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WESIS

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UNCERTAINTY ASSESMENT AND VALIDATION OF
PREDICTIVE MICROBIAL GROWTH MODELS

presented by

KARINA G. MARTINO

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UNCERTAINTY ASSESSMENT AND VALIDATION OF PREDICTIVE
MICROBIAL GROWTH MODELS

By

Karina G. Martino

A DISSERTATION
Submitted to
Michigan State University

in partial fulﬁllment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Biosystems and Agricultural Engineering

2006

ABSTRACT

UNCERTAINT Y ASSESSMENT AND VALIDATION OF PREDICTIVE
MICROBIAL GROWTH MODELS

By
Karina G. Martino

Microbial models enable a proactive approach, conveniently used by the food
industry and risk assessors, to predict microbial food safety. However, the validity,
reliability, and uncertainty of these models in application to real food products are rarely
well known. Therefore, the overall goals of this study encompassed validation of
predictive microbial growth models, assessment of the uncertainty related to those
models, and deconstruction of the different errors that contribute to the total uncertainty
of a microbial growth model.

For illustration purposes, Listeria monacytogenes growth data from laboratory
broth and meat and poultry products were used throughout. The primary and secondary
models used in the US. Department of Agriculture — Agricultural Research Service
(U SDA-ARS) Pathogen Modeling Program (PMP), a widely used tool by the food
industry to estimate pathogen growth/survival/inactivation in food, were the principal
models analyzed throughout this study.

Robustness of the broth-based growth models was evaluated using a Robustness
Index (RI). Inside the calibration domain of the PMP, the best R1 for application to meat
products was 0.37; the worst was 3.96. Outside the domain, the best R1 was 0.40, and the
worst was 1.22. Meat product type inﬂuenced the RI values (P<0.01).

Two different microbial modeling procedures, using the broth-based data, were

compared and validated against independent data for microbial growth in meat and

poultry products. A global regression method yielded a lower root mean squared error,
0.95 log(CFU/ml) for aerobic and 1.21 log(CFU/ml) for anaerobic conditions, than did a
two-step procedure, which yielded errors of 1.35 log(CFU/ml) for aerobic and 1.62
log(CFU/ml) for anaerobic conditions. Validating with data from meat and poultry, the
global regression was more robust than the two-step procedure for 65% of the cases
studied. However, the predictions were overestimated (fail-safe) in more cases for the
two-step than for the global regression.

In deconstructing the overall model error, the total uncertainty was assumed to be
an aggregated contribution of errors due to organism, substrate, laboratory
methodologies, replications, and primary and secondary regressions. The total
uncertainties for aerobic and anaerobic conditions, with the PMP L. monocytogenes
growth models, were 1.35 and 1.62 log(CFU/ml), respectively. The errors from the
primary regression were 1.02 and 1.22 log(CFU/ml), for aerobic and anaerobic
conditions, respectively. The errors from the secondary regression were 1.48 and 1.42
log(CFU/ml), for aerobic and anaerobic conditions respectively. The variability due to
replications was 0.26 log(CFU/ml) for aerobic and 0.21 log(CFU/ml) for anaerobic
conditions.

Following the methodologies described here could lead to better informed and

more reliable decisions, for ensuring food safety and for evaluating consumer risk.

Cepyright by
KARINA G. MARTINO
2006

To mfamily

ACKNOWLEDGMENTS

Funding for this study and for my degree completion came from the US.
Department of Agriculture, Agricultural Research Service, cooperative agreement No.
58-1935-2-238, via the National Alliance for Food Safety and Security; the MSU
Quantitative Biology and Modeling Initiative; and the MSU Graduate School Dissertation
Completion Fellowship.

My sincere appreciation goes to my major professor, Dr. Bradley Marks, for his
endless support, encouragement, and motivation. I would also like to thank my
committee members, Doctors D. Gilliland, K. Dolan, and E. Ryser; the members of our
research team; and my friends.

Finally a very special thanks to my beloved husband and daughter, and my family

for all their support throughout this journey.

TABLE OF CONTENTS

 

 

 

 

 

 

 

LIST OF FIGURES ix

LIST OF TABLES xii

CHAPTER 1 - - _ - 1

INTRODUCTION .......................................................................................................... 1

1.1 STRUCTURE AND SCOPE OF THE DISSERTATION .................................... 1

1.2 IMPACT OF PREDICTIVE MICROBIOLOGY IN FOOD SAFETY ................ 2

1.3 JUSTIFICATION ................................................................................................. 4

1.4 OBJECTIVES ....................................................................................................... 5

CHAPTER 2 - -- 6

LITERATURE REVIEW ............................................................................................... 6

2.1 PREDICTIVE MICROBIOLOGY ....................................................................... 6

2.1.1 Primary growth models .................................................................................. 7

2.1.2 Secondary growth models ............................................................................ 10

2.1.3 Tertiary growth models ................................................................................ 12

2.3 GROWTH MODELS .......................................................................................... 15

2.4 MODEL LIMITATIONS .................................................................................... 17

CHAPTER 3 20

ROBUSTNESS OF MICROBIAL GROWTH MODELS ........................................... 20

3.1 SUMMARY ........................................................................................................ 20

3.2 INTRODUCTION .............................................................................................. 21

3.3 MATERIALS AND METHODS ........................................................................ 24

3.3.1 Data sources ................................................................................................. 24

3.3.2 Predictive models. ........................................................................................ 26

3.3.4 Robustness Index (RI) .................................................................................. 29

3.4 RESULTS AND DISCUSSION ......................................................................... 31

CHAPTER 4 40
EFFECT OF DIFFERENT MODELING PROCEDURES ON MICROBIAL

GROWTH MODEL PERFORMANCE ....................................................................... 40

4.1 SUMMARY ........................................................................................................ 40

4.2 INTRODUCTION .............................................................................................. 41

4.3 MATERIALS AND METHODS ........................................................................ 44

4.3.2 Predictive models ......................................................................................... 45

4.3.3 Robustness Index (RI) .................................................................................. 49

4.4 RESULTS AND DISCUSSION ......................................................................... 50

CHAPTER 5 60

UNCERTAINTY ASSESSMENT IN BROTH-BASED MICROBIAL ...................... 60

GROWTH MODELS .................................................................................................... 60

5.1 BACKGROUND ................................................................................................ 60

5.2 METHODS ......................................................................................................... 63

vii

5.2.1 Data sources. ................................................................................................ 63

 

 

 

 

 

5.2.2 Error sources ................................................................................................ 63
5.2.3 Error calculations ......................................................................................... 64
5.2.4 Calculation of the prediction limits and parameter errors ........................... 66
5.3 RESULTS AND DISCUSSION ......................................................................... 68
5.3.1 Deconstruction of the model uncertainty ..................................................... 68
5.3.2 Prediction intervals and errors of the parameters ........................................ 71
CHAPTER 6 74
CONCLUSIONS AND FUTURE WORK ................................................................... 74
6.1 CONCLUSIONS ................................................................................................. 74
6.2 SUGGESTIONS FOR FUTURE WORK ........................................................... 77
APPENDIX A 80
COMPUTER PROGRAMS USED FOR DATA ANALYSIS AND DATA SOURCE
....................................................................................................................................... 80
APPENDIX B 83
BROTH-BASED AND MEAT-BASED DATA DESCRIPTION AND
ORGANIZATION ........................................................................................................ 83
APPENDIX C 110
STANDARD ERROR ANALYSIS ............................................................................ 110
APPENDIX D 1 15
DJ COMPARISON OF EXPERIMENTAL VARIABILITY BETWEEN BROTH
AND MEAT-BASED DATA ..................................................................................... 115

D2 STANDARD ERROR OF PREDICTION AND ROBUSTNESS INDEX
VALUES OBTAINED AFTER NON-SIGNIFICANT TERMS WERE ELIMINATED

 

..................................................................................................................................... 120
APPENDIX E 121
SCRIPTS USED FOR NONLINEAR REGRESSION AND ..................................... 121
DATA ANALYSIS IN JMP ....................................................................................... 121
E1. SCRIPT USED FOR NONLINEAR REGRESSION. .................................... 121
E2. SCRIPT USED FOR NONLINEAR REGRESSION WITHOUT SECOND
DERIVATIVE. ....................................................................................................... 122
E3. SCRIPT USED FOR NONLINEAR REGRESSION WITH SECOND
DERIVATIVE. ....................................................................................................... 122
EA SCRIPT USED FOR REITERATIVE ANALYSIS. ....................................... 122
REFERENCES 123

 

viii

LIST OF FIGURES

F IGURE.1.1. Illustration of the impact that uncertainty in process calculations can have
on product safety. ........................................................................................................ 5
FIGURE 2.1. Microbial growth curve. ............................................................................... 9
FIGURE 3.1. Comparison of the predicted (solid line) and actual (full squares) growth
log counts ﬁom the data set (No. 13) resulting in the best RI value (0.37) inside the
PMP model domain (95% conﬁdence intervals, broken lines). ................................ 34
FIGURE 3.2. Comparison of the predicted (solid line) and actual (full squares) growth
log counts from the data set (No. 23) resulting in the worst RI value (3.96) inside the
PMP model domain (95% conﬁdence intervals, broken lines) ................................. 34
FIGURE 3.3. Comparison of the predicted (solid line) and actual (full squares) growth
log counts from the data set (No. 48) resulting in the best RI value (0.40) outside the
PMP model domain (95% conﬁdence intervals, broken lines) ................................. 38
FIGURE 3.4. Comparison of the predicted (solid line) and actual (full squares) growth
log counts from the data set (No. 46) resulting in the worst RI value (1.22) outside
the PMP model domain (95% conﬁdence intervals, broken lines). .......................... 38
FIGURE 4.1. Observed versus predicted L. monocytogenes counts in broth ﬁ'orn global
regression, showing a randomly selected 10% of the total 3,680 data points and the
1:1 line (aerobic conditions). .................................................................................... 53
FIGURE 4.2. Observed versus predicted L. monocytogenes counts in broth from two-step
regression, showing a randomly selected 10% of the total 3,680 data points and the

1:1 line (aerobic conditions). .................................................................................... 54

ix

FIGURE 4.3. Robustness Index values of global regression applied to meat and poultry
data. ........................................................................................................................... 57
FIGURE 4.4. Robustness Index values of two-step regression applied to meat and poultry
data. ........................................................................................................................... 57
FIGURE 4.5. Relative error values of global regression applied to meat and poultry data.
................................................................................................................................... 58
FIGURE 4.6. Relative error values of two-step regression applied to meat and poultry
data. ........................................................................................................................... 58
FIGURE 5.1. Conﬁdence (small dashed lines) and prediction (wide dashed lines)
intervals for global regression (data set No. 623, aerobic conditions, treatment No.
26: pH = 6, T =19°C, nitrite = 0 ppm, salt = 0 g/liter). Solid line: predicted curve;
full squares: observed data. ....................................................................................... 71
FIGURE 5.2. Conﬁdence (small dashed lines) and prediction (wide dashed lines)
intervals for global regression (data set No. 646, aerobic condition, treatment No.
80: pH = 7, T = 19°C, nitrate = 0 ppm, salt = 25 g/liter). Solid line: predicted curve;
full squares: observed data. ....................................................................................... 72
FIGURE A.l. Printed screen of JMP (SAS Institute Inc, Cary, NC. Version 4.0.4)
formula box that contains the global model. ............................................................. 80
FIGURE A.2. Screen picture of PMP (US. Department of Agriculture, 2003b), which
shows the growth curve for L. monocytogenes, the growth parameters, and the

experimental variables. ............................................................................................. 81

FIGURE A.3. Screen picture of ComBase (US. Department of Agriculture, 2003a).
Experimental conditions, microorganism, environment, and authors can be chosen,
so related literature can be found. ............................................................................. 82
FIGURE C. 1 . Standard error (SE) versus time, treatment 1 for L. monocytogenes growth

in broth (pH= 4.5, T= 19 °C, salF 0 g/l, and nitrite= 0 ppm) ................................. 114

xi

LIST OF TABLES

TABLE 3.1. References and keys in ComBase for meat and poultry data used in this

study. ......................................................................................................................... 26
TABLE 3.2. Coefﬁcient values for secondary models ..................................................... 27
TABLE 3.3. R1 values for conditions inside the PMP model domain .............................. 32
TABLE 3.4. ANOVA results for R1 vs. product/process variables .................................. 36
TABLE 3.5. Overall RI values for species inside PMP domain ....................................... 36
TABLE 3.6. R1 values for conditions outside the PMP model domain ............................ 37

TABLE 4.1. References and keys in ComBase for meat and poultry products used for
model validation in this study. .................................................................................. 45
TABLE 4.2. Coefﬁcient values of secondary models for L. monocytogenes growth, ﬁtted
to broth-based data via two-step regression. ............................................................. 52
TABLE 4.3. Coeﬁicient values computed after global regression for L. monocytogenes
growth, ﬁtted to broth-based data. ............................................................................ 53

TABLE 4.4. SEC (uncertainty) of broth-based L. monocytogenes growth predictions

resulting from different modeling regression procedures. ........................................ 55
TABLE 4.5. R1 and RE values with experimental conditions (assumed aw = 0.997). ..... 56

TABLE 5.1. RMSE of the parameters B (h") and M (h) estimated from the secondary
regression. ................................................................................................................. 65

TABLE 5.2. Relative contributions of the different sources of error to the total model

uncertainty for aerobic and anaerobic conditions. .................................................... 68
TABLE 5.3. RMSE of the model parameters. ................................................................. 72

xii

TABLE B.l. Keys to ComBase for aerobic (data set No. 386-938) and anaerobic (data set
No. 1-385) conditions. .............................................................................................. 83
TABLE 82. No-growth broth-based data sets (L. monocytogenes) eliminated for
anaerobic and aerobic conditions. ............................................................................. 91
TABLE B.3. B (h") and M (h) estimated ﬁcm the primary regression for anaerobic
conditions. ................................................................................................................. 92
TABLE B.4. B (h") and M (h) estimated ﬁom the primary regression for aerobic
conditions. ................................................................................................................. 95
TABLE 35. Treatment (treat) and data set numbers for anaerobic conditions used to
calculate error due to replication calculations ......................................................... 100
TABLE B.6. Treatment (treat) and data set numbers for aerobic conditions used to
calculate error due to replication calculations ......................................................... 102
TABLE B.7. Experimental variables each treatment (aerobic conditions, (U .S.
Department of Agriculture, 2003a)) ........................................................................ 105

TABLE B.8. Experimental variables in each treatment (anaerobic conditions, (U .S.

Department of Agriculture, 2003a)) ........................................................................ 108
TABLE C. 1 . Effects tests for the experimental variables ............................................... 110
TABLE C.2. Effects test for the experimental variables with interactions. ................... 111
TABLE C.3. Data sets included in treatment No. 1 ........................................................ 112
TABLE D.1.1. Broth-based L. monogdogenes growth data. ......................................... 116
TABLE D.1.2. Meat-based L. monogdogenes growth data. .......................................... 118

xiii

CHAPTERl

INTRODUCTION

1.1 STRUCTURE AND SCOPE OF THE DISSERTATION

The overall subject of this study was the inﬂuence of predictive microbiology on the
food safety system. Chapter 1 is comprised of an introduction to this subject, followed by
a description of the need and speciﬁc research objectives of this dissertation. In order to
conduct a quantitative analysis, the growth of Listeria monocytogenes in broth was used
as the case study for the dissertation.

A general literature review, with respect to the basic topics that are covered in this
study, is presented in Chapter 2, including an overview of predictive microbiology, L.
monocytogenes, and growth models. This overview encompasses the basic concepts that
were used throughout this study.

Chapters 3, 4, and 5 are “stand-alone” manuscripts. Chapter 3 is a growth model
validation study, which was published in the Journal of Food Protection (Martino et a1. ,
2005). An introduction of the currently used validation methods is presented. The
Robustness Index (RI) was applied in order to validate broth-based models against data
collected from actual food products. The results quantiﬁed model robustness inside and
outside their original domain.

Chapter 4 is a paper that addresses model ﬁtting procedures (submitted for
publication in August 2006). This chapter analyzes how different ﬁtting procedures affect
the overall uncertainty of a growth model and its robustness when applied to a food
system. Knowing the inﬂuence of the ﬁtting methods on overall uncertainty of the model

can help improve the accuracy of model predictions by choosing the methodology that

gives lower prediction errors.

Chapter 5 presents the deconstruction of model uncertainty by identifying and
quantifying the different sources of error that contributed to it. These insights give a
better understanding of how these errors can be quantiﬁed and segregated, so efforts to
reduce them can be prioritized.

Chapter 6 presents overall conclusions and recommendations for future work.

1.2 IMPACT OF PREDICTIVE MICROBIOLOGY IN FOOD SAFETY

Current food safety tools used by food processors and risk assessors rely on
predictive tools. These predictive tools (e.g., predictive microbial software programs)
utilize models that describe the behavior of microorganisms under different physical or
chemical conditions, such as temperature, pH, and water activity. An example of this
kind of tool is the USDA-ARS Pathogen Modeling Program (PMP, (U .S. Department of
Agriculture, 2003b)), which is a widely used too] in the food industry. Model predictions
enable a proactive approach to avoid undesirable results or consequences (mainly human
illness or death). They allow the prediction of microbial food safety or shelf life of
products, the detection of critical parts of the production and distribution process, and the
optimization of production and distribution chains (Zwietering et al., 1990). In general, in
the food area, prediction comes from mathematical models that were developed from
broth-based data; however, some tools are being updated with models developed using
data from actual food systems. The food safety and food microbiology community is
working to account for variables that inﬂuence growth or death of foodbome pathogens

of concern. Unfortunately, the accuracy, or uncertainty of the predictions is often not well

known.

The predictive tools that are currently used in the food industry are still empirically
managed. There is no systematic or standardized approach to develop a model, conduct a
speciﬁc laboratory analysis for a particular microorganism, conduct statistical analysis, or
determine the uncertainty of the outcomes of a particular model.

Furthermore, regulatory agencies are putting increasing pressure on industries. For
example, the USDA Food Safety and Inspection Service (F SIS, 2003), following several
outbreaks, conﬁrmed that it will maintain its “zero tolerance” policy with respect to L.
monocytogenes in ready-to—eat meat and poultry products; no minimum lethality or
maximum growth will be allowed, on a 25 gram sample, in any of these products (F SIS,
2003). Predictive tools represent a vital element in food processing, in order to enable
rapid determination of potential pathogen growth.

Therefore, validation of these models is a critical component of predictive
microbiology; however, validation is usually done in terms of parameters (growth rates,
lag phase duration, etc.), forgetting that the true measure of product safety is actual
microbial counts, not model parameters. Once predicted values are calculated, their
uncertainty must also be established in order to determine limits, and to ensure that these
limits do not present unacceptable risk to consumers.

In order to perform an accurate quantitative analysis of microbial growth, predictive
microbiology needs to become a reliable tool, that is, a valid and systematic approach that
can be directly used by the food industry, regulators, and/or risk assessors. Moreover, an
increase in demand for predictive microbial software programs with application to food

systems is expected (T amplin et al., 2004), which includes the use of predictive models

for research, HACCP development, product formulation, and risk assessment.

1.3 JUSTIFICATION

From the farm to the table, there are unavoidable risks to consumers; a very
signiﬁcant one is microbial contamination, either intentional or not, of the food system.

Pathogens are present across the entire harvesting-processing—distribution chain;
therefore, it is necessary to take a proactive approach to minimize/prevent their survival.
Predictive tools enable this type of approach, and are especially useful for processors who
must comply with government regulations. However, problems can arise if processors
rely on these tools without questioning their validity.

For example, food processors calculating microbial growth have no reliable estimate
of uncertainty in those calculations; therefore, given normal process variability over time,
and an unknown uncertainty in the calculations, processors may under- or over-process
the product. Even though their prediction outcome is above the regulatory target, their
uncertainty limits could extend below the target, which could represent a risk to the
consumer (Fig. .1.1). However, if that processor had a tool that produced a reliable
estimate of the uncertainty in the process survival or growth calculations, then the degree
of over or under-processing could be based on real statistical information (prediction
error + normal process variability), and the safety of the product could be better ensured
(Fig. 1.1).

Therefore, there is a need to improve microbial prediction tools in order to help food
processors, academia, and regulatory agencies to better validate processes, conduct

microbial risk assessments, and better assess the uncertainty behind these predictions.

outcome :1: error (reported)

 

 

 

 

 

 

 

 

 

E- outcome :t: error (unremrted) g.
“-4 ---------------------- “d l- --------------------------
8 5i
‘6 '6 "' ‘“
:3 Regulatory :: Regulatory
g """""""""""""" target E- target
a) b)
operational time operational time

FIGURE] .1. Illustration of the impact that uncertainty in process calculations can
have on product safety.
1.4 OBJECTIVES
A quantitative assessment of errors in predictive microbiology has not been
conducted previously. Furthermore, acceptable limits of prediction have not been
established, meaning that conﬁdence of the actual prediction values have not been
deﬁned or calculated. Therefore, the overall goal of this study was to provide background
information with respect to overall model uncertainty, in order to improve current tools
used in predictive microbiology.
To advance toward that goal, the speciﬁc objectives of this study were:
1. To validate L. monocytogenes broth-based growth models in terms of microbial
counts.
2. To assess the performance of different ﬁtting procedures in predictive
microbiology.
3. To identify and quantify sources of error that contribute to microbial model
uncertainty.
4. To demonstrate that microbial food safety limits should be represented by

prediction intervals instead of conﬁdence intervals.

CHAPTER 2

LITERATURE REVIEW

2.1 PREDICTIVE MICROBIOLOGY

Predictive food microbiology was deﬁned by Schaffner and Labuza (Schaffner
and Labuza, 1997) as the gathering of “the disciplines of food microbiology, engineering,
and statistics to provide useful predictions about microbial behavior in food systems.”
Currently, predictive microbiology is considered an essential element of modern food
microbiology; furthermore, in the future, it could be accepted as a mature subdiscipline of
microbiology (McMeekin and Ross, 2002).

Microorganisms are primarily characterized by their adaptation to and
exploitation of change. They can colonize almost every habitat on earth, such as brine
ponds in the frozen wastes of polar regions, boiling water of hot springs, thermal volcanic
vents, and the bottom of the deep ocean (Adams and Moss, 1995). A rich microﬂora of
bacteria, yeasts and ﬁrngi can be found in structures such as leaves, fruits and roots,
which could be used as raw ingredients for food processing. Therefore, assurance of food
safety requires proactive and adaptive strategies. Generally, challenge tests are used to
describe the relationship between pathogens and the inﬂuence of environmental
conditions in their growth or decline. However, this traditional approach is typically
expensive and slow, and requires specialized facilities and microbiological skills
(Baranyi and Roberts, 1995). Therefore, with predictive microbiology, time and effort
can be minimized by quickly giving the ranges of concern for a factor and thereby
guiding the design of challenge tests, storage trials, and other conventional techniques to

assess the probability of pathogen growth (Whiting, 1995). However, speciﬁc

interactions between the microorganisms and their environment have to be known, the
predictive models have to be validated, and their related uncertainty should be carefully
assessed.

Predictive microbiology mathematically describes, using microbial models, the
growth or decline of foodbome microbes under speciﬁc environmental conditions,
allowing the prediction of microbial food safety or shelf life of products, the detection of
critical parts of the production and distribution process, and the optimization of
production and distribution chains (Zwietering et al., 1990).

Microbial models can be classiﬁed as primary, secondary, or tertiary (Whiting,
1995). Primary models describe how the number of microorganisms in a population
changes with time under speciﬁc conditions. Secondary models relate the primary model
parameters to environmental or intrinsic variables. Tertiary models combine primary and

secondary models with a computer interface, providing a complete prediction tool.

2.1.1 Primary growth models

Primary growth models can be classiﬁed as follows.

van Gerwen and Zwietering (1998) stated that assuming ﬁrst-order kinetics was
the simplest way to describe microbial growth, such that:

InN=lnN0+/.tt Eq. 2.1

where:

In N = microbial counts at time t

In N0 = initial microbial counts

,1: = growth rate

t= time

In contrast, the lag-exponential function includes the lag time (van Gerwen and

Zwietering, 1998):

1nN=lnN0,fort</t Eq. 2.2
lnN=lnN0+,u(t—A),fortzl Eq.2.3
it=lagtime

The growth curve also can be empirically described by several sigrnoidal
functions, such as logistic (eq. 2.4), Gompertz (eq. 2.5), or Richards, Schnute, and

Stannard (van Gerwen and Zwietering, 1998).

 

a
lnN = lnN + . 2.4
0 [1 + exp(b — cx)] Eq
where a, b, and c are ﬁt parameters.
0r,
lnN=lnN0 +Aexp{-exp|:ﬂga;—xi(l—t)+l]} Eq. 2.5
where:

,umax= maximum speciﬁc growth rate (h'l)

A = maximum level of increase: ln(Nw/No)

A typical bacterial growth curve is shown in Figure 2.1.

Stationary

Lag
‘ / Exponential

Log (cell numbers)

 

 

 

 

 

Time
FIGURE 2.1. Microbial growth curve.

