beef
Pork
-1

*Parameter units: bref = min , β1 =K, β2 = K·min

Table 30. Standard and relative errors for path-dependent model parameters.
Parameter*

Estimate

Standard
error

Relative
error (%)

bref

0.9071

0.0214

2.35

β1

50,787

636

1.25

β2
bref

0.0017
0.9389

0.0001
0.0328

5.47
3.49

β1

44,710

878

1.96

β2

0.0018

0.0001

7.56

bref

Species

0.7040

0.0248

3.52

β1

54,713

836

1.52

β2

0.0016

0.0001

8.39

Turkey

Beef

Pork
-1

*Parameter units: bref = min , β1 =K, β2 = K·min

134

6.3

6.3.1

MATLAB and R programming codes for multi-product multi-factor model
parameter estimation (section 4)
MATLAB code example (OLS method)

%thank you to Dr. Dolan for providing the file template
%modified by: Isabel Tenorio
% units:
% *time: minutes!!!
% *log reductions: CFU/g or CFU/ml, (-) in model function, and (-) in
% experimental observations (Excel file)
% *Temperature in Excel file in Celsius!!
%Column order in excel: T(C)--time(min)--logN/N0--muscle--turkey--beef—pork,
% (last 4 are either 0 or 1)
%This program will estimate parameters and produce confidence intervals and
%prediction intervals for a generic data set
clear
clear all
nlinfitcheck = statset('nlinfit');
nlinfitcheck.FunValCheck='off';
%read in data
%data format must be first column is temperature (K), second column is time
%(sec), 3rd column is log reductions (log N/No).
gt = xlsread('MatLab File.xls','Ground Turkey');
wt = xlsread('MatLab File.xls','Whole Turkey');
%initial estimates
initialparams=xlsread('MatLab File.xls','initial parameters');
%initial estimates for T-only model
param0gtT(1)= initialparams(1,1); %bref
param0gtT(2)= initialparams(2,1); %B1
param0wtT(1)= initialparams(1,4); %bref
param0wtT(2)= initialparams(2,4); %B1
%---Set up for T-only model
%---set up independent variables, ground turkey
xgtT(:,1)=gt(:,1);
xgtT(:,2)=gt(:,2);

135

xgtT(:,3)=initialparams(7,1); %Tref
%set up dependent variable, ground turkey
YobsgtT = gt(:,3);
%---set up independent variables, whole turkey
xwtT(:,1)=wt(:,1);
xwtT(:,2)=wt(:,2);
xwtT(:,3)=initialparams(7,4); %Tref
%set up dependent variable, whole turkey
YobswtT = wt(:,3);
%estimate the parameters
%---only-T model
[paramgtT,residsgtT,JgtT,covargtT,msegtT] = nlinfit(xgtT,YobsgtT,'modelT',param0gtT);
[paramwtT,residswtT,JwtT,covarwtT,msewtT] = nlinfit(xwtT,YobswtT,'modelT',param0wtT);
%parameter 95% asymptotic confidence intervals
cigtT = nlparci(paramgtT,residsgtT,JgtT);
ciwtT = nlparci(paramwtT,residswtT,JwtT);
%RMSE of fits
RMSE(1,1)=sqrt(msegtT);
RMSE(1,4)=sqrt(msewtT);
%bias of fits
bias(1,1)=mean(residsgtT);
bias(1,4)=mean(residswtT);
%asymptotic simultaneous CONFIDENCE intervals for Y
% T-only model
[ypredgtT, deltagtT] = nlpredci('modelT',xgtT,paramgtT,residsgtT,JgtT,0.05,'on','curve');
asyCIupgtT=ypredgtT+deltagtT;
asyCIdogtT=ypredgtT-deltagtT;
[ypredwtT, deltawtT] = nlpredci('modelT',xwtT,paramwtT,residswtT,JwtT,0.05,'on','curve');
asyCIupwtT=ypredwtT+deltawtT;
asyCIdowtT=ypredwtT-deltawtT;
%asymptotic simultaneous PREDICTION intervals for Y (number 3 in ypredgtT3 and such is just
to differentiate from CI's parameter).
% T-only model
[ypredgtT3, deltaobgtT] =
nlpredci('modelT',xgtT,paramgtT,residsgtT,JgtT,0.05,'on','observation');
asyPIupgtT=ypredgtT+deltaobgtT;
asyPIdogtT=ypredgtT-deltaobgtT;
136

tgtT=xgtT(:,2); %for plotting later
[ypredwtT3, deltaobwtT] =
nlpredci('modelT',xwtT,paramwtT,residswtT,JwtT,0.05,'on','observation');
asyPIupwtT=ypredwtT+deltaobwtT;
asyPIdowtT=ypredwtT-deltaobwtT;
twtT=xwtT(:,2); %for plotting later
% Correlation between parameters
%R is the correlation matrix for the parameters, sigma is the standard error vector
[RgtT,sigmagtT]=corrcov(covargtT);
[RwtT,sigmawtT]=corrcov(covarwtT);
%Relative standard error for parameters
%RSE=zeros(6,9);
RSE(1,1)=sigmagtT(1)/finalparams(1,1);
RSE(2,1)=sigmagtT(2)/finalparams(2,1);
RSE(1,4)=sigmawtT(1)/finalparams(1,4);
RSE(2,4)=sigmawtT(2)/finalparams(2,4);
%Add B3 to parameter matrices for TF models
B3s=xlsread('MatLab file.xls','B3s');
paramgtTF=[paramgtT,B3s(1,:)];
paramwtTF=[paramwtT,B3s(1,:)];
%Add B2 to parameter matrices for TTau models
B2s=xlsread('MatLab file.xls','B2s');
paramgtTTau=[paramgtT,B2s(1,:)];
paramwtTTau=[paramwtT,B2s(1,:)];
%parameters with B2 and B3 for TFTau models
paramgtTFTau=[paramgtT,B2s(1,:),B3s(1,:)];
paramwtTFTau=[paramwtT,B2s(1,:),B3s(1,:)];

6.3.2

R code example (mixed-effects method)

library(lme4)
#change data file for species/muscle data desired: GT, GB, GP, WT, WB, WP
dat=read.csv(file="WPmixeffdata, TFw.csv",header=F)
#dat = read.csv("C:/USER/sb/sm.csv",header=F)
colnames(dat)=c("Fat","Temp","Time","Y","Group")
137

