MITIGATING ANTHOCYANINS AND COLOR DEGRADATION IN PASTEURIZED CRANBERRY JUICE FORTIFIED WITH VITAMIN C  By Sunisa Roidoung        A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of  Food Science  Doctor of Philosophy 2016    ABSTRACT MITIGATING ANTHOCYANINS AND COLOR DEGRADATION IN PASTEURIZED CRANBERRY JUICE FORTIFIED WITH VITAMIN C By Sunisa Roidoung Color degradation in cranberry juice during storage is the most common consumer complaint for this juice. To enhance nutritional quality, juice is typically fortified with vitamin C. Although vitamin C is an effective antioxidant, vitamin C fortification increases degradation of color in cranberry juice during storage. The color degradation is not only an appearance attribute, but also reflects the degradation of health beneficial components, anthocyanins (ACY), because ACY are natural pigments as well as antioxidant compounds. The overall goal of this study was to preserve endogenous ACY in cranberry juice with a feasible solution for the food industry. This study included two specific aims: 1) to evaluate the effectiveness of different antioxidants on ACY retention in cranberry juice and assess the effect on vitamin C retention, color intensity, and browning index (BI) during storage; and 2) to estimate the kinetic parameters and model predictive equations for color and ACY retention in cranberry juice during storage. Three natural phenolic compounds (hesperidin, catechin, and gallic acid) were tested for their protective effect against anthocyanins and color degradation. Cranberry juice was fortified with 40-80 mg/100 mL ascorbic acid and potential protective agents were added at different concentrations. The juice was then pasteurized at 85±2°C for 1 minute and stored in the dark at 23±2°C for 16 days.  Juice ACY, vitamin C, color intensity, and BI were evaluated at 2-day intervals. Among the three phenolic compounds, gallic acid showed the most effective protection against ACY degradation. Addition of gallic acid significantly increased red color intensity (37%) (p < 0.01) and ACY concentration (41%) (p < 0.03) during storage, compared to control juice samples.  At the end of 16-day storage,   the BI of gallic acid-added juice was significantly lower than that of the control juice (0.80 vs 1.00), confirming the protective effect of gallic acid on juice color. Therefore, the experimental data with gallic acid addition were used for the kinetic study, in order to develop predictive equations for the parameters and the dependent variables. Measurements of total monomeric anthocyanins and red color intensity were used to determine degradation rate constants (k values) and order of reaction (n) of ACY and color. Due to high correlation, k and n could not be estimated simultaneously. To overcome this difficulty, both n and k were held at different constant values in separate analyses to allow accurate estimation of each. Parameters n and k were modeled empirically as functions of vitamin C, and of vitamin C and gallic acid, respectively. Reaction order n ranged from 1.2 to 4.4, and decreased with increasing vitamin C concentration. The final models offer an effective tool that could be used for predicting ACYs and color retention in cranberry juice during storage. The outcome of this research not only provided a potential solution of using gallic acid to address color degradation in commercial cranberry juice, but also proposed models for predicting color and ACY retention.                  Copyright by SUNISA ROIDOUNG 2016           v    ACKNOWLEDGEMENTS  My journey to Doctor of Philosophy would never happen without an opportunity from department of Technology, Maha Sarakham University, Thailand. The department has awarded me with full financial scholarship from Royal Thai government for five consecutive years from 2011 to 2016. I also appreciate care from Office of Educational Affairs, Royal Thai Embassy, in Washington DC regarding  life and wellness during studying in USA. As an aspect of Ph.D. student at Michigan State University, I am blessed working with Dr. Kirk Dolan, my supervisor, who has shared motivation in both academic and life perspective. His expertise on parameter estimation has taught me the sophisticated method beyond my food science background to solve problems in food industries. My gratitude is going to my committee as well; Dr. Gale Strasburg, Dr. Leslie Bourquin, and Dr. Bradley Marks, for their knowledgeable guidance and a lot of patience in me. I also acknowledge Dr. Ferhan Ozadali for guiding me with a research interest in product quality in a food industry. He challenged me to work on the consumer complaint regarding color degradation in cranberry juice.  I am grateful for unconditional love from my family, especially Suttida Roidoung and Phanida Roidoung, who are always by my side and share every moment of my happiness, joy, and stress. Thanks are also to friends, colleagues, and staffs in Food Science and Human Nutrition department for all assistance throughout my graduate program. It is a delightful experience pursuing Ph.D. in a friendly and supportive atmosphere.   vi  TABLE OF CONTENTS  LIST OF TABLESviii LIST OF FIGURESx CHAPTER 1  1 INTRODUCTION   1 1.1 Overview of the dissertation  2 1.2 Statement of problem  3 1.3 Significance of the study 1.4 Objectives of the study  CHAPTER 2 .. 6 LITERATURE REVIEW  6  2.1 Cranberries  7  2.2 Anthocyanins as a natural colorant   2.3 Cranberry juice production  13   2.3.1 Vitamin C fortification  14   2.3.2 Juice pasteurization (Hot-filling versus aseptic technique)  17  2.4 Protective effect of polyphenols against ACY degradation  17   2.4.1 Hesperidin . 19   2.4.2 Catechin . 20   2.4.3 Gallic acid  21  2.5 Kinetics of ACY and color degradation... 22   2.5.1 Estimation of parameters  23   2.5.2 Scaled sensitivity coefficients  25 CHAPTER 3  OBJECTIVE ONE  Determination of protective effect of selected phenolic compounds against degradation  of anthocyanins and color in vitamin C-fortified cranberry juice .  26  3.1 Materials and Methods    3.1.1 Juice preparation  27   3.1.2 Juice pasteurization and storage  29 3.1.3 Anthocyanin content . 30 3.1.4 Color intensity and browning index (BI) . 31 3.1.5 L-ascorbic acid quantification .  31 3.1.6 Statistical analysis  32 3.2 Results and Discussion  32   3.2.1 Anthocyanins retention  32   3.2.2 Color intensity 5   3.2.3 Browning index    3.2.4 Vitamin C retention  vii   3.3 Conclusions  42  3.4 Limitations of the study  43 CHAPTER 4  44 OBJECTIVE TWO  44 Estimating kinetic parameters of anthocyanins and color degradation during storage  by using an inverse method of ordinary least squares  44  4.1 Materials and Methods 45   4.1.1 Mathematical Modeling  45    4.1.1.1 Estimation of the kinetic parameters  45    4.1.1.2 Parameter correlation  46    4.1.1.3 Error of parameters  47   4.1.2 Developing secondary models for parameters as a function           of vitamin C and gallic acid  48  4.2 Results and Discussion  49   4.2.1 Parameter estimation  49    4.2.1.1 Determination of parameter correlation  50    4.2.1.2 Determining error of parameters  56   4.2.2 Developing secondary models.... 57  4.3 Conclusions  66  4.4 Limitations of the study  67 CHAPTER 5  68 OBJECTIVE THREE  68 Demonstrating practical application of results for the use in food industry, especially juice processor  68  5.1 Introduction 69  5.2 Example case study#1 ... 71  5.3 Example case study#2  74  5.4 Conclusions ...... 77 CHAPTER 6 .. 79 OVERALL CONCLUSIONS AND FUTURE DIRECTIONS . 79  6.1 Overall concl. 80 6.2 Future directions ........ 81 APPENDICES ..... 83  APPENDIX A: Experimental measurements for red color intensity, anthocyanin,             and L-ascorbic acid content ... 84  APPENDIX B: Example MATLAB syntax and additional results for Chapter 4 ..... 97   APPENDIX C: MATLAB syntax for Chapter 5 .... 114 REFERENCES 118  viii  LIST OF TABLES  Table 2.1 Nutritional values of cranberries (per 100 g product)  Table 2.2 Chemical profile of American cranberry juice (Vaccinium macrocarpon Ait.)  9 Table 2.3 In vitro activity of cranberries against bacteria  Tabel 2.4 Summary of protective effect of polyphenol on stability of anthocyanins..... 