In the lag-phase, there is no apparent growth, while the inoculum adjusts to the
new environment, synthesizes the enzymes required for its exploitation, and repairs any
lesions resulting from earlier injury (e. g., freezing, drying, heating) (Adams and Moss,
1995). The next phase, the exponential or logarithmic, is characterized by an increase in
cell numbers; the slope corresponds to the organism’s speciﬁc grth rate, p, which
depends on the experimental conditions. When key nutrients become depleted (aﬁer the
exponential growth), or inhibitory metabolites accumulate, the culture moves to the
stationary phase (Adams and Moss, 1995).

There are two levels of detail that can be used, depending on the necessity, for
microbial growth estimation (van Gerwen and Zwietering, 1998). For rough risk
assessments (level 1), orders of magnitude for growth can be estimated easily using
equation 1. Neglecting it results in fail-safe predictions, and stationary growth is
generally not relevant in risk assessments. However, if growth is one of the main

determinants of risk (level 2), the description of the entire growth curve (equation 5, for

example) would be useful.

2.1.2 Secondary growth models

The three approaches most frequently used with secondary growth models are the
response surface model (regression equation that is ﬁtted using standard regression
techniques, which may contain linear, quadratic, cubic, or reciprocal terms and includes
interaction or cross product terms, (Whiting, 1995)), the Arrhenius relationship
(logarithm of the rate versus the reciprocal of the temperature (K)), and the square root
model.

The square root model (eq. 2.6) is a simple empirical model suggested by
Ratkowsky et a1. (1982), in which the data (experimental growth rate) were transformed,
by taking their square root, to stabilize their variance (making the variance of the
distribution independent of its mean, and close to normal distribution, (Ratkowsky,
2004)). This model and its numerous expansions are called square-root-type, Ratkowsky-

type, or Béleradek-type models (Ross and Dalgaard, 2004).
pm =bx(T—Tmin) Eq. 2.6
where:

b = constant

T = temperature

Tm in = parameter, a theoretical minimum temperature for growth.

Growth rates of microorganisms are described less appropriately by the

Arrhenius-type equations than by square-root-type models (Ross and Dalgaard, 2004).

10

However, the Arrhenius-type models remain useful as secondary kinetic models for less

extensive ranges of storage temperatures. The empirical .Arrhenius-van’t Hoff

relationship is (Ross and Dalgaard, 2004):

E
rate = Aexp[— 72%] Eq. 2.7
where:
A = constant

Ea = activation energy

R = the gas constant (8.314 J/K/mol)

T = temperature in Kelvin

The most common secondary models applied within predictive microbiology are
the polynomial models. They are relatively easy to ﬁt to experimental data by multiple
linear regression, and they allow virtually any of the environmental variables and their
interactions to be taken into account (Ross and Dalgaard, 2004); however, higher order
polynomial models (e.g., cubic or quadratic) have been criticized for being too ﬂexible
and for attempting to model, rather than eliminate, experimental error; furthermore, these
models also include coefﬁcients with no biological interpretation (Baranyi et al. , 1996b;
Ross and Dal gaard, 2004). F mthermore, because of their ﬂexibility, they should only be
used to provide predictions by interpolation(Baranyi et al., 1996a; Baranyi et al. , 1996b).
A quadratic equation is an illustration of such polynomial models:

lny = £1 + ,6le + [33x2 + 64x3 + ﬂsxlxz + [36.7ch3

Eq. 2.8
+ ﬂ7x2x3 + ﬂ8x12 + [99x3 + 6101:? + e

11

where:
1n y = natural logarithm of the modeled growth responses (y = umax, lag time, or

maximum population density [MPD]).

ii,- (i = I, 10) = coefﬁcients to be estimated.
x1, x2, x3 = environmental variables (e.g., temperature, pH, salt)

e = random error

2.1.3 Tertiary growth models

In tertiary models, environmental values of interest are entered into secondary
models to obtain speciﬁc parameter values for the primary model. The primary model is
then solved for increasing periods of time to obtain the growth or inactivation curve
expected ﬁom that combination of environmental values. The primary and secondary
models are used in conjunction with spreadsheets or other software programs, which
avoids the reentering of equations, takes advantage of graphics capabilities of the
software, and allows performance of other calculations. Tertiary systems vary in
complexity from an equation on a spreadsheet to expert systems or risk assessment
simulations (Doyle et al., 2001).

An example of the tertiary model is the USDA Pathogen Modeling Program
(PMP, (U .S. Department of Agriculture, 2003b)). The PMP (Fig. A2) is a free software
package of microbial models that describes growth, survival, inactivation, or toxin
production of several pathogens, under various conditions deﬁned by the user. Depending
on the speciﬁc model, environmental inputs include atmosphere (aerobic or anaerobic),

temperature, pH, water activity, ionizing radiation, varying concentrations of lactic acid,

12

sodium chloride, nitrite, and sodium pyrophosphate, or all of these. Lag phase duration,
generation time, and time are displayed either in hours or days. Growth/inactivation
curves are displayed in both graphical and tabular formats, with their respective
conﬁdence intervals. In addition, the PMP contains dynamic temperature models for the
growth of Clostridium perﬁ'ingens and Clostridium botulinum (McKellar and Lu, 2004;
Tamplin et al., 2004; Whiting, 1995).

Once these predictive tools are validated and tested, they can be used in a variety
of ways. For example, quantitative risk assessment for the fate of pathogens in food
products relies on predictive models for growth, survival, and inactivation, in order to
determine strategies to improve food safety. In the same way, predictive microbiology
assists in identifying hazards and critical control points, and in specifying limits for
corrective actions in the formulation of hazard analysis critical control point (HACCP)
plans (McMeekin and Ross, 2002). Furthermore, the consequences of reformulating a
food product can be evaluated. Alternative formulations can be evaluated using models,
the inﬂuence of different factors can be obtained, and formulations with similar or
enhanced resistance to growth can be identiﬁed (Whiting, 1995).

Computer technology and advances in computational power make predictive
microbiology readily available for the ultimate user. However, this user (i.e., a food
processor) generally operates these tools “as is”, conﬁdent that the elements provided
were already validated and tested, because he/she does not have the time, facilities, or
knowledge to corroborate the validity of that too]. Therefore, it is necessary to improve
currently available tools, to avoid unknown predictions that could lead to dangerous

outcomes, causing an increasing risk for consumers.

13

2.2 LISTERIA MONOCYTOGENES

Listeria is a small, regular Gram-positive rod with round ends. The genus Listeria
contains six species: L. monocytogenes, L. ivanovii, L. innocua, L. welshimeri, L.
seeligeri, and L. grayi (Ryser and Marth, 1999).

L. monocytogenes, in particular, represents a threat to the food industry, due to its
resistance (it can survive adverse environmental conditions), its ability to colonize,
multiply, and persist on processing equipment, and its ability to multiply at refrigeration
temperatures. It can be found in soil, water, and in plant material. L. monocytogenes also
has been recognized as a human pathogen since 1929, and in 2000 the Centers for
Disease Control and Prevention (CDC) reported that of all the foodbome illnesses tracked
by CDC, L. monocytogenes had the second highest case fatality rate (21%) and the
highest hospitalization rate (90.5%) (U. S. Department of Health and Human Service,
2003). This disease usually occurs in high-risk groups, including pregnant women,
neonates, and immunocompromised adults (Ryser and Marth, 1999; U. S. Department of
Health and Human Service, 2003).

Foodbome listeriosis outbreaks could be dated back to 1914, even before Murray
isolated it in 1926 (Ryser and Marth, 1999). Vehicles of infection, reported since 1949,
can range from raw milk, sour milk, cream, pork, raw vegetables, shellﬁsh, raw eggs,
processed meats or pate, sweet corn, cheese, to rice salad (Ryser and Marth, 1999). These
foods are usually preserved by refrigeration and offer an appropriate environment for the
multiplication of L. monocytogenes during manufacturing, aging, transportation, and
storage.

Accurate prediction via mathematical models is becoming more important, not

14

only for risk assessors, but also for food processors, to evaluate the production-
processing-consumption chain for food products that can be suitable substrates for L.
monocytogenes growth. There are several models currently used to predict L.
monocytogenes growth, which were developed either ﬁom broth-based data or actual

food systems. A description of some of the more widely used models follows.

2.3 GROWTH MODELS

The most ﬁequently used primary growth models are the Gompertz (modiﬁed
model, eq. 2.7, (Gibson et al., 1987)) and Baranyi equations (eq. 2.8, (Doyle et al., 2001))
the ﬁrst being a sigmoidal relationship, and the second being based in part on the concept
that the rate of bacterial growth is controlled by the rate of a “bottleneck” biochemical
reaction.

l-Blt-Mll
L(t) = A + Ce—e Eq. 2.9

where:
L(t) = log counts of bacteria at time t (log(CFU/ml))
A = Asymptotic log count of bacteria as t decreases indeﬁnitely (log(CFU/rnl))
C = Asymptotic, incremental increase in log count of bacteria as t increases
indeﬁnitely (log(CFU/ml))
M = Time at which the absolute growth rate is maximum (h)
B = Relative growth rate at M (1/h)
t = time (h)

and,

15

Y, = Y0 + pmax A(t) - ln{1 + [exp(,umaxa(t) -—l)exp(Y1mm —— Y0 )]} Eq. 2.10
where:

Y,, Y 0, I’max = log cell concentration (log(CFU/ml)) at time t, at inoculation, and
at maximum cell density (stationary phase), respectively.

,umax = maximum speciﬁc growth rate (l/h)

A(t) = integral of the adjustment function cr(t).

Several authors compared the performance of these two models, plus some other
models, applied to different microorganisms (Baty and Delignette-Muller, 2004;
Buchanan et al., 1997; Juneja et al., 1999). For example, Baty et al. (2004) found that
the Gompertz model seems to be inﬂuenced more by the quality of the data set than is the
Baranyi model. They also concluded that the Baranyi model provided the best ﬁt for the
majority of their data and gave reasonably precise estimates of the lag time. Another
study also found that the Gompertz equation can overestimate the model parameters,
which could bias the comparison with a diﬂerent model (Membre et aI. , 2004).

There are a numerous studies on secondary models (Augustin et al., 1999;
Augustin et al., 2000; Buchanan and Phillips, 1990; Buchanan et al., 1989; Cheroutre-
Vialette et al., 1998; Dalgaard and Jorgensen, 1998; Delignette-Muller et al., 1995;
Farber et al., 1996; Fernandez et al., 1997; George et al., 1996; Whiting and Bagi, 2002;
Wijtzes et al., 1993), which include square-root, polynomial, Arrhenius, etc., applied to
L. monocytogenes. These studies focused mainly on how the lag phase and/or growth rate
is inﬂuenced by pH, water activity, carbon dioxide, lactic and acetic acids, salt, nitrite,

and/or temperature. These parameters are important mainly because they give an estimate

16

of the shelf life (how long a product will last under its storage conditions without L.
monocytogenes growing) of food products.

L. monocytogenes is a pathogen that has one of the highest rates of mortality;
fortunately, predictive microbiology is becoming a very signiﬁcant tool to limit risks
associated to this pathogen. Research done so far has extensively focused on the
inﬂuence of experimental conditions on growth parameters, most of them in broth
(Augustin et al., 1999; Augustin et al., 2000; Breand et al., 1999; Buchanan and Phillips,
1990; Buchanan et al., 1989; Comu et al., 2002; Delignette-Muller et al., 1995; Farber et
al., 1996; Fernandez et al., 1997; George et al., 1996; Whiting and Bagi, 2002; Wijtzes et
al., 1993), and some in food systems (Bovill et al., 2000; Cheroutre-Vialette et al., 1998;
Dalgaard and Jorgensen, 1998). However, to be conﬁdent about the results that users are
currently getting from models, more research is needed on validation of these models, in
terms of actual microbial counts; the previous cited literature only focused on validation
of models for growth parameters. Moreover, an overall assessment of the accuracy and
uncertainty of these predictions will further contribute to the true estimates of either

growth or inactivation of this foodbome pathogen.

2.4 MODEL LIMITATIONS

Models in general have inevitable limitations, either statistical or biological, that
need to be addressed before they are applied. For instance, these limitations could be the
inherent uncertainty of the model itself, the lack of ﬁt of the model to a speciﬁc data, the
type of ﬁtting, and the domain in which the model was developed.

Before analyzing a model, it is important to understand that the growth process is

17

both variable and uncertain. Growth is variable, because the growth curve of one
population will never be exactly the same as that of another population, not even for the
same strain under identical circumstances; this variability is reﬂected by a spread of
values for any particular property of the individuals that make up the population (Barker
et al., 2005; Nauta, 2000). Growth is uncertain, because growth progress will never be
exactly known; microbiological measurements used to construct the growth curve will
always be somehow imperfect (N auta, 2000). Both uncertainty and variability greatly
inﬂuence the performance of a model.

The goal is to use the simplest model that describes the response of an organism
to a speciﬁc food system. Ideally, the model should give parameters that help to
understand the system and design new experiments. Linear and nonlinear regressions can
be used to ﬁt a mathematical model to the data in order to determine the best-ﬁt values of
the model parameters (Motulsky and Christopoulus, 2004). Reliable prediction of risk
can be made by incorporating uncertainty in the predictions; in order to do that, the
accountability of its limits is necessary. Various methods can be used to characterize
uncertainty in statistics, such as the mean or standard deviation, including analytical
solutions and numerical solutions. Some transformations are used to homogenize the
variances for ﬁtting the models, so normal distribution can be attained. Although the
process of reparameterization can be controversial, it makes variances more uniform and
normally distributed and makes the parameters more interpretable (Ross and McMeekin,
2003; Whiting, 1995; Zheng and Frey, 2004).

When using nonlinear regression to ﬁt a model to raw data, resulting conﬁdence

and prediction intervals are only approximations. When a model is nonlinear in the

18

parameters, no explicit analytical solutions are available for the parameters or the
conﬁdence intervals, and a solution must be found by linear approximation; meaning,
iteratively, starting from initial values supplied by the analyst, or estimated by the
computer program. Determination of the prediction limits is difficult; however the easiest
method is to use asymptotic standard errors (SE), which are found ﬁom the matrix of
variances and covariances of the parameters (Van Boekel, 1996). Once the SE is
calculated, the conﬁdence intervals can be computed using the t-parameter for a
conﬁdence level (1-0.5a) and degrees of ﬁeedom v = n-p, where n is the number of data
points and p is the number of parameters. The Monte Carlo method is considered the best
method (Motulsky and Christopoulus, 2004; Van Boekel, 1996), in which synthetic data
are produced based on the model function and the obtained parameters, but with addition
of random errors. The synthetic data set is then analyzed again with the same model
function to obtain new parameter estimates. Doing this many times yields a distribution
of parameter values, from which a conﬁdence interval of the parameter and/or the
dependent variable can be formd.

Besides the uncertainty, lack of ﬁt, and type of ﬁtting of a model, the domain of
the model has to be clearly speciﬁed, meaning, what microorganisms, what factors, the
ranges of each factor, and what combination of factors give valid answers (biological
limitations); otherwise, cautious interpretation of the predictions is required (Whiting,
1995). The overall uncertainty of a model accounts for both the statistical and biological
limitations. Therefore, its assessment and understanding, either inside or outside the

model domain, would contribute to the improvement of risk and food safety predictions.

l9

CHAPTER 3

ROBUSTNESS OF MICROBIAL GROWTH MODELS

3.1 SUMMARY

Given the importance of Listeria monocytogenes as a risk factor in meat and
poultry products, there is a need to evaluate the relative robustness of predictive growth
models applied to meat products. The US. Department of Agriculture — Agricultural
Research Service (U SDA-ARS) Pathogen Modeling Program (PMP) is a tool widely
used by the food industry to estimate pathogen growth/survival/inactivation in food.
However, the robustness of the PMP broth-based L. monocytogenes growth model in
meat and poultry application has not been speciﬁcally evaluated. In the present study, this
model was evaluated against independent data in terms of predicted microbial counts
covering a range of conditions inside and outside the original model domain. The
Robustness Index (RI) was calculated as the ratio of the standard error of prediction, SEP
(root mean square error [RMSE] of the model against an independent data set not used to
create the model), to the standard error of calibration, SEC (RMSE of the model against
the data set used to create the model). Inside the calibration domain of the PMP, the best
R1 for application to meat products was 0.37; the worst was 3.96. Outside the domain, the
best R1 was 0.40, and the worst was 1.22. Product type inﬂuenced the RI values
(P<0.01). In general, the results indicated that broth-based predictive models should be
validated against independent data in the domain of interest; otherwise, signiﬁcant
predictive errors can occur.

This paper was published in the Journal of Food Protection ((Martino et a1. ,

2005).

20

3.2 INTRODUCTION

Quantitative risk assessments for the fate of pathogens in food products depend
heavily on the validity of predictive models for pathogen growth, survival, and
inactivation. An accurate prediction may have to consider whether a model is easy to use
(the simplest one for a given purpose and data quality), whether it is robust and accurate
(it must reﬂect reality), and whether it is validated against independent data sets
(Ratkowsky, 2004). The validation or performance evaluation of a model can also be
referred to as the robustness of the model (Campos et al. , 2004). The robustness indicates
how well a model predicts future independent results across a wide domain of conditions.
However, experimental data and associated models are rarely available to account for all
of the relevant variables and range of conditions for a speciﬁc pathogen, product, and
process being analyzed. Therefore, a risk assessment might extrapolate the predictive
models, either in terms of the process parameters (e.g., temperature) or product
parameters (e. g., fat content). Even though this practice is fundamentally undesirable, it
might be the only means to completing a risk assessment for a given product/system;
therefore, it is desirable and necessary to fully understand the implications of this
practice.

In particular, given the importance of Listeria monocytogenes as a risk factor in
meat and poultry products, there is a need to evaluate the relative robustness of predictive
microbial growth models for this speciﬁc pathogen. Previous research has shown that
product/process variables (e.g., pH, water activity) signiﬁcantly affect L. monocytogenes
response (Cheroutre-Vialette et al. , 1998). However, knowing that an effect exists is not

sufﬁcient to account for that effect quantitatively in predictive models. Some previous

21

studies only reported descriptive models (meaning that experimental data are generated
and a model is ﬁt to those data), which described the combined effect of temperature, pH,
water activity, and C02 concentrations (Farber et al., 1996; Wijtzes et al., 1993). Other
investigators compared their mathematical models (of the effect of C02, pH, temperature,
NaCl, organic acids and modiﬁed atmospheres) against independent data sets (Buchanan
and Phillips, 1990; Fernandez et al. , 1997; George et al. , 1996). Unfortunately, they made
only qualitative comparisons between observed and predicted values and did not present
a quantitative validation for their models.

On the other hand, Ross (Ross, 1996) presented the bias and accuracy factors as
indices to evaluate the performance of predictive models in food microbiology, in terms
of growth parameters (i.e., growth rates, lag phase duration). The bias factor is an overall
average of the ratio of discrete model predictions to observations and assesses whether or
not the model is “fail-safe”, “fail-dangerous”, or perfect. The accuracy factor is similar to
the bias factor, except that it is the absolute value of the ratio of predictions to
observation, thus providing an accumulated measure of overall model accuracy.
However, these factors have only been used to evaluate the performance of secondary
models in predicting growth parameters; actual log counts (ﬁom primary + secondary
models) were not considered.

The true measure of product safety is actual microbial counts, not model
parameters. Campos et al. (Campos et al. , 2004) introduced a new methodology to
evaluate the robustness of a microbial growth model in terms of microbial counts. The
robustness index (R1) was deﬁned as the ratio of the standard error of prediction (SEP) to

the standard error of calibration (SEC). The SEC and SEP are the root mean squared

22

errors (RMSE) calculated from the original and independent data sets, respectively. The
RMSE is one of the most useful and informative measures of the goodness-of—ﬁt against
the model prediction for linear and nonlinear regressions. Moreover, it is a way to
estimate the discrepancy between the observed and predicted data, which reﬂects whether
a model truly ﬁts the data well (Lammerding and McKcllar, 2004). A robust model will
have an R1 value near or less than 1, meaning that the overall performance of a microbial
model tested against an independent data set is within the expected error (SEC) of the
model. Campos et al. (2004) also stated that the RI value alone does not tell whether
observed values are above or below the predicted values; the mean relative error (RE) is
used with the R1 in order to provide this information.

The US. Department of Agriculture (USDA) -— Agricultural Research Service
(ARS) Pathogen Modeling Program (PMP, (2003b) and the UK. Food MicroModel are
tools used by the food industry to estimate pathogen growth/survival/inactivation in food.
The majority of these models were developed from pure-culture, broth-based data.
Because these models are based on pure-culture systems containing high level of
nutrients and no competitive microbial ﬂora, they are generally assumed to provide
conservative estimates of pathogen growth.

Several authors have considered the performance of the PMP and other microbial
growth models. For instance, te Giﬂ‘el and Zwietering (1999) evaluated the prediction of
L. monocytogenes growth rates in foods, including meat, by general models (e. g.,
Gamma-concept, PMP, Food MicroModel) and by speciﬁc models (e.g., modiﬁed
Arrhenius equation, third-order polynomial model, quadratic equation, etc.). They tested

these models against independent data sets and validated the models by graphical

23

comparison and mathematical/statistical comparison (mean squared error, regression
coefﬁcient, bias and accuracy factors). They recommended the use of a set of criteria to
evaluate the performance of models, because the use of one criterion may fail to reveal
some forms of systematic deviation between observed and predicted behavior. Again, the
prior study evaluated the performance of only a secondary model; actual log count
predictions were not evaluated.

Additionally, the evaluation of the models mentioned here did not include data
outside their original domain, which is critically important if they are to be applied to
broader risk analyses for foodbome pathogens in ready-to—eat food products.
Furthermore, the prior studies only evaluated secondary models for growth parameters
(i.e., growth rate, generation time, lag phase duration). They did not evaluate the
robustness of the complete model (primary + secondary), which predicts the actual
growth values, and gives the complete behavior (lag phase, exponential growth, and
stationary phase) of the pathogen of interest.

Therefore, the objective of the present study was to evaluate the robustness,
against independent data, of the PMP broth-based growth model for L. monocytogenes in
meat and poultry products, in terms of predicted microbial counts covering a range of

conditions inside and outside the original model domain.

3.3 MATERIALS AND METHODS
3.3.1 Data sources
ComBase ((Baranyi and Tamplin, 2004; US. Department of Agriculture, 2003a),

Fig. A.3) was used as the main source of independent data sets. ComBase pro-distribution

24

version 2002, was searched for all records that included microbial counts with organism:
“L. monocytogenes/innocua”, and broth or food category: meat or meat products. In total,
65 data sets were found; 41 were within the domain of the PMP L. monocytogenes
growth model, and 24 were outside the domain of the model (Table 3.1).

The original data sets used to develop the PMP broth-basedl L. monocytogenes
growth model were also obtained from ComBase ((U.S. Departrrrent of Agriculture,
2003a), source: “Buchanan_90”, organism: “L. monocytogenes/innocua ”, environment:
“culture medium”, pH: “0.1 to 14”, temperature: “-25 to 120”, and aw: “0.01 to 1”). These
data sets were assumed to be the original ones used in the PMP, because they were in the
same range of experimental conditions (pH: 4.5-7.5, nitrite: 50-1000 ppm, salt: 15-50 g/l,
and temperature: 5-37 °C), and had a similar number of data sets (385 for anaerobic, and

553 for aerobic). The no-growth data were eliminated (Buchanan and Phillips, 1990;
Buchanan et al., 1989). The remaining data sets (aw: 291, “points: 2,302 for anaerobic,
and ”sets: 476, “points: 3,680 for aerobic) were used to calculate the SEC of the PMP

growth model.

 

' “Inoculum: Strains were maintained in BHI broth at 4° C and transferred monthly. Procedure:
The test medium was 50 ml TPB in a 250 ml Erlenmeyer ﬂask (aerobic) or a 250 trypsinizing
ﬂask ﬁtted with a side arrn (anaerobic, nitrogen ﬂushed). Flasks were inoculated with 0.5 ml of a
24 h culture, shaken at 150 rpm. Enumeration: aliquots were plated on Tryptose Phosphate agar,
and incubated for 24 h at 37 C”, from Buchanan and Phillips (1990).

25

TABLE 3.1. References and keys in ComBase for meat and poultry data used in this

 

 

 

study.
Data set Key (ComBase) Product type Reference
No.
1-6 1206_Lm to 1211_Lm Ground Beef (Nissen et al., 2000)
7-12 JZ32_Lm to J237_Lm Cooked Chicken (Barakat and Harris,
1999)
13 M007 Pate (Bovill er al., 2000)
14-18 M200_LM to M204_LM Cooked Pork (Fang and Lin, I994)
19 M263_LM Precooked Beef (Cooksey et al., I 993)
20-24 M263_Lma to M263_Lme Precooked Beef (Cooksey et al., I 993)
25-26 M656_LM to M657_Lm Cooked Beef w/ Gravy (Grant er al., I 993)
27-28 M660_Lm to M661_Lm Cooked Beef w/ Gravy (Grant er al., 1993)
29 M921_LM Home-style salad (chicken with (Erickson et al., 1993)
no mayonnaise added)
'30 M921_LMa Home-style chicken salad (Erickson et al., 1993)
31 M921__LMb Home-style salad (real (Erickson et al., 1993)
mayonnaise + chicken)
32 M921_LMd Home-style salad (reduced (Erickson er al., 1993)
calorie mayonnaise + chicken)
33 M921_LMg Home-style salad (real (Erickson et al., I 993)
mayonnaise + chicken)
34 M921_Lmi Home-style salad (reduced (Erickson et al., 1993)
calorie mayonnaise + chicken)
35 SL1 13 Turkey (Mano et al., 1995)
36 SL1 18 Turkey (Mano et al., 1995)
37 SL123 Turkey (Mano er al., I 995)
38-41 SL59 to SL62 Pork (Mano et al., 1995)
42-52 M122_133 to M122_l44 Pate or ham (Mano et al., 1995)
54-65 M122_37 to M122_48 Pate or ham (Mano et al., I995)
3.3.2 Predictive models.