dat2 = dat[dat$Time!=0, ]
model = function(bref,beta1,beta3,Time,Temp,Fat){
const = -bref*exp(-beta1*(1/Temp-1/333.15)-beta3*Fat)*Time
model = const
gradient <- cbind(const/bref,-const*(1/Temp-1/333.15),-const*Fat)
attr(model, "gradient") <- gradient
model
}
res = nlmer( Y ~ model(bref,beta1,beta3,Time,Temp,Fat)~(bref|Group),data=dat2,
start=c(bref=1.0818162302473487,beta1=43951.966921578,beta3=0.352765575011846))
res
fixef(res)
fitted(res)
resid(res)
# to get RMSE, change nobs as needed
SSE=resid(res)*resid(res)
SSE=sum(SSE)
RMSE=sqrt(SSE/(105-3))
#Function Coding# with assistance from CSTAT at MSU
pred<-function(res,n.sim,temp,time,fat,level=0.95) {
#Retrieve the standard deviation estimate for the random effect term bref
brefstdev<-sqrt(VarCorr(res)$Group[1,1])
#Retrieve the standard deviation estimate of the model error
errstdev<-attr(VarCorr(res),"sc")
#Retrive the fixed effect coefficient estimates
beta1<-fixef(res)[2]
beta3<-fixef(res)[3]
#Generate n.sim random effect of bref
bref<-rnorm(n.sim,fixef(res)[1],brefstdev)
#Number of time lags
n.time<-length(time)
Y<-rep(NaN,n.sim)
for(j in 1:n.sim) {
#Compute the mean of cumulative predicted log reduction value
cplr.mean<-0
b=bref[j]*exp(-beta1*(1/(temp+273.15)-1/333.15)-beta3*fat)
for(i in 1:(n.time-1)) {
cplr.mean<-cplr.mean-(b[i+1]+b[i])/2*(time[i+1]-time[i])
138

}
#Sample a new observation Y based on mean=cplr.mean and st.dev=model's standard
deviation
Y[j]<-rnorm(1,cplr.mean,errstdev)
}
#Obtain the simulated sample mean of new Y
Y.mean=mean(Y)
#Obtain the simulated sample median of new Y
Y.median=median(Y)
#Obtain the Predictive interval of new Y
Y.predictive=quantile(Y,c((1-level)/2,(1+level)/2))
Y.output=list(Y.mean=Y.mean,Y.median=Y.median,Y.predintv=Y.predictive,n.sim=n.sim)
return(Y.output)
}
fat=1.53
validdata=read.csv(file="Tasha PR.csv",header=F)
#To tell R how many sets you have in total in the csv file, you can use col(validdata) to tell how
many columns in total then divide it by 2
n.set=ncol(validdata)/2
predInt<-matrix(NaN,n.set,2)
for(i in 1:n.set) {
set=na.omit(validdata[,(2*i-1):(2*i)])
colnames(set)=c("temp","t.min")
predInt[i,]<-pred(res,1000,set$temp,set$t.min,fat)$Y.predintv
}

139

6.4
6.4.1

Hot dog methods and data (section 4)
Preparation of Salmonella

An 8-serovar Salmonella cocktail consisting of S. Thompson FSIS 120 (chicken isolate),
S. Typhimurium DT 104 H3380 (human isolate), S. Hadar MF60404 (turkey isolate), S.
Copenhagen 8457 (pork isolate), S. Montevideo FSIS 051 (beef isolate), and S. Heidelberg
F5038BGI (human isolate), was previously obtained from V.K. Juneja (Agricultural Research
Service, Eastern Regional Research Center, USDA-ARS, Wyndmoor, PA). Each serovar was
maintained separately at -80°C in vials containing tryptic soy broth (Difco Laboratories, Sparks,
MD) with yeast extract (TSBYE) and 20% glycerol. Cultures were grown separately in TSBYE
at 37°C with a minimum of two consecutive 24-hour transfers prior to inoculation.

6.4.2

Preparation of inoculated frankfurters

Emulsified beef and turkey frankfurter batter was obtained from a federally inspected
commercial supplier. Emulsion was vacuum packaged in 1000 g packages, frozen (-10°C).
Twenty-four hours prior to experiment, two packages of either beef or turkey batter were thawed
in a refrigerator (4°C) until day of experiment.
Concentrated inoculum was prepared by combining 36 ml of each culture to yield a total
of 288 ml. This cocktail was centrifuged (6000xg, 15 minutes) and the pellet was re-suspended
in 14 ml of 1% sterile buffered peptone water (Difco Laboratories, Sparks, MD). In order to
enumerate the inoculum, a 1 ml sample was taken from the 14 ml inoculum and was serially
diluted and plated on modified Tryptic Soy Agar (mTSA*) (Difco Laboratories, Sparks, MD)
9
plates (37°C, 48 hours). Meat and 13 ml of marinade (10 CFU/g) was added to a Kitchen Aid
mixer and mixed for 180s at setting 1 and using the paddle.
6
Prior to stuffing, the inoculated emulsion (10 CFU/g) was vacuumed (101.325kPa of
vacuum, 10 s) to reduce air bubbles. The emulsion was stuffed into cellulose casing using a
hand-crank stuffer system to make three two-link hot dog chains with a mass of about 60 g
(~15.5 cm long, 2 cm diameter) per hot dog. Frankfurters were tied in the center and on each end
to form links. In order to place a probe in the center of each hot dog, a jig was created to insert a
hypodermic needle (16G x 12.7cm) at a height of 1 cm into the length of the frankfurter, along
its center axis. This needle was pushed all the way through the hot dog and out the other end; a
wire thermocouple was inserted half-way (~7.75cm) into the length of the frankfurter, the needle
was pulled out, and the thermocouple remained. This procedure was repeated for each of the six
frankfurters per treatment.
*mTSA recipe: 2 L deionized distilled water, 80 g TSA, 12 g yeast extract, 1g ammonium iron
citrate, 0.6g sodiun thiosulfate.

140

6.4.3

Frankfurter cooking and survivor recovery

The hot dogs were cooked in a pilot scale, moist air convection oven (Cres Cor, Mentor,
OH) using a cook schedule for low-fat frankfurters similar to that used in industry. This cycle
increased temperature and humidity over a period of ~140 min (Table 31). A predicted lethality
of 4 log and 6 log was calculated real-time using a data logger (Datapaq Inc., Wilmington, MA)
with inputs of D- and z- values from previously published laboratory data (Table 30) (Breslin
2009; Tuntivanich et al. 2008). When the frankfurter reached the end lethality, it was taken out
of the oven and quenched in liquid nitrogen to terminate cooking. Additional experiments were
run to end temperatures of 160°C and 165°C for beef and turkey, respectively. The goal of these
cooks was to ensure no bacterial survival at these end temperatures, often used in industry. The
center 5 cm of length of each hot dog was cut and cored (1.2 cm diameter), serially diluted, and
plated on mTSA (37°C, 48 h) to enumerate survivors.

Table 31. Commercial cooking schedule for frankfurters.
Cumulative time
(min)

Dry bulb temperature
(°C)

% Relative
Humidity

0 – 20

60

38

20 – 35

71.1

37

35 – 50

76.7

39

50 – 140

82.2

79

Table 32. D- and z- values from previous research used to predict real-time lethality.
Parameter

Ground Beef

Ground Turkey

Dref (sec)

60.55

62.66

z (°C )

5.48

5.14

Tref (°C )

60

60

141

6.4.4

Results

For each sample, the log reduction predicted with the D- and z- values from Table 32,
along with the experimental lethality are presented in Table 33 and Table 34.