18 Table 2.5 Examples of n-value for degradation kinetics of anthocyanins and color . 23 Table 4.1 Data analysis design for parameter estimation (k, C0, n) for color and anthocyanins       at different concentration of vitamin C and gallic acid (12 treatments*...  Table 4.2 Three parameter (k, C0, n) estimation for anthocyanins and color in cranberry       juice fortified with vitamin C (60 mg/100 mL) and gallic acid (80 mg/100 mL),       during 16-day storage a Table 4.3 Two parameter (k, C0) estimation for anthocyanins and color in cranberry       juice fortified with vitamin C (60 mg/100 mL) and gallic acid (80 mg/100 mL),        during 16-day storage at 23 °C  Table 4.4 Two parameter (n, C0) estimation for anthocyanins and color in cranberry       juice fortified with vitamin C (60 mg/100 mL) and gallic acid (80 mg/100 mL),       during 16-day storage at 23 °C  Table A1 Data of red color intensity, anthocyanins, and vitamin C in cranberry juice       at day 0 after pasteurizing at 85 °C, the juice was not fortified with vitamin C        Table A2 Data of red color intensity, anthocyanins, and vitamin C in cranberry juice       pasteurizing at 85 °C, the juice was not fortified with vitamin C and no        Table A3 Data of red color intensity in cranberry juice fortified with vitamin C       (4080 mg/100 mL) and gallic acid addition (0 Table A4 Data of anthocyanin content in cranberry juice fortified with vitamin C       (4080 mg/100 mL) and gallic acid addition (0 Table A5 Data of L-ascorbic acid content in cranberry juice fortified with vitamin C       (4080 mg/100 mL) and gallic acid addition (03  ix  Table B1 Summary of n-values with smallest RMSE of all 12 treatments for color and      anthocyanins ... 110 Table B2 The nACY and ncolor calculated from Eq. (4.5) and E Table B3 Two parameter (k, C0) estimation for anthocyanins in cranberry juice      fortified with vitamin C (40-80 mg/100 mL) and gallic acid (0-320 mg/100 mL),     during 16-day storage at 23 °C  110 Table B4 Two parameter (k, C0) estimation for color in cranberry juice      fortified with vitamin C (40-80 mg/100 mL) and gallic acid (0-320 mg/100 mL),      during 16-day storage at 23 °C  112 Table B5 Model comparison using p-value  113                 x  LIST OF FIGURES  Figure 2.1 Anthocyanins formation through glycosylation of anthocyanidins 12 Figure 2.2 Process flow diagram for juice production . 14 Figure 2.3 Oxidation mechanism of L-ascorbic acid . Figure 2.4 Pro-oxidant effect of ascorbic . 16 Figure 2.5 Chemical structures of hesperidin (a) and hesperetin (b) . 19 Figure 2.6 Chemical structure of catechin (a) and  20 Figure 2.7 Chemical structure of gallic acid . 21 Figure 3.1 Diagram of sample preparation with addition of vitamin C and selected        phenolic compounds Figure 3.2 Juice pasteurization in water bath equipped with a shaker, and a hand-held         thermometer for monitoring temperature of water bath and cranberry juice Figure 3.3 Average anthocyanin content during 16-day storage, at 23 °C.         Treatment levels per 100 mL were: vitamin C fortification (80 mg)         without adding antioxidant compounds (Ctrl); Hesperidin Trt-1 (5 mg),         Trt-2 (9 mg), and Trt-3 (18 mg); Catechin Trt-1 (5 mg) and Trt-2 (15 mg);         and Gallic Acid Trt-1 (80 mg), Trt-2 (160 mg), and Trt-3 (320 mg). .... 33   Figure 3.4 Effect of gallic acid (0-320 mg/100 mL) on the average anthocyanin content         of cranberry juice, fortified with 40-80 mg/100 mL vitamin C,         during 16-days storage at 23 °C  34 Figure 3.5 Average red color intensity during 16-day storage, at 23 °C.        Treatment levels per 100 mL were: vitamin C fortification (80 mg)              without adding antioxidant compounds (Ctrl); Hesperidin Trt-1 (5 mg),        Trt-2 (9 mg), and Trt-3 (18 mg); Catechin Trt-1 (5 mg) and Trt-2 (15 mg);  and Gallic Acid Trt-1 (80 mg), Trt-2 (160 mg), and Trt-3 (320 mg)  Figure 3.6 Effect of gallic acid (0-320 mg/100 mL) on the average color intensity of         cranberry juice, fortified with 40-80 mg/100 mL vitamin C,         during 16-days storage at 23 °C    xi  Figure 3.7 Effect of antioxidants on the browning index of cranberry juice         fortified with vitamin C (80 mg/100 mL) during 16-day storage at 23 °C.          Treatment levels per 100 mL: Catechin Trt-1 (5 mg) and Trt-2 (15 mg);         and Gallic Acid Trt-1 (80 mg), Trt-2 (160 mg), and Trt-3 (320 mg).         Standard deviation of browning index varied from 0.00016 to 0.053 .. 39 Figure 3.8 Effect of gallic acid (0-320 mg/100 mL) on browning index of cranberry juice,         fortified with 40-80 mg vitamin C, during 16-day storage at 23 °C .. 39   Figure 3.9 Effect of gallic acid (0-320 mg/100 mL) on the vitamin C content of cranberry juice,         fortified with 40-80 mg vitamin C, during 16 day storage at 23 °C . 41 Figure 4.1 Representative SSC plots from three-parameter estimation of anthocyanins (A)         and color (B) in cranberry juice fortified with vitamin C (60 mg/100 mL)         and gallic acid (80 mg/100 mL), during 16 day storage at 23 °C  51  Figure 4.2 Representative plots of RMSE versus n (reaction order) of anthocyanins (A)         and color (B) in cranberry juice fortified with vitamin C (40, 60, 80 mg/100 mL)         and gallic acid (0 mg/100 mL), during 16-day storage at 23 °C ... 54  Figure 4.3 Effect of gallic acid (0-320 mg/100 mL) on order of degradation reaction         of anthocyanins (nACY) and color (ncolor) in cranberry juice,         fortified with 40-80 mg/100 mL vitamin C, during 16-day storage at 23 °C ... 55  Figure 4.4 Degradation rate constant of anthocyanins (A) and color (B)         in cranberry juice fortified with vitamin C (4080 mg/100 mL) and         gallic acid (0320 mg/100 mL),  during 16-day storage at 23 °C.. 59  Figure 4.5 The 3D plots of logkACY and logkcolor from experiment ()         and predictive surface calculated from Eq. 4.11 (R2adjusted = 0.9994)         and Eq. 4.12 (R2adjusted = 0.9948), respectively ..... 61 Figure 4.6 Representative model fittings of anthocyanins (A) and color (B) retention         in cranberry juice fortified with vitamin C (60 mg/100 mL) and                gallic acid (0320 mg/100 mL), during 16-day storage at 23 °C,         while GA and Pred refer to gallic acid, and predicted values, respectively.... 62 Figure 4.7 Representative residual plots showing difference between observed and         predicted values of anthocyanins (A) and color (B) in cranberry juice         fortified with vitamin C (60 mg/ 100 ml) and gallic acid (80 mg/100 mL),         during 16-day storage at 23 °C .............................................................................. 64      xii   Figure 4.8 Representative histograms, plotted by dfittool in MATLAB,         showing normal distribution of residuals in prediction of anthocyanins (A)         and color (B) retention in cranberry juice fortified with vitamin C (60 mg/100 mL)             and gallic acid (80 mg/100 mL), during 16-day storage at 23 °C ......................... 65  Figure 5.1 Retention of color (A) and anthocyanin (B) in pasteurized cranberry juice after        storage at 23 °C, 16 days, with addition of vitamin C (4080 mg/100 mL)        and gallic acid (0320 mg/100 mL)  78 Figure B1 Plot of RMSE versus n (reaction order) of anthocyanins (A) and color (B)        in cranberry juice with vitamin C (4060 mg/100 mL) and gallic acid        (80 mg/100 mL), during 16-.. 