The robustness of the broth-based L. monocytogenes growth models (aerobic and
anaerobic) in PMP version 7.0 ((U.S. Department of Agriculture, 2003b) was determined
by testing the model predictions against the independent data sets. Because the PMP, as it

is distributed, cannot run outside the calibration domain, the secondary models used to

26

calculate generation time (GT) and lag phase duration (LPD) within the PMP domain
(Table 3.2) were implemented in a spreadsheet (source: A. Pickard, USDA-ARS Eastern
Regional Center), to generate predictions for the data sets that were outside the original

domain.

TABLE 3.2. Coefﬁcient values for secondary models

 

 

Aerobic Anaerobic
Variable Ln GT Ln LPD Ln GT Ln LPD
Int 21.45832 26.86796 13.51036 19.82645
T -0.26798 -021535 -0.10334 -0.20281
pH -5.29657 -6.5596 -3.34632 4.34946
NaCl 0.012824 0.051605 0.042326 0.031356
N02 0.020202 0.019974 0.021956 0.024464
1"po 0.00757 0.003684 -0.01424 -0.0032
TxNaCl 7.945-06 0.000223 -3.55-05 7.065-05
TxNOz -5.15-07 1.935-05 4.83E—06 1.625-05
pHxNaCl -0.00137 0.00686 -0.0036 -0.00181
pHxNOz -0.00278 -0.0028 0.00282 -0.00321
NaClxNOZ 5.28E—06 -3.75-06 4.14E-06 -2.75-06
T2 0.002666 0.001918 0.002725 0.003123
sz 0.384181 0.487334 0.262941 0.310527
NaClz 0.000122 0.000102 -0.00027 2.6505
N022 5.91507 7.365-07 -8.65-07 -4.8E-07

 

The primary model was the Gompertz equation:

L(t) =A+Ce'e
where:

L(t) = log counts of bacteria at time t (log (CFU/ml))

[‘3 (“Mil

Eq. 3.1

A = Asymptotic log count of bacteria as t decreases indeﬁnitely (log (CFU/ml))

C = Asymptotic log count of bacteria as t increases indeﬁnitely (log (CFU/ml))

27

M = Time at which the absolute growth rate is maximum (h)
B = Relative growth rate at M (l/h)
t = time (h),

and where (Buchanan and Phillips, 1990):

 

B = “’ng Eq. 3.2
GT-C
M = urn-11} Eq. 3.3

Using the Gompertz primary model and the response surface secondary model
from the PMP, log counts were predicted for conditions and times matching every
experimental data point from the described data sources, given the initial log counts for

the respective experimental growth curve.

Conﬁdence intervals2 (95%) were generated based on the following equation

(adapted ﬁom (N eter et al. , 1992)):

C1 = y j r (zXSEC) Eq. 3.4
where:

C] = conﬁdence interval

5)}. = Predicted value of j‘h data point (log (CFU/ml))

z = 2(1-(1/2), o=0.05

SEC = Standard error of calibration (formula below)

 

2 For the subsequent chapters (4 and 5) these conﬁdence intervals were more accurately described
as “prediction intervals”, given that SEC is the root mean squared error between the observed
values and the actual predicted values, not the mean responses.

28

3 .3 .4 Robustness Index (RI)
The R1 for the PMP broth-based L. monocytogenes growth model was calculated
based on the following equation (Campos et al. , 2004):

R1_ SEP

__ 5 .35
SEC q

where:

SEC = Standard error of calibration3

Fin-9.?
= '-‘ Eq. 3.6

 

 

n
where:

9,. = Predicted value of i'h data point (log(CFU/ml))
yi = Observed value of the i"‘ data point ﬁom the original data sets used to

develop the model (log(CFU/ml)), assuming 1 CFU/ml = 1 CFU/ g

n = number of observed data points ﬁorn the original data set

and

SEP = Standard error of prediction

 

 

= Eq. 3.7

where:

 

3 In the following chapters, the denominator for SEC is more accurately represented as n-p, where
p = number of parameters of the model. For this section p=2 and n~3000, so the ﬁnal outcome
was not signiﬁcantly affected.

29

j» j : Predicted value ofjtil data point (log(CFU/g))
y j = Observed value of the j‘h data point from an independent data set

(log(CFU/g», assuming 1 CFU/ml = 1 CFU/g

n = number of observed data points from an independent data set

The overall R1 for each product type was calculated using the combined observed

data from all independent sources:

11
REE-lb)" 4"?
'1

R1 = . 3.8
SEC Eq

 

 

 

where:

52k = Predicted value of ltth data point (log(CFU/g))
y k = Observed value of the k"I data point ﬁom all independent data sets

corresponding to each product type (log(CFU/g)), assuming 1 CFU/ml
= 1 CFU/ g,
n = total number of observed data points from all independent data sets

corresponding to each meat product type

Additionally, the mean relative error (RE) was calculated based on the following

formula (Campos et al. , 2004):

A

 

ﬁ J’j-J’j
j=l I’j
RE: Eq.3.9
n

30

where,

y]. = Predicted value of j‘11 data point (log(CFU/g))

y j 2 Observed value of the jd' data point from an independent data set

(log(CFU/g)), assuming 1 CFU/ml = l CFU/g
n = number of data points
To evaluate whether any of the product/process variables affected the RI, an
analysis of variance (AN OVA) was conducted using JMP (SAS Institute Inc., Cary, NC.

Version 4.0.4).

3.4 RESULTS AND DISCUSSION

The SEC values for the PMP growth models for anaerobic and aerobic conditions
were 1.49 log(CFU/ml) and 1.15 log(CFU/ml), respectively. This means that the model
accuracy was :l:1.32 log(CFU/ml) accuracy, on average, for the broth-based data for both
atmospheric conditions.

The RI values for all the meat and poultry products that were inside the PMP
domain were between 0.37 and 3.96 (Table 3.3). The mean relative error shows that the
PMP growth model over-predicted (i.e., fail-safe) the log counts for 85% of the cases.

For the data set yielding the best RI value (Fig. 3.1), predicted and actual log
counts were within the conﬁdence levels, which implies that the model performed better
than expected. On the other hand, for the data set yielding the worst RI value (Fig. 3.2),
the actual log counts were outside the conﬁdence bands predicted. This particular data set

presented no growth in the total period that was studied.

31

TABLE 3.3. R1 values for conditions inside the PMP model domain

 

 

Data set Product type Temp (°C) H a a 9 Atmosphere RI Mean
No. p w Relative
Error (RE)
1 Ground Beef 4 5.8 0.997 Anaerobic 2.44 -0.33
2 Ground Beef 4 5.8 0.997 Aerobic 2.21 -0.27
3 Ground Beef 4 5.8 0.997 Aerobic 2.18 -025
4 Ground Beef 10 5.8 0.997 Anaerobic 2.57 -0.39
5 Ground Beef 10 5.8 0.997 Aerobic 2.65 -0.25
6 Ground Beef 10 5.8 0.997 Aerobic 2.52 ~0.28
7 Cooked Chicken 3.5 6 0.997 Anaerobic 1.15 -O. 16
8 Cooked Chicken 3.5 6 0.997 Anaerobic 2.10 -0.32
9 Cooked Chicken 6.5 6 0.997 Anaerobic 0.45 0.02
10 Cooked Chicken 6.5 6 0.997 Anaerobic 0.78 —0.07
1 1 Cooked Chicken 10 6 0.997 Anaerobic 0.98 -0.08
12 Cooked Chicken 10 6 0.997 Anaerobic 1.52 -0.18
13 Pate 6.8 5.6 0.997 Aerobic 0.37 -0.03
14 Cooked Pork 4 6.3 0.997 Anaerobic 1.40 -0.21
15 Cooked Pork 4 6.2 0.997 Aerobic 1.20 -0.15
16 Cooked Pork 20 6.3 0.997 Anaerobic 0.95 -0.12
17 Cooked Pork 20 6.2 0.997 Aerobic 1.41 -0.14
18 Cooked Pork 20 6.3 0.997 Aerobic 0.66 -0.07
19 Precooked Beef 4 6 0.997 V C 1.40 -0.16
acuum
20 Precooked Beef 4 6 0.997 V c 0.41 -0.04
acuum
21 Precooked Beef 4 6 0.997 V C 2.76 040
acuum
22 Precooked Beef 4 6 0.997 Vacuum 1.94 -0.27
c
23 Precooked Beef 4 6 0.997 Vacuum 3.96 -0.58
c
24 Precooked Beef 4 6 0.997 Vacuum 2.36 -0.34
25 Cooked Beef w/ Gravy 5 6 0.997 Aerobic 0.48 0.06
26 Cooked Beef w/ Gravy 10 6 0.997 Aerobic 0.93 -0.08
27 Cooked Beef w/ Gravy 5 6 0.997 Aerobic 1.41 -0.26
28 Cooked Beef w/ Gravy 10 6 0.997 Aerobic 1.95 -0.24
29 Home-style salad 4 6 0.997 Aerobic 0.53 0.11
(chicken with no
mayonnaise added)

 

32

TABLE 3.3. Continuation.

 

 

Data Product type Temp pH a 3w b Atmosphere RI Mean
set No. (°C) Relative
Error (RE)
30 Home-style chicken 4 6 0.997 Aerobic 0.56 0.08
salad
31 Home-style salad 4 6 0.997 Aerobic 1.29 -0.23
(real mayonnaise +
chicken)
32 Home-style salad 4 6 0.997 Aerobic 1.47 -0.27
(reduced calorie
mayonnaise +
chicken)
33 Home-style salad 12.8 5 0.997 Aerobic 2.08 0.52
(real mayonnaise +
chicken)
34 Home-style salad 12.8 5 0.997 Aerobic 1.28 0.31
(reduced calorie
mayonnaise +
chicken)
35 Turkey 7 6 0.99 Anaerobic 1 . 16 -0.19
36 Turkey 7 6 0.99 Aerobic 1.52 -0.19
3 7 Turkey 7 6 0.99 Aerobic 1 .78 -0.21
38 Pork 7 6 0.99 Aerobic 2.45 -0.30
39 Pork 7 6 0.99 Anaerobic 2.87 -0.45
40 Pork 7 6 0.99 Aerobic 3.45 -0.41
41 Pork 7 6 0.99 Aerobic 3.44 -0.40

 

a Assumed values for data sets 7-12.
b Assumed values for data sets 1-34.

0 Assumed anaerobic for calculations

33

log (CPU/ml)

 

 

 

FIGURE 3.1. Comparison of the predicted (solid line) and actual (full squares)
growth log counts from the data set (No. 13) resulting in the best RI value (0.37) inside
the PMP model domain (95% conﬁdence intervals, broken lines).

log (CPU/ml)

 

 

 

 

 

I I -
500 1000 1500 2000 2500
Time (h)

FIGURE 3.2. Comparison of the predicted (solid line) and actual (full squares)
growth log counts from the data set (No. 23) resulting in the worst RI value (3.96) inside
the PMP model domain (95% conﬁdence intervals, broken lines).

34

Among the variables tested, only product type affected the RI value (T able 3.4)4.
Therefore, the data were grouped into classes of similar product type, and an overall
performance (RI) was calculated for each group (Table 3.5). The RI values between 0 and
2 (i.e., pate, cooked chicken, cooked pork and turkey) indicated satisfactory robustness
for the PMP in the application. RI=2 is an arbitrary criterion; it means that the actual log
counts were generally within the range described by standard error of calibration (SEC)
of the model. The RI values above 2 (i.e., pork, ground beef, and precooked beef) suggest
that actual log counts are more likely to fall outside the conﬁdence limits of the model for
this particular type of meat, and under these speciﬁc conditions, the model did not
perform as expected. The differences in RI values might be due to the variability of the
laboratory methods. For each product type, and even within the same product type, it is
possible to ﬁnd different inoculation and enumeration methods; different treatment or
sample preparation; and different L. monocytogenes strains. All of these variables
affected the error in the prediction. Controlling or coordinating the challenge tests could
decrease the effects of those variables on the uncertainty of the model; however, these

data sets represent the current state-of-data in this domain.

 

‘ In order to make a fair comparison (based on meat data) the chicken salad data were not
included.

35

TABLE 3.4. ANOVA results for R1 vs. product/process variables

 

 

Variable P value
Product type 0.0004
pH 0.1484
Temp 0.0975
Atmospherea 0.2856

 

a Atmosphere: aerobic or anaerobic

TABLE 3.5. Overall RI values for species inside PMP domain.

 

Product Atmosphere R1

We
Ground Anaerobic 2.20
beef

Aerobic 3.10
Cooked Anaerobic l .25

 

 

 

 

 

Chicken

Pate Aerobic 0.30
Cooked Anaerobic 1.10

Pork
Aerobic 1 .08
Precooked Vacuum 2.40

Beef
Turkey Anaerobic 1 .07
Aerobic 1 .63

 

Pork Anaerobic 2.80
Aerobic 3.40

 

For data sets under experimental conditions outside the PMP domain (i.e., low
temperature), RI values were between 0.40 and 1.22 (Table 3.6). Again, for these data
sets, the PMP growth model over-predicted (i.e., fail-safe) the log counts for the majority
of the cases (83%). The ANOVA of these data showed no signiﬁcant inﬂuence of the

experimental conditions on the RI values, probably due to lack of variation in those

36

variables. As was the case for data within the model domain, for the best RI value, actual
and predicted log counts fell within the conﬁdence intervals (Fig. 3.3). For the data set
yielding the worst RI value, the PMP growth model still performed as expected; most of
the actual log counts fell within its conﬁdence bands (Fig. 3.4), because the R1 was still
~1.20. It should be noted that this evaluation of the model performance in an extrapolated
domain was very limited, both in terms of the number and domain of the data.
Extrapolation of predictive microbial models is always undesirable and not
recommended; however, the R1 is one possible method for evaluating the performance of
models both within and outside the original calibration domain.

TABLE 3.6. R1 values for conditions outside the PMP model domain.

 

 

Data set Produt type pH Temp A Nitrite Salt (%) R1 Mean
No. (°C) w (ppm) Relative
Error (RE)
42 Pate or ham 6.2 2 0.991 81.2 1.6 0.78 .003
43 Pate or ham 62 2 0.991 81.2 1.6 0.93 -0.05
44 Pate or ham 6.2 2 0.991 81.2 1.6 0.70 -0.07
45 Pate or ham 6.2 2 0.991 81.2 1.6 0.70 -0.06
46 Pate or ham 6.2 2 0.991 81.2 1.6 0.91 .0,] 1
47 Pateorham 6.2 2 0.991 81.2 1.6 1.22 .0,10
48 Pate or ham 6.2 0 0.991 81.2 1.6 1.15 —0.03
49 Pate or ham 6.2 0 0.991 81.2 1.6 0.40 -0.06
50 Pate or ham 6.2 0 0.991 81.2 1.6 0.69 .0,09
51 Pate or ham 6.2 0 0.991 812 1.6 0.85 -0. 10
52 Fate or ham 6.2 0 0.991 81.2 1.6 0.69 -0.10
53 Pate or ham 6.2 0 0.991 81.2 1.6 0.81 .o, 14
54 Pate or ham 6.3 2 0.989 103 2.0 1.10 .0,03
55 Pate or ham 6.3 2 0.989 103 2.0 0.70 0,01
56 Pate or ham 6.3 2 0.989 103 2.0 0.78 -0.05
57 Pate or ham 6.3 2 0.989 103 2.0 0.57 .0,04
58 Pate or ham 6.3 2 0.989 103 2.0 0.87 .007
59 Pate or ham 6.3 2 0.989 103 2.0 0.93 -0.02
60 Pate or ham 6.3 0 0.989 103 2.0 0.47 .o_02
61 Pate or ham 6.3 0 0.989 103 2.0 0.62 0,00
62 Pate or ham 6.3 0 0.989 103 2.0 0.85 0,07
63 Pate or ham 6.3 0 0.989 103 2.0 0.70 0.07
64 Pate or ham 6.3 0 0.989 103 2.0 0.86 -0.11
65 Pate or ham 6.3 0 0.989 103 2.0 0.97 .007

 

37

 

 

 

 

log (CFU/ml)

 

 

 

Time 01)

FIGURE 3.3. Comparison of the predicted (solid line) and actual (full squares)
growth log counts ﬁom the data set (No. 48) resulting in the best RI value (0.40) outside
the PMP model domain (95% conﬁdence intervals, broken lines).

 

 

    
 

 

 

 

12 A”- -
10 i ‘1
A 8 . .. ~ I

a . ................. ,
D 6 r ‘ ' " I
Ln .
8 4 I
on , . .
.9. 2 """""" ‘:
O ' i ____ "_T ’MA’ I ’ ' ""' T" ‘—:

0 200 400 600 800
Time (h)

FIGURE 3.4. Comparison of the predicted (solid line) and actual (full squares)
growth log counts from the data set (No. 46) resulting in the worst RI value (1.22) outside
the PMP model domain (95% conﬁdence intervals, broken lines).

To avoid dangerous errors when utilizing growth models for risk assessment (or
other application), predictive models should be validated against independent data
relevant to the application. In prior studies, the broth-based PMP growth model for E.

coli 0157:H7 under-predicted (i.e., fail-dangerous) microbial counts when compared to

38

data in ground beef (Campos etal., 2004; Tamplin M., 2005; Tamplin, 2002; Tamplin et
al. , 2005). Similar results were reported in the PMP for the Clostridium perﬁingens
growth model against data from broth (Smith and Schaffner, 2004). In the present study,
the broth-based PMP growth model for L. monocytogenes performed reasonably well
overall for meat and poultry products, both inside and outside its original domain. In
other words, it is a robust model for growth predictions that can be applied to meat and
poultry products. Moreover, in some cases the model performed better than expected
(RI~ 0-1). In general, microbial counts for L. monocytogenes were over-predicted by the
PMP growth model. However, the data outside the model domain were limited to a very
small range (i.e., low temperature, and just one product type); future work should further
evaluate models in a broader domain of extrapolation and/or generate more experimental

data to widen the validated domain of predictive models.

39

CHAPTER 4
EFFECT OF DIFFERENT MODELING PROCEDURES ON MICROBIAL

GROWTH MODEL PERFORMANCE

4.1 SUMMARY

Two different microbial modeling procedures were compared and validated
against independent data for microbial growth. The most generally used method is two
consecutive regressions: growth parameters are estimated from a primary regression of
microbial counts, and then a secondary regression relates the growth parameters to
experimental conditions. A global regression is an alternative method, in which the
primary and secondary models are combined, giving a direct relationship between
experimental factors and microbial counts. The Gompertz equation was the primary
model, and a response surface model was the secondary model. Independent data from
meat and poultry products were used to validate the modeling procedures. The global
regression yielded the lower standard errors of calibration (SEC), 0.95 log(CFU/ml) for
aerobic and 1.21 log(CFU/ml) for anaerobic conditions. The two-step procedure yielded
errors of 1.35 log(CFU/ml) for aerobic and 1.62 log(CFU/ml) for anaerobic conditions.
For food products, the global regression was more robust than the two-step procedure for
65% of the cases studied. Robustness Index (RI) values for the global regression ranged
ﬁom 0.27 (performed better than expected) to 2.60. For the two-step method, RI values
ranged from 0.42 to 3.88. The predictions were overestimated (fail-safe) in more than
50% of the cases using the global regression, and in more than 70% of the cases using the
two-step regression. Overall, the global regression performed better compared to the two-

step procedure, for this speciﬁc application.

40

4.2 INTRODUCTION

In general, predictive microbiology models are ﬁtted to observed data in a two-
step regression process (Baranyi et al., 1999; Baty and Delignette-Muller, 2004;
Buchanan et al., 1989; Dalgaard and Jorgensen, 1998; Delignette-Muller et al., 1995; te
Giffel and Zwietering, 1999): 1) The ﬁrst step is to ﬁt a primary growth model to
observed experimental data, which yields estimated parameters, and 2) The second step is
to independently ﬁt a secondary model to each of these estimated parameters, as a
function of experimental factors (e.g., temperature, pH, water activity, or nitrite). These
two steps are usually not linked, and inherent 1mcertainty associated with the primary
model is subsequently neglected. The lack of ﬁt to an individual growth curve is not
considered in the secondary model, and all parameters estimated ﬁom observed values
are generally given the same weight in the second step, regardless of the goodness of ﬁt
of the primary mode], potentially leading to poor estimates of the parameters (Pouillot et
al. , 2003).

The Pathogen Modeling Program (U .S. Department of Agriculture, 2003b)) is a
widely used tool in the food industry. This program (PMP) uses models developed via the
two-step procedure to estimate, for example, the growth of Listeria monocytogenes. Its
primary model is the Gompertz equation, and its secondary model is a response surface
model. The L. monocytogenes growth models behind the PMP are based on the work of
Buchanan et al. (1990), who evaluated the inﬂuences that several experimental conditions
(pH, temperature, salt, and nitrite) have on L. monocytogenes growth.

In 2001, Claeys et al. (2001) studied the kinetics of hydroxymethylfurfural,

lactulose, and furoxine formation using global and the two-step regression methods. They

41

agreed with Pouillot et al. (2003), regarding the two-step procedure; errors of the ﬁrst
regression inﬂuenced the exactness of the second regression, resulting in less accuracy
and precision of the estimated parameters. By doing a global ﬁt, the data set is considered
as a whole, increasing the number of degrees of freedom and decreasing the conﬁdence
intervals for the model parameters. However, Claeys et al. (2001) concluded that when
calculating kinetic parameters using both two-step linear (zeroth order reaction, and
Arrhenius equation) and one-step nonlinear regression approaches, the results were
comparable, superiority of either method was not clear, and the performance of a speciﬁc
regression approach depends on the data set to which it is applied. However, these
conclusions were based on zero“ order chemical kinetics, so the analysis may not
necessarily be relevant to microbiological responses that follow ﬁrst order or some other
nonlinear behavior.

Few predictive microbiology studies have focused on the use of a one-step
(global) method. Breand et a1. (1999) used a primary model describing the lag time
duration as a function of two different parameters related to stress temperature and
duration. By replacing the two parameters with their corresponding secondary models,
the lag time was directly related to stress temperature and duration. They found that the
precision of the parameters improved with the global regression.

Membre et al. (2004) combined primary and secondary models that directly
related the growth of L. monocytogenes to cooling temperature, simulating post-process
contamination of packaged pork meats. Optimal growth rate was one of the parameters
estimated hour this regression, which was the main focus of the study. By using global

regression, they lowered the bias factor of the growth rate compared to PMP results

42

(~40% lower).

In order to predict microbial inactivation under dynamic conditions (dynamic
temperature proﬁles mimicking hot air treatments on a fully wetted and on a lean meat
product), Valdramidis et al. (2005) used the global approach. They used the Bigelow

model as their secondary model, which relates the speciﬁc inactivation rate to

temperature. The model parameters (Dref, z, and Tref) were estimated by one-step

regression. The authors mainly focused on the description of microbial inactivation under
dynarrric conditions. In a different study, Fernandez et al. (2002) reported the joint effect
of pH and temperature on thermal resistance of Bacillus cereus in vegetable substrate,
using the Weibull distribution. The parameters were described as a function of
experimental conditions using Arrhenius-type relationships, which were replaced into the
Weibull model. They concluded that the one-step analysis increased the precision of the
estimated parameters, because it avoided estimation of intermediate parameters and used
all the raw data. However, validation or comparison of this approach versus the two-step
approach was not considered in either of these studies.

The previous literature (Breand et al., 1999; Fernandez et al., 2002; Membre et
al., 2004; Valdramidis et al., 2005) that studied the global regression procedure focused
on the estimation of model parameters, and evaluated different models that used the two-
step modeling procedure. To our knowledge, no published studies have directly
compared global versus two-step modeling procedures in terms of microbial counts.
Therefore, the objective of this study was to compare different ﬁtting procedures applied
to broth-based growth models for L. monocytogenes, and validate them using

independent data from meat and poultry products.