Table 33. Predicted and experimental log reductions for turkey hot dogs.
Sample

Log reduction predicted

Log reduction experimental

A1

3.83

1.10

A2

3.95

0.96

A3

3.65

1.09

A4

3.49

0.97

A5

2.13

1.36

A6

2.03

0.85

B1

5.23

2.89

B2

5.95

2.97

B3

5.65

3.62

B4

1.73

4.04

B5

1.87

3.41

B6

5.54

3.49

142

Table 34. Predicted and experimental log reductions for beef hot dogs.
Sample

Log reduction predicted

Log reduction experimental

A1

2.47

1.44

A2

3.59

1.74

A3

2.20

1.71

A4

3.62

1.56

A5

3.11

1.36

A6

2.90

1.39

B1

5.28

2.81

B2

5.79

3.48

B3

6.34

2.82

B4

5.04

2.62

B5

4.01

2.55

B6

4.82

2.80

143

6.5

6.5.1

Standard and relative errors for the multi-product, multi-factor model parameters
(section 4)
OLS method

Table 35 shows the standard and relative errors for the parameters obtained when doing
the regression on the fat-relevant data (Table 6), which would then lead to the use of solely β3 in
the multi-product multi-factor model (see section 4.2.3.1.2 for details).
Table 35. Parameter standard and relative errors for β3 estimation.
Model Species

G
turkey
T (g)

G beef

G pork

Parameter

Estimate
Std error
% Rel error
Estimate
Std error
% Rel error
Estimate
Std error
% Rel error

β1

-1

min

0.461
0.010
2.19
0.396
0.011
2.80
0.344
0.009
2.62

β3

K

bref

fat %

39,231
542
1.38
39,885
509
1.28
39,877
721
1.81

-1

0.030
0.003
8.87
0.023
0.002
7.73
0.014
0.001
10.93

Since β2 and β3 were obtained separately, they cannot be included with the values from
the bref and β1 regressions. β2’s errors from its corresponding estimation are in Table 30 (section
6.2). β3’s errors are on Table 35 (above).
Table 36. Parameter standard and relative errors for OLS models (continued next page).
Model Species

G
turkey
T (g)

G beef

G pork

Parameter

Estimate
Std error
% Rel error
Estimate
Std error
% Rel error
Estimate
Std error
% Rel error
144

bref
-1

min

1.11
0.038
3.43
0.83
0.016
1.98
0.63
0.020
3.20

β1
K
52,269
1,673
3.20
44,242
800
1.81
41,750
1,206
2.89

Table 36 (cont’d). Parameter standard and relative errors for OLS models.
Model Species

Parameter

Estimate
Std error
% Rel error
Estimate
W beef Std error
% Rel error
Estimate
W pork Std error
% Rel error
W
turkey

T (w)

6.5.2

β1

bref
-1

K

min

0.37
0.014
3.78
0.44
0.011
2.44
0.45
0.016
3.54

48,589
1,696
3.49
44,799
859
1.92
47,164
1,337
2.84

Mixed-effects method

Table 37 shows the parameter standard and relative errors for the mixed-effects models.
Notice that because β2 was estimated from a different regression (section 4.2.3.1.2), its errors
cannot be presented in the same table. However, β2’s errors from its corresponding estimation
are in Table 30 (section 6.2).
Table 37. Parameter standard and relative errors for mixed-effects models (continued next
page).
Model Species

G
turkey
T (g)

G beef

G pork

Parameter

Estimate
Std error
% Rel error
Estimate
Std error
% Rel error
Estimate
Std error
% Rel error

bref
-1

min

0.672
0.052
7.74
0.555
0.034
6.13
0.430
0.042
9.77

β1
K
50,750
4,131
8.14
44,710
2,260
5.05
53,950
2,591
4.80

145

β3
fat %
-

β4
-1

β5

β6

β7

•

•

•

•

-

-

-

-

Table 37 (cont’d). Parameter standard and relative errors for mixed-effects models
(continued next page).
Model Species

W
turkey
T (w)

W beef

W pork
G
turkey
TF
(g)

G beef

G pork
W
turkey
TF
(w)

W beef

W pork

TFS
(g)

TFS
(w)

G
turkey,
beef,
pork
W
turkey,
beef,
pork

Parameter

bref
-1

min

β1

β3

K

fat %

β4

β5

β6

β7

-1

•

•

•

•

Estimate
Std error
% Rel error
Estimate
Std error
% Rel error
Estimate
Std error
% Rel error
Estimate
Std error
% Rel error
Estimate
Std error
% Rel error
Estimate
Std error
% Rel error
Estimate
Std error
% Rel error
Estimate
Std error
% Rel error
Estimate
Std error
% Rel error
Estimate
Std error

0.593
0.024
4.05
0.538
0.020
3.72
0.654
0.045
6.88
1.582
0.089
5.63
0.949
0.057
6.01
0.742
0.045
6.06
1.363
0.133
9.76
0.796
0.092
11.56
1.045
0.311
29.76
1.534
0.172

50,750
1,761
3.47
44,710
1,500
3.35
53,950
2,637
4.89
36,470
1,944
5.33
36,320
1,833
5.05
35,940
1,713
4.77
44,710
1,658
3.71
44,710
1,424
3.18
53,950
3,076
5.70
49,800
1,778

0.2094
0.007
3.34
0.0627
0.004
6.38
0.0635
0.004
6.30
0.897
0.098
10.93
0.128
0.037
28.91
0.188
0.121
64.36
0.065
0.004

-

0.832 0.776 1.089
0.087 0.100 0.106

% Rel error

11.21

3.57

6.15

-

10.46 12.89

Estimate
Std error

5.200
0.877

45,690
1,166

0.127
0.046

-

2.145 1.881 1.791
0.190 0.211 0.200

% Rel error

16.87

2.55

36.22

-

8.86

146

9.73

11.22 11.17

Table 37 (cont’d). Parameter standard and relative errors for mixed-effects models.
Model Species

G+W
turkey
TFM

G+W
beef
G+W
pork

TFM
S

6.6
6.6.1

G+W
turkey,
beef,
pork

Parameter

β1

bref
-1

K

min

β3
fat %

β4
-1

β5

β6

β7

•

•

•

•

Estimate
Std error
% Rel error
Estimate
Std error
% Rel error
Estimate
Std error
% Rel error
Estimate
Std error

0.516
0.034
6.59
0.612
0.034
5.56
0.674
0.044
6.53
1.094
0.064

36,860
1,723
4.67
37,490
1,433
3.82
40,300
1,575
3.91
49,410
1,160

0.210
0.008
3.81
0.062
0.004
6.45
0.066
0.004
6.06
0.062
0.003

-1.140
0.075
-6.58
-0.453
0.071
-15.67
-1.355
0.083
-6.13
-0.506
0.062

0.652 0.651 0.855
0.085 0.074 0.073

% Rel error

5.85

2.35

4.84

-12.25

13.04 11.37

8.54

Correlation coefficients for multi-product multi-factor model parameters (section 4)
OLS models
Table 38. Parameter correlation coefficients for β3 estimation.
Model

Species

G turkey
T (g)