107 Figure B2 Plot of RMSE versus n (reaction order) of anthocyanins (A) and color (B)        in cranberry juice with vitamin C (4060 mg/100 mL) and gallic acid        (160 mg/100 mL), during 16-8 Figure B3 Plot of RMSE versus n (reaction order) of anthocyanins (A) and color (B)        in cranberry juice with vitamin C (4060 mg/100 mL) and gallic acid        (320 mg/100 mL), during 16-day storage at 23 9       1        CHAPTER 1 INTRODUCTION             2   1.1 Overview of the dissertation Cranberry products are widely consumed due to their potential health benefits. American cranberries (Vaccinium macrocarpon) are a good source of phytochemicals, which potentially provide health benefits not only in reducing risks of cancer and heart disease, but also in protecting -induced obesity, insulin resistance, and intestinal inflammation (Anhe and others 2015; Howell 2007; Zafra-Stone and others 2007). Cranberries have also gained interest among patients with urinary tract infections. Proanthocyanidins in cranberries are active compounds that help preventing bacteria adhesion to the urothelial cell wall (Mansour and others 2014; Durham and others 2015; Foxman and others 2015). Due to these benefits, American cranberries are nowadays consumed not only during traditional holidays, but also throughout the year as a regular diet in different forms (e.g., juice, sauce, dried fruits).  Commercial juice products are commonly fortified with vitamin C (ascorbic acid) in order to extend shelf-life and enhance nutritional quality. Mostly, the fortification levels follow dietary guideline for vitamin C daily intake; 75 (women) and 90 (men) mg/day (NIH, 2016). As a result of fortification, manufacturers claim 100% vitamin C of recommended daily intake on the product label. However, vitamin C fortification could cause anthocyanin (ACY) degradation via oxidation. Cranberry ACYs are natural pigments that provide a bright red color to cranberry juice, and also are well-known for their antimicrobial and antioxidant functionalities. The ACY degradation, besides lowering antioxidant compounds, results in an unpleasant brownish color in juice products, which is a major consumer complaint.  The detrimental effect of vitamin C on ACY is well-known, it is a common commercial practice to fortify fruit juices with vitamin C, which is added either to prevent oxidation in juices, . 3   Attempts to overcome ACY degradation resulting from vitamin C fortification have been previously researched. Since oxidation plays a role in ACY and color degradation, polyphenols, as natural antioxidants, have been studied for their protective effect. The application of polyphenols in juices is rather limited due to their relatively low water solubility. Encapsulation or ACY structure modification has been suggested in order to improve stability of bioactive compounds (Matsufuji and others 2007). However, these methods may increase the cost of juice production, and hence could increase the retail price.  In fact, among the tremendous variety of phenolic compounds in nature, there are some compounds that are soluble in water (e.g., catechin and gallic acid) (Srinivas and others 2010b), or the water solubility substantially increases at elevated temperature (e.g., hesperetin) (Liu and Chen 2008). These compounds had never been investigated for their antioxidant effect in juices with fortified vitamin C. Therefore, it is desirable to investigate these potential compounds, which could give a feasible solution to the juice industry and, hence, avoid adding artificial colorants such as Red #40 to maintain the color.  1.2 Statement of problem 1. There are no reports of applicable phenolic compounds that could be practically applied in the commercial juice industry to overcome color degradation in vitamin C-fortified pigment-rich juices. 2. The effect of vitamin C fortification on degradation kinetics of ACY and color in juices is rarely discussed in most studies. 4   3. There are no publications that have reported the use of advanced parameter estimation techniques to determine degradation rate constant (k value) and order of degradation reaction (n value) for ACY and color in vitamin C-fortified juices.  1.3 Significance of the study 1. This work investigated natural phenolic compounds for their potential protective effect against ACY and color degradation in cranberry juice fortified with vitamin C. The measurements of ACY, color intensity, vitamin C retention, and browning index during storage were used to determine the efficacy of phenolic compounds. The data will help explain the antioxidant mechanism of phenolic compounds in protecting ACY degradation. 2. Degradation kinetics of anthocyanins and color in cranberry juice containing different concentrations of vitamin C and phenolic compounds were determined by using a nonlinear inverse method with ordinary least squares, and the effect of vitamin C levels on ACY and color degradation was determined in this study. Moreover, the predictive equations developed in this study can serve as a guideline for process design in industry.  1.4 Objectives of the study  1. To determine the ability of natural antioxidant compounds (hesperidin, catechin, and gallic acid) as protective agents against ACY and color degradation in pasteurized cranberry juice fortified with vitamin C. 2. To estimate the kinetic parameters (k, C0, n) of ACY and of color degradation during storage using an inverse method of ordinary least squares and to develop predictive equations for color and ACY in cranberry juice during storage. 5   3. To demonstrate practical applications of results from objective 2 for the use in the food industry, especially by juice processors.                   6        CHAPTER 2 LITERATURE REVIEW              7   2.1 Cranberries Cranberries (Family: Ericaceae) are low bush plants, growing in fresh water and mountain soil with layers of sand, peat, gravel, and clay. There are two types of cranberries, American cranberry (Vaccinium macrocarpon Ait.) and European cranberry (Vaccinium oxycoccus L.). European cranberries are grown in Finland and Germany. Although the European variety possesses an anthocyanin profile similar to the American variety, the fruits of European cranberries are smaller in size and contain different levels of malic, citric, and quinic acid (Cape Cod Cranberry Grow, 2016). American cranberries have been more widely consumed more than European cranberries, and hence have been extensively studied. Native Americans consumed cranberries as fresh and dried fruits. Owing to the large amount of benzoic acid in cranberries, the dried cranberries were used as a natural food preservative as well (Davidson, 1999). The American cranberries are mostly cultivated in North America (Wisconsin, Massachusetts, New Jersey, Oregon, Washington) during April to November. The majority of cranberries are harvested between September and October. Nutrient compositions of cranberry products are shown in Table 2.1. Cranberries contain a complex mixture of organic acids (~14 types, i.e., ferulic acid, vanillic acid, caffeic acid, benzoic acid), and flavonoids (~22 types; i.e., catechin, quercetin, myricetin, kaempferol, prunin), with quercetin and myricetin predominating (Guay 2009; Pappas and others 2009). The chemical profile of cranberry juice was evaluated by Hummer and others (2014) as shown in Table 2.2. Polyphenols in cranberries, especially anthocyanins and proanthocyanidins, provide natural defensive functions in plants against microbes, which believe to reduce risks of cancer and heart disease, to -induced obesity, insulin resistance, and intestinal inflammation (Anhe and others 2015; Howell 2007).  8   Table 2.1 Nutritional values of cranberries (per 100 g product)  Nutrient Cranberry product Frozena Concentrate Sweetened/Driedb Flavored SDCc Powder Calories (kcal) 48 198 298367 337342 360 Saturated fat (g) 0 0 0 0 0 Cholesterol (mg) 0 0 0 0 0 Sodium (mg) 3 14 34 23 29 Potassium (mg) 73 500 4090 11 734 Sugar (g) 4 22 6469 6768 37 Total Carbohydrate (g)  10  49  8288  8384  89 Dietary fiber (g) 4 <0.5 69 56 6 Protein (g) 0.6 <0.5 <0.5 <0.5 <0.5 Vitamin A (IUd) 0 0 70e 16,200f 0 Vitamin C (mg) 18 58 0 1 5 Calcium (mg) 10 39 1018 4 184 Iron (mg) 0.6 1.7 0.5 0 4 Source: Girard and Sinha (2012) aWhole or sliced. bRegular, soft and moist, and glycerated forms. cOrange, blueberry, cherry, or raspberry flavored sweetened dried cranberries (SDCs). dAs provitamin A. eValue for glycated forms of sweetened dried cranberries. fValue for orange flavored sweetened dried cranberries      9   Table 2.2 Chemical profile1 of American cranberry juice (Vaccinium macrocarpon Ait.)  