43

4.3 MATERIALS AND METHODS
4.3.1 Data sources

Brotlr-based L. monocytogenes growth data sets were obtained ﬁom ComBase
(U .S. Department of Agriculture, 2003a), using source: “Buchanan_90”, organism: “L.
monocytogenes/innocua ”, environment: “culture medium”, pH: “0.1 to 14”, temperature:

“-25 to 120”, and aw: “0.01 to 1”. In total, 385 data sets for anaerobic conditions and 553

for aerobic conditions were found (Table B.1). The no-growth data were eliminated

(Buchanan and Phillips, 1990; Buchanan et al., 1989)(Table B2). The remaining data
sets (nsets= 291 , “points: 2,302 for anaerobic, and nsets= 476, upoims= 3,680 for aerobic)

were considered for this study. The number of data points within each growth curve

ranged from 4 to 19 for anaerobic conditions, and 4 to 21 for aerobic conditions.
To validate these ﬁtting procedures, independent data (nsets= 23, npoims= 174)

from meat and poultry products were used (pH range: 5—6.3, and temperature range: 4-20
°C). The number of data points within each growth curve was between 3 and 15 points.
These data sets were obtained from a pre—distribution version of ComBase 2002 (U .S.
Department of Agriculture, 2003a), and all records found using the above search criteria

were included; keywords and references are reported in Table 4.1.

TABLE 4.1. References and keys in ComBase for meat and poultry products used for

 

 

 

model validation in this study.
Data set No. Key (ComBase) Product type Reference
1-4 J207_Lm, 1208_Lm, Ground Beef (Nissan et
1210_Lm, and 1211_Lm al., 2000)
5 M007 Pate (Bovill et
al., 2000)
6-8 M201_LM, M203_LM, Cooked Pork (Fang and
and M204_LM Lin, 1994)
9-10 M656_LM to M657_Lm Cooked Beef w/ Gravy (Grant et
al., I 993)
1 l M660_LM Cooked Beef w/ Gravy (Grant et
al., I993)
12 M661_Lm Cooked Beef w/ Gravy (Grant et
al., I993)
13 M921_LM Home-style salad (chicken (Erickson
with no mayonnaise added) et al.,
I 993)
14 M921_LMa Home-style chicken salad (Erickson
et al.,
I 993)
15 M921_LMb Home-style salad (real (Erickson
mayonnaise + chicken) et al.,
I993)
16 M921_LMd Home-style salad (reduced (Erickson
calorie mayonnaise + chicken) er al.,
. I993)
17 M921_LMg Home—style salad (real (Erickson
mayonnaise + chicken) et al.,
I 993)-
18 M921_LMi Home-style salad (reduced (Erickson
calorie mayonnaise + chicken) et al.,
I993)
19 SL1 18 Turkey (Mano et
al., 1995)
20 SL123 Turkey (Mano et
al., I 995)
21-23 SL59, SL6], and SL62 Pork (Mano et
al., I995)
4.3.2 Predictive models

The total uncertainty was calculated based on the root mean squared error
(RMSE) between the observed and predicted log counts, identiﬁed as the standard error

of calibration, SEC (eq. 4.1).

45

SEC = Standard error of calibration

its-iii
___ I-l Eq.4.1
n-p

 

 

where:

52,. = Predicted value of ith data point (log(CFU/ml))

yi = Observed value of the im data point ﬁ'om the original data sets used to

ﬁt the model (log(CFU/ml)), assuming 1 CFU/ml = 1 CFU/g
n = number of observed data points ﬁom the original data set

p = number of parameters

The two-step ﬁtting procedure was as follows: ﬁrst, the growth curves were ﬁtted
to the Gompertz equation (eq. 4.2), and growth parameters were obtained (B and M,

(Martino et al., 2005)).

—B t—M
L(t) = A + Ce‘e[ H Eq. 4.2

where:

L(t) = log counts of bacteria at time t (log(CFU/ml)), 9i
A = Asymptotic log count of bacteria as t decreases indeﬁnitely (log(CFU/ml))
= No
C = Asymptotic, incremental increase in log count of bacteria as t increases
indeﬁnitely (log(CFU/ml)) = N.Jo - No

M = Time at which the absolute growth rate is maximum (h)

46

B = Relative growth rate at M (1/11)
t = time (11)

N0 = initial microbial concentration (log(CFU/m1)), at t = 0

Furthermore, N00 was not estimated as a parameter; instead, the N00 values were

assumed to be the same ﬁxed values as reported by Buchanan and Phillips (1990). For

aerobic conditions, N00 was 9.57 log(CFU/ml), and for anaerobic conditions, N00 was

9.32 log(CFU/ml). The initial concentration (No) was a single ﬁxed value for each data

set (i.e., the reported count 3 t = 0).

Then, response surface models (eq. 4.3) were ﬁtted to the natural logarithms (In)
of the B and M parameters, as functions of experimental factors (pH, temperature, salt,
and nitrite) via multiple linear regression. The secondary models were then used to
calculate B and M for each combination of experimental factors. These B and M

parameters were then replaced in equation 2, and microbial counts were computed. The
observed data (yi, broth-based) and the predicted value (9,, from 2-step) were replaced in

equation 1, and the total uncertainty was calculated.

lnBorlnM =131 + [3sz +p3r + B4NaC1+BSN + 136ml2 +

57a x pH) + psrz + 09(NaC1x pH) +
[310(NaClx r) +011NaC12 + 012(Nx pH) +
013(Nx T)+Bl4(NxNaCl)+815N2

Eq. 4.3

where:

3,. = coefﬁcients

47

T = temperature (°C)
NaCl = salt (g/liter)

N = nitrite (ppm)

The global model was the combination of the Gompertz equation and the response
surface models. Parameters B and M in equation 4.2 were replaced by the response

surface model (eq. 4.3), and a global regression was performed (Fig. A.l). The observed
data (yi, broth-based) and the log predictions ﬁ'om the global regression (9i) were

replaced in equation 4.1, and the total uncertainty was calculated.

To calculate the uncertainty resulting ﬁom the primary regression itself
(calibration of the regression), ﬁrst the Gompertz equation (eq. 4.2) was ﬁtted to each
growth curve, and the parameters B and M were obtained ﬁom each individual
regression. The ﬁtting procedure converged for most of the data sets using an
optimization algorithm that included “second derivatives method” (N ewton-Raphson
method, it uses second derivatives as well as ﬁrst derivatives in the iteration method to
ﬁnd a solution); for the cases where this procedure did not converge, a procedure without
using “second derivatives meth ” always converged (Scripts used in JMP for the
nonlinear regression can be found in appendix E). For aerobic conditions, 476 pairs of the
parameters B and M were obtained (T able BA), and for anaerobic conditions, 291 pairs
were obtained (Table 8.3). Second, each pair of parameters (obtained ﬁom the primary
regression) was replaced in equation 4.2, and predicted microbial counts were computed
for each individual growth curve. The difference ﬁ'om the previous procedures is that the

obtained parameters were neither ﬁtted to the response surface model (as a function of

48

the experimental conditions) nor replaced in the primary model by their respective
secondary models. Rather, the values obtained from each individual regression were

directly used in the primary model, and rrricrobial counts were calculated independently

for each growth curve. The observed data (yi, broth-based) and the predicted values (9i)

were replaced in equation 4.1, and the total uncertainty was calculated.
All the statistical analyses and nonlinear regressions were performed using JMP

IN (SAS Institute Inc., Cary, NC. Version 5.1.2).

4.3.3 Robustness Index (RI)
The predictions ﬁom the two-step and global procedures were validated using the
RI (Martino et al., 2005), based on:

_ SEP
SEC

Eq. 4.4

where,

SEC = Standard error of calibration. The SEC used in these calculations was the
one obtained from either the two-step or global regression.

SEP = Standard error of prediction

n A

1% (Y j ‘ 3’ j )2

: Eq. 4.5
n

 

 

where:

j» j = Predicted value of jtll data point (log(CFU/g)), from either two-step

or global regression

49

y j = Observed value of the jth data point from an independent data set

(log(CFU/g)), assuming 1 CFU/ml =1 CFU/g

n = number of observed data points from an independent data set (from 3 to 15)

Additionally, the mean relative error (RE), which determines if the predictions are
over/under estimated, was calculated based on the following formula (Martino et al. ,

2005):

n y ° ‘5’ °
J I

Z ___.
j=1 y j

= Eq. 4.6

n

 

where,

j) j = Predicted value ofj“I data point (log(CFU/g)), ﬁom either two-step or

global regression.

y J. = Observed value of the j'” data point from an independent data set

(log(CFU/g», assuming 1 CFU/ml =1 CFU/g

n = number of data points (from 3 to 15)

4.4 RESULTS AND DISCUSSION

Evaluation of the main effects and interactions among the experimental
conditions, based on P values (Table 4.2), showed the following: temperature showed a
signiﬁcant inﬂuence (P < 0.05) in all cases (for both parameters and for both atmospheric
conditions); salt and nitrite were only signiﬁcant for the M parameter estimation, for both

atmospheric conditions; pH was signiﬁcant for all models, except aerobic In B. In the

50

case of the B parameter (aerobic), the P values for the interactions ranged from <0.0001
to 0.9200, with r x pH, r2, NaCl x pH, NaClz, and N x pH having a signiﬁcant
inﬂuence. For the M parameter (aerobic), the P value for the interactions ranged ﬁom
<0.0001 to 0.6590, with sz, T x pH, T2, N x pH, and N x T having a signiﬁcant
inﬂuence. For the anaerobic condition, P values for the interactions for the B parameter
ranged from 0.0013 to 0.6630, with T x pH being the only second order term having
signiﬁcant inﬂuence; for the M parameter, P values ranged from <0.0001 to 0.8800, with
sz, T2, T x pH, and N2 having a signiﬁcant inﬂuence. The coefﬁcient values for the
secondary model are presented in Table 4.2 for the two-step regression, and Table 4.3 for
the global regression. Even though some of the terms were not signiﬁcant (or = 0.05), all
terms were included in the model, because they all were included in the PMP, which was
the case study of this research Therefore, comparisons and analyses needed to use the
same model as is used in the PMP.

However, when the non-signiﬁcant terms ﬁ'om the secondary models (Table D2)
were eliminated, the SEP values decreased for 56% of the data sets. The SEP values
increased in 44% of the cases, which directly affected the RI values. However, an overall
SEP (including all the data for meat and poultry products, aerobic conditions) decreased
ﬁom 1.78 (when including all terms) to 1.40 (when excluding the non-signiﬁcant terms),
indicating that the reduced model actually was slightly more robust than the complete
model for these cases.

Overall, the global regression (Fig. 4.1) predicted the broth-based data better than

the two-step (Fig. 4.2) procedure did, for both aerobic and anaerobic conditions.

51

to broth-based data via two-step regression.

TABLE 4.2. Coefﬁcient values of secondary models for L. monocytogenes growth, ﬁtted

 

 

 

 

Variable In B P value In M P value
(In h") (In h)
Aerobic
Intercept Br -5.015+00 <.0001 6.42E+00 <.0001
pH I32 5.90502 0.3470 -2.26E-01 <.0001
T(°C) B; 1.44501 <.0001 -1.081'£-01 <.0001
Nac1(g/l) B4 1.69503 0.5590 7.72503 <.0001
Nitrite (ppm) 85 -5.75504 0.4210 6.07504 0.0446
(pH-6.58876)x(pH-6.58876) Br, -1.95502 0.7510 2.63501 <.0001
(T-19.3929)x(pH-6.58876) I37 -l.68502 0.0048 8.96503 0.0004
(T-19.3929) x (12193929) [33 -2.59503 <.0001 2.85503 <.0001
(NaCl-l7.5525) x (pH-6.58876) B9 -1.55502 <.0001 -5.42504 0.6590
(NaC1-17.5525)x(T-19.3929) I310 -2.56E-04 0.1950 9.65505 0.2470
(NaCl-17.5525)x(NaCl-l7.5525) Bu 0.53504 0.0025 1.15504 02070
(Nitrite-103.151)x(pH-6.58876) 1312 1.40503 0.0016 -4.83E—04 0.0099
(Nitrite-103.151)x(T-l9.3929) 1313 -2.10505 0.1920 1.38505 0.0433
(Nitrite-103.151)x(NaC1-17.5525) I314 7.40E-06 0.4060 -5.00506 0.1600
(Nitrite-103.151)xQIitrite-103JSI) [315 -9.24E-08 0.9200 -2.54507 0.5120
Anaerobic

Intercept I31 -6.105+00 <.0001 6.7lE+00 <.0001
pH 32 3.49501 0.0027 -3.32501 <.0001
T(°C) B3 9.13502 <.0001 -7.88502 <.0001
Nac1(g/I) Be 6.33503 0.3450 9.92503 0.0003
Nitrite (ppm) 05 -1.70503 0.1670 1.15503 0.0215
(pH-6.66621)x(p1-I-6.66621) Be -l.4650l 02300 1.95501 <.0001
(T-21.269)x(pH-6.6662l) I37 2.89502 0.0013 -1.62E-02 <.0001
(T-21.269) x (T-21.269) Br -5.56E-04 0.4640 2.90503 <.0001
(NaC1-18.1724) x (pH-6.66621) I39 7.86503 0.1590 3.22503 0.1550
(NaCl-l8.1724)x(T-21.269) Bio -1.88E-04 0.5890 -2.10505 0.8800
(Ned-18.1724)x(NaCl-18.l724) Bu -2.69E-04 0.6630 0.74504 0.1370
(Nitrite-142.759)x(pH-6.66621) BIZ 520504 0.2790 4.35505 0.8230
(Nitrite-l42.759)x(r-2l.269) 1313 -1.50505 0.5960 2.03505 0.0753
(Nitrite-142.759)x(NaCl-l8.1724) I314 -l.60505 0.2510 -2.00506 0.7100
QIitrite-l42.759)x(Nitrite-142.759L515 2.50506 0.1300 -1.00506 0.0353

 

52

TABLE 4.3. Coefﬁcient values computed after global regression for L. monocytogenes

 

 

 

 

 

 

growth, ﬁtted to broth-based data.
Aerobic Anaerobic

Variable In B In M In B In M

(In ’13 (bill (In h'1) (In h)

31 Int -3.195+01 2.925+01 -l265+01 1.905+01

32 pH 7.335+00 6.765+00 l.845+00 4.135+00

[33 r 2.80501 2.65501 1.32501 -7.58E-02

B4 NaCl 2.00502 3.93502 9.07504 5.16502

[35 N 2.00502 1.33502 2.89502 1.36502

[36 pli2 -5.06E-01 4.93501 423501 321501

B7 1*po 4.10502 5.57503 2.18502 2.53502

Ba 1‘2 2.18503 2.84503 -5.45E-03 4.13503

Bo pHxNaCl 1.37503 4.89503 -3.85E—03 2.30503

Bio TxNaCl 4.14504 -3.69E-05 7.98504 -3.28E-04

Bit NaC12 0.57504 1.06504 2.93504 2.74504

1312 pHxN 2.69503 -1.82E-03 3.72503 -1.56E-03

[313 TxN 1.76505 3.56506 2.05504 -3.25E-05
014 NaClxN -9.48E-06 625507 6.14505 -1.07E-06

ﬁts N2 -1.63E-06 3.15507 6.97507 2.42507

12 A - 4 — —— -— - 4 —— — -

f" 10 l
E’- s l
i l
E“ 4“ l
0 . . #1

 

 

Log Nobs (CFU/ml)

FIGURE 4.1. Observed versus predicted L. monocytogenes cormts in broth ﬁom
global regression, showing a randomly selected 10% of the total 3,680 data points and the
1:1 line (aerobic conditions).

53

12 a - - ~- -—-- - -- —-—— “___ —

Log N2-step (CFU/ml)

 

 

Log Nobs (CFU/rnl)

FIGURE 4.2. Observed versus predicted L monocytogenes counts in broth from
two-step regression, showing a randomly selected 10% of the total 3,680 data points and
the 1:1 line (aerobic conditions).

Uncertainty of the model from the global regression was smaller than that ﬁ'om
the two-step regression by approximately 30% for both atmospheric conditions (Table
4.4). Compared to the SEC of the primary regression itself (i.e., no secondary model), the
two-step regression gave higher values, by ~25% for both atmospheric conditions. In the
global regression case, the SEC values did not differ considerably from the primary
regression itself, only 1% for anaerobic and 7% for aerobic; in this case, the global

regression had lower uncertainty than the primary regression itself.

54

TABLE 4.4. SEC (uncertainty) of broth-based L. monocytogenes growth predictions
resulting from different modeling regression procedures.

Log(CFU/ml)
Aerobic Anaerobic
Primary regression (no secondary model) 1.02 1.22

2-step 1.35 1.62
Global 0.95 1.21

 

 

 

These results reﬂect how the secondary regression in the two-step procedure can
affect the overall performance of a model, in this case by increasing the uncertainty
related to that model. Not only is uncertainty from the primary regression affecting the
results, but the uncertainty related to the secondary regression also contributes to the
overall error. On the other hand, by doing a global regression, the data set is considered
as a whole, increasing the degrees of freedom, and making the conﬁdence intervals of the
parameters smaller (Claeys et al., 2001); the estimation of intermediate parameters and
giving the same weight to each data set used to estimate the parameters are avoided
(Claeys et al., 2001; Pouillot et al., 2003).

Validation of these procedures was done using aerobic meat and poultry data. In
65% of the cases studied (Table 4.5), the global regression was more robust, compared to
the two-step procedure, meaning that the global approach performed as expected, with
respect to the model calibration, in the majority of the cases.

RI values for the global regression ranged ﬁom 0.27 (performed better than
expected) to 2.60. In approximately 80% of the cases, RI values were lower than 2.0 (Fig.
4.3). For the two-step method, RI values ranged ﬁom 0.42 to 3.88, with almost 60% of
the cases yielding RI values lower than 2 (Fig. 4.4). An RI less than 2.0 indicates that the

model was reasonably robust, with errors less than double that expected, based on the

55

original model ﬁtting (Martino et al., 2005).

TABLE 4.5. R1 and RE values with experimental conditions (assumed aW = 0.997).

 

Data set No RIJlobal RE global RI 2-step RE 2— pH T °C
1 1.03 0.21 2.49 —0.24 4 5.8
2 1 0.19 2.46 -0.23 4 5.8
3 0.27 0.01 2.65 -0.2 10 5.8
4 0.46 0.07 2.85 -0.24 10 5.8
5 0.6 -0.07 0. 42 0.03 6.8 5.6
6 1.62 -0.23 1.35 -0.07 4 6.2
7 0. 72 0 1.59 -0.13 20 6.2
8 1.35 —0.1 1 0.74 -0.06 20 6.3
9 2.6 -0.61 0.54 0.16 5 6
10 2.54 -0.38 1.05 -0.04 10 6
11 1.91 —0.18 1.59 —0.2 5 6
12 1.65 -0.14 2.2 -0.23 10 6
13 1.71 -0.33 0.6 0.18 4 6
14 1.78 -0.33 0.63 0.17 4 6
15 0.72 0.12 1.45 -0.18 4 6
16 0.92 0.16 1.66 -0.22 4 6
17 2.18 -0.42 2.35 0.57 12.8 5
18 1.28 «0.23 1.44 0.35 12.8 5
19 0.87 0.11 1.71 -0.14 7 6

20 1.19 0.14 2.01 -0.17 7 6
21 1.38 0.2 2.76 -0.23 7 6
22 2.51 0.33 3.89 —0.35 7 6
23 2. 51 0.33 3.88 —0.35 7 6

 

 

 

 

 

The predictions were overestimated (fail-safe) in more than 50% of the cases
using the global regression (Fig. 4.5). On the other hand, for the two-step regression,
predictions were overestimated in more than 70% of the cases (Fig. 4.6). Even though the
global regression represents a more robust procedure, the two-step regression is on the
fail-safe side of the curve in the majority of the cases, for these particular types of

products.

56

 

 

4.50 1— , —A ~ -
tool
3.50 I
3.00 I
2.50 4
2.00 4

unhIMhhu

2 3 4 5 6 7 8 91011121314151617181920212223

 

 

 

 

Data set No.

FIGURE 4.3. Robustness Index values of global regression applied to meat and
poultry data.

 

4.50 . w_ __._____ _..
4.00 i
3.50 3
3.00 *1
2.50 a
2.00 a
1.50

1.00 a
0.50 '
0.00 l

 

 

 

 

12 3 4 5 6 7 8 91011121314151617181920212223
DatasetNo.

FIGURE 4.4. Robustness Index values of two—step regression applied to meat and
poultry data.

57

 

 

 

0.6
0.4 —+ l
0.2 ~ [I
RE 0 tllﬁlv'vr l 1 T l
-624 l
0.4 ~ ‘
-0.6 1 I
-08 LL -______ . _.____ l
12 3 4 5 6 7 8 91011121314151617181920212223

Data setNo.

FIGURE 4.5. Relative error values of global regression applied to meat and
poultry data.

 

 

I

l

l
.03 44- _L_. -_ __LL- --..-_. ___--.” LL
12 3 4 5 6 7 8 91011121314151617181920212223

Data set No.

 

 

FIGURE 4.6. Relative error values of two-step regression applied to meat and
poultry data.

58

When a model is considered, it is important not only to validate it against
independent data (related to the speciﬁc application), but also to assess the modeling
procedure used to obtain the predicted values. For L monocytogenes, the broth-based
growth model uncertainty is lowered by using a global regression, instead of applying the
generally used method of two consecutive regressions, one to ﬁt the raw data, and the
second to ﬁt the parameters to experimental conditions. By doing a one-step regression,
the parameters can still be described as a frmction of the experimental factors (in this
study: pH, temperature, salt, and nitrite).

In conclusion, in order to improve microbial growth (or inactivation) predictions,
an overall knowledge of the model is needed, meaning its performance related to the
application, and the best ﬁtting procedure. In the present study, an approximation of the
total uncertainty of different modeling procedures was estimated, and a global regression
yielded a lower error than did the two-step regression. The Gompertz equation was used
as a primary model and the response surface models as secondary models; however, this

procedure can be applied to any combination of primary and secondary models.

59

CHAPTER 5
UNCERTAINTY ASSESSMENT IN BROTH-BASED MICROBIAL

GROWTH MODELS

5.1 BACKGROUND

Uncertainty and variability of model parameters are increasingly important in
several ﬁelds of risk analysis. Uncertainty is deﬁned as the lack of perfect knowledge of
the parameter value, and variability is the true population heterogeneity that is a
consequence of the physical system. For example, some authors have quantiﬁed
microbial growth variability among strains of a single species (Delignette-Muller, 2000).
They found that biological variability has a great impact on the accuracy of the results
and should not be systematically neglected.

Baranyi and Roberts (1995) described ﬁve different types of errors in predictive
growth models: (a) the assumption that a bacterial population is homogeneous
(homogeneity error), (b) the restriction of bacterial responses only to a few experimental
factors, for instance, pH, salt, temperature, etc. (completeness error), (e) the error due to
parameter estimations (model ﬁmction error), and (d) the error due to laboratory
replications (measurement error). The ﬁfth error mentioned is the one due to ﬁtting the
model to the observed data (numerical procedure error). An example application was
presented for growth parameters (maximum growth rate and generation time) based on
broth-based data, where homogeneity, completeness, and numerical procedure errors
were neglected, and only experimental and model function errors were considered. They

stated that in real food systems, that are varied and complex structures, “the microbial

60

response is biochemically complex”; therefore, detailed information (cell distribution,
heterogeneity, microbial population, etc.) is rarely available, too difficult, or too costly to
acquire. Consequently, certain assumptions and simpliﬁcations must be made, that affect
the uncertainty and variability of the predictions. For example, Pouillot et al. (2003)
applied the Bayesian inference to quantify growth parameter 1mcertainty and growth
variability between strains(Pouillot et al., 2003); however, due to the nature of their data
(collected ﬁom published literature), the physiological state of the strain, the exact nature
of the isolate used and the counting method, were not considered, which can signiﬁcantly
inﬂuence the overall uncertainty of their results.

Users of predictive models should understand how various sources of uncertainty
and variability contribute to potential predictive errors. These sources can ultimately
underestimate and/or overestimate total uncertainty of a model, which can lead to
signiﬁcant problems in commercial food processing applications. If uncertainty is
underestimated, processes might be designed or operated in a way that results in
unacceptable risk to consumers. In contrast, if uncertainty is overestimated, processes
will be over—designed, resulting in unnecessary expenditure of energy and/or a decrease
in the quality and nutritional value of food products.

For example, the uncertainty of microbial growth models found in computer
programs (e.g., Pathogen Modeling Program, PMP) is typically represented by 95%
conﬁdence intervals (CI, obtained ﬁom the 95% CI of the growth parameters (Juneja et
al., 1999; US. Department of Agriculture, 2003b)). However, these intervals reﬂect only
the uncertainty of the mean response and are based only on the secondary regression,

neglecting uncertainty of the actual predicted values, uncertainty ﬁom the primary

6]

regression, and experimental variability.

In contrast, if prediction intervals are used, uncertainty of the curve itself, scatter
of the data around the curve (Motulsky and Christopoulus, 2004), and total uncertainty
related to that model and data are reported. Furthermore, using this simultaneous
conﬁdence intervals method (conﬁdence intervals of the parameters used to calculate
conﬁdence intervals of the prediction), the actual 95% conﬁdence intervals for the
microbial counts end up to be ~ 90% conﬁdence interval of the mean response (Neter et
aL,1992)

Determination of conﬁdence and prediction intervals in nonlinear regression can
be difﬁcult, and these could be underestimated by a factor of 2-3 (Dolan et al., 2006; Van
Boekel, 1996). Dolan et al. (2006) proposed a technique to be used in nonlinear
regression to estimate conﬁdence and prediction intervals, which is among the ﬁrst
reported methodologies for the prediction intervals applied to actual survival of
microorganisms, not just the model parameters.