G beef
G pork
-1

Parameter*

bref

β1

β1

-0.042

-

β3

0.848

-0.046

β1

0.0005

-

β3

0.922

-0.0002

β1

-0.061

-

0.825

-0.073

β3

*Parameter units: bref = min , β1 =K, β3 = fat %

-1

147

Table 39. Correlation coefficients for OLS models parameters.
Model

bref

β1

2.84e-06

G beef

β1

5.47e-06

G pork

β1

1.41e-05

W turkey

β1

-6.77e-06

W beef

β1

-1.33e-05

W pork

T (w)

Parameter

G turkey
T (g)

Species

β1

-1.61e-05

-1

*Parameter units: bref = min , β1 =K

6.6.2

Mixed-effects models

Table 40 shows the parameter correlation coefficients for the mixed-effects models.
Notice that because β2 was estimated from a different regression (section 4.2.3.1.2), its
correlation with the other parameters cannot be presented in the same table. Table 29 (section
6.2) shows the correlation of β2 with the parameters from its original estimation.
Table 40. Correlation coefficients for mixed-effects models parameters (continued next
page).
Model

Species

Parameter

β1

bref
-1

min

K

β3
fat %

β4
-1

β5

β6

•

•

•

G turkey

-

-

-

-

-

G beef

β1

0.058

-

-

-

-

-

β1

0.185

-

-

-

-

-

W turkey

β1

0.318

-

-

-

-

-

W beef
T (w)

0.065

G pork

T (g)

β1

β1

-0.011

-

-

-

-

-

W pork

β1

0.251

-

-

-

-

-

G pork

β1

0.063

-

-

-

-

-

β3

0.669

0.295

-

-

-

-

148

Table 40 (cont’d). Correlation coefficients for mixed-effects models parameters (continued
next page).
Model

Species

Parameter

β1

β3

min

-1

K

fat %

bref

β4
-1

β5

β6

•

•

•

G turkey

β1

0.003

-

-

-

-

-

β3

0.658

0.331

-

-

-

-

G beef

β1

-0.194

-

-

-

-

-

β3

0.689

-0.248

-

-

-

-

W turkey

β1

~0

-

-

-

-

-

β3

1.000

~0

-

-

-

-

W beef

β1

-0.067

-

-

-

-

-

β3

0.955

-0.062

-

-

-

-

W pork

β1

-0.451

-

-

-

-

-

β3

0.973

-0.515

-

-

-

-

β1

-0.362

-

-

-

-

-

β3

-0.004

-0.062

-

-

-

-

β5

0.614

-0.245

-0.092

-

-

-

β6

0.668

-0.395

-0.490

-

0.460

-

β7

0.624

-0.331

-0.411

-

0.424 0.644

β1

0.032

-

-

-

-

-

β3

0.072

-0.050

-

-

-

-

β5

0.846

-0.032

-0.176

-

-

-

β6

0.739

0.059

-0.596

-

0.779

-

β7

0.783

0.031

-0.571

-

0.801 0.952

β1

0.116

-

-

-

-

-

β3

0.111

0.267

-

-

-

-

β4

0.669

0.079

-0.477

-

-

-

β1

-0.113

-

-

-

-

-

β3

0.152

-0.232

-

-

-

-

β4

0.589

0.059

-0.515

-

-

-

β1

-0.038

-

-

-

-

-

β3

0.094

0.252

-

-

-

-

β4

0.626

-0.142

-0.519

-

-

-

TF
(g)

TF
(w)

TFS
(g)

TFS
(w)

G turkey,
beef, pork

W turkey,
beef, pork

G+W
turkey

G+W
TFM
beef

G+W
pork

149

Table 40 (cont’d). Correlation coefficients for mixed-effects models parameters.

β5

β6

•

•

•

-

-

-

-

-0.103

-

-

-

-

0.013

-0.020

-0.389

-

-

-

β5

0.495

-0.169

0.209

-0.610

-

-

0.627

-0.273

-0.138

-0.449

0.621

-

β7

Species

β4

β6

Model

0.611

-0.259

-0.085

-0.372

0.559 0.683

Parameter

β1

bref

6.7.1

-0.306

-

0.053

β4

β3

K

β3

6.7

min

β1
G+W
TFM
turkey,
S
beef, pork

-1

fat %

-1

OLS method prediction interval calculations (section 4)
Ellipse, bootstrapping, and Monte Carlo (PI methods) contours for each calibration
product

1

bref (min- )

As described in section 4.2.3.1.3, contours were developed for each calibration data set
with the PI methods to determine the best approach for calculating PIs. The contours for the
ground turkey data set are shown in the main text (section 4.3.2, Figure 10).

β1x10

-4

-1

(K )

Figure 32. PI methods parameter contours for whole turkey calibration set.

150

-1

bref (min )

β1x10

-4

-1

-1

bref (min )

(K )
Figure 33. PI methods parameter contours for ground beef calibration set.

β1x10

-4

-1

(K )

Figure 34. PI methods parameter contours for whole beef calibration set.

151

-1

bref (min )

β1x10

-4

-1

(K )

-1

bref (min )

Figure 35. PI methods parameter contours for ground pork calibration set.

β1x10

-4

-1

(K )
Figure 36. PI methods parameter contours for whole pork calibration set.

152

6.7.2

Prediction intervals for each PI method and each calibration product.

Just as shown for the ground turkey data set in section 4.3.2 (Figure 11 and Figure 12),
the following plots represent the PIs generated for the rest of the data sets.
Asymptotic PI

Prediction line

Zoom-in area,
Figure 38

Ellipse,
bootstrapping,
Monte Carlo PIs

Figure 37. PIs with all methods for whole turkey calibration data set.

Asymptotic PI

Monte Carlo PI
Bootstrap PI

Figure 38. Zoom-in section from Figure 37.

153

Ellipse PI

Asymptotic PI

Prediction line

Zoom-in area,
Figure 40

Ellipse,
bootstrapping,
Monte Carlo PIs

Figure 39. PIs with all methods for ground beef calibration data set.

Ellipse PI

Asymptotic PI

Monte Carlo PI
Bootstrap PI

Figure 40. Zoom-in section from Figure 39.

154

Asymptotic PI
Zoom-in area,
Figure 42

Prediction line
Ellipse, bootstrapping,
Monte Carlo PIs

Figure 41. PIs with all methods for whole beef calibration data set.

Asymptotic PI

Monte Carlo PI
Bootstrap PI

Figure 42. Zoom-in section from Figure 41.

155

Ellipse PI

Asymptotic PI
Zoom-in area,
Figure 44

Prediction line
Ellipse, bootstrapping,
Monte Carlo PIs

Figure 43. PIs with all methods for ground pork calibration data set.

Asymptotic PI
Bootstrap PI
Monte Carlo PI

Ellipse PI

Figure 44. Zoom-in section from Figure 43.

156

Asymptotic PI
Zoom-in area,
Figure 46

Prediction line
Ellipse,
bootstrapping,
Monte Carlo PIs

Figure 45. PIs with all methods for whole pork calibration data set.

Asymptotic PI

Monte Carlo PI

Bootstrap PI

Figure 46. Zoom-in section from Figure 45.