Parameters Content ºBrix 9.58 pH 2.7 Titratable acidity (as citric) (g kg1) 0.25 Sugar (g kg1)      Glucose      Fructose      Sucrose  0.352 0.108 0.002 Anthocyanins (mg kg1)      Cyanidin-3-galactoside      Cyanidin-3-glucoside      Cyanidin-3-arabinoside      Peonidin-3-galactoside      Peonidin-3-glucoside      Peonidin-3-arabinoside  27.1 1.3 14.9 38.5 4.4 13.0 Antioxidants      Vitamin C (mg/kg)      ORAC (µmol L1 Trolox equivalent kg1)      FRAP (µmol L1 Trolox equivalent kg1)      Total Phenolics (mg gallic acid equivalent kg1)  0.930 86.5 107.6 12.8 1Chemical analysis was done in 2009, while the results were published in 2014.  Source: Adapted from Hummer and others (2014)  In addition, cranberry extracts have been studied as a potential treatment for urinary tract infections (UTIs) in humans (Mansour, 2014; Durham 2015; Foxman 2015). Studies conducted over a five-year period (20062010) examining anti-bacterial properties in cranberries were summarized by Hisano and others (2012), as shown in Table 2.3. D-mannose and distinctive containing A-type proanthocyanidins in cranberries have been reported to exhibit significant anti-adhesion effects on P-fimbriated Escherichia coli bacteria to bladder cell receptors (Blumberg and others 2013; Han and others 2010; Hummer and others 2014; Tao and others 2011). Urinary tract infection is a common disease among sexually active women (2055 year old) due to short urethras 10   that allow bacteria adhering to bladder cell wall (Foxman 2003). It is a long-term and recurrent disease, which could cause economic burden in UTI patients.  Cranberries have been used as an herbal medicine to treat urinary tract infections. Patients with UTIs take dry cranberry juice extract at a dose of 500 to 2,000 mg, 3 times a day, to reduce bacterial adhesion to uroepithelial cells (Jeske 2014). However, mild side effects (stomach upset, diarrhea, and occasional allergy) were reported regarding the use of cranberry concentrate. 11   Table 2.3 In vitro activity of cranberries against bacteria.  Study Study design and patients (N) Cranberry preparation Micro-organism (s) and Results Pinzon-Arango et al., (2009) In vitro bacteria cultured in medium and human uroepithelial cell culture PAC of 0, 64, 128 and 345.8 µg/ml E.coli: Anti-adhesion (E. coli) from 50.2 to 7.9 bacteria/cell (p<0.01); dose dependence effect Lee et al., (2010) In vitro urine activity after cranberry consumption in volunteers.  (16 women, 4 men); phase  275 mg of whole, dry cranberries and 25 mg of concentrated, dry cranberries E. coli, K. pneumonia and C. albicans: Phase 1: anti-adhesion activity in 35% (E. coli), 65% (K. pneumoniae) and 45% (C. albicans). Phase 2: anti-adhesion activity in 23% (E. coli), 33% (C. albicans) and 67% (K. pneumoniae). Lavigne et al., (2007) In vitro urine activity after cranberry consumption in volunteers with crossover.  36 mg cranberry capsules of; daily dosage was 36 or 108 mg or placebo E. coli: Anti-adhesion activity (p<0.001). Dose dependence effect Gupta et al. (2007) In vitro anti-adhesion activity against bladder and vaginal epithelial cells Cranberry capsule with 2.7 mg of PAC diluted from 0 to 75 µg/ml E. coli: Anti-adhesion activity of PAC against E. coli from 6.9 to 2.2 and 1.6 bacteria/cell following PAC at 0, 25 and 50 µg/ml, respectively (p<0.001). Howell et al., (2010) Multicentric, randomized, double-blind in vitro urine activity after cranberry consumption in volunteers. females Cranberry capsule of 0, 18, 36 or 72 mg of PAC E. coli:  Anti-adhesion activity increasing with the amount of PAC. Virulence was also reduced with PAC in a dose-dependent fashion. Valentova et al., (2007) Double-blind, placebo-controlled in vitro urine. 200 mg.  400 mg or 1200 mg per day of dried cranberry juice S. aureus, E. faecalis, E. coli, P. aeruginosa, E. faecium and K. pneumoniae:  Anti-adhesion activity in a dose-dependent fashion (p<0.05); highest activity observed against P. aeruginosa. Di Martino et al., (2006) Double-blind, randomized, placebo-controlled in vitro urine activity after cranberry consumption in volunteers. males, 10 females) 250 or 750 ml of 27% cranberry juice E. coli:  Dose-dependent decreases in bacterial adhesion to human epithelial cell line of 45% and 62% for 250 and 750 ml of cranberry juice, respectively (p<0.05), independent of antibiotic resistance. Source: Hisano and others (2012) 12   2.2 Anthocyanins as a natural colorant Apart from antimicrobial and antioxidant properties, anthocyanins (ACY) provide natural red, purple, and blue colors in fruits, leaves, or flowers (McGhie and Walton 2007). ACY change colors according to pH, thus ACY are somewhat a pH indicator. ACY are glycosylation of anthocyanidins, which known as color pigments. There are six types of anthocyanidins; cyanidin, delphinidin, malvidin, peonidin, petunidin, and pelargonidin. The anthocyanidins basically compose of 15 carbons of 2 aromatic rings (ring A, ring B) and one heterocyclic ring (ring C), as shown in Figure 2.1. In nature, anthocyanidins are not commonly found by themselves, but rather presented as ACY by glycosylation on the 3-carbon of the C-ring through an O-linkage (Glover and Martin 2012). Major ACY in cranberry juice are 3-O-galactosides and 3-O-arabinosides of cyanidin and peonidin. Monomeric ACY are major compounds that provide bright red characteristic. However, in a solution with high ACY concentration, self-aggregation of monomeric ACY could occur and results in polymeric ACY. The polymeric ACY contribute deep brownish-red color (Pappas and Schaich 2009; Brownmiller and others 2008), which is a favorable appearance in red wine (Pina and others 2015; He and others 2012).     Figure 2.1 Anthocyanins formation through glycosylation of anthocyanidins. 13   ACY could switch among colors (red purple blue) regard to alteration of OH group position in anthocyanidins. In contrast, color degradation in juice occurs via reactions that detach sugars (deglycosylation) from ACY structures. The ACY then degrade to chalcone, which is a colorless phenolic compound (Sun and others 2011; Sadilova and others 2007). ACY are water-soluble and susceptible to degradation via oxidation by oxygen, light, and high temperatures. White and others (2011) reported that heat degraded cranberry ACY, and resulted in flavonol aglycone formation (quercetin and myricetin) that could impact color of the product.  2.3 Cranberry juice production Due to the seasonal cultivation of cranberries from April to November, commercial cranberry juices are typically produced from juice concentrate rather than fresh fruits. Upon harvesting, fresh cranberries are ground and macerated with pectinase and /or cellulase enzyme to extract juice. The juice is then concentrated by mild evaporation, membrane filtration or reverse osmosis (Brownmiller and others 2008). The standard cranberry concentrate is 50 °Brix (Girard and Sinha, 2012). Pasteurization could be applied to juice concentrate to prevent microbial growth during storage. Even though cranberry fruits are a good source of vitamin C, cranberry juice is rarely consumed fresh owing to its extreme tart flavor. Thus, cranberry juice products are mostly diluted from juice concentrate with water, and sweetened, or blended with other juices to overcome the tart flavor. Juice production is illustrated as a flow diagram in Figure 2.2.  14    Figure 2.2 Process flow diagram for juice production. (Source: Alfa Laval membrane filtration)  2.3.1 Vitamin C fortification Vitamin C, also known as ascorbic acid, is an effective antioxidant that prevents cell damage caused by free radicals. Vitamin C is also crucial for protein synthesis, wound healing, bone growth, and mineral absorption. Vitamin C is an essential nutrient that cannot be synthesized or stored in the human body. Therefore, it is necessary to include vitamin C in the daily diet. Examples of vitamin C-rich foods are citrus fruits and juices, berries, cantaloupe, broccoli, spinach, peppers. In addition, vitamin C is widely fortified in many food products in order to extend product shelf-life and enhance nutritional levels.  