Quantitative microbial risk assessors, food processors, and food regulators should
utilize knowledge of the total uncertainty behind a model, in terms of prediction intervals
of the actual predicted value, in order to improve ultimate estimates of risk. Moreover,
once sources of errors are identiﬁed and quantiﬁed, they can be prioritized and targeted
to reduce uncertainty.

Therefore, the objective this part of the study was to assess sources of uncertainty
and variability in broth-based growth models, applied to L. monocytogenes, to better
understand how the data, model, parameters, and methodologies contribute to the total

uncertainty of a model.

62

5.2 METHODS
5.2.1 Data sources.

The same data sets described in Chapter 4, section 4.3.1 were used.

Additionally, the data sets were also classiﬁed based on the treatment (pH,
temperature, salt, and nitrite). For aerobic conditions, 132 different treatments were
identiﬁed (Tables B6, B7); 19 treatments that were not replicated were excluded from
the analyses. There were 118 different anaerobic treatments (Tables B5, B8); 44

unreplicated treatments were excluded from the analyses.

5.2.2 Error sources

Total uncertainty was assumed to be an aggregated contribution of the
uncertainties due to the primary and secondary regressions, and variability due to
replications, substrate, organism, and laboratory methodologies, conceptually represented

as:

8Total = f(aorganism ’ 8substrate ’ 8111b methods ’ sreplic’ 8 primary model’ 8secondary model’ 8random )
where:

310ml: Total uncertainty

Spﬁmary model: Error ﬁom the primary regression

Esmndary model: Error from the secondary regression
Sorganism = Error due to variability in microorganisms or strains

Ssubstrate = Error due to variability in substrate

63

Slab methods = Error due to different laboratory methodologies

Sreplic = Error due to replications

Brandon, = Random error

For this study, the errors due to organisms (Seganism), substrate (Swim-ate), and

laboratory methodologies (slab methods) were neglected, because the same

rnicroorganisrrr, substrate, and methodology were used for all of the data, and the data

were all generated in a single laboratory.

5.2.3 Error calculations
All statistical analyses and nonlinear regressions were performed using JMP IN

(SAS Institute Inc., Cary, NC. Version 5.1.2).

Total uncertainty (smell) was calculated based on the root mean squared error

(RMSE) between the observed and predicted log counts, identiﬁed as the standard error
of calibration, SEC (reported in Chapter 4, equation 1). In order to obtain the predicted
microbial counts, the two-step procedure described in Chapter 4, section 4.3.2 was

followed.

To quantify the contribution from the secondary regression (chondary model): the

upper and lower limits of the primary models parameters (UPIBM, and LPIBM) were
computed using equation 5.1 (adapted from (Neter et al., 1992)). The RMSE for both
parameters was obtained ﬁom the secondary regression (Table 5.1). Then, the upper and

lower limits of the log predictions (UPI, and LPI) were calculated (with equation 4.2,

64

Chapter 4), using the upper and lower values of the parameters B and M (obtained from

equations 5.1 and 5.2). Having the upper and lower limits of the log count predictions,
the error from the secondary regression (swam model) in terms of microbial count

was calculated with equation 5.3.

TABLE 5.1. RMSE ofthe parameters B (h!) and M (h) estimated from the secondary

 

 

regression.
RMSE ﬁom secondary
regression
B M
Aerobic 1.6 1.4
Anaerobic 4.1 1.8
UPIB,M=5’J.+(ZXRMSEB.M) Eq.5.1
LPIBM =yj—szRMSEBM) Eq.5.2

where:

)3 j : predicted B or M
UPI B, M: Upper prediction limit for B or M

LPI B, M: Lower prediction limit for B or M
2 = z(1-01/2), 01:0.05
RMSE B, M = root mean squared error estimated ﬁom secondary regression, for B

orM

Subsequently,

65

_ UPI — LPI

a . __
secondary regressron z x 2

Eq. 5.3

where,
UPI = Upper prediction limit for log(CFU/ml)

LPI = Lower prediction limit for log(CFU/ml)

To quantify the error contribution from the primary regression (8!,de model),

the method described in Chapter 4, section 4.3.2, was followed (calibration of
regression), in which the B and M parameters estimated ﬁ'om the primary regression were
directly replaced in the Gompertz equation to calculate the log predictions (without using

secondary models).
The error due to replications (areplic) was computed as follows. A two-way

AN OVA (factors: time and treatment, general linear model, and 2-factor factorial design)
was performed on the whole set of treatments, using JMP. Effects between factors were
neglected, and the error from replications was then obtained (represented in J MP by

“pure error”).

5.2.4 Calculation of the prediction limits and parameter errors

The asymptotic (or approximate) standard error (SE) accounts for uncertainty in
both parameter estimates and error in the model. It was calculated from the variance-
covariance matrix associated with the parameter estimates, the composition of the partial
derivatives of the model with respect to each parameter, and the estimate of error

variance, represented in JMP by SQRT(VecQuadratic(matrixI, vector1)+mse (JMP,

66

2006). The diagonal elements of the variance-covariance matrix represent the asymptotic
SE, and the off-diagonal elements represent covariance between parameters (Van Boekel,
1996).

This SE is used to calculate the prediction limit values for an individual predicted
value, adding the variance of the error term to the variance of prediction involving the
estimates, to form the interval. A speciﬁc SE corresponds to each data point, which is
used to construct the prediction intervals (JMP, 2006).

To calculate the conﬁdence intervals, the SE accounts for all of the uncertainty in
the parameter estimates, but does not account for the uncertainty in predicting individual
responses (N eter et al. , 1992). The SE for the conﬁdence intervals is calculated from the
covariance matrix associated with the parameter estimates, and the composition of the
partial derivatives of the model with respect to each parameter. The command in JMP is
Sqrt(V ecQuadratic(matrix1 ,vectorl , (JMP, 2006)).

An illustration of how the speciﬁc SE of each data point varies with time, and
how SE of each data set varies with pH, temperature, nitrite, and salt is presented in
appendix C.

The uncertainty of the parameters was also calculated using the RMSE between
the B and M parameters estimated from the primary regression (observed values), and the
predicted values, which were obtained ﬁom the response surface model (Chapter 4,
equation 4.3, and the coefficients from Table 4.2) for the two-step procedure. For the
global procedure, the coefﬁcients from Table 4.3 were replaced in equation 4.3 (Chapter

4), the B and M parameters were computed.

67

5.3 RESULTS AND DISCUSSION
5.3.1 Deconstruction of the model uncertainty

Although the total uncertainty for anaerobic conditions was almost 20% higher
than for aerobic conditions (Table 5.2), the relative contributions of the various
components of the total uncertainty were fairly consistent across the two conditions. The
largest contribution to the total uncertainty came from the secondary regression
uncertainty (1.48 log(CFU/ml) for aerobic, and 1.42 log(CFU/ml) for anaerobic).
Experimental variability was the smallest component (0.26 log(CFU/ml) for aerobic, and
0.21 log(CFU/rnl) for anaerobic). The uncertainty due to the primary regression also had
a high relative contribution to the total uncertainty (1.02 log(CFU/ml) for aerobic, and
1.22 log(CFU/ml) for anaerobic). As stated at the beginning of section 5.2.2, even though
these errors represent the aggregated contribution to the total uncertainty, they cannot be
directly added (N eter et al. , 1992), so the numbers calculated in this section represent
how much each source is contributing to the total uncertainty, separately and

independently.

TABLE 5.2. Relative contributions of the different sources of error to the total model
uncertainty for aerobic and anaerobic conditions.

 

 

2g (CFU/ml)
Errors Aerobic Anaerobic
8total 1.35 1.62
t’zprimary regression 1.02 1.22
8secondary regression 1.48 1.42
8reolic 0.26 0.21

 

68

When experiments are conducted under controlled conditions (microorganisms,
substrate, temperature, etc.), variability due to experiments is assumed to have less effect
on the overall uncertainty of a model, as was found in this study. However, when
experiments are run with food matrices, larger contribution of the experimental
variability could be expected. A test of this is presented in appendix D.1.

Uncertainty due to the primary regression is assumed to contain not only the error

of the regression itself, but also, the variability that comes with the data (in this study:
Steppe). For this particular study, error due to the primary regression was smaller than

error due to the secondary regression. When food matrices are considered, the results
could be the opposite, because of the natural variability of food matrices and how
microorganisms react to these systems. However, given the experimental control for the
data used here, this error did not have a great impact on the primary regression
uncertainty.

The uncertainty due to the secondary regression had the largest contribution to the
overall uncertainty, probably because the response surface model that was ﬁtted to the
parameters, as a function of the experimental conditions, was not the model with the best
performance for this speciﬁc application, meaning that a more robust model could lead to
a smaller uncertainty.

These results demonstrated that the impact that the various sources of error had on
the total uncertainty of a model cannot be ignored. This assessment of the relative
magnitude of the various sources of uncertainty can help prioritize efforts to minimize
uncertainty in model development, by choosing a different model, redesigning the

experimental methodology, and/or improving the statistical analyses.

69

For example, if two-step regression is used, different secondary models should be
tested to determine which yields the smallest secondary model uncertainty.
Alternatively, a global regression (combining primary and secondary models, giving a
direct relationship between experimental factors and microbial counts) might be advised.
By doing a global regression, only the error due to this regression contributes to total
uncertainty, avoiding errors due to the addition of more parameters (Fernandez et al. ,
2002), and avoiding giving the same weight to each data set used to estimate the
parameters (Pouillot et al. , 2003). In Chapter 4, it was found that by performing a global
regression instead of a two-step regression, the total uncertainty was reduced ~30%, with
respect to predicted microbial counts. Another study showed that by ﬁtting the response
surface model to the natural logarithms (In) of the parameter, uncertainty of the estimated
parameter increased by 20%, compared to the error estimated hour the primary ﬁt
(Baranyi and Roberts, 1995).

Experimental variability could greatly depend on personnel, which is controllable
to some degree, through training and management. Furthermore, other factors can
contribute to this error, including the materials and equipments used for the experiments,
laboratory/ambient conditions, etc. All these factors, either together or separately, can be
controlled to minimize the error due to experiments.

Uncertainty from the primary regression gave an idea of how well the primary
model represented the growth pattern of the experimental data. This uncertainty would be
affected by the particular choice of primary model, assuming that one model form would

ﬁt a given data set better than another.

70

5.3.2 Prediction intervals and errors of the parameters

Two examples of PI versus C1 are shown in Figures 5.1 and 5.2. Even though the
model ﬁtted the data better on Figure 5.2, prediction intervals in both cases still contain
most of the data. However, this was not the case for the conﬁdence intervals; in both

cases, the majority of the data were outside the limits.

 

 

 

 

 

 

14 _ , __ ________ _____ L L- A
12 .
E 10 —~l ':
E
e 8—
z 6
w
.3
4
2 ..
0 I _ I —" ___T _—' I T I” ‘I—‘_'— -—'
0 20 4o 60 80 100 120 140
Time(h)

FIGURE 5.1. Conﬁdence (small dashed lines) and prediction (wide dashed lines)
intervals for global regression (data set No. 623, aerobic conditions, treatrrrent No. 26: pH
= 6, T =19°C, nitrite = 0 ppm, salt = 0 g/liter). Solid line: predicted curve; full squares:
observed data.

71

 

Log N (CFU/ml)

 

 

0 l '_" ' I" ‘ﬂ—‘_ I —_ “ "—1T ”___—"7 __“—T— _‘—_ _l '————' ___”—
0 10 20 30 40 50 60 70 80
Time (h)

FIGURE 5.2. Conﬁdence (small dashed lines) and prediction (wide dashed lines)
intervals for global regression (data set No. 646, aerobic condition, treatment No. 80: pH
= 7, T = 19°C, nitrate = 0 ppm, salt = 25 g/liter). Solid line: predicted curve; full squares:
observed data.

The natural logarithm of B and M calculated with the coefﬁcients obtained ﬁom
the global regression gave higher RMSEs, for both atmospheric conditions, than did the
values calculated from the response surface model on the two-step regression (Table 5.3).

TABLE 5.3. RMSE of the model parameters.

RMSE
Aerobic Anaerobic

In B In M In B In M

(In 1Q (1n h) (1n 11") (In h)

Global 1.58 0.48 2.14 0.77

2-step 0.91 0.38 1.42 0.59
For In B, aerobic conditions, the global regression values had ~42% higher error

compared to the two-step regression. For anaerobic conditions, it was ~33% higher.

For In M, aerobic and anaerobic conditions, the global regression had ~20%

higher error for the global regression compared to the two-step regression.

72

These values showed that the methodologies used for model development do not
always have the same impact on the results. For instance, in terms of microbial counts,
the global regression performed better than the two-step regression; however, in terms of
model parameters (e.g., B, and M), the two-step procedure gave a lower prediction error.

To conclude, by assessing the different sources of error that might contribute to
the total uncertainty of a model, a better understanding of the model and the data used to
parameterize it is also attained. Therefore, the techniques illustrated here can lead to
improvement of a model, by reducing uncertainty, and/or improving the way that a model

is utilized for process validation or risk analysis.

73

CHAPTER6

CONCLUSIONS AND FUTURE WORK

6.1 CONCLUSIONS

1.

The growth models used in the Pathogen Modeling Program (PMP) for L.
monocytogenes (aerobic and anaerobic conditions) were validated, inside and
outside their domain with growth data from meat and poultry products, and an
acceptable robustness for this speciﬁc application was established. A Robustness
Index (RI) between 0 and 2.0 indicated satisfactory robustness for the PMP
growth models. RI values above 2.0 suggested that the model did not perform as
expected. The type of food product (e.g., pété, cooked chicken, turkey)
signiﬁcantly affected (P < 0.05) RI values. The standard errors of calibration for
the growth models used in the PMP were i 1.30 log(CFU/rnl), on average, for the
broth-based data. Inside the model domain, RI values for the meat and poultry
products were between 0.37 and 3.96; the model overestimated the log counts for
85% of the cases. Outside the model domain, RI values were between 0.40 and
1.22. In most cases (83%), the log counts were overestimated. Overall, for meat
and poultry products, both inside and outside the model domain, the PMP growth
models for L. monocytogenes were robust models, although they were developed

ﬁom broth-based data.

The uncertainty related to the global (one-step) regression, in terms of microbial

counts, was almost 30% smaller than from two-step regression for both

74

atmospheric conditions. For the primary regression itself (using B and M
estimated from the primary regression, instead of those calculated from the
secondary regression), the standard error of calibration (SEC) was about 25%
smaller than when using the two-step regression. By doing a global regression,
instead of two consecutive regressions, the process avoids giving the same weight
to each data set used to estimate the parameters. Moreover, with the global
regression, the data set is considered as a whole, increasing the degrees of
freedom, which makes the conﬁdence intervals smaller. When validating against
data ﬁom food products (meat and poultry), global regression gave more robust
results, compared to two-step regression. RI values for global regression ranged
from 0.27 to 2.60; in 80% of the cases, RI values were lower than 2.0. R1 values
for the two-step regression ranged from 0.42 to 3.88, with ~60% of the cases
yielding RI values lower than 2.0. The predictions of microbial counts were
overestimated in more than 50% of the cases using the global regression, and in
more than 70% of the cases using the two-step regression. For this particular type
of food product, global regression represented a more robust procedure; however,
the two-step regression was on the fail-safe side of the curve in the majority of the
cases. However, for the primary model parameters (B and M), global regression
results gave higher errors compared to those generated ﬁom the two-step
regression, for both atmospheric conditions. When assessing model performance,

it is important to consider both the model form and the ﬁtting procedure.

75

3. The sources of errors that contributed to the total uncertainty of a bacterial growth
model were identiﬁed and quantiﬁed. The overall uncertainty of a model was
assumed to be the result of the relative contributions of the error due to
experimental variability, error due to primary regression, and error due to
secondary regression. For the L. monocytogenes broth-based data, the secondary
regression had the highest relative contribution, followed by the primary
regression uncertainty, and the experimental variability. The lower relative
contribution of the experimental variability might be due to controlled
experimental conditions (microorganisms, laboratory procedures, and substrate
did not change). On the other hand, the secondary regression uncertainty had the
higher relative contribution, possibly due to lack of a good ﬁt for the secondary
model. Choosing a secondary model that better ﬁts the data could possibly lower
this contribution. The experimental variability was assumed to form part of the
primary regression uncertainty. In this study, its contribution did not greatly affect
the error due to the primary regression; different results might be expected for

models developed ﬁom microbial data in food products.

4. When the true risk to the consumers, based on food safety, is assessed by the food
industry, the uncertainty of the actual predicted value should be reported. It is
common to ﬁnd in the literature the uncertainty reported as conﬁdence intervals
of the mean prediction, not the individual prediction. However, by reporting the
prediction intervals of the predicted values, the uncertainty of an individual

outcome is known, and the risk related to the speciﬁc prediction can be assessed.

76

 

For example, for a processor of liquid eggs, it is important to know whether each
individual package is safe (environmental conditions are controlled, so
Salmonella growth is inhibited); only predicting that the mean package is safe is
not sufﬁcient control. Therefore, it is important that predictive microbial tools

predict the risk limits for each speciﬁc package/serving that is processed.

6.2 SUGGESTIONS FOR FUTURE WORK

1.

First, a simple summation of sources of error was assumed to be the total
uncertainty of the model; however, later it was found that direct summation of
errors is not statistically possible. The purpose at the beginning of the study was
to quantify the direct contribution of each error to total uncertainty. The relative
contribution of each error was assessed; however, assessment of the exact

contribution of each error to total uncertainty of a model is still needed.

The magnitude of each source of error varies with time, which could be explained
with the growth curve. For example, during the lag phase duration, only error due
to experimental variability was assumed to affect overall uncertainty; during
exponential growth, errors due to primary and secondary regression were assumed
to affect total uncertainty. Then, during the stationary phase, once again the
experimental variability was assumed to affect overall uncertainty. These
assumptions could be explained with the variation of the asymptotic SE with time.
This variation was small at the beginning and end of the curve, and slightly

increased in the middle. However, this trend needs more analysis in order to

77

speciﬁcally quantify and conﬁrm that the changes in magnitude of the different

error sources vary with the different phases of the growth curves.

. In the present study, the relative contribution of errors to the model uncertainty
based on broth-based data was assessed. Further analysis of the relative
contribution of errors to model uncertainty based on data ﬁom food products is

still needed.

. For the same data used in this study, evaluation of different secondary models and
the impact on the secondary regression uncertainty, and later the total uncertainty,

would be interesting.

. Development of a “model diagnostic tool” would be interesting and very useful
for academia and government. For example, assume ground beef data had been
collected from a laboratory, and a model to describe the growth of E. coli is
needed. The data sets could be downloaded to the software, including all the
experimental variables; then the primary and secondary models could be chosen,
the software runs the models, and calculates the relative contribution of each error
and the total uncertainty. From the results, it would be possible to evaluate and
compare model performance, so informed decisions could be made. Such
decisions could involve model choice (primary or secondary), subsequent
experimental design, data collection methodology, or number of replications.

Once these changes are done, accuracy of the predictions could be improved, and

78

the uncertainty related to those predictions could be decreased.

79

APPENDIX A

COMPUTER PROGRAMS USED FOR DATA ANALYSIS AND DATA SOURCE

 

 

 

 

 

 

 

 

 

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4.0.4) formula box that contains the global model.

80

 

 

 

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‘ which shows the growth curve for L. monocytogenes, the growth parameters, and the

experimental variables.

81

 

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2003a). Experimental conditions

82

APPENDIX B

BROTH-BASED AND MEAT-BASED DATA DESCRIPTION AND
ORGANIZATION

TABLE 8.]. Keys to ComBase for aerobic (data set No. 386-938) and anaerobic (data set
No. 1-385) conditions.

 

Key to Data set Key to Data set Key to Data set Key to Data set
ComBase No. ComBase No. ComBase No. ComBase No.
Ll_A_l 1 L15_A_1 33 L17_A_3 65 L21_A_l 97
L 1_A_2 2 L15_A_l l 34 L1 7_A__4 66 L21_A__2 98
Ll_A_3 3 L15_A_6 35 Ll7_A_5 67 L22_A_l 99
Ll_A_4 4 L15_C_12 36 L17_A__6 68 L22_A_ll 100
Ll_A_5 5 L15_C_2 37 Ll8_A_l 69 L22_A_l6 101
Ll_A_6 6 Ll 5_C_7 38 Ll8_A_2 70 L22_A_17 102
Ll_A_7 7 L15_D_l3 39 Ll8_A_3 7l L22_A_18 103
Ll_A_8 8 L15_D_3 40 L19_A_l 72 L22_A_6 104
L l _A_9 9 L15_D_8 41 L l 9_A_1 l 73 L22_C_12 105
L 10_A_l 10 L 15_F_14 42 L19_A_16 74 L22_C__2 106
L10_A_1 1 l l L15_F_4 43 L19_A_18 75 L22_C_7 107
L10_A_6 12 L15_F_9 44 L19_A__l9 76 L22_D_13 108
L10_C_12 l3 L15_G_10 45 L19_A_6 77 L22_D_3 109
L10_C_2 14 L1 5_G_15 46 L l 9_C_1 2 78 L22_D_8 1 10
L10_C_7 15 L15_G_5 47 L19_C_2 79 L22_F_14 111
L10_D_l 3 l6 L16_A_1 48 L l 9_C_7 80 L22_F__4 l 12
L10_D_3 17 Ll6_A_ll 49 L19_D_13 81 L22_F_9 113
L10_D_8 l 8 L16_A_6 50 L1 9_D__l 7 82 L22_G_10 1 l4
L10_F_14 l9 L16_C_12 51 L19__D_3 83 L22_G__15 1 15
L10_IF_4 20 L16_C_2 52 L19_D_8 84 L22_G_5 116
L10_F_9 21 L16_C_7 53 L19_F_14 85 L23_A_l l l 7
L10_G_10 22 L16_D_l3 54 L19_F_4 86 L24_A_l 1 1 8
L10_G_15 23 L16_D_3 55 Ll9_F_9 87 L24_D_2 119
L10_G_5 24 L16_D_8 56 L19_G_10 88 L24_F_3 120
L1 1_C_l 25 L16_F_14 57 L19_G_15 89 L25_D_l 121
L11_E_2 26 L16_F_4 58 L19_G_5 90 L25_D_10 122
L 12_C_l 27 L1 6_F_9 59 L2_A_l 91 L25_D_2 123
L12_E_2 28 L16_G_10 60 L2_A_2 92 L25_D_3 124
L13_C_1 29 L16_G_15 61 L2_A_3 93 L25_D_4 125
Ll 3_E_2 30 L16_G_5 62 L20_A_l 94 L2 5_D_5 126
L14_C_l 31 L17_A_1 63 L20_A_2 95 L25_D_6 127
L l 4_E_2 32 Ll LA_2 64 L20_A_3 96 L2 5_D_7 128

 

 

 

 

83

TABLE B.1. Continuation.

 

Key to Data set Key to Data set Key to Data set Key to Data set
ComBase No. ComBase No. ComBase No. ComBase No.
L25_D_8 129 L3_A_1 162 L4_A_6 195 L55_A_l 228
L25_D_9 130 L3_A_11 163 L4_C_12 196 L56_A_1 229
L26_D_1 131 L3_A_6 164 L4_C_2 197 L57_A_1 230
L27_A_1 132 L3_C_12 165 L4_C_7 198 L58_A_l 231
L27_A_11 133 L3_C_2 166 L4_D_13 199 L59_B_1 232
L27_A_6 134 L3_C_7 167 L4_D_3 200 L6_A_1 233
L27_C_12 135 L3_D_l3 168 L4_D_8 201 L6_A_11 234
L27_C_2 136 L3_D_3 169 L4_F_l4 202 L6_A_6 235
L27_C_7 137 L3_D_8 170 L4_F_4 203 L6_C_12 236
L27_D_13 138 L3_F_14 171 L4_F_9 204 L6_C_2 237
L27_D_l6 139 L3_F_4 172 L4_G_10 205 L6_C_7 238
L27_D_3 140 L3_F_9 173 L4_G_15 206 L6_D_13 239
L27_D_8 141 L3_G_10 174 L4_G_5 207 L6_D_3 240
L27_F_14 142 L3_G_15 175 L40_A_1 208 L6_D_8 241
L27_G_5 143 L3_G_5 176 L41_A_1 209 L6_F_14 242
L28_A_l 144 L30_A_l 177 L42_A_1 210 L6_F__4 243
L28_A_11 145 L30_A_2 178 L43_A_l 211 L6_F_9 244
L28_A_6 146 L30_A_3 179 L44_A_l 212 L6_G_lO 245
L28_C_12 147 L31_C_l 180 L45_A_1 213 L6_G_15 246
L28_C_2 148 L31_E_2 181 L46_A_l 214 L6_G_5 247
L28_C_7 149 L32_C_l 182 L47__A_1 215 L60__A_1 248
L28_D_l3 150 L32_E_2 183 L48_A_1 216 L61_B_1 249
L28_D_3 151 L33_C_1 184 L49_A_1 217 L61_C_2 250
L28_D_8 152 L33_E_2 185 L5_D_1 218 L62_A_1 251
L28_F_14 153 L34_A_1 186 L5_D_2 219 L63_A_l 252
L28_F_4 154 L35__A_1 187 L5_D_3 220 L65_B_1 253
L28_F_9 155 L36_A_1 188 L50_A_1 221 L65_C_2 254
L28_G_10 156 L37_A_1 189 L51_A_1 222 L66_B_1 255
L28_G_15 157 L38_A_1 190 L52_A_l 223 L66_C_2 256
L28_G_5 158 L39_C_1 191 L52_A_2 224 L67_A_1 257
L29_A_1 159 L39_E_2 192 L52_A_3 225 L68_A_1 258
L29_A_2 160 L4_A_l 193 L53_A_1 226 L69_A_l 259
L29_A_3 161 L4_A_1 1 194 LS4_B_1 227 L69_A_1 1 260

 

 

 

 

84

TABLE B. 1 . Continuation.