157

Ellipse PI

6.8

Random effects variance for the mixed-effects models (section 4)

Table 41. Random effects variance for mixed-effects models.
Model

Species

G turkey
T (g)

Random effect
variance
0.13198

G beef

0.08530

G pork

0.078252

W turkey
T (w)

0
0.0074619

W pork

0.023054

G turkey

0.19477

G beef

0.12499

G pork

T F (g)

W beef

0.047815

W turkey
T F (w)

4.9773e-14

W beef

0.012638

W pork

0.058894

T F S (g)

G turkey, beef, pork

1.36663

T F S (w)

W turkey, beef, pork

0.64025

G + W turkey

0.018597

G + W beef

0.040271

G + W pork

0.039966

G + W turkey, beef, pork

0.20553

TFM
TFMS

158

6.9
6.9.1

Validation against pilot-scale data (section 4)
Minimum and maximum errors for each model

6.9.1.1 Impingement oven data
Table 42. OLS models prediction errors chicken fillets and turkey patties (impingement
cooked).
Data source for
validation

Model*

Max error

Min error

log (CFU/g)

T (g)

-0.12

-9.50

T F (g)

0.05

-9.15

-0.04

-9.29

0.13

-8.95

T (g) + T (w)

2.20

-9.50

T F (g) + T F (w)

2.23

-9.15

T τ (g) + T τ (w)

2.23

-9.29

T F τ (g) + T F τ (w)

2.25

-8.95

T (g)

-0.12

-9.50

Ground turkey + T τ (g)
Whole chicken T F τ (g)
nobs = 44

Ground turkey

T F (g)

0.05

-9.15

nobs = 23
fat = 1.05%

T τ (g)

-0.04

-9.29

τavg =
8.77 K·min

T F τ (g)

0.13

-8.95

T (g)

-0.61

-8.72

T F (g)

-0.57

-8.62

T τ (g)

-0.55

-8.59

T F τ (g)

-0.51

-8.50

T (w)

2.20

-2.89

T F (w)

2.23

-2.86

T τ (w)

2.23

-2.85

Whole chicken
nobs = 21
fat = 0.33%
τavg =
7.92 K·min

2.25
-2.81
T F τ (w)
*where (g)+(w) models appear, ground-muscle calibrated models predicted lethality for groundmuscle data and whole-muscle calibrated models did for whole-muscle data.
159

Table 43. OLS models prediction errors on beef steaks and patties (impingement cooked).
Model*

Data source
for validation

Max error

Min error

log (CFU/g)

T (g)

Whole beef
nobs = 25
fat = 2.68%
τavg =
8.18 K·min

2.84

-5.39

T F τ (g)

3.03

-4.99

2.76

-3.39

T F (g) + T F (w)

2.96

-3.13

2.84

-3.29

3.03

-3.03

T (g)

2.76

-3.39

T F (g)

2.96

-3.13

T τ (g)

2.84

-3.29

T F τ (g)

3.03

-3.03

T (g)

τavg =
9.28 K·min

T τ (g)

T F τ (g) + T F τ (w)

nobs = 19
fat = 2.32%

-5.05

T τ (g) + T τ (w)
Ground beef

2.96

T (g) + T (w)
nobs = 44

-5.45

T F (g)
Ground +
Whole beef

2.76

0.87

-5.45

T F (g)

1.10

-5.05

T τ (g)

0.95

-5.39

T F τ (g)

1.17

-4.99

T (w)

2.47

-2.68

T F (w)

2.60

-2.43

T τ (w)

2.51

-2.64

2.65
-2.40
T F τ (w)
*where (g)+(w) models appear, ground-muscle calibrated models predicted lethality for groundmuscle data and whole-muscle calibrated models did for whole-muscle data.

160

Table 44. OLS models prediction errors on pork steaks and patties (impingement cooked).
Model*

Data source
for validation

Max error

Min error

log (CFU/g)

T (g)

2.91

-6.12

T F (g)

2.42

-5.01

T τ (g)

2.02

-6.00

T F τ (g)

2.52

-4.91

T (g) + T (w)

2.69

-6.12

T F (g) + T F (w)

2.77

-5.01

T τ (g) + T τ (w)

2.72

-6.00

T F τ (g) + T F τ (w)

2.77

-4.91

T (g)

1.61

-6.12

T F (g)

2.42

-5.01

T τ (g)

1.73

-6.00

T F τ (g)

2.52

-4.91

T (g)

2.91

-3.66

T F (g)

2.05

-4.78

T τ (g)

2.02

-4.85

nobs = 29
fat = 1.53%

T F τ (g)

2.09

-4.71

T (w)

2.69

-3.97

τ avg =
8.21 K·min

T F (w)

2.77

-3.77

T τ (w)

2.72

-3.91

Ground +
Whole pork
nobs = 56

Ground pork
nobs = 27
fat = 10%
τavg =
9.34 K·min

Whole pork

2.77
-3.78
T F τ (w)
*where (g)+(w) models appear, ground-muscle calibrated models predicted lethality for groundmuscle data and whole-muscle calibrated models did for whole-muscle data.

161

Table 45. OLS models prediction errors on ALL impingement oven data.
Data source
for validation

ALL
impingement
oven data
nobs = 144

Max error

Model*

Min error

log (CFU/g)

T (g) + T (w)

2.76

-9.50

T F (g) + T F (w)

2.96

-9.15

T τ (g) + T τ (w)

2.84

-9.29

T F τ (g) + T F τ (w)

3.03

-8.95

*In all cases, ground-muscle calibrated models predicted lethality for ground-muscle data and
whole-muscle calibrated models did for whole-muscle data. In addition, models used were
species specific, that is, turkey models predicted for turkey data, and so on.

Table 46. Mixed-effects models prediction errors on chicken steaks and turkey patties
(impingement cooked) (continued next page).
Data source for
validation

Max error

Model*

Min error

log (CFU/g)

T (g)

1.52

-5.55

T F τ (g)

1.58

-5.46

T F S (g)

2.97

-3.58

T F τ S (g)

3.00

-3.52

T (g) + T (w)

2.73

-3.62

T F (g) + T F (w)

1.52

-5.03

T F τ (g) + T F τ (w)

1.58

-5.46

T F S (g) + T F S (w)

2.97

-3.09

T F τ S (g) + T F τ S (w)

3.00

-3.00

TFM

3.05

-5.22

TFMτ

3.07

-5.09

TFMS

nobs = 44

-3.75

T F (g)

Ground Turkey
+
Whole Chicken

2.73

2.21

-5.04

TFMτS
2.24
-4.91
*where (g)+(w) models appear, ground-muscle calibrated models predicted lethality for groundmuscle data and whole-muscle calibrated models did for whole-muscle data.
162

Table 46 (cont’d). Mixed-effects models prediction errors on chicken steaks and turkey
patties (impingement cooked).
Data source for
validation

Max error

Model*

Min error

log (CFU/g)

T (g)

2.73

-3.62

T F (g)

1.52

-5.03

T F τ (g)

1.58

-4.90

T F S (g)

2.97

-3.09

T F τ S (g)

3.00

-3.00

TFM

1.42

-5.22

τavg = 8.77 K·min T F M τ

1.48

-5.09

TFMS

1.98

-5.04

TFMτS

2.03

-4.91

T (g)

1.79

-3.75

T F (g)

0.02

-5.55

T F τ (g)

0.07

-5.46

T F S (g)

1.84

-3.58

T F τ S (g)

1.87

-3.52

T (w)

2.05

-3.22

T F (w)

0.97

-4.81

T F τ (w)

1.01

-4.73

T F S (w)

2.29

-2.70

T F τ S (w)

2.32

-2.66

TFM

3.05

-1.50

TFMτ

3.07

-1.47

TFMS

2.21

-2.90

Ground turkey
nobs = 23
fat = 1.05%

Whole chicken
nobs = 21
fat = 0.33%
τavg = 7.92 K·min

TFMτS
2.24
-2.86
*where (g)+(w) models appear, ground-muscle calibrated models predicted lethality for groundmuscle data and whole-muscle calibrated models did for whole-muscle data.