15   L-ascorbic acid is the form of vitamin C found in nature, which is easily converted to dehydroascorbic acid (DHA) upon oxidation. However, DHA can revert back to L-ascorbic acid with the addition of two hydrogen atoms. Meanwhile, further hydrolysis reaction will result in irreversible foundation of diketogulonic acid. Oxidation mechanism of L-ascorbic acid is shown in Figure 2.3.   Figure 2.3 Oxidation mechanism of L-ascorbic acid (Source: Levine and others, 1996)  Despite being a potent antioxidant, fortification of vitamin C in pigment-rich juices has been found to induce free radical formation and accelerate color and ACY degradation (Starr and Francis 1968; Remini and others 2015; Li and others 2014; Choi and others 2002).  The mechanism of ACY degradation as a result of vitamin C fortification was explained via two pathways (Ozkan and others 2005; Poei-Langston and Wrolstad 1981; Garcia-Viguera and Bridle 1999; Sun and others 2011):  16   1) Condensation of ascorbyl radical: Anion of ascorbyl radical could react with flavylium cation at carbon-4 of ACY. The reaction damages ACY by cleaving the structure. 2) Hydrogen peroxide formation during oxidation of ascorbic acid: Hydrogen atom donation from ascorbic acid could result in a formation of hydrogen peroxide, which suddenly decomposes to highly reactive compounds; peroxyl anion (HOO), perhydroxyl radical (HOO), and hydroxyl radicals (HO). Among these three, HO is the main reactive specie to cleave benzene ring in ACY. It is also known that high doses of vitamin C can act as a pro-oxidant, which induces more free radical formation (Paolini and others 1999). At high doses of vitamin C, another pathway of Figure 2.4), where ascorbate reduces metal ions in juice, such as Fe3+ (Rietjens and others 2002)reaction, Fe2+ is oxidized to Fe3+ by hydrogen peroxide (H2O2) and resulted in a hydroxyl radical ion (OH). Meanwhile, another molecule of H2O2 reduces Fe3+ to Fe2+ and +).   Figure 2.4 Pro-oxidant effect of ascorbic   17   2.3.2 Juice pasteurization (Hot-filling versus aseptic process) Cranberry juices are available in grocery stores across the United States. Ready-to-drink juices are pasteurized to extend their shelf-life, as pasteurization is an effective method to inhibit growth of spoilage microorganisms. In the food industry, juice pasteurization can be done by two techniques; hot fill or aseptic processing. Hot filling is a low cost operation compared to the aseptic technique. In hot filling, the juice is rapidly heated to 8595°C in a heat exchanger and held at that temperature to assure product safety from microbial growth, then hot-filled into containers. Cooling is done immediately afterward. However, temperature of the juice does not decrease rapidly, thus the heat remaining in the juice during cooling could cause damage to the and color. In contrast, the aseptic method comes with modern system controls that provide rapid continuous heat and cooling before packaging. The processes are also handled under sterile conditions, hence the operational cost of the aseptic system could be more than US $1,000,000 (Bates and others 2001). Therefore, hot-fill processing is mostly installed in small- and medium-scale juice businesses, which often encounter quality losses.    2.4 Protective effect of polyphenols against ACY degradation Free radicals are produced in the presence of oxygen, radiation, sunlight, or pollution.  The phenol group (or OH group) in phenolic compounds helps prevent degradation by scavenging free radicals. Flavonols are a subclass of polyphenols, and exhibit a powerful radical scavenging by donating H-atoms from their phenol groups to the radicals. Therefore, the free radicals in the food system could not attack health-beneficial components, such as ACY. Quercetin is the most abundant flavonol distributed in many plants, and hence it is the focus of interest among researchers. An early study in 1974, Shrikhan and Francis (1974) found that quercetin helped 18   decrease degradation of cranberry ACY in the  juice fortified with vitamin C. However, the authors reported the limitation of quercetin as it precipitated out immediately after pasteurization. Protective effects of different polyphenols on ACY degradation are summarized in Table 2.4.   Table 2.4 Summary of protective effect of polyphenol on stability of anthocyanins. Study Polyphenols Degradation factors Blackcurrant juice (Clegg and Morton 1968)  Quercetin Quercitrin Flavonol aglycone Ascorbic acid Cranberry juice (Shrikhan and Francis 1974) Quercetin Quercitrin Ascorbic acid Strawberry and blackcurrant juice (Skrede and others 1992) Strawberry flavonoid extract Ascorbic acid Grape juice (Brenes and others 2005) Rosemary extract Ascorbic acid Blood orange juice (Cao and others 2009)  Hesperidin Narirutin Naringin Neohesperidin Ascorbic acid  Glucose Sucrose Fructose Grape skin anthocyanins (Yan and others 2013) Enzymatically modified isoquercitrin Heat and light Plum juice (Hernandez-Herrero and Frutos 2015) Rutin Ascorbic acid  Overall, polyphenols exhibited a protective effect on ACY against degradation caused by ascorbic acid, sugar, heat, and light. Therefore, phenolic compounds are potential agents to protect ACY from degradation via scavenging activity, and copigmentation, which occurs via chelation between ACY and other non-colored phenolic compounds (Hernandez-Herrero and Frutos 2015; Boulton 2001). Although the studies by Brenes and others (2005) and Cao and others (2009), as shown in Table 2.4, were done by pasteurizing juices, the process durations were at 30 minutes, and 3 hours, respectively, which are not a regular practice for commercially pasteurized juices. 19   2.4.1 Hesperidin A common limitation of most polyphenols is low water solubility, which limits their use in juice processing. Thus, water-soluble phenolic compounds were considered in this study. Liu and Chen (2008) reported an increase of water solubility in hesperetin with increasing temperature (15 to 50 °C). Thus, hesperetin possibly dissolves in aqueous solutions under pasteurization conditions (85 °C). Hesperidin is mostly found in nature rather than hesperetin, since the glycosylated hesperidin is a flavonoid in citrus fruits (Figure 2.5). The natural form of flavonoids are mostly glycosylated; sugar linkage. Glycosides help increase water solubility of the flavonoids. Hesperidin is a flavonol-diglycoside, which is composed of rhamnose and glucose. The aglycone form (no sugar) of hesperidin is then called hesperetin. Therefore, hesperidin could be a potential water soluble polyphenol that might be used in juice processing.    (a)     (b) Figure 2.5 Chemical structures of hesperidin (a) and hesperetin (b)    20   2.4.2 Catechin Catechin is similar to quercetin, in both natural availability and chemical structure (Figure 2.6). Therefore, catechin is also a potent antioxidant, and has been extensively studied for its copigmentation ability to stabilize ACY (Hidalgo and others 2010; Gordillo and others 2015; Kopjar and Pilizota 2009). Unlike quercetin, catechin possesses higher water solubility (2.26 g/L at 25°C (Srinivas and others 2010b), than quercetin (0.00215 g/L at 25°C, (Srinivas and others 2010a), which makes catechin more feasible to apply directly in juiceproduction. Although structures of catechin and quercetin look very similar (Figure 2.6), the difference in structural orientations could possibly affect chemical properties, including water solubility of these two compounds. The stick notation represents plane bonding, wedged notation means one bond is coming out of the plane, and dashed line shows the bond is going below the plane.              Figure 2.6. Chemical structure of catechin (a) and quercetin (b).    21   2.4.3 Gallic acid Gallic acid, an organic acid with a benzene ring as a core structure (Figure 2.7), is a phenolic compound, which is commonly available in many plant parts such as leaf, bark, wood, root, fruit, and seed. Besides antifungal and antiviral properties, gallic acid exhibits remarkable antioxidant activities. Gallic acid has been reported to possess anti-allergic, anti-inflammatory, anti-mutagenic and anti-carcinogenic activities (Gali and others 1992). Gallic acid is not only a potent water-soluble antioxidant, but also exhibits effective antioxidant in emulsions (Cholbi and others 1991). Gallic acid has been studied in many applications, including juice production due to sweetness inducer.  