Keyto Dataset Keyto Dataset Keyto Dataset Keyto Dataset
ComBase No. ComBase No. ComBase No. ComBase No.

 

L69_A_16 261 L73_D_2 294 L85_D_8 327 L88_D_8 360
L69_A_6 262 L74_B_1 295 L85_F_l4 328 L88_F_14 361
L69_C_12 263 L75_A_1 296 L85_F_4 329 L88_F_4 362
L69_c_2 264 L76_F_l 297 L85_F_9 330 L88_F_9 363
L69_C_7 265 L77_A__l 298 L85_G_10 331 L88_G_10 364
L69_D_13 266 L78_A_1 299 L85_G_15 332 L88_G_15 365
L69_D_3 267 L78_A_2 300 L85_G_5 333 L88_G_5 366
L69_D_8 268 L78_A__3 301 L86_D_1 334 L89_A_1 367
L69_F_14 269 L78_A_4 302 L86_D_2 335 L9_A_1 368
L69_F_4 270 L78_A_5 303 L86_D_3 336 L9_A_1 1 369
L69_F_9 271 L78_A_6 304 L87_A_1 337 L9_A_6 370
L69_G_10 272 L78_A_7 305 L87_A_11 338 L9_C_12 371
L69_G_15 273 L78_A_8 306 L87_A_6 339 L9_c_2 372
L69_G_5 274 L78_A_9 307 L87_C_12 340 L9_C_7 373
L7_A_1 275 L79_A_l 308 L87_c_2 341 L9_D_13 374
L7_A_ll 276 L8_A_l 309 L87_c__7 342 L9_D_3 375
L7_A_6 277 L8_A_2 310 L87__D_13 343 L9_D_8 376
L7_c_12 278 L8_A_3 311 L87_D_3 344 L9_F_l4 377
L7_C_2 279 L80_F_l 312 L87_D_8 345 L9_F_4 378
L7_C_7 280 L81_A_1 313 L87_F_14 346 L9_F_9 379
L7_D_13 281 L82_F_1 314 L87_F_4 347 L9_G_10 380
L7_D_3 282 L83_A_1 315 L87_F_9 348 L9_G_15 381
L7_D_8 283 L83_A_2 316 L87_G__10 349 L9_G_5 382
L7__F__14 284 L84_A_1 317 L87_G_15 350 L90_F_1 383
L7_F_4 285 L85_A_l 318 L87_G_5 351 L91_A_1 384
L7_F_9 286 L85_A_11 319 L88_A_1 352 L92_F_l 385
L7_G_10 287 L85_A_16 320 L88_A_11 353 LM002_1 386
L7_G_15 288 L85__A_6 321 L88_A__6 354 LM002_2 387
L7_G_5 289 L85_c_12 322 L88_C_12 355 LM002_3 388
L70_B_l 290 L85_C_2 323 L88_C_2 356 LM002_4 389
L71_C_1 291 L85_C_7 324 L88_C_7 357 LM002_5 390
L72_A_1 292 L85_D_l3 325 L88_D_13 358 LM002_6 391
L73_c_1 293 L85_D_3 326 L88_D_3 359 LM003_1 392

 

 

 

 

85

Key to Data seﬂ

ComBase

No.

TABLE B. l . Continuation.

Keyto Dataset

ComBase

No.

Keyto Dataset

ComBase

No.

Key to Data set

ComBase

No.

 

LM003_10
LM003_1 1
LM003_12
LM003_13
LM003_14
LM003_15
LM003_2
LM003_3
LM003_4
LM003_5
LM003_6
LM003_7
LM003_8
LM003_9
LM004_1
LM004_2
LM004_3
LM005_1
LM005_10
LM005_1 1
LM005_12
LM005_13
LM005_14
LM005_15
LM005_2
LM005_3
LM005_4
LM005_5
LM005_6
LM005_7
LM005_8
LM005_9
LM006_1

393
394
395
396
397
398
399
400
401
402
403

405
406
407
408

410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425

 

LM006_2
LM006_3
LM007_1
LM007_10
LM007_1 1
LM007_12
LM007_13
LM007_14
LM007_15
LM007_2
LM007_3
LM007_4
LM007_5
LM007_6
LM007_7
LM007 8

LM007_9
LM008_1
LM008_10
LM008_1 1
LM008_12
LM008_13
LM008_14
LM008_15
LM008_2
LM008_3
LM008_4
LM008_5
LM008_6
LM008_7
LM008_8
LM008_9

LM0091

426
427
428
429
430
43 1
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
45 l
452
453
454
455
456
457
458

 

LM009_10
LM009_1 1
LM009_12
LM009_13
LM009_14
LM009_15
LM009_2
LM009_3
LM009_4
LM009_5
LM009_6
LM009_7
LM009_8
LM009_9
LM010_1
LM010_2
LM010_3
LM01 1_1
LM01 1_10
LM01 1_1 1
LM01 1_12
LM01 1_13
LM01 1_14
LM01 1_15
LM01 1_2
LM01 1_3
LM01 1_4
LM01 1_5
LM01 1_6
LM01 1_7
LM01 1_8
LM01 1_9
LM012_1

459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491

 

LM012_10
LM012_1 1
LM012_12
LM012_13
LM012_14
LM012_15
LM012_16
LM012_17
LM012_18
LM012_19
LM012_2
LM012_20
LM012_21
LM012_22
LM012_23
LM012_24
LM012_25
LM012_26
LM012_27
LM012_3
LM012_4
LM012_5
LM012_6
LM012_7
LM012_8
LM012_9
LM013_1
LM013_2
LM013_3
LM013__4
LM014_1
LM014_2
LMO 14 g

492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
51 1
512
513
514
515
516
517
518
519
520
521
522
523
524

 

86

TABLE B.1. Continuation.

 

Keyto Dataset Keyto Dataset Keyto Dataset Keyto Data set
ComBase No. ComBase No. ComBase No. ComBase No.
LM014_4 525 LM018_4 558 LM024_1 591 LM030_2 624
LM015_1 526 LM018_5 559 LM024_2 592 LM031_1 625
LM015_2 527 LM018_6 560 LM025_1 593 LM031_2 626
LM015_3 528 LM018_7 561 LM025_2 594 LM032_1 627
LM015_4 529 LM018_8 562 LM026_1 595 LM032_2 628
LM016_1 530 LM018_9 563 LM026_2 596 LM033_1 629
LM01 6_2 53 1 LM019_1 564 LM026_3 597 LM033_2 630
LM016_3 532 LM019_10 565 LM026_4 598 LM034_1 631
LM016_4 533 LM019_1 l 566 LM027_1 599 LM034_2 632
LM017_1 534 LM019_12 567 LM027_10 600 LM035_1 633
LM017_10 535 LM019_13 568 LM027_1 1 601 LM035_2 634
LM017_1 l 5 36 LM019_14 569 LM027_12 602 LM036_1 635
LM017_12 537 LM019_15 570 LM027_13 603 LM036_2 636
LM017_13 538 LM019_16 571 LM027_14 604 LM036_3 637
LM017_14 539 LM019_17 572 LM027_15 605 LM036_4 638
LM017_15 540 LM019_18 573 LM027_16 606 LM037_1 639
LM017_2 541 LM019_19 574 LM027_17 607 LM03 7_2 640
LM017_3 542 LM019_2 575 LM027_18 608 LM037_3 641
LM017_4 543 LM019_3 576 LM027_19 609 LM038_1 642
LM017_5 544 LM019_4 577 LM027_2 610 LM039_1 643
LM01 7_6 545 LM019_5 578 LM027_20 61 l LM039_2 644
LM017_7 546 LM019_6 579 LM027_3 612 LM039_3 645
LM017_8 547 LM019_7 580 LM027_4 613 LM040_1 646
LM017_9 548 LM019_8 581 LM027_5 614 LM041_1 647
LM018_1 549 LM019_9 582 LM027_6 615 LM041_2 648
LM018_10 550 LM020_1 583 LM027_7 616 LM042__1 649
LM018_11 551 LM020_2 S84 LM027_8 617 LM043_1 650
LM018_12 552 LM021_1 585 LM027_9 618 LM043__2 651
LM018_13 553 LM021_2 586 LM028_1 619 LM044_1 652
LM018_14 554 LM022_1 587 LM028_2 620 LM044_2 653
LM018_15 555 LM022_2 588 LM029_1 621 LM045_1 654
LM018_2 556 LM023_1 589 LM029_2 622 LM045_2 655
LM01 8_3 5 57 LM023_2 590 LM03041 623 LM046_1 656

 

 

 

 

87

TABLE B. 1 . Continuation.

 

Keyto named Keyto Dataset Keyto Dataset Keyto Dataset
ComBase No. ComBase No. ComBase No. ComBase No.
LM046_2 657 LM050_2 690 LM055_2 723 LM065_2 756
LM047_1 658 LM051_1 691 LM056_1 724 LM066_1 757
LM047_10 659 LM051_10 692 LM056_10 725 LM066_2 758
LM047_11 660 LM051_11 693 LM056_11 726 LM067_1 759
LM047_12 661 LM051_12 694 LM056_12 727 LM067_2 760
LM047_13 662 LM051_2 695 LM056_13 728 LM068_1 761
LM047_14 663 LM051_3 696 LM056__14 729 LM068_2 762
LM047_15 664 LM051_4 697 LM056_15 730 LM069_1 763
LM047_16 665 LM051_5 698 LM056_2 731 LM069_2 764
LM047_17 666 LM051_6 699 LM056_3 732 LM070_1 765
LM047_18 667 LM051_7 700 LM056_4 733 LM070_2 766
LM047_19 668 LM051_8 701 LM056_5 734 LM071_1 767
LM047_2 669 LM051_9 702 LM056_6 735 LM071_2 768
LM047_20 670 LM052_1 703 LM056_7 736 LM072_1 769
LM047_21 671 LM053_1 704 LM056_8 737 LM072_2 770
LM047_22 672 LM053_10 705 LM056__9 738 LM073_1 771
LM047_23 673 LM053_11 706 LM057_1 739 LM073_2 772
LM047_24 674 LM053_12 707 LM057_2 740 LM074_1 773
LM047_25 675 LM053_13 708 LM058_1 741 LM074_2 774
LM047_26 676 LM053_14 709 LM058_2 742 LM075_1 775
LM047_27 677 LM053_15 710 LM059_1 743 LM075_2 776
LM047_3 678 LM053_16 711 LM059_2 744 LM076_1 777
LM047_4 679 LM053_2 712 LM060_1 745 LM076_2 778
LM047_5 680 LM053_3 713 LM060_2 746 LM077_1 779
LM047_6 681 LM053_4 714 LM061_1 747 LM077_2 780
LM047_7 682 LM053_5 715 LM061_2 748 LM078_1 781
LM047_8 683 LM053_6 716 LM062_1 749 LM078_2 782
LM047_9 684 LM053_7 717 LM062_2 750 LM080_1 783
LM048_1 685 LM053_8 718 LM063_1 751 LM081_1 784
LM049_1 686 LM053_9 719 LM063_2 752 LM082_1 785
LM049_2 687 LM054_1 720 LM064_1 753 LM083_1 786
LM049_3 688 LM054_2 721 LM064_2 754 LM084_1 787
LM050_1 689 LM05 5_1 722 LM065 1 755 LM084_2 788

 

 

 

 

88

TABLE B.l. Continuation.

 

Key to Data setl Key to Data set Key to Data set Key to Data set
ComBase No. ComBase No. ComBase No. ComBase No.
LM085_1 789 LM099_4 823 LM 1 13_5 857 LM127_14 891
LM085_2 790 LM099_5 824 LMl 13_6 858 LM127_15 892
LM086_1 791 LM099_6 825 LM113_7 859 LM127_16 893
LM087_1 792 LM099_7 826 LMl 13_8 860 LM127_2 894
LM088_1 793 LM099_8 827 LMl 13_9 861 LM127_3 895
LM089_1 794 LM099_9 828 LMl 14_l 862 LM127_4 896
LM089_2 795 LM100__I 829 LM 1 15_1 863 LM127_5 897
LM089_3 796 LM100_2 830 LMl 15_2 864 LM127_6 898
LM090_1 797 LM101_1 83 1 LMl 16_1 865 LM127_7 899
LMO90_2 798 LM101_2 832 LMl l6_2 866 LM127_8 900
LM091_1 799 LM102_1 833 LMl 17_1 867 LM127_9 901
LM091_2 800 LM102_2 834 LMl 17__2 868 LM128_1 902
LM092_1 801 LM103_1 835 LMl 18_1 869 LM128_2 903
LM093_1 802 LM103_2 836 LM118_2 870 LM129_1 904
LM093_2 803 LM104_1 837 LM119_1 871 LM129_2 905
LM094_1 804 LM104_2 838 LMl 19_2 872 LM129_3 906
LM094_2 805 LM105_1 839 LM120_1 873 LM130_1 907
LM095_1 806 LM105_2 840 LM120_2 874 LM130_10 908
LM095_2 807 LM106_1 841 LM121_1 875 LM130_1 1 909
LM096_1 808 LM107_1 842 LM121_2 876 LM130_12 910
LM096__2 809 LM108_1 843 LM122_1 877 LM130_13 911
LM097__1 810 LM108_2 844 LM122__2 878 LM130_14 912
LM097_2 81 1 LM109_1 845 LM123_1 879 LM130_15 913
LM098_1 812 LM109_2 846 LM123_2 880 LM130_2 914
LM099_1 813 L114] 10_1 847 LM124_1 881 LM130_3 915
LM099_10 814 LM110_2 848 LM124_2 882 LM130_4 916

LM099_11 815 LM111_1 849 LM125_1 883 LM130_5 917
LM099_12 816 um 1 1_2 850 LM125_2 884 LM130_6 918
LM099_13 817 LM 1 12_1 851 LM126_1 885 LM130_7 919
LM099_14 818 LMl 12_2 852 LM127_1 886 LM130_8 920
LM099_15 819 LMl 13_1 853 LM127_10 887 LM130_9 921
LM099_16 820 LM 1 13_2 854 LM127_1 1 888 LM13 1_1 922
LM099_2 821 LMl 13_3 855 LM127_12 889 LM13 1_2 923
LM099_3 822 LM] 13_4 856 LM12L1 3 890 LMl 32_1 924

 

 

 

 

89

TABLE B. l . Continuation.

Key to Data set Key to Data set Key to Data set
ComBase No. ComBase No. ComBase No.

 

LM132_10 925 LM132_15 930 LM132_6 935
LM132_11 926 LM132_2 931 LM132_7 936
LM132_12 927 LM132_3 932 LM132_8 937
LM132_13 928 LM132_4 933 LM132_9 938
LM132_14 929 LM132_5 934

 

 

 

9O

TABLE 82. No-growth broth-based data sets (L. monocytogenes) eliminated for
anaerobic and aerobic conditions.

Anaerobic I Aerobic
Data set No
1 211 293 386 741 912
4 212 294 387 742 913
9 213 295 388 745 916
26 223 331 389 746 917
29 224 332 390 747 921
30 225 333 391 748
45 226 340 398 749
46 227 341 402 750
47 228 342 444 780
63 229 343 449 781
64 230 344 453 782
82 231 345 482 794
85 232 346 486 795
86 233 347 527 796
87 245 348 529 797
88 248 349 535 798
89 249 350 540 799
90 250 351 544 800
91 253 364 600 801
117 254 365 605 802
119 255 366 607 803
120 256 380 614 804
139 263 381 685 805
142 264 382 686 806

 

 

 

143 265 687 807
160 266 688 808
161 267 705 809
1 74 268 709 814
l 75 269 710 818
178 270 71 1 819
179 271 714 823
1 80 272 715 824
l 81 273 719 85 1
1 84 274 739 852
185 290 740 908

 

91

TABLE B.3. B (h'l) and M (h) estimated ﬁ'om the primary regression for anaerobic

 

conditions.
Data set Data set Data set
No. B M No. B M No. B M

2 0.016 1 82.517 41 0.046 43 .205 80 0.042 32.080
3 0.956 10.666 42 0.950 10.670 81 0.036 36.910
5 1.710 23.193 43 0.031 51.691 83 0.039 35.301
6 0.205 80. 1 77 44 0.045 39.534 84 0.028 47.227
7 0.106 302.736 48 3.060 23.094 92 0.016 183.557
8 0.484 201 .306 49 0.889 10.654 93 0.280 98.601
10 0.209 6.622 50 2.858 22.809 94 0.184 19.073
1 1 0.203 6.932 51 0.195 10.324 95 0.190 19.1 18
12 0.195 7.086 52 0.162 11.838 96 0.107 13.118
13 0.203 6.774 53 0.684 19.824 97 0.088 16.009
14 0.871 10.996 54 0.228 9.961 98 0.090 16.605
15 0.861 10.915 55 3.209 23.183 99 0.097 14.935
16 0.870 1 1.034 56 3.965 7.179 100 0.128 1 1.787
17 0.867 10.808 57 0.761 20.187 101 0.101 13.946
18 1.136 8.871 58 1.151 21.657 102 0.104 14.103
19 0.169 8.090 59 0.561 21.222 103 0.105 14.102
20 0.176 7.831 60 0.582 22.527 104 0.1 15 12.831
21 0.164 8.120 61 0.125 14.657 105 0.108 15.200
22 0.109 1 1.606 62 0.471 19.653 106 0.084 16.792
23 0.147 9.1 15 65 0.421 94.777 107 0.176 1 1.394
24 0.122 10.194 66 0.020 80.855 108 0.103 13.586
25 1.327 57.243 67 0.019 80.491 109 0.104 15.160
27 1 .062 6.968 68 0.020 76.174 1 10 0.074 19.583
28 l .093 7.052 69 0.921 45.057 1 1 1 0.097 14.779
31 1.990 7.127 70 0.117 29.015 112 0.103 14.170
32 1.500 7.350 71 0.097 29.061 1 13 1.643 6.394
33 0.260 17.849 72 0.109 13 .803 1 14 1.855 8.884
34 0.230 10.649 73 0.244 19.947 1 15 0.952 10.703
35 0.922 10.170 74 0.156 19.483 1 16 4.164 23.700
36 0.962 10.512 75 0.1 14 15.284 1 18 0.822 29.874
37 0.058 26.198 76 0.182 19.079 121 0.041 47.743
38 0.078 29.784 77 0.1 76 19.963 122 0.035 48.385
39 0.090 25.531 78 0.074 24.201 123 0.034 51.268
40 0.234 20.497 79 0.064 29.694 124 0.046 44.100

 

 

 

92

TABLE B.3. Continuation.

 

 

 

Data set Data set Data set

No. B M No. B M No. B M

125 0.027 59.629 164 1.115 9.545 206 0.912 8.184
126 0.041 47.975 165 0.177 11.823 207 3.904 16.487
127 0.040 46.321 166 7.804 7.105 208 0.024 57.146
128 0.036 48.095 167 0.514 20.642 209 0.023 56.600
129 0.042 48.345 168 0.933 10.175 210 0.022 61.343
130 0.039 41.167 169 0.096 13.451 214 0.975 20.753
131 1.929 47.204 170 0.148 13.157 215 0.978 10.863
132 0.085 21.158 171 0.986 9.934 216 0.014 129.406
133 0.087 19.229 172 2.501 23.648 217 0.017 85.001
134 0.094 21.453 173 0.970 9.981 218 0.1 19 13 .246
135 0.053 43.004 176 0.982 9.946 219 0.100 16.191
136 0.024 85.145 177 0.003 494.122 220 0.105 15.305
137 0.082 44.055 182 0.033 41.969 221 0.011 155.611
138 0.064 36.615 183 0.033 44.217 222 0.052 149.320
140 0.049 38.124 186 0.020 70.745 234 1.585 7.330
141 0.031 57.037 187 0.025 56.194 235 0.185 10.074
144 0.138 20.848 188 0.027 54.055 236 0.050 30.102
145 4.599 26.821 189 0.023 59.368 237 0.076 26.661
146 0.093 21.540 190 0.026 57.652 238 0.070 28.490
147 0.087 22.303 191 0.023 65.533 239 0.080 26.354
148 0.093 21 .884 192 0.065 72.028 240 0.067 27. 105
149 0.088 21.224 193 0.240 6.650 241 0.067 26.575
150 0.952 10.076 194 0.271 6.103 242 0.065 37.214
151 0.919 10.140 195 0.256 6.446 243 0.957 10.446
152 0.062 24.221 196 2.089 8.945 244 0.955 10.563
1 53 0.106 23 .297 197 0.281 6.122 246 0.063 55.242
154 0.141 25.824 198 1.744 8.570 247 1.013 11.762
155 0.641 29.639 199 0.274 6.066 251 0.011 183.021
1 56 0.549 23.887 200 0.300 6.304 252 0.005 296.848
157 0.605 24.053 201 0.264 6.1 14 257 0.010 232.661
158 0.181 23.577 202 1.833 8.829 258 0.005 477.574
159 0.003 494.198 203 0.249 6.174 259 0.353 48.701
162 0.291 7.092 204 0.243 6.287 260 25 .923 120.120
163 0.783 7.270 205 1 .372 7.744 261 0.989 10.697

 

93

TABLE B.3. Continuation.

 

Data set Data set Data set
No. B M No. B M No. B M
262 0.975 10.945 307 0.010 143.720 339 0.954 200.734
275 0.426 16.715 308 0.045 239.062 352 0.01 1 142.644
276 1.345 7.479 309 0.054 41.014 353 0.014 135.248
277 0.929 10.825 310 0.036 49.552 354 0.009 161.137
278 0.998 10.304 31 1 0.039 46.982 355 0.016 137.446
279 0.927 10.831 312 0.012 138.531 356 0.011 144.186
280 1.148 8.899 313 0.007 236.1 16 357 0.012 136.504
281 1.343 7.498 314 0.993 154.908 358 0.010 157.094
282 0.577 15.293 315 0.239 164.455 359 0.010 157.447
283 1.332 7.583 316 0.276 164.132 360 0.006 217.585
284 1.144 8.924 317 0.009 169.041 361 0.004 413.335
285 1.361 7.350 318 0.010 142.339 362 0.012 248.843
286 1.147 8.904 319 0.009 146.859 363 0.266 284.1 16
287 0.951 10.620 320 0.009 192.921 367 0.012 139.988
288 1.143 8.937 321 0.015 172.845 368 2.937 22.552
289 1.145 8.928 322 0.009 156.028 369 0.193 8.077
291 0.020 271.282 323 0.009 159.316 370 1.01 1 9.921
292 0.010 171.758 324 0.009 165.278 371 0.070 17.900
296 0.012 166.289 325 0.009 145.480 372 0.056 22.260
297 0.164 179.709 326 0.010 144.412 373 0.096 16.181
298 0.009 168.174 327 0.01 1 141.319 374 0.077 21.589
299 0.010 169.396 328 0.009 171.077 375 0.066 22.340
300 0.017 123.855 329 0.007 178.235 376 0.069 22.020
301 0.014 120.746 330 0.009 184.649 377 0.071 22.860
302 0.018 102.843 334 0.012 391.181 378 0.064 24.001
303 0.016 1 1 1.003 335 0.010 380.690 379 0.067 23.108
304 0.015 1 10.348 336 0.968 382.834 383 0.009 191.736
305 0.015 1 14.322 337 0.957 200.690 384 0.009 205.806
306 0.010 152.000 338 0.952 200.764 385 0.004 427.784

 

 

 

94

TABLE B.4. B (h") and M (h) estimated from the primary regression for aerobic

 

 

 

conditions.
Data set Data set Data set

No. B M No. B M No. B M

392 0.280 5.993 428 0.865 1 1.083 465 0.41 1 19.463
393 0.109 24.392 429 1.079 9.793 466 0.3 14 18.451
394 0.901 5.834 430 0.344 7.414 467 0.467 19.811
395 0.226 6.573 43 1 0.284 7.343 468 0.396 17.928
396 0.213 6.960 432 0.282 7.145 469 0.631 21.198
397 2.5 10 9.275 433 0.268 7.759 470 0.548 20.685
399 1.155 8.857 434 2.805 22.617 471 0.418 18.915
400 0.220 7.418 435 0.266 7.485 472 0.119 13.124
401 1 .630 8.697 436 0.247 7.922 473 1 .239 24.686
403 1 .423 9.913 43 7 0.870 10.860 474 1 . 156 24.698
404 0.225 6.094 438 0.286 8.824 475 1.124 24.702
405 0.225 6.984 439 0.969 10.446 476 0.463 16.616
406 0.208 9.535 440 0.809 7.443 478 0.925 10.848
407 0.922 10.012 441 0.289 7.484 479 1.143 8.937
408 0.979 9.960 442 0.285 7.590 480 0.440 18.853
409 0.982 9.949 443 1.474 7.034 481 0.517 21.074
410 0.472 16.292 445 1.195 7.878 483 1.660 5.137
41 1 0.242 5.849 446 1 .037 7.892 484 1 .843 7.422
412 0.253 5.239 447 1.104 9.434 485 0.544 21.718
413 0.232 5.796 448 1.164 23.548 487 0.928 10.830
414 1.142 8.851 450 1.033 7.906 488 1.145 8.921
415 0.250 5.829 451 1.157 9.604 489 0.412 18.073
416 0.984 10.236 452 0.934 23.318 490 0.505 21.491
4 1 7 0.240 5.389 454 0.903 8.273 491 1 . 183 8.668
41 8 1.300 7.923 455 0.903 10.434 492 0.902 10.205
419 0.991 9.972 456 0.438 15.754 493 2.157 7.272
420 0.251 6.217 457 1.087 23.664 494 2.141 7.253
421 0.422 16.565 458 0.270 8.128 495 1.249 9.217
422 0.262 5.496 459 0.098 15.751 496 4.000 7.133
423 1.130 8.933 460 0.147 10.734 497 0.906 10.195
424 0.239 5.749 461 0.651 21.594 498 0.183 8.709
425 0.922 10.012 462 0.928 10.050 499 0.200 8.807
426 0.979 9.960 463 0.123 12.661 500 3.221 22.775
427 0.982 9.949 464 0.950 10.276 501 0.197 8.289

 

95

TABLE B.4. Continuation.