163

Table 47. Mixed-effects models prediction errors on beef steaks and patties (impingement
cooked) (continued next page).
Data source for
validation

Max error

Model*

Min error

log (CFU/g)

T (g)

-3.14

3.56

-3.10

T F S (g)

3.82

-4.02

T F τ S (g)

3.88

-3.97

T (g) + T (w)

4.24

-2.67

T F (g) + T F (w)

3.49

-2.88

T F τ (g) + T F τ (w)

3.56

-2.84

T F S (g) + T F S (w)

3.82

-3.03

T F τ S (g) + T F τ S (w)

3.88

-2.99

TFM

3.41

-2.08

TFMτ

3.47

-2.01

TFMS

2.91

-3.65

TFMτS

2.98

-3.54

T (g)

4.24

-1.40

T F (g)
Ground beef

3.49

T F τ (g)

nobs = 44

-2.80

T F (g)

Ground
+
Whole beef

4.24

3.49

-1.91

T F τ (g)

3.56

-1.84

T F S (g)

3.82

-2.30

nobs = 19
fat = 2.32%

T F τ S (g)

3.88

-2.22

τavg =
9.28 K·min

TFM

3.41

-2.08

TFMτ

3.47

-2.01

TFMS

2.91

-3.65

TFMτS
2.98
-3.54
*where (g)+(w) models appear, ground-muscle calibrated models predicted lethality for groundmuscle data and whole-muscle calibrated models did for whole-muscle data.

164

Table 47 (cont’d). Mixed-effects models prediction errors on beef steaks and patties
(impingement cooked).
Data source for
validation

Max error

Model*

Min error

log (CFU/g)

T (g)

2.39

-2.80

T F (g)

1.61

-3.14

T F τ (g)

1.67

-3.10

T F S (g)

2.07

-4.02

T F τ S (g)

2.12

-3.97

T (w)

2.47

-2.67

nobs = 25
fat = 2.68%

T F (w)

2.35

-2.88

T F τ (w)

2.40

-2.84

τavg =
8.18 K·min

T F S (w)

2.33

-3.03

T F τ S (w)

2.37

-2.99

TFM

2.70

-1.64

TFMτ

2.74

-1.61

TFMS

2.58

-2.95

Whole beef

TFMτS
2.62
-2.90
*where (g)+(w) models appear, ground-muscle calibrated models predicted lethality for groundmuscle data and whole-muscle calibrated models did for whole-muscle data.

165

Table 48. Mixed-effects models prediction errors on pork steaks and patties (impingement
cooked) (continued next page).
Data source for
validation

Max error

Model*

Min error

log (CFU/g)

T (g)

4.08

-4.68

T F (g)

5.51

-1.65

T F τ (g)

5.56

-1.61

T F S (g)

5.59

-1.75

T F τ S (g)

5.64

-1.76

T (g) + T (w)

3.73

-4.68

T F (g) + T F (w)

5.51

-5.82

T F τ (g) + T F τ (w)

5.56

-5.75

T F S (g) + T F S (w)

5.59

-3.04

T F τ S (g) + T F τ S (w)

5.64

-2.97

TFM

5.23

-1.70

TFMτ

5.28

-1.66

TFMS

4.50

-3.09

TFMτS

4.44

-3.01

T (g)

3.73

-4.68

T F (g)

5.51

-0.66

T F τ (g)

5.56

-0.66

nobs = 27
fat = 10%

T F S (g)

5.59

-1.18

T F τ S (g)

5.64

-1.13

τavg =
9.34 K·min

TFM

5.23

-1.07

TFMτ

5.28

-1.02

TFMS

4.37

-3.09

Ground
+
Whole pork
nobs = 56

Ground pork

TFMτS
4.44
-3.01
*where (g)+(w) models appear, ground-muscle calibrated models predicted lethality for groundmuscle data and whole-muscle calibrated models did for whole-muscle data.

166

Table 48 (cont’d). Mixed-effects models prediction errors on pork steaks and patties
(impingement cooked).
Data source for
validation

Max error

Model*

Min error

log (CFU/g)

T (g)

4.08

-2.06

T F (g)

3.90

-1.65

T F τ (g)

3.92

-1.61

T F S (g)

4.41

-1.75

T F τ S (g)

4.12

-1.76

T (w)

3.08

-4.44

nobs = 29
fat = 1.53 %

T F (w)

2.50

-5.82

T F τ (w)

2.54

-5.75

τavg =
8.21 K·min

T F S (w)

3.97

-3.04

T F τ S (w)

4.00

-2.97

TFM

3.95

-1.70

TFMτ

3.98

-1.66

TFMS

4.50

-1.36

Whole pork

TFMτS
4.32
-1.37
*where (g)+(w) models appear, ground-muscle calibrated models predicted lethality for groundmuscle data and whole-muscle calibrated models did for whole-muscle data.

167

Table 49. Mixed-effects models prediction errors on ALL impingement oven data.
Data source for
validation

Max error

Model*

Min error

log (CFU/g)

T (g) + T (w)

5.51

-5.68

T F τ (g) + T F τ (w)

5.56

-5.75

T F S (g) + T F S (w)

6.03

-3.09

T F τ S (g) + T F τ S (w)

5.64

-3.00

TFM

5.23

-5.22

TFMτ

5.28

-5.09

TFMS

nobs =144

-4.68

T F (g) + T F (w)
ALL
impingement
oven data

4.24

4.50

-5.04

TFMτS
4.44
-4.91
*In all cases, ground-muscle calibrated models predicted lethality for ground-muscle data and
whole-muscle calibrated models did for whole-muscle data. In addition, models used were
species specific, that is, turkey models predicted for turkey data, and so on.