The evidence supporting feasibility and benefits of using gallic acid in commercial juice production is well documented. First of all, gallic acid is soluble in water (up to 20 g/L) and considered a non-toxic substance to humans (Rajalakshmi and others 2001), and humans seem to be able to absorb gallic acid better than polyphenols (Daglia and others 2014). There is no upper limit of gallic acid usage in the diet, as it is safe (Rajalakshmi and others 2001). Secondly, food grade gallic acid is relatively inexpensive ($2629/kg), as compared some other antioxidants. Lastly, gallic acid induces long-lasting and non-caloric sweetness, and a mildly sour taste is developed at concentrations greater than 0.05 M, or 850 mg/100 mL (Srinivas and others 2010b; Verhagen and others 2002).  Figure 2.7. Chemical structure of gallic acid 22   2.5 Kinetics of ACY and color degradation Kinetics are often studied to determine food quality and food safety during processing or storage. Degradation kinetics are also a basic requirement for shelf-life prediction. The nth-order degradation kinetic model, Eq. (2.1), consists of three parameters: degradation rate constant (k, [conc]1-n time-1), initial concentration (C0, conc), and reaction order (n, dimensionless). Concentration at any time t is given by C(t). The n value is mostly varied in a range from 0 to 2, and typically assumed to be 0 or 1 for nutritional degradation reactions, including degradation of color and ACY (Ozkan and others 2005; Wang and Xu 2007).     (2.1) In addition, Peleg and others (2015) reported experimental evidence that a 1st-order reaction represented thermal degradation of ACY at various studied conditions, i.e., processing and storage.  Examples of n-order determination in previous studies are presented in Table 2.5. As summarized in Table 2.5, n = 1 has been widely reported and assigned in many studies, thus most researchers assumed 1st-order without proving it from the data. According to the literature review, determining actual n value of degradation reaction is not a common practice among researchers, even though n could play significant impact on k and C0 values. When n = 0 or 1, the integrated kinetic model (Eq. (2.1)) becomes a linear model (n = 0) or can be log-transformed to a linear model (n = 1), hence it is easy to estimate parameters k and C0 by linear regression. Some studies, without initially assigning n = 1, determined the most suitable n using a nonlinear method by fixing n at 0, 0.5, 1, and 2 (Remini and others 2015; Wibowo and others 2015), or varying n from 1 to 1.8 in 0.05 increments (Buckow and others 2010), then chose the n that gave the lowest 23   Root Mean Square Error (RMSE) between predicted and experimental values to represent the reactions.   Table 2.5 Examples of n-value for degradation kinetics of anthocyanins and color  Kinetics n-order n determining method Reference Anthocyanins: Pomegranate juice  Strawberry juice  Cherry nectar Blackberry juice Blueberry juice Blueberries  1 1  1 1 1.4 1  assigned 1st order assigned 1st order  assigned 1st order assigned 1st order varied from 1 to 1.8 assigned 1st order  Ozkan and others (2005) Ozkan and others (2005); Garzon and Wrolstad (2002) Ozkan and others (2005) Wang and Xu (2007) Buckow and others (2010) Martynenko and Chen (2016) Color: Blood orange juice Chestnuts Bayberry juice Apple slices  1 1 1 0 and 1  compared 0, 0.5, 1, 2 compared 0, and 1 assigned 1st order compared 0, and 1  Remini and others (2015) Hou and others (2015) Guangming and others (2016) Qian-yu and others (2015)  2.5.1 Estimation of parameters One of the most common methods of estimating kinetic parameters (k, C0, n) is fixing n as an integer (0, 1, and 2), and then estimating parameters k and C0 through curve-fitting or optimization. However, this approach does not provide sufficient statistical information that could interpret physical meaning of estimated parameters, especially parameter errors (van Boekel 1996). Unlike curve-fitting or optimization, parameter estimation considers parameter errors, scaled sensitivity coefficients, the sensitivity matrix, and confidence intervals (CIs), which inform whether parameters are accurate, can be estimated, are correlated, or are significantly different 24   from zero and can be removed from model (Dolan and Mishra 2013).  Linear and nonlinear models can be evaluated by examining the sensitivity coefficient (Xi) (Eq. (2.2)):   (2.2) where  is the dependent variable, and are the true values of the parameters, and the ith parameter is i.  The model is linear if all the sensitivity coefficients (Xi) are not a functions of any parameter(s) i. For example, the explicit form of the kinetic model (Eq. (2.1)) is , the first derivatives of are a function of parameter n, k, and C0, respectively. Therefore, the degradation kinetic model is a nonlinear model. Due to the complication of a non-linear method, non-linear equations are generally transformed to linear models. The transformation impacts error structure, which could result in incorrect estimated parameters (Chowdhury and Das Saha 2011). Therefore, in recent years, non-linear methods have gained interest in food quality research (Wibowo and others 2015).  In order to estimate parameters in nonlinear models, an initial guess of parameter(s) and iteration are required. Solver® in Excel can also be used to estimate parameters in nonlinear models from the nonlinear algorithm, while the sensitivity matrix X (Eq. (2.3)) is needed to compute errors of parameters by matrix multiplication (Dolan 2003).  25        (2.3) where X is an n-by-p matrix of the sensitivity coefficients; n is number of data; p is number of parameters.  2.5.2 Scaled sensitivity coefficients  Scaled sensitivity coefficients, SSC  are a useful statistical criteria because they not only provide information regarding the ease and accuracy of estimating parameters, but also illustrate parameter correlation. The  i by its Xi (Eq. (2.2)). The parameter with large means the small change of parameter creates a large response, thus such parameter can be estimated easily with a small relative error. The estimation of a parameter with a small size of would be very difficult, and hence could give huge error. Correlation of parameters is determined by identical shape of  plots, and verified by a constant ratio of  , meaning those correlated parameters cannot be estimated simultaneously (Dolan and Mishra 2013).       26        CHAPTER 3 OBJECTIVE ONE Determination of protective effect of selected phenolic compounds against degradation of anthocyanins and color in vitamin C-fortified cranberry juice           27   3.1 Materials and Methods 3.1.1 Juice preparation Cranberry juice concentrate, processed in August 2014, was purchased from Dynamic health Laboratories Inc. (Brooklyn, NY, USA), and kept frozen at 18 °C until used, within about six months. The juice concentrate was diluted with HPLC-grade water at a ratio 1:14 (v:v) (dilution factor, DF= 15) to obtain 3.8 °Brix, which is typical of the commercial cranberry juice. The diluted juice was centrifuged at 7,600 g for 10 minutes, and the supernatant was filtered through Whatman® No.1 filter paper. A preliminary study was conducted at 80 mg/100 mL vitamin C fortification, with varying concentration of hesperidin, catechin, and gallic acid. The juice (100 mL) was fortified with 80 mg of L-ascorbic acid (CAS no. 50-81-7, Sigma Aldrich, St. Louis, MO, USA). Hesperidin (CAS no. 520263, Sigma Aldrich) was added to the fortified juice. The concentration of hesperidin was varied from 5 to 18 mg/100 mL. Catechin hydrate (C1251, CAS no. 225937-10-0, Sigma Aldrich, St. Louis, MO, USA) and gallic acid (CAS no. 149917, Sigma Aldrich, St. Louis, MO, USA) were added at 5, 15 mg/100 mL and 0, 80, 160, or 320 mg/100 mL, respectively. Juice was mixed until well-dissolved using a magnetic stirrer. Samples were prepared in two replicates. A final selection was conducted with regard to result from preliminary study. The juice (100 mL) was fortified with 40, 60, or 80 mg of L-ascorbic acid. Then, gallic acid was added to the juice at 0, 80, 160, or 320 mg/100 mL. Samples were prepared in four replicates. Figure 3.1 is a diagram illustrating sample preparation for vitamin C and phenolic compound addition in juice. 