 

Data set Data set Data set
No. B M No. B M No. B M
502 1 .676 7.995 539 0.075 17.337 575 0.024 57 .438
503 0.199 8.237 541 0.267 10.737 576 0.060 41.956
504 0.195 8.232 542 0.234 9.891 577 0.050 41.660
505 2.392 22.359 543 0.059 19.965 578 0.007 206.641
506 0.877 10.822 545 0.247 10.542 579 0.008 223.769
507 0.205 7.526 546 0.312 10.347 580 0.007 229.171
508 1.555 8.858 547 0.220 10.994 581 1 .260 50.272
509 0.233 6.837 548 0.059 17.829 582 0.823 49.679
510 1.087 9.683 549 0.137 12.335 583 0.018 122.431
51 1 0.418 14.524 550 0.103 14.584 584 0.020 97.915
512 1.178 9.362 551 0.128 12.729 585 0.022 162.688
513 1.156 9.754 552 0.132 12.790 586 0.023 154.398
514 2.006 7.312 553 0.112 15.275 587 1.911 49.752
515 2.105 7.261 554 0.121 14.493 588 0.956 101.016
516 2.899 7.196 555 0.112 13.921 589 0.984 100.707
517 0.882 10.345 556 0.128 12.622 590 0.984 100.695
518 0.039 43.714 557 0.123 13.336 591 0.937 101.444
519 0.975 10.940 558 0.142 12.127 592 1.447 100.814
520 1.473 49.964 559 0.100 14.445 593 1.482 100.538
521 0.957 10.609 560 0.1 19 13.678 594 0.481 101.654
522 0.982 9.946 561 0.129 12.954 595 0.125 15.604
523 5.939 20.080 562 0.120 13.580 596 0.123 15.563
524 1.158 33.893 563 0.115 13.215 597 0.191 13.036
525 2.708 16.673 546 0.024 59.836 598 0.156 1 1.397
526 0.948 10.299 565 0.029 94.308 599 0.185 20.993
528 0.935 9.967 566 0.030 89.682 601 0.200 21.125
530 0.895 10.429 567 0.028 90.237 602 0.168 21.471
53 1 1.529 8.862 568 0.031 90.823 603 0.163 22.492
532 2.994 22.695 569 0.955 30.808 604 0.047 29.714
533 2.690 16.699 570 0.955 30.812 606 0.210 15.327
534 0.236 10.733 571 0.026 101.757 608 0.135 13.140
536 0.262 10.415 572 0.019 104.605 609 0.190 8.177
537 2.210 23.419 573 0.019 103.627 610 0.538 23.267
538 0.273 10.599 574 0.022 98.502 61 1 0.191 8.730

 

 

 

96

TABLE B.4. Continuation.

 

Data set Data set Data set

No. B M No. B M No. B M

612 0.114 22.521 647 0.096 23.616 681 0.096 18.908
613 0.046 30.887 648 0.101 23.974 682 0.095 21.094
615 0.147 20.920 649 0.090 27.089 683 0.106 20.548
616 0.177 22.075 650 0.053 32.597 684 0.106 21.326
617 0.124 22.082 651 0.062 30.938 689 0.095 22.623
618 0.048 27.651 652 0.054 35.564 690 0.094 22.883
619 0.090 22.241 653 0.052 31.600 691 0.091 25.817
620 0.088 21.319 654 0.091 18.297 692 0.049 30.625
621 0.054 26.517 655 0.097 19.038 693 0.069 28.162
622 0.064 25.864 656 0.105 14.385 694 0.060 31.361
623 0.064 31.213 657 0.090 17.792 695 0.063 31.651
624 0.062 31.636 658 0.107 20.765 696 0.080 26.272
625 0.072 27.744 659 0.436 23 .419 697 0.063 27.354
626 0.079 26.829 660 0.1 12 20.224 698 0.079 27.947
627 0.095 22.624 661 0.114 20.313 699 0.061 28.405
628 0.094 22.883 662 0.109 21.359 700 0.036 39.850
629 0.058 31.491 663 0.092 20.954 701 0.052 33.447
630 0.061 31.259 664 0.575 23.620 702 0.075 27.975
631 0.041 52.874 665 0.124 17.736 703 0.098 15.782
632 0.052 40.657 666 0.132 17.944 704 0.054 28.739
633 0.052 31.988 667 0.135 17.507 706 0.059 29.444
634 0.054 34.239 668 0.114 19.835 707 0.047 45.365
635 0.126 11.866 669 0.091 19.993 708 0.056 65.442
636 0.129 11.756 670 0.098 18.412 712 0.044 45.737
637 0.180 8.479 671 0.101 19.231 713 0.061 62.339
638 0.197 8.070 672 0.098 20.699 716 0.061 28.584
639 0.087 18.832 673 0.094 20.620 717 0.045 47.243
640 0.065 24.620 674 0.092 21 .029 718 0.090 59.420
641 0.071 24.617 675 0.074 21.066 720 0.052 31.988
642 0.091 19.003 676 0.084 22.582 721 0.054 34.239
643 0.094 22.543 677 0.080 22.745 722 0.059 21 .229
644 0.068 26.983 678 0.106 21.025 723 0.906 10.187
645 0.063 26.352 679 0.089 21.431 724 0.063 27.375
646 0.098 20.402 680 0.964 23.734 725 0.867 24.372

 

 

 

97

TABLE B.4. Continuation.

 

Data set Data set Data set

No. B M No. B M No. B M

726 0.073 29.178 770 0.019 75.165 829 0.011 173.799
727 0.068 26.773 771 0.676 121.559 830 0.013 169.092
728 0.061 27.959 772 0.022 65.201 83 1 0.007 206.964
729 0.064 29.057 773 0.022 76.079 832 0.009 204.1 1 1
730 0.594 24.515 774 0.020 64.748 833 0.009 400.774
731 0.063 26.304 775 0.019 90.197 834 0.006 383.934
732 0.069 26.91 1 776 0.020 88.856 835 0.009 213.803
733 0.072 28.891 777 0.977 71.195 836 0.009 213.626
734 0.637 24.390 778 0.036 62.503 837 0.012 171.105
735 0.061 21.279 779 0.021 87.858 838 0.013 171.990
736 0.063 28.365 783 0.033 48.043 839 0.006 235.330
737 0.096 26.562 784 0.024 57.570 840 0.006 239.583
738 0.063 29.973 785 0.024 56.813 841 0.008 228.468
743 0.892 104.334 7 86 0.022 61.451 842 0.01 1 158.868
744 0.030 648.047 787 0.057 30.320 843 0.005 253.157
751 0.951 10.739 788 0.033 61.952 844 0.007 229.541
752 0.020 75.468 789 0.040 43.258 845 0.007 212.71 1
753 1 .449 72.305 790 0.042 42.770 846 0.007 232.976
754 0.03 1 63.782 791 0.023 60.749 847 0.006 276.770
755 0.016 100.126 792 0.021 65.139 848 0.006 297.124
756 0.016 99.919 793 0.023 57.227 849 0.004 557.919
757 0.05 8 31.353 810 0.01 1 232.032 850 0.003 596.528
758 0.061 31.1 16 81 1 0.01 1 235.292 853 0.009 187.027
759 0.014 138.300 812 0.010 197.185 854 0.013 140.772
760 0.013 146.489 813 0.012 129.769 855 0.014 128.213
761 0.009 152.598 815 0.012 138.558 856 0.016 112.864
762 0.088 57.583 816 0.01 1 163.508 857 0.013 127.059
763 0.024 63 .408 817 0.007 257.826 858 0.013 123 .159
764 0.020 66.861 820 0.009 257.083 859 0.013 128.762
765 0.388 25.823 821 0.009 180.254 860 0.012 158.229
766 0.035 50.620 822 0.007 262.008 861 0.010 144.320
767 0.034 47.421 825 0.012 135.966 862 0.015 135.187
768 0.037 46.798 826 0.009 196.525 863 0.012 153.132
769 0.021 75.590 827 0.005 288.501 864 0.010 173.042

 

 

 

98

TABLE B.4. Continuation.

 

 

Data set Data set

No. B M No. B M

865 0.007 200.274 899 0.010 163.798
866 0.009 204.733 900 0.010 164.632
867 0.01 1 165.480 901 0.01 1 152.368
868 0.010 176.612 902 0.006 235.456
869 0.015 96.338 903 0.006 239.690
870 0.016 89.720 904 0.016 151.365
871 0.010 172.599 905 0.013 178.187
872 0.009 168.587 906 0.015 174.290
873 0.016 160.084 907 0.008 198.709
874 0.013 156.457 909 0.007 227.3 15
875 0.008 224.263 910 0.007 297.308
876 0.009 217.646 91 1 0.007 386.786
877 0.007 240.956 914 0.009 284.062
87 8 0.006 265.994 915 0.004 439.285
879 0.009 198.395 918 0.009 209.478
880 0.009 179.296 919 0.006 332.618
881 0.010 129.998 920 0.007 335.046
882 0.010 141.609 922 0.012 171.121
883 0.013 1 14.840 923 0.013 172.005
884 0.016 1 10.004 924 0.010 179.316
885 0.012 146.337 925 0.014 146.925
886 0.010 155.764 926 0.01 1 179.223
887 0.01 1 144.304 927 0.012 163.802
888 0.010 156.268 928 0.012 173.334
889 0.010 156.438 929 0.012 170.474
890 0.010 154.515 930 0.013 147.530
891 0.009 156.153 931 0.01 1 177.617
892 0.01 1 145.476 932 0.014 161.775
893 0.01 1 156.722 933 0.01 1 159.964
894 0.010 160.651 934 0.014 149.552
895 0.010 156.1 18 935 0.01 1 176.798
896 0.01 1 155.369 936 0.013 178.495
897 0.011 141.126 937 0.012 169.827
898 0.010 168.463 938 0.01 1 155.331

 

99

TABLE B.5. Treatment (treat) and data set numbers for anaerobic conditions used to
calculate error due to replication calculations.

Data TreatiDataset Treat Data Treat. Data Treat. Data Treat.
setNo. No. No. No. setNo. No. setNo. No. No. No.

48 99 84 24 128 26 168 41
49 99 92 39 129 26 169 41
50 99 93 39 130 26 170 41
51 100 94 62 132 27 171 42
52 100 95 62 133 27 172 42
53 100 96 62 134 27 173 42
10 98 54 101 97 73 135 28 177 12
11 98 55 101 98 73 136 28 186 12
12 98 56 101 99 87 137 28 193 104
13 98 57 102 100 87 138 29 194 104
14 98 58 102 101 87 140 29 195 104
15 98 59 102 102 87 141 29 196 105
16 98 60 103 103 87 144 93 197 105
17 98 61 103 104 87 145 93 198 105
18 98 62 103 105 88 146 93 199 106
19 98 65 2 106 88 147 94 200 106
20 98 66 2 107 88 148 94 201 106
21 98 67 2 108 89 149 94 202 107
22 98 68 2 109 89 150 95 203 107

6 89

6

6

 

Ab-b-bah-h

23 98 69 1 10 151 95 204 107
24 98 70 l 1 1 90 152 95 205 108
33 35 71 112 153 96 206 108
34 35 72 22 l 13 90 154 96 207 108
96
97

8

35 35 73 22 114 91 155 214 54
36 36 74 22 115 91 156 215 54
37 36 75 22 116 91 157 97 218 44
38 36 76 22 121 26 158 97 219 44
39 37 77 22 122 26 162 39 220 44
40 37 78 23 123 26 163 39 221 1 18
41 37 79 23 124 26 164 39 222 1 18
42 38 80 23 125 26 165 40 234 45
43 38 81 24 126 26 166 40 235 45
44 38 83 24 127 26 167 40 236 46

 

 

 

 

 

100

TABLE B.5. Continuation.

Data set Treat. Data set Treat. Data set Treat.
No. No. No. No. No. No.

 

237 46 304 57 359 83
238 46 305 57 360 83
239 47 306 57 361 83
240 47 307 57 362 84
241 47 308 57 363 84
242 48 310 3 369 30
243 48 311 3 370 30
244 48 312 3 371 30
246 49 316 72 372 31
247 49 317 72 373 31
251 8 319 76 374 31
252 8 320 76 375 32
260 13 321 76 376 32
261 13 322 76 377 32
262 13 323 77 378 33
276 109 324 77 379 33
277 109 325 77
278 109 326 78
279 1 10 327 78
280 1 10 328 78
281 1 10 329 79
282 11 1 330 79
283 1 l 1 335 17
284 1 1 1 336 17
285 1 12 337 17
286 1 12 338 18
287 112 339 18
288 113 353 81
289 113 354 81
300 57 355 81
301 57 356 82
302 57 357 82
303 57 358 82

 

 

 

101

TABLE B.6. Treatment (treat) and data set numbers for aerobic conditions used to
calculate error due to replication calculations.

Data set Treat. Data Treat. Data Treat. Data set Treat. Data set Treat.
No. No. set No. No. set No. No. No. No. No. No.
392 44 428 123 464 132 500 108 536 40
394 44 429 127 465 129 501 108 537 41
395 45 430 123 466 130 502 109 538 42
396 46 431 124 467 131 503 108 539 43
397 47 432 125 468 132 504 108 541 41
399 45 433 126 469 128 505 108 542 42
400 46 434 127 470 129 506 108 543 43
401 47 435 124 471 130 507 108 545 40
403 44 436 125 472 131 508 108 546 41
404 45 437 126 473 2 509 108 547 42
405 46 438 127 474 2 510 108 548 43
406 47 439 123 475 2 51 1 1 10 549 1 13
407 49 440 124 476 36 512 111 550 1 17
408 49 441 125 478 36 513 112 551 113
409 49 442 126 479 37 514 108 552 1 14
410 118 443 50 480 38 515 109 553 115
411 122 445 50 481 39 516 110 554 116
412 118 446 51 483 37 517 111 555 117
413 119 447 52 484 38 518 9 556 114
414 120 448 53 485 39 519 10 557 115
415 121 450 51 487 36 520 9 558 116
416 122 451 52 488 37 521 10 559 117
417 1 19 452 53 489 38 522 68 560 1 13
418 120 454 50 490 39 523 69 561 1 14
419 121 455 51 491 108 524 68 562 115
420 122 456 52 492 1 12 525 69 563 1 16
421 1 18 457 53 493 108 526 1 1 564
422 1 19 458 128 494 109 528 1 1 565
423 120 459 132 495 1 10 530 70 566
424 121 460 128 496 111 531 71 567
425 125 461 129 497 1 12 532 70 568
426 125 462 130 498 108 533 71 569
427 125 463 131 499 108 534 40 570

 

 

 

 

 

u—Ih—I—OI—II-dI—II—l

 

102

TABLE 86. Continuation.

 

Data set Treat. Data Treat. Data Treat. Dataset Treat. Dataset Treat.
No. No. set No. No. set No. No. No. No. No. No.
57 l 1 606 26 641 79 674 97 720 32
572 1 608 26 642 79 67 5 97 721 32
573 l 609 26 643 80 676 97 722 32
574 1 610 27 644 80 677 97 723 32
575 1 61 1 26 645 80 678 99 724 103
57 6 1 612 28 646 80 679 100 725 107
577 1 613 29 647 81 680 101 726 103
578 1 615 26 648 81 681 97 727 104
579 1 616 27 649 81 682 98 728 105
580 1 617 28 650 81 683 99 729 106
581 1 618 29 651 81 684 100 730 107
582 1 619 26 652 81 689 30 731 104
583 4 620 26 653 81 690 30 732 105
S84 4 621 26 654 83 691 31 733 106
585 4 622 26 655 83 692 31 734 107
586 4 623 26 656 83 693 3 1 735 103
587 5 624 26 657 83 694 31 736 104
588 5 625 26 658 97 695 3 l 737 105
589 5 626 26 659 101 696 3 1 738 106
590 5 627 30 660 97 697 31 743 3
591 6 628 30 661 98 698 31 744 3
592 6 629 30 662 99 699 3 1 751 23
593 6 630 30 663 100 700 31 752 23
594 6 631 26 664 101 701 31 753 24
595 8 632 26 665 97 702 3 1 7 54 24
596 8 633 26 666 97 704 32 755 24
597 8 634 26 667 97 707 32 7 56 24
598 8 635 67 668 97 708 33 757 24
599 26 636 67 669 98 712 33 758 24
601 26 637 67 670 97 713 34 759 25
602 27 638 67 671 97 716 32 760 25
603 28 639 79 672 97 717 33 761 25
604 29 640 79 673 97 718 34 762 25

 

 

 

 

 

103

TABLE B.6. Continuation.

Dataset Treat. Data Treat. Data TreatPDatasetTreat.
No. No. set No. No. setNo. No. No. No.

 

763 76 833 13 869 72 903 17
764 76 834 13 870 72 904 18
765 76 835 13 871 73 905 18
766 76 836 13 872 73 906 18
767 76 837 13 873 73 907 19
768 76 838 13 874 73 909 19
769 77 839 13 875 74 910 20
770 77 840 13 876 74 911 21
771 77 842 56 877 74 914 20
772 77 843 56 878 74 915 21
773 77 844 56 879 82 918 19
774 77 845 57 880 82 919 20
775 78 846 57 881 82 920 21
776 78 847 58 882 82 922 19
777 78 848 58 883 82 923 19
778 78 849 59 884 82 924 91
784 55 850 59 886 86 925 95
791 75 853 61 887 90 926 91
810 13 854 61 888 86 927 92
811 13 855 61 889 87 928 93
813 13 856 61 890 88 929 94
815 13 857 61 891 89 930 95
816 14 858 61 892 90 931 92
817 15 859 61 893 86 932 93
821 14 860 61 894 87 933 94
822 15 861 61 895 88 934 95
825 13 862 72 896 89 935 91
826 14 863 72 897 90 936 92
827 15 864 72 898 86 937 93
829 13 865 72 899 87 938 94
830 13 866 72 900 88
831 13 867 72 901 89
832 13 868 72 902 17

 

 

 

 

104

TABLE B.7. Experimental variables each treatment (aerobic conditions, (U .S.

 

Department of Agriculture, 2003a)).
pH T °C NaCl Nitrite No. of No. of pH T °C NaCl Nitrite No. of No. of
(an) (rpm) a... treat. (8") (ppm) data treat.
Jomts Jomts
4.5 19 0 0 200 l 6 19 25 0 64 30
4.5 28 0 0 30 2 6 19 25 100 92 31
5 12 25 0 28 3 6 19 45 0 60 32
5 19 0 0 66 4 6 19 45 50 24 33
5 19 25 0 38 5 6 19 45 100 16 34
5 19 45 0 50 6 6 19 45 1000 7 35
5.25 10 15 50 7 7 6 28 0 0 12 36
5.25 19 0 0 32 8 6 28 0 50 12 37
5.25 28 15 50 14 9 6 28 0 100 12 38
5.25 28 15 150 17 10 6 28 0 200 12 39
5.25 28 35 50 15 1 1 6 28 45 0 24 40
5.5 5 0 0 7 12 6 28 45 50 22 41
6 5 0 0 151 13 6 28 45 100 24 42
6 5 0 50 21 14 6 28 45 200 21 43
6 5 0 100 21 15 6 37 0 0 16 44
6 5 0 200 7 16 6 37 0 50 17 45
6 5 25 0 18 17 6 37 0 100 18 46
6 5 25 100 24 18 6 37 0 200 18 47
6 5 45 0 36 19 6 37 0 1000 6 48
6 5 45 50 21 20 6 37 25 100 15 49
6 5 45 100 21 21 6 37 45 0 15 50
6 10 0 0 14 22 6 37 45 50 15 51
6 12 0 0 27 23 6 37 45 100 15 52
6 12 25 0 66 24 6 37 45 200 15 53
6 12 45 0 42 25 6.25 5 0 0 7 54
6 19 0 0 177 26 6.25 10 O 0 13 55
6 19 0 50 18 27 6.5 5 0 0 25 56
6 19 0 100 18 28 6.5 5 20 0 18 57
6 19 0 200 18 29 6.5 5 40 0 20 58

 

 

105

TABLE B.7. Continuation.

 

pH T °C NaCl Nitrite No. of No. of pH T °C NaCl Nitrite No. of No. of
(8/1) (PP!!!) data treat. (8") (PP!!!) data twat-
points points
6.5 5 60 0 26 59 7.5 5 0 100 24 88
6.5 10 0 0 13 60 7.5 5 0 200 24 89
6.75 5 0 0 75 61 7.5 5 0 1000 24 90
6.75 10 0 0 13 62 7.5 5 45 0 18 91
6.75 10 15 50 7 63 7.5 5 45 50 18 92
6.75 10 15 150 8 64 7.5 5 45 100 18 93
6.75 10 35 50 7 65 7.5 5 45 200 18 94
6.75 10 35 150 7 66 7.5 5 45 1000 15 95
6.75 19 0 0 32 67 7.5 10 0 0 13 96
6.75 28 15 50 13 68 7.5 19 0 0 1 14 97
6.75 28 15 150 16 69 7.5 19 0 50 15 98
6.75 28 35 50 14 70 7.5 19 0 100 15 99
6.75 28 35 150 15 71 7.5 19 0 200 15 100
7 5 0 0 84 72 7.5 19 0 1000 13 101
7 5 25 0 42 73 7.5 19 25 100 6 102
7 5 45 0 38 74 7.5 19 45 0 18 103
7 10 0 0 13 75 7.5 19 45 50 18 104
7 12 0 0 46 76 7.5 19 45 100 18 105
7 12 25 0 54 77 7.5 19 45 200 18 106
7 12 45 0 34 78 7.5 19 45 1000 18 107
7 19 0 0 38 79 7.5 28 0 0 90 108
7 19 25 0 39 80 7.5 28 0 50 18 109
7 19 45 0 68 81 7.5 28 O 100 18 110
7 2 5 0 0 46 82 7.5 28 0 200 18 111
72 19 0 0 34 83 7.5 28 0 1000 18 112
7.25 5 0 0 7 84 7.5 28 45 0 21 113
7.25 10 0 0 13 85 7.5 28 45 50 21 1 14
7.5 5 0 0 31 86 7.5 28 45 100 21 115
7.5 5 0 50 24 87 7.5 28 45 200 21 1 16

 

 

106

TABLE B.7. Continuation.

 

pH T°C NaCl Nitrite No. of No. of pH T°C NaCl Nitrite No. of No. of
(all) (ppm) aaa treat. (all) (ppm) data treat.
pomts pomts
7.5 28 45 1000 21 117 7.5 37 25 100 33 125
7.5 37 0 0 17 118 7.5 37 25 200 18 126
7.5 37 0 50 17 119 7.5 37 25 1000 18 127
7.5 37 0 100 18 120 7.5 37 45 0 18 128
7.5 37 0 200 16 121 7.5 37 45 50 18 129
7.5 37 0 1000 17 122 7.5 37 45 100 17 130
7.5 37 25 0 18 123 7.5 37 45 200 18 131
7.5 37 25 50 18 124 7.5 37 45 1000 16 132

 

107

TABLE B.8. Experimental variables in each treatment (anaerobic conditions, (U .S.
Department of Agriculture, 2003a)).