168

6.9.1.2 Big roasts

Table 50. OLS models prediction errors on roasts.
Data source for
validation

Max error

Model

Min error

log (CFU/g)

T (g)

nobs = 13
fat = 2.68%
τrange =
100 – 480 K·min

Pork roasts
nobs = 20
fat =1.53%
τrange =
100 – 600 K·min

-14.19

T τ (g)

-9.08

-13.16

T F τ (g)

-8.96

-13.04

T (w)

-2.06

-5.30

T F (w)

-2.00

-5.25

T τ (w)

-1.21

-4.79

-1.16

-4.74

-2.23

-15.19

T F (g)

-1.75

-14.14

T τ (g)

-1.78

-13.87

T F τ (g)

-1.33

-12.90

T (w)

1.15

-6.12

T F (w)

1.45

-7.14

T τ (w)

1.43

-6.98

1.69

-6.41

T (g)

Beef roasts

-10.83

T F τ (w)

τrange =
200 – 500 K·min

T F (g)

T (g)

nobs = 9
fat = 0.27%

-14.32

T F τ (w)

Turkey roasts

-10.96

-0.11

-4.64

T F (g)

-0.01

-4.50

T τ (g)

0.03

-3.80

T F τ (g)

0.14

-3.68

T (w)

1.60

-1.77

T F (w)

1.71

-1.65

T τ (w)

1.70

-1.30

T F τ (w)

1.81

-1.16

169

Table 51. Mixed-effects models prediction errors on roasts (continued next page).
Data source for
validation

Max error

Model

Min error

log (CFU/g)

T (g)

-9.06

-15.21

T F τ S (g)

1.03

-3.54

-1.78

-5.21

T F (w)

-15.90

-22.66

T F τ (w)

-1.98

-7.22

T F S (w)

-2.39

-5.72

T F τ S (w)

1.26

-3.13

-1.58

-5.93

TFMτ

1.45

-2.62

TFMS

-3.56

-8.17

TFMτS

1.64

-2.75

T (g)

1.00

-8.13

T F (g)

-1.97

-13.48

T F τ (g)

0.89

-8.32

T F S (g)

0.92

-8.81

T F τ S (g)

3.33

-5.21

T (w)

τrange =
100 – 480 K·min

-12.02

TFM

nobs = 13
fat = 2.68%

-6.51

T (w)

Beef roasts

-22.66

T F S (g)

τrange =
200 – 500 K·min

-15.90

T F τ (g)

nobs = 9
fat = 0.27%

-6.28

T F (g)

Turkey roasts

-2.97

-1.97

-13.47

T F (w)

0.91

-8.32

T F τ (w)

3.08

-4.62

T F S (w)

0.97

-8.29

T F τ S (w)

3.15

-4.60

TFM

0.89

-7.68

TFMτ

2.66

-4.21

TFMS

1.83

-6.67

TFMτS

3.65

-3.72

170

Table 51 (cont’d). Mixed-effects models prediction errors on roasts.
Data source for
validation

Max error

Model

Min error

log (CFU/g)

T (g)

-4.47

1.46

-1.21

T F S (g)

2.58

-0.56

T F τ S (g)

3.27

0.71

T (w)

1.81

-1.02

T F (w)

1.14

-1.77

T F τ (w)

1.46

-1.21

T F S (w)

1.09

-2.53

T F τ S (w)

2.02

-0.48

TFM

1.47

-2.84

TFMτ

2.15

-0.42

TFMS

2.81

-0.36

TFMτS

τrange =
100 – 600 K·min

0.88

T F τ (g)

nobs = 20
fat = 1.53%

-0.02

T F (g)

Pork roasts

2.96

3.44

0.82

Table 52. OLS models prediction errors on ALL roast data.
Data source for
validation

Max error

Model

Min error

log (CFU/g)

T (w)
ALL roast data
nobs = 42

1.60

-6.12

T F (w)

1.71

-7.14

T τ (w)

1.70

-6.98

T F τ (w)

1.81

-6.41

171

Table 53. Mixed-effects models prediction errors on ALL roast data.
Data source for
validation

Max error

Model

Min error

log (CFU/g)

T (w)

1.14

-22.66

T F τ (w)

3.08

-7.22

T F S (w)

1.09

-8.29

T F τ S (w)

3.15

-4.60

TFM

1.47

-7.68

TFMτ

2.66

-4.21

TFMS

2.81

-8.17

TFMτS

nobs = 42

-13.48

T F (w)
ALL roast data

1.81

3.65

-3.72

6.9.1.3 Hot dogs
Table 54. OLS models prediction errors on hot dogs.
Data source for
validation

Turkey hot dogs
nobs = 12
fat = 4.28%
τrange =
50 – 125 K·min
Beef hot dogs
nobs = 12
fat =15.42%
τrange =
100 – 275 K·min

Model

Max error

Min error

log (CFU/g)

T (g)

1.88

-4.85

T F (g)

2.14

-3.90

T τ (g)

2.22

-3.61

T F τ (g)

2.44

-3.06

T (g)

-0.49

-3.50

T F (g)

0.16

-1.64

T τ (g)

0.35

-1.80

T F τ (g)

0.75

-0.46

172

Table 55. OLS models predictions errors on ALL hot dog data.
Data source
for validation

Max error

Model

Min error

log (CFU/g)

nobs = 24

T (g)

1.88

-4.85

T F (g)

2.14

-3.90

T τ (g)

ALL hot dog
data

2.22

-3.61

T F τ (g)

2.44

-3.06

Table 56. Mixed-effects models prediction errors on hot dogs.
Data source for
validation

Model

Max error

Min error

log (CFU/g)

T (g)

Beef hot dogs
nobs = 12
fat = 15.42%
τrange =
100 – 275 K·min

2.17

-3.05

T F τ (g)

5.92

-2.67

T F S (g)

3.16

-1.19

T F τ S (g)

5.04

-0.89

TFM

6.13

-3.06

TFMτ

2.47

-2.68

5.82

-2.00

5.23

-1.73

T (g)

τrange =
50 – 125 K·min

T F (g)

TFMτS

nobs = 12
fat =4.28%

-1.68

TFMS

Turkey hot dogs

2.90

0.41

-0.96

T F (g)

0.63

-0.37

T F τ (g)

2.72

-0.04

T F S (g)

2.07

-1.01

T F τ S (g)

3.81

0.77

TFM

3.72

-0.33

TFMτ

3.07

0.0

TFMS

2.83

0.35

TFMτS

2.33

0.52

173

Table 57. Mixed-effects models prediction errors on ALL hot dog data.
Data source
for validation

Model

Max error

Min error

log (CFU/g)

T (g)

2.17

-3.05

T F τ (g)

2.47

-2.67

T F S (g)

3.16

-1.08

T F τ S (g)

3.30

-0.89

TFM

2.17

-3.06

TFMτ

2.47

-2.68

TFMS

2.76

-2.00

TFMτS

nobs = 24

-1.68

T F (g)
ALL hot dog
data

2.90

2.96

-1.73

6.9.1.4 Overall model performance on pilot-scale data

Table 58. OLS models prediction errors on ALL pilot-scale data.
Data source
for validation

ALL
pilot-scale
data
nobs = 210

Max error

Model

Min error

log (CFU/g)

T (g) + T (w)

2.76

-9.50

T F (g) + T F (w)

2.96

-9.15

T τ (g) + T τ (w)

2.84

-9.29

T F τ (g) + T F τ (w)