28           Figure 3.1 Diagram of sample preparation with addition of vitamin C and selected phenolic compounds.Cranberry juice (Dilution factor =15)  Hesperidin:      5 (mg/100 mL)   9            18          Preliminary study  Final selection  Vitamin C 80 mg/100 mL   Catechin:          5 (mg/100 mL)   15          Gallic acid:       0 (mg/100 mL)    80             160             320  Vitamin C 40 mg/100 mL    Vitamin C 60   Vitamin C 80   Gallic acid:       0 (mg/100 mL)    80             160             320  Gallic acid:       0 (mg/100 mL)    80             160             320  Gallic acid:       0 (mg/100 mL)    80             160             320  29   3.1.2 Juice pasteurization and storage Eight milliliters of juice were pipetted into each 15-mL glass test tube and capped (polyethylene cap). One tube was fitted with a Type-K thermocouple (Digi-Sense Type-K, Std Pen Probe, Cole-Parmer®, Vernon Hills, IL, USA), by puncturing the cap to monitor juice temperature during pasteurization. All juice samples were pasteurized at 85±2 °C for 1 minute. Pasteurization was performed in a thermostatic water bath equipped with a shaker (Figure 3.2). Temperatures of sample and water bath were monitored using a thermocouple meter (Digi-Sense Dual J-T-E-K, Model 91100-40, Cole-Parmer®, Vernon Hills, IL, USA). Pasteurizing time started when the thermocouple inside an assigned tube reached 85 °C. The come up time was 11.5 minutes. After one minutes of pasteurizing, the juice was then hot-filled into a 5-mL polypropylene vial, screw-capped and cooled in ice bath for 2 hours. Samples were kept in the dark at room temperature (23±2 °C) for 16 days. Samples were drawn every 2 days to analyze for ascorbic acid content, red color intensity, browning index, and ACY content.  Figure 3.2 Juice pasteurization in water bath equipped with a shaker, and a hand-held thermometer for monitoring temperature of water bath and cranberry juice.    30   3.1.3 Anthocyanin content Monomeric ACY are major compounds that provide bright red color in cranberry juice (Pappas and Schaich 2009). The total monomeric ACY content were determined by the pH differential method of Lee and others (2005), which is a rapid and simple assay. This study determined total ACY content as peonidin-3-galactoside equivalent, rather than individual anthocyanins since a total of 13 ACY in cranberry juice have been reported (Blumberg and others 2013, Milbury and others 2010), and the three major ACY are peonidin-3-O-galactosides (~32%), cyanidin-3-O-galactosides (23%), and peonidin-3-O-arabinosides (18%). Cranberry juice (1 mL) was diluted separately with 1 mL each of pH 1.0 and pH 4.5 buffers. The absorbance values of the solution were determined spectrophotometrically at 520 and 700 nm (Genesys10S UV-vis spectrophotometer, Thermo Scientific, Waltham, MA, USA). ACY content was calculated by following equation:  Where  A = (A520  A700)pH1.0  (A520  A700)pH4.5;    MW (molecular weight) of peonidin-3-galactoside = 463.41 g/mol;   DF = dilution factor;   L = pathlength in cm;   ,900 L×mol1×cm1);   103 = factor for conversion of g to mg.     31   3.1.4 Color intensity and browning index (BI) Two milliliters of each sample were pipetted into a 2-mL polycarbonate cuvette. A dual beam Genesys 10S UV-vis spectrophotometer was used to measure absorbance values (AU) at 520 nm (AU520), which is the maximum absorbance of red color from monomeric ACY (Vegara and others 2013). The absorbance of juice was also measured at 430 nm (AU430) and the browning index was expressed as a ratio of AU430/AU520. 3.1.5 L-ascorbic acid quantification L-ascorbic acid content was quantified using high performance liquid chromatography assay and BreezeTM software (Waters Corporation, Milford, MA, USA), with mobile phase of 65:35 mixture of 30 mM KH2PO4 buffer pH 2.5 (adjusted by adding H3PO4) and acetronitrile. Cranberry juice samples (1 mL) were filtered through a 25-mm diameter nylon syringe filter with  (Water Corporation)each juice sample was injected into the HPLC and separated was conducted using a C18 SORBAX column (dimension 4.6 x 250 mm, size) equipped with a guard column (dimension 4.6 x 12.5 mm).  The mobile phase was pumped through the columns at flow rate of 1 mL/min. Absorbance was measured at 245 nm was set for  L-ascorbic acid standard stock was prepared by dissolving 10 mg L-ascorbic acid in HPLC-grade water and the volume was adjusted stock solution was diluted in 10 mL HPLC-grade water to obtain final concentrations of 10, 20, 40, 60, 80, and 100 mg/100 mL. All samples were prepared in duplicate to prepare a standard curve. The regression equation according to the standard curve, was as follows:  32   3.1.6 Statistical analysis All data were analyzed using SAS software (SAS Institute, Inc., Cary, NC, USA). Effect of vitamin C and antioxidant treatments were analyzed using one-way analysis of variance , and the statistical significant level was defined as  = 0.05.  3.2 Results and Discussion 3.2.1 Anthocyanins retention The use of antioxidants (hesperidin, catechin, gallic acid) had variable effect on the retention of anthocyanins (ACY) in cranberry juice, stored 16 days at 23 °C (Figure 3.3). Control juice (80 mg/100 mL vitamin C, with no added antioxidants) had average ACY content of 9.63 mg/L during storage. Generally, increasing levels of different antioxidants showed an increasing protective effect on ACY. 20.).  The relatively high protective effect of hesperidin was possibly due to its copigmentation with ACY, as has been reported previously and Ozkan 2014). Copigmentation is a result of hydrophobic interactions among phenols and ACY (Mazza and Brouillard 1990).  However, hesperidin was found to precipitate somewhat in the juice during storage. Catechin addition at both concentrations (5 and 15 mg/100mL) showed no significant difference (p > 0.05) on ACY retention. In contrast to catechin, addition of gallic acid significantly increased ACY retention. 33    Figure 3.3 Average anthocyanin content during 16-day storage, at 23 °C. Treatment levels per 100 mL were: vitamin C fortification (80 mg) without adding antioxidant compounds (Ctrl); Hesperidin Trt-1 (5 mg), Trt-2 (9 mg), and Trt-3 (18 mg); Catechin Trt-1 (5 mg) and Trt-2 (15 mg); and Gallic Acid Trt-1 (80 mg), Trt-2 (160 mg), and Trt-3 (320 mg).  Treatments sharing same letters (a, b, c) are not significantly different from each other within  = 0.05).  Hesperidin showed better ACY retention than catechin and gallic acid; however, its rather poor solubility in water limits its commercial application. Based on the results of this experiment, gallic acid was selected to further evaluate its protective effect on ACY in vitamin C fortified cranberry juice. Increasing levels of gallic acid from 0 to 320 mg/100 mL showed a consistent positive effect on ACY retention at all vitamin C fortification levels (Figure 3.4). ACYs were protected by scavenging activity of gallic acid. Gallic acid scavenges radicals by donating H-atoms from its phenol groups. The proton donation neutralizes free radicals to be less active, and thus limits the interaction with ACY (Daglia and others 2014; Yen and others 2002). However, increasing the level of vitamin C fortification had a negative impact on ACY, thereby negating the 39.6cacbabbaabaa0.05.010.015.020.025.030.035.040.045.050.0No vitamin CHesperidinCatechinGallic AcidAnthocyanin content (mg/L)CtrlTrt-1Trt-2Trt-334   protective effect of gallic acid.  Ascorbic acid showed a detrimental effect on ACY in cranberry juice, as was evidenced by decreasing ACY values from 40 mg vitamin C juice to 80 mg vitamin C juice.   Figure 3.4 Effect of gallic acid (0-320 mg/100 mL) on the average anthocyanin content of cranberry juice, fortified with 40-80 mg/100 mL vitamin C, during 16-day storage at 23 °C.  Treatments sharing same letters (a, b, c) and (A, B, C) are not significantly different from each other within the same vitamin C fortification level and across different levels, reHSD test,  = 0.05).   Starr and Francis (1968) showed that addition of vitamin C reduced ACY in cranberry juice by 925% depending on oxygen level in the headspace. Similarly, Li and others (2014) reported that adding ascorbic acid at 360 mg/L increased the degradation rate constant of ACY in purple sweet potatoes from 7.2410-2 to 8.5710-2 (hr1). Ozkan and others (2005) and Sun and others (2011) explained that oxidation of vitamin C-induced ACY loss occurs via two mechanisms. The first mechanism is a direct condensation of ascorbate radicals to ACY structure. The second ACY. The radicals AcBbCcAbBbCbcAaBaCabAaBaCa0481216202440 mg Vit C60 mg Vit C80 mg Vit CAnthocyanin content (mg/L)GA-0mgGA-80mgGA-160mgGA-320mg35   are byproducts from H2O2 decomposition upon vitamin C oxidation. The radicals are known as reactive oxygen species (ROS), which are damaging to ACY.  3.2.2 Color intensity  Juice color was determined to correlate with ACY retention. Control juice had average red color intensity of 0.93 AU during storage. Catechin and gallic acid were shown to have significant (p < 0.05) protective effect on red color retention in cranberry juice during 16-day storage at 23 °C (Figure 3.5)all concentrations (5, 9, and 18 mg/100 mL) showed no effect on color intensity. The possible reason for this inverse effect is that co-pigmentation of hesperidin with anthocyanins might have resulted in polymeric anthocyanins, which contributed dark red color in the juice, and hence lower the bright red color intensity.  Although Liu and Chen (2008) reported an increase in the water solubility of hesperetin on heating, the time and temperature used during pasteurization in the present study might have been insufficient to significantly increase the solubility. In contrast to the results for ACY retention, catechin was found to have a significant positive effect on the color intensity at both concentrations (5 and 15 mg/100 mL). However, the red color intensity at both catechin concentrations were not significantly different.  In addition, measurement of color intensity with catechin addition could be misleading due to yellow color formation regarding oxidized catechin. Bark and others (2011) reported yellow color formation by catechin upon dissolving in aqueous solution due to catechin oxidation.  This study confirmed yellow color formation by dissolving catechin (5 and 15 mg/100 mL) in water at ambient temperature for 2 hours, and found that the higher the catechin concentration, the more intense the yellow color in solution (visual observation). In contrast to hesperidin and catechin, we observed that gallic acid 36   solubilized well in water and did not form yellow compound upon dissolving in water. Moreover, gallic acid significantly (p < 0.05) increased red color retention in cranberry juice. The higher color intensity during storage represented better retention of ACY.    Figure 3.5 Average red color intensity during 16-day storage, at 23 °C. Treatment levels per 100 mL were: vitamin C fortification (80 mg) without adding antioxidant compounds (Ctrl); Hesperidin Trt-1 (5 mg), Trt-2 (9 mg), and Trt-3 (18 mg); Catechin Trt-1 (5 mg) and Trt-2 (15 mg); and Gallic Acid Trt-1 (80 mg), Trt-2 (160 mg), and Trt-3 (320 mg).  Treatments sharing same letters (a, b, c) are not significantly different from each other within  = 0.05).  The effect of gallic acid was also assessed in cranberry juice at various concentration levels of vitamin C fortification. The negative impact of vitamin C on color retention (Figure 3.6) was found to be similar to the effect that was observed in the case of ACY retention. With an increase of vitamin C fortification level from 40 to 80 mg/100 mL, the color intensity of cranberry juice kept decreasing. As the fortification of vitamin C was increased to 60 and 80 mg/100 mL, the color abbaabaaaaa0.00.20.40.60.81.01.21.41.61.82.02.2No vitamin CHesperidinCatechinGallic AcidColor intensity (AU)CtrlTrt-1Trt-2Trt-337   intensity decreased by 15.7% and 29.6%, respectively. Even though gallic acid showed significant increase in color intensity, as its concentration was increased up to 320 mg/100 mL, such increases could not bring back the original red color intensity (2 AU) due to the corresponding detrimental effect of added vitamin C. These results were consistent with the negative impact of vitamin C addition on ACY, as reported by Li and others (2014) and Sadilova and others (2007). Since the characteristic red color of cranberry juice is due to presence of ACY, any detrimental effect on ACY would have directly translated to a corresponding decrease in color intensity.   Figure 3.6 Effect of gallic acid (0-320 mg/100 mL) on the average color intensity of cranberry juice, fortified with 40-80 mg/100 mL vitamin C, during 16-day storage at 23 °C.  Treatments sharing same letters (a, b, c) and (A, B, C) are not significantly different from each other within the same vitamin C fortification level and across different levHSD test,  = 0.05).   AcBdCdAbBcCcAbBbCbAaBaCa0.00.30.60.91.21.51.840 mg Vit C60 mg Vit C80 mg Vit CColor intensity (AU)GA-0mgGA-80mgGA-160mgGA-320mg38   3.2.3 Browning index Degradation of polyphenols, including ACY, could result in brownish color in juices. The browning color represents chalcone formation, which is a byproduct regarding ACY degradation.  Browning index (BI) represents color changes from reddish to yellowish or brownish. The BI greater than 1.0 is unacceptable, since brownish (AU430) is predominant over reddish (AU520) shades, according to BI evaluation criteria in pomegranate juices (Vegara and others 2013). Due to the poor solubility, BI regarding hesperidin addition was not determined. The BI observed with catechin and gallic acid addition are shown in Figure 3.7. BI consistently increased over 16-day storage at 23 °C.  Compared to the control juice (vitamin C 80 mg/100 mL, no antioxidant), addition of gallic acid, at 80320 mg/100 mL, showed protective effect as exhibited by lower BI, whereas catechin had no protective effect as evidenced by higher BI. On day 16, gallic acid at 80, 160, and 320 mg/100 mL showed 7.44%, 14.68%, and 20.49% decrease in the BI of cranberry juice. This demonstrated that gallic acid addition had a protective effect on color, as was exhibited by BI of <1.0.  It was also found that vitamin C fortification tended to increase BI, as shown in Figure 3.8, whereas gallic acid addition tended to decrease BI. Although gallic acid addition showed no significant difference in BI of juices at 40 and 60 mg/100 mL vitamin C levels, addition of gallic acid significantly (p < 0.0001) decreased BI at 80 mg/100 mL vitamin C level. However, the maximum gallic acid concentration at 320 mg/100 mL could not bring BI to be equal to juice without vitamin C fortification (BI = 0.57).   39    Figure 3.7 Effect of antioxidants on the browning index of cranberry juice fortified with vitamin C (80 mg/100 mL) during 16 day storage at 23 °C.  Treatment levels per 100 mL: Catechin Trt-1 (5 mg) and Trt-2 (15 mg); and Gallic Acid Trt-1 (80 mg), Trt-2 (160 mg), and Trt-3 (320 mg). Standard deviation of browning index varied from 0.00016 to 0.053.  Figure 3.8 Effect of gallic acid (0-320 mg/100 mL) on browning index of cranberry juice, fortified with 40-80 mg/100 mL vitamin C, during 16-day storage at 23 °C.  Treatments sharing different letters (a, b, c) and (A, B, C) are significantly different from each HSD test,  = 0.05).  0.50.60.70.80.91.01.10246810121416Browning Index (AU430/AU520nm)Storage Time (days)CtrlCatechin (Trt-1)Catechin (Trt-2)Gallic Acid (Trt-1)Gallic Acid (Trt-2)Gallic Acid (Trt-3)BaBaAaAaAaAbAaAaAbAaAbAc0.00.20.40.60.81.040 mg Vit C60 mg Vit C80 mg Vit CBrowning Index (AU430/AU520nm)GA-0mgGA-80