 

pH T °C NaCl Nitrite No. of No. of pH T °C NaCl Nitrite No. of No. of

 

(8”) (Wm) data treat. (8") (13131“) data treat.
pomts pomts
4.5 10 0 0 10 28 0 0 18 30
4.5 19 0 0 43 28 0 50 25 31
4.5 28 0 0 31 28 0 100 22 32
4.5 37 0 0 57 28 0 200 12 33
5.25 10 15 50 6 28 0 1000 9 34

5.25 19 0 0 30 28 45 0 16 35

“O“QO‘UI-BWNH

6

6

6

6

6

6
5.3 28 15 50 8 6 28 45 50 17 36
5.5 5 0 0 26 6 28 45 100 22 37
5.5 5 0 50 15 6 28 45 200 23 38
5.5 5 5 0 5 10 6 37 0 0 42 39
5 .5 5 25 0 14 11 6 37 0 50 16 40
5.5 10 0 0 23 12 6 37 0 100 17 41
6 5 0 0 19 13 6 37 0 200 14 42
6 5 0 25 10 14 6 37 0 1000 4 43
6 5 0 50 10 15 6 37 25 100 27 44
6 5 0 1000 5 16 6 37 45 0 16 45
6 5 25 100 26 17 6 37 45 50 26 46
6 5 45 0 16 18 6 37 45 100 28 47
6 5 45 25 7 19 6 37 45 200 24 48
6 5 45 1000 8 20 6 37 45 1000 20 49
6 10 0 0 13 21 6.25 5 0 0 6 50
6 19 0 0 53 22 6.25 10 0 0 13 51
6 19 0 50 29 23 6.5 5 0 0 7 52
6 l9 0 100 33 24 6.5 5 0 200 7 53
6 19 25 0 8 25 6.5 8 0 0 19 54
6 19 25 100 100 26 6.5 8 50 O 9 55
6 19 45 0 21 27 6.5 10 0 0 13 56
6 19 45 50 24 28 6.75 5 0 0 75 57
6 19 45 100 24 29 6.75 10 0 0 13 58

 

 

108

TABLE B.8. Continuation.

 

 

pH T °C NaCl Nitrite No. of No. of pH T °C NaCl Nitrite No. of No. of
(8/1) (Ppm) data twat» (8") (mm!) data treat.
points points

6.75 10 15 150 6 59 7.5 19 0 100 24 89
6.75 10 35 50 6 60 7.5 19 0 200 27 90
6.75 10 35 150 7 61 7.5 19 0 1000 27 91
6.75 19 0 0 30 62 7.5 19 25 100 7 92
6.75 28 35 50 7 63 7.5 19 45 0 24 93
6.75 28 35 150 7 64 7.5 19 45 50 24 94
6.8 28 15 50 7 65 7.5 19 45 100 18 95
6.8 28 15 150 6 66 7.5 19 45 200 18 96
7 5 0 0 11 67 7.5 19 45 1000 16 97
7 5 0 200 10 68 7.5 28 0 0 105 98
7 5 5 0 7 69 7.5 28 45 0 18 99
7 5 5 200 6 70 7.5 28 45 50 18 100
7 10 0 0 13 71 7.5 28 45 100 18 101
7.2 5 0 0 13 72 7.5 28 45 200 18 102
7.2 19 0 0 16 73 7.5 28 45 1000 21 103
7.25 5 0 0 8 74 7.5 37 0 0 19 104
7.25 10 0 0 13 75 7.5 37 0 50 21 105
7.5 5 0 0 31 76 7.5 37 0 100 20 106
7.5 5 0 50 24 77 7.5 37 0 200 21 107
7.5 5 0 100 24 78 7.5 37 0 1000 20 108
7.5 5 0 200 16 79 7.5 37 45 0 14 109
7.5 5 0 1000 9 80 7.5 37 45 50 15 110
7.5 5 45 0 24 81 7.5 37 45 100 15 111
7.5 5 45 50 24 82 7.5 37 45 200 15 112
7.5 5 45 100 24 83 7.5 37 45 1000 10 113
7.5 5 45 200 15 84 8 5 0 0 6 114
7.5 5 45 1000 9 85 8 5 0 200 10 115
7.5 10 0 0 13 86 8 5 5 0 14 116
7.5 19 0 0 51 87 8 8 0 0 8 117
7.5 19 0 50 21 88 8 8 50 0 19 118

 

 

109

APPENDIX C

STANDARD ERROR ANALYSIS

After the nonlinear regression was performed to the broth-based data, the
asymptotic standard error (SE) of each data set was obtained as described in chapter 5,
section 5.2.4. A trend or signiﬁcant relationship between SE and experimental conditions
(pH, temperature, salt, or nitrite), time, microbial counts, and treatments (speciﬁc
combination of the experimental variables) was assessed (aerobic conditions).

An analysis of variance for SE versus pH, temperature, salt, or nitrite, showed
that, assuming there was no interaction between the variables, only pH had a signiﬁcant

inﬂuence on SE (Table C.1).

TABLE C. 1 . Effects tests for the experimental variables.

 

Variable P value
pH 0.0017
T °C 0.1346

NaCl (g/L) 0.1952
Nitrite (ppm) 0.3796

An analysis of variances with the same variables, but including interactions

showed that 1i and pit2 had a signiﬁcant inﬂuence on SE (Table c2).

110

TABLE C.2. Effects test for the experimental variables with interactions.

 

Source P value
pH 0.15 16
T °C 0.1066
NaCl (g/L) 0.0741
Nitrite (ppm) 0.1889
pprH 0. 0215
pHx T °C 0.0677
T °C x T °C 0.0005
pHxNaCl (g/L) 0.3955
T °C xNaC1(g/L) 0.3137
NaCl (g/L) xNaC1(g/L) 0.5090
pHxNitrite (ppm) 0.4696
T °C xNitrite (ppm) 0.7862

NaCl (g/L) xNitrite (ppm) 0.9335
Nitrite (ppm) xNitrite (ppm) 0.3758

 

No signiﬁcant relationship was found between treatment and SE, meaning that
speciﬁc combination of the variables did not affect SE.

The analysis to ﬁnd the relationship between SE versus time and microbial counts
was done separately because the experimental variables do not change within the same
data set.

Both time and microbial counts had a signiﬁcant inﬂuence (P <0.0001) on the SE,
for all treatments.

To illustrate the relationship of SE with time and microbial counts, treatment No.
1 was chosen as an example, because of its high number of replications (19 data sets,
table C.3).

Additionally, it was found that SE had a unique trend as a function of time. SE
slightly increased in value at the middle of the time period, decreasing at the beginning

and end of the period (Fig. C.1). The same trend was found for all treatments.

111

TABLE C.3. Data sets included in treatment No. 1.

 

Data Time Log SE Data Time Log SE Data Time Log SE

 

No. (h) (CPU/ml) No. (b) (CFU/ml) No. (b) (CPU/ml)

564 0 1.97 1.1987 568 0 3.63 1.1965 573 0 3.26 1.197
564 3 1.92 1.1989 568 24 3.65 1.1973 573 24 3.31 1.1979
564 20 2.76 1.1998 568 48 3.74 1.1976 573 48 3.44 1.1981
564 24 2.76 1.2 568 72 4.48 1.1974 573 72 4.13 1.198
564 27 2.92 1.2001 568 96 5.43 1.1975 573 96 4.81 1.198
564 44 3.98 1.2003 568 168 7.69 1.2005 573 168 6.67 1.2014
564 48 4.21 1.2003 568 216 8.31 1.2026 573 216 7.66 1.2039
564 51 4.39 1.2003 568 264 8.26 1.2029 573 264 8.13 1.2041
564 69 5.44 1.2002 568 336 7.64 1.2004 573 336 7.8 1.2014

 

564 75 5.97 1.2001 569 0 3.56 1.1966 574 0 3.2 1.197
564 93 7.2 1.2002 569 24 3.57 1.1974 574 24 3.26 1.198
564 99 7.61 1.2003 569 48 3.56 1.1977 574 48 3.4 1.1982
564 121 8.39 1.2012 569 72 4.25 1.1975 574 72 4.1 1.1981

 

565 0 3.59 1.1966 569 96 5.24 1.1976 574 96 4.92 1.1981
565 24 3.6 1.1974 569 168 7.23 1.2006 574 168 6.8 1.2016
565 48 3.35 1.1976 569 216 8.21 1.2029 574 216 8.23 1.2041
565 72 4.6 1.1975 569 264 8.13 1.2031 574 264 8.15 1.2043
565 96 5.08 1.1975 569 336 7.7 1.2006 574 336 7.7 1.2015

 

565 168 7.63 1.2006 570 0 3.53 1.1967 575 0 2.01 1.1986
565 216 8 1.2028 570 24 3.55 1.1975 575 3 2.27 1.1988
565 264 8.32 1.203 570 48 3.49 1.1977 575 20 2.63 1.1998
565 336 7.64 1.2005 570 72 4.4 1.1976 575 24 2.69 1.1999

 

566 0 3.61 1.1966 570 96 5.07 1.1976 575 27 2.85 1.2
566 24 3.66 1.1974 570 168 7.15 1.2007 575 44 3.99 1.2003
566 48 3.74 1.1976 570 216 8.19 1.203 575 48 4.93 1.2003
566 72 4.6 1.1975 570 264 8.09 1.2032 575 51 4.4 1.2003
566 96 5.38 1.1975 570 336 7.84 1.2007 575 69 5.39 1.2001
566 168 7.68 1.2005 571 0 3.47 1.1967 575 75 5.9 1.2001
566 216 8.29 1.2027 571 24 3.49 1.1976 575 93 7.15 1.2001
566 264 8.14 1.203 571 48 3.55 1.1978 575 99 7.61 1.2002
566 336 7.64 1.2005 571 72 4.06 1.1977 575 121 8.35 1.2011

 

 

 

567 0 3.63 1.1965 571 96 4.87 1.1977 576 0 4.29 1.1958
567 24 3.55 1.1973 571 168 7.18 1.2009 576 3 4.06 1.1959
567 48 3.65 1.1976 571 216 8.15 1.2032 576 20 3.63 1.1964
567 72 4.54 1.1974 571 264 8.1 1.2034 576 24 4.93 1.1965
567 96 5.31 1.1975 571 336 7.77 1.2008 576 27 5.01 1.1965
567 168 7.61 1.2005 572 0 3.26 1.197 576 44 5.96 1.1966
567 216 6.96 1.2026 572 24 3.35 1.1979 576 48 6.24 1.1966
567 264 8.29 1.2029 572 48 . 3.4 1.1981 576 69 8 1.1966
567 336 7.72 1.2004 572 72 4.19 1.198 576 75 8.09 1.1965

 

 

572 96 4.76 1.198 576 93 8.54 1.1966
572 168 6.73 1.2014 576 99 8.58 1.1966
572 216 7.86 1.2039 576 121 8.25 1.1971

 

572 264 8.27 1.2041
572 336 7.88 1.2014

 

 

 

112

TABLE C.3. Continuation.

 

Data Time Log s13 Data Time Log SB
No. (b) (CFU/ml) No. (b) (CFU/ml)

 

577 0 3.97 1.1962 580 0 3.05 1.1972
577 3 4.02 1.1963 580 6 2.99 1.1975
577 20 3.55 1.1968 580 24 3.08 1.1982
577 24 4.74 1.19691 580 48 3.09 1.1985
577 27 4.96 1.1969 580 54 326 1.1984
577 44 5.92 1.1971 580 72 3.55 1.1983
577 48 5.94 1.1971 580 78 3.65 1.1983
577 69 7.77 1.197 580 96 3.73 1.1984
577 75 7.96 1.197 580 100 3.74 1.1984
577 93 8.52 1.197 580 192 4.6 1.2035
577 99 8.71 1.197 580 240 5.16 1.205
577 121 8.3 1.1975 580 336 7.16 1.2019

 

578 0 3.01 1.1973 580 408 7.77 1.1982
578 6 3.12 1.1976 580 504 8.16 1.1951

 

578 24 3.14 1.1983 581 0 3.61 1.1966
578 48 3.25 1.1985 581 24 3.67 1.1974
578 54 3.33 1.1985 581 48 3.55 1.1976
578 72 3.53 1.1984 581 72 4.1 1.1975
578 78 3.63 1.1984 581 96 4.51 1.1975
578 96 3.75 1.1984 581 168 6.97 1.2005
578 100 3.83 1.1985 581 216 8.2 1.2027
578 192 4.74 1.2036 581 264 8.31 1.203
578 240 5.33 1.2052 581 336 7.97 1.2005

 

578 336 7.54 1.202 582 0 3.58 1.1966
578 408 7.93 1.1983 582 24 3.54 1.1974
578 504 8.04 1.1951 582 48 3.64 1.1976

 

579 0 3.05 1.1972 582 72 4.06 1.1975
579 6 2.98 1.1975 582 96 4.89 1.1976
579 24 2.85 1.1982 582 168 7.39 1.2006
579 48 3.13 1.1985 582 216 8.3 1.2028
579 54 2.98 1.1984 582 264 8.35 1.2031
579 72 3.42 1.1983 582 336 7.76 1.2005

 

579 78 3.55 1.1983
579 96 3.64 1.1984
579 100 3.69 1.1984
579 192 4.63 1 .2035
579 240 4.99 1.205

579 336 7.1 1.2019
579 408 7.77 1.1982
579 504 7.94 1.1951

 

 

113

 

1.206 l—m—w— *mv ~ , - 4—4
1.204 -- c 8
1.202 .

1.2 .

 

1.198

SE, Log N (CF U/ml)

1.196

 

1.194 -

 

Time (11)

FIGURE C. 1 . Standard error (SE) versus time, treatment 1 for L. monocytogenes
growth in broth (pH= 4.5, T= 19 °C, salt= 0 g/l, and nitrite= 0 ppm).

114

APPENDIX D

D.l COMPARISON OF EXPERIMENTAL VARIABILITY BETWEEN BROTH
AND MEAT-BASED DATA

In order to perform this analysis on the same basis, broth and meat-based data that
had the same number of treatments and the same number of replications within those
treatments, were randomly selected.

Three different treatments with two replications for each one were selected from
the L. monocytogenes broth-based growth data, anaerobic conditions (Table D. 1 . 1). The
experimental variability was calculated as described in Chapter 5, section 5.2.3, and was
found to be 0.05 log(CFU/ml).

L. monocytogenes growth data in cooked chicken were obtained ﬁom ComBase.
Three different treatments with two replications for each one (anaerobic conditions) were
selected (Table D.1.2). Again the experimental variability was calculated as described in
Chapter 5, section 5.2.3, which was found to be 0.97 log(CFU/ml).

The experimental variability due to replications was approximately 95% higher

for the food-based data than for the broth-based data.

115

TABLE D.1.1. Broth-based L. monocytogenes growth data.

Data set Time (h) Log Treat. No. pH T °C NaC1(g/1) Nitrite

 

 

 

 

No. (CPU/ml) (ppm)
378 0 2.94 33 6 28 0 200
378 3 3.14 33 6 28 0 200
378 7 3.35 33 6 28 0 200
378 24 5.23 33 6 28 0 200
378 48 8.33 33 6 28 0 200
378 54 8.33 33 6 28 0 200
379 0 3.01 33 6 28 0 200
379 3 3.15 33 6 28 0 200
379 7 3.46 33 6 28 0 200
379 24 5.38 33 6 28 0 200
379 48 8.48 33 6 28 0 200
379 54 8.37 33 6 28 0 200
234 0 3.72 45 6 37 45 0
234 3 3.72 45 6 37 45 0
234 7 4.69 45 6 37 45 0
234 24 9.15 45 6 37 45 0
234 27 9.1 1 45 6 37 45 0
234 3 1 9.23 45 6 37 45 0
234 48 9.07 45 6 37 45 0
234 54.5 8.26 45 6 37 45 0
235 0 3.67 45 6 37 45 0
235 3 3.66 45 6 37 45 0
235 7 4.59 45 6 37 45 0
235 24 8.42 45 6 37 45 0
235 27 8.26 45 6 37 45 0
235 31 8.92 45 6 37 45 0
235 48 8.69 45 6 37 45 0
235 54.5 8.76 45 6 37 45 0

 

116

TABLE D.1.1. Continuation

 

 

Data set Time (h) Log Treat. No. pH T °C NaCl (g/l) Nitrite
No. (CPU/m1) (ppm)
246 0 3.64 49 6 37 45 1000
246 3 3.52 49 6 37 45 1000
246 7 3.56 49 6 37 45 1000
246 24 3.21 49 6 37 45 1000
246 27 3.22 49 6 37 45 1000
246 31 3.66 49 6 37 45 1000
246 48 4 49 6 37 45 1000
246 54.5 4.13 49 6 37 45 1000
246 72 4.53 49 6 37 45 1000
246 79 4.85 49 6 37 45 1000
247 0 3.63 49 6 37 45 1000
247 3 3.57 49 6 37 45 1000
247 7 3.48 49 6 37 45 1000
247 24 3.21 49 6 37 45 1000
247 27 3.42 49 6 37 45 1000
247 31 3.44 49 6 37 45 1000
247 48 4.03 49 6 37 45 1000
247 54.5 4.29 49 6 37 45 1000
247 72 5.11 49 6 37 45 1000
247 79 5.21 49 6 37 45 1000

 

117

TABLE D. 1 .2. Meat-based L. monocytogenes growth data.

 

 

 

 

Key to Time (h) Log Treat. No. pH T °C NaCl (g/l) Nitrite
ComBase (CFU/ml) (ppm)
J232_Lm 0 2.5 1 3.5 6 5 0
1232_Lm 96 3.1 l 3.5 6 5 0
J232_Lm 168 3.4 1 3.5 6 5 0
1232_Lm 264 4.9 1 3.5 6 5 0
J232_Lm 360 5.9 1 3.5 6 5 0
.1232_Lm 432 6.7 1 3.5 6 5 0
.1232_Lm 552 8 1 3.5 6 5 0
J232_Lm 600 8.4 1 3.5 6 5 0
.1232_Lm 696 9.2 1 3.5 6 5 0
1232_Lm 768 9.6 1 3.5 6 5 0
1232LLm 840 9.4 1 3.5 6 5 0
.1233_Lm 0 2.5 l 3.5 6 5 0
1233_Lm 96 2.7 l 3.5 6 5 0
J233_Lm 168 2.7 1 3.5 6 5 0
J233_Lm 264 2.5 1 3.5 6 5 0
.1233_Lm 360 4 1 3.5 6 5 0
J233_Lm 432 4.9 1 3.5 6 5 0
1233_Lm 552 6.4 1 3.5 6 5 0
.1233_Lm 600 6.9 1 3.5 6 5 0
1233_Lm 696 7.8 l 3.5 6 5 0
J233_Lm 840 9 1 3.5 6 5 0
J234_Lm 0 2.5 2 6.5 6 5 0
1234_Lm 96 5 .5 2 6.5 6 5 0
.1234_Lm 168 7.4 2 6.5 6 5 0
.1234_Lm 264 10 2 6.5 6 5 0
.1234_Lm 336 10.3 2 6.5 6 5 0
JZ3S_Lm 0 2.7 2 6.5 6 5 0
J235_Lm 96 3.9 2 6.5 6 5 0
J235_Lm 168 6.5 2 6.5 6 5 0
J23 5_Lm 264 7.7 2 6.5 6 5 0
1235_Lm 336 9.8 2 6.5 6 5 0
.1235_Lm 432 9.8 2 6.5 6 5 0
1235_Lm 504 10 2 6.5 6 5 0

 

118

TABLE D. 1 .2. Continuation.

 

 

Key to Time (h) Log Treat. No. pH T °C NaCl (g/l) Nitrite
ComBase @U/ml) (ppm)
.1236_Lm 0 2.5 3 10 6 5 0
.1236_Lm 72 4.2 3 10 6 5 0
.1236_Lm 120 6.4 3 10 6 5 0
J236_Lm 240 9.2 3 10 6 5 0
.1236_Lm 288 10.1 3 10 6 5 0
.1236_Lm 360 10.2 3 10 6 5 0
.123 7_Lm 0 2.4 3 10 6 5 0
.123 7_Lm 72 2.8 3 10 6 5 0
1237_Lm 120 4.5 3 10 6 5 0
.123 7_Lm 240 7.3 3 10 6 5 0
.123 7_Lm 264 9.1 3 10 6 5 0
1237_Lm 360 9.6 3 10 6 5 0
.123 7_Lm 480 9.5 3 10 6 5 0

 

119

D.2

STANDARD ERROR OF PREDICTION

AND ROBUSTNESS

 

 

 

120

INDEX

VALUES OBTAINED AFTER NON-SIGNIFICANT TERMS WERE
ELINIINATED
TABLE D.2
Without non- Without non-
Aerobic srgnrﬁcant terms srgnrﬁcant terms
SEP SEP
Data No log(CFU/ml) log(CFU/ml) R1 R1

1 1.8920 1. 1821 1.4015 0.8956
2 1.8193 1.2326 1.3476 0.9338
3 2.1808 0.9916 1.6154 0. 7512
4 2.4599 1.3240 1.8221 1.0030
5 0.2097 0.7492 0.1553 0.5676
6 0.6434 0.981 1 0. 4766 0.7433
7 1.4425 0.6687 1.0685 0.5066
8 0. 7040 1.0798 0.5215 0.8180
9 0. 9086 1 .9229 0. 6 731 1 .4568
10 1.0226 1.2218 0. 7575 0.9256
11 1.1478 1.3936 0.8502 1.0558
12 1.9816 0.8186 1.4678 0.6202
13 1.0640 1.4165 0. 7882 1.0731
14 1.1691 1.4804 0.8660 1.1215
15 0.9486 0.8692 0.7027 0.6585
16 1.1806 1.0934 0.8745 0.8284
17 2.5245 2.5351 1.8700 1.9205
18 1.5938 1.6042 1. 1806 1.2153
19 1.1776 0. 7699 0.8723 0.5833
20 1.5308 1.0991 1.1339 0.8326
21 1.8256 1.0491 1.3523 0. 7948
22 3.0180 2.2340 2.2356 1.6924
23 3.0185 2.2407 2.2360 1.6975

 

 

 

APPENDIX E

SCRIPTS USED FOR NONLINEAR REGRESSION AND
DATA ANALYSIS IN JMP

E. 1 . SCRIPT USED FOR NONLINEAR REGRESSION.

This script was applied to each data set in order to perform nonlinear regression,
with second derivative, and the results were given in a separate table:

“ columnS << set formula (Parameter({B =1, M =10, C = 5}, C * Exp(-Exp(-B "‘
( :Time - M)))));nlin=Non1inear(Y( :Name("Log(N/No)")), X( :Column 5), Second Deriv
Method(1),ﬁnish, plot ( 1), save estimates);
nParameters = n row(report(nlin)["Solution"] [table box(2)][1] << get as matrix);
errorTable << add row(l);
nRowsErrorTable = n row(errorTable);
colSSE[nRowsErrorTab1e] = report(nlin)["Solution"] [table box(1)][1][1];
colDFE[nRowsErrorTab1e] = report(nlin)[”Solution"] [table box(1)][2][1];
colMSE[nRowsErrorTable] = report(nlin)["Solution"] [table box(1)] [3][1];
colRMSE[nRowsErrorTab1e] = report(nlin)[" Solution"] [table box(1)][4][1];
for(i = 1, j <= nParameters, jH,
parameterTable << add row( 1 );
nRowsParamTable = n row(parameterTable);
colParameter[nRowsParamTab1e] = report(nlin) [" Solution"][table
b0X(2)][1]1j];
colEst[nRowsParamTable] = report(nlin)["Solution"][table box(2)][2][i];
colApproxStdErr[nRowsParamTab1e] = report(nlin)["Solution"] [table
b01(0)] [311i];
colLowerCL[nRowsParamTable] = report(nlin)[" Solution"][table
b0X(2)][4][i];
colUpperCL[nRowsParamTab1e] = report(nlin)["Solution"][table
b0X(2)][5][i]); “-

121

E.2. SCRIPT USED FOR NONLINEAR REGRESSION WITHOUT SECOND
DERIVATIVE.

“col=new column ("Model"); col<<set formula (Parameter({B = 0.009, M = 80},
:Log C[l] + (9.57 - :Log C[1]) “‘ Exp(—Exp(-B " ( :Name("Time (11)") - M)))))”.

E.3. SCRIPT USED FOR NONLINEAR REGRESSION WITH SECOND
DERIVATIVE.

“col=new column ("Model"); col<<set formula (Parameter({B = 0.02, M = 114},
:Log C[l] + (9.57 - :Log C[1]) “ Exp(-Exp(-B “' ( :Name("Time (H)") - M)))));
Nonlinear(Y( :Log C), X( :Model), Second Deriv Method(1));”.

E.4 SCRIPT USED FOR REITERATIVE ANALYSIS.

“for (i=1, i<=9, i++, columns << set formula (Parameter( {B = 0.02, M = 114, C =
5}, C * Exp(-Exp(-B "‘ ( :Time - M)))));nlin=Nonlinear(Y( :Name("Log(N/No)")), X(
:Column 5), Second Deriv Method(1),ﬁnish, plot (1), save estimates;report (nlin)
["Solution"][tab1e box (1)] << make into data table; report (nlin) ("Solution"][table box
(2)] << make into data table));”.

122

K.

 

 

 

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