3.03

-8.95

174

Table 59. Mixed-effects models prediction errors on ALL pilot-scale data.
Data source for
validation

Model

Max error

Min error

log (CFU/g)

T (g) + T (w)

-22.66

5.56

-7.22

T F S (g) + T F S (w)

6.03

-8.29

T F τ S (g) + T F τ S (w)

5.64

-4.60

TFM

5.23

-7.68

TFMτ

5.28

-5.09

TFMS

4.50

-8.17

TFMτS

6.9.2

5.51

T F τ (g) + T F τ (w)

nobs = 210

-13.48

T F (g) + T F (w)
ALL
pilot-scale data

4.24

4.44

-4.91

Plots showing model predictions and PIs for representative data sets and models

Given the large number of validated models and data sets, this project produced ~320
plots (one for each validation). However, it was decided to present the plots of only a
representative data set across all models, in addition to any other that may have shown special
features.
6.9.2.1 OLS models

To show the effect of the addition of the fat and sublethal injury parameters (β3 and β2,
represented by F and τ in model names) and the importance of using whole-muscle calibrated
models (represented by (w)) to predict lethality in whole-muscle data (instead of ground-muscle
calibrated models, represented by (g)), the impingement-cooked whole-muscle pork steaks were
chosen. Notice the slight improvement created by the addition of β2 and β3, but the significant
difference caused by the use of the whole-muscle-calibrated models (w) instead of the groundmuscle-calibrated versions (g).

175

95% PI widthavg= ±2.12 log CFU/g
RMSE = 1.95 log CFU/g
bias = -0.82 log CFU/g
% above low PI = 83%

Figure 47. OLS T (g) model validated with whole-muscle pork steaks.

95% PI widthavg= ±2.04 log CFU/g
RMSE = 1.92 log CFU/g
bias = -0.78 log CFU/g
% above low PI = 69%

Figure 48. OLS T F (g) model validated against whole-muscle pork steaks.

95% PI widthavg= ±2.05 log CFU/g
RMSE = 1.94 log CFU/g
bias = -0.82 log CFU/g
% above low PI = 69%

Figure 49. OLS T τ (g) model validated against whole-muscle pork steaks.

176

95% PI widthavg= ±2.04 log CFU/g
RMSE = 1.89 log CFU/g
bias = -0.72 log CFU/g
% above low PI = 69%

Figure 50. OLS T F τ (g) model validated against whole-muscle pork steaks.

95% PI widthavg= ±2.12 log CFU/g
RMSE = 1.71 log CFU/g
bias = -0.04 log CFU/g
% above low PI = 93%

Figure 51. OLS T (w) model validated against whole-muscle pork steaks.
95% PI widthavg= ±2.11 log CFU/g
RMSE = 1.69 log CFU/g
bias = 0.08 log CFU/g
% above low PI = 93%

Figure 52. OLS T F (w) model validated against whole-muscle pork steaks.

177

95% PI widthavg= ±2.11 log CFU/g
RMSE = 1.71 log CFU/g
bias = 0.00 log CFU/g
% above low PI = 93%

Figure 53. OLS T τ (w) model validated against whole-muscle pork steaks.

95% PI widthavg= ±2.11 log CFU/g
RMSE = 1.69 log CFU/g
bias = 0.08 log CFU/g
% above low PI = 93%

Figure 54. OLS T F t (w) model validated against whole-muscle pork steaks.

6.9.2.2 Mixed-effects models

For the mixed-effect models, the impingement-cooked whole-muscle chicken breast
samples were chosen because of the significant changes noted with the addition of parameters
and with the use of the ground-muscle-calibrated model (represented by (g)) versus the wholemuscle-calibrated versions (represented by (w)).

178

95% PI widthavg= ±3.30 log CFU/g
RMSE = 1.34 log CFU/g
bias = -0.16 log CFU/g
% above low PI = 100%

Figure 55. Mixed-effects T (g) model validated against whole-muscle chicken breasts.

95% PI widthavg= ±3.05 log CFU/g
RMSE = 2.72 log CFU/g
bias = -2.25 log CFU/g
% above low PI = 71%

Figure 56. Mixed-effects T F (g) model validated against whole-muscle chicken breasts.

95% PI widthavg= ±3.02 log CFU/g
RMSE = 2.65 log CFU/g
bias = -2.18 log CFU/g
% above low PI = 71%

Figure 57. Mixed-effects T F τ (g) model validated against whole-muscle chicken steaks.

179

95% PI widthavg= ±4.82 log CFU/g
RMSE = 1.33 log CFU/g
bias = -0.04 log CFU/g
% above low PI = 100%

Figure 58. Mixed-effects T F S (g) model validated against whole-muscle chicken breasts.

95% PI widthavg= ±4.73 log CFU/g
RMSE = 1.32 log CFU/g
bias = 0.01 log CFU/g
% above low PI = 100%

Figure 59. Mixed-effects T F S τ (g) model validated against whole-muscle chicken breasts.
95% PI widthavg= ±0.83 log CFU/g
RMSE = 1.31 log CFU/g
bias = 0.23 log CFU/g
% above low PI = 81%

Figure 60. Mixed-effects T (w) model validated against whole-muscle chicken breasts.

180

95% PI widthavg= ±0.85 log CFU/g
RMSE = 1.86 log CFU/g
bias = -1.20 log CFU/g
% above low PI = 38%

Figure 61. Mixed-effects T F (w) model validated against whole-muscle chicken breasts.

95% PI widthavg= ±0.85 log CFU/g
RMSE = 1.82 log CFU/g
bias = -1.14 log CFU/g
% above low PI = 38%

Figure 62. Mixed-effects T F τ (w) model validated against whole-muscle chicken breasts.

95% PI widthavg= ±1.66 log CFU/g
RMSE = 1.36 log CFU/g
bias = 0.50 log CFU/g
% above low PI = 95%

Figure 63. Mixed-effects T F S (w) model validated against whole-muscle chicken breasts.

181

95% PI widthavg= ±1.66 log CFU/g

RMSE = 1.38 log CFU/g
bias = 0.53 log CFU/g
% above low PI = 95%

Figure 64. Mixed-effects T F S τ (w) model validated against whole-muscle chicken breasts.

95% PI widthavg= ±1.35 log CFU/g

RMSE = 1.73 log CFU/g
bias = 1.22 log CFU/g
% above low PI = 100%

Figure 65. Mixed-effects T F M model validated against whole-muscle chicken breasts.

95% PI widthavg= ±1.34 log CFU/g

RMSE = 1.75 log CFU/g
bias = 1.25 log CFU/g
% above low PI = 100%

Figure 66. Mixed-effects T F M τ model validated against whole-muscle chicken breasts.

182

95% PI widthavg= ±2.56 log CFU/g

RMSE = 1.36 log CFU/g
bias = 0.45 log CFU/g
% above low PI = 100%

Figure 67. Mixed-effects T F M S model validated against whole-muscle chicken breasts.

Figure 68. Mixed-effects T F M S τ model validated against whole-muscle chicken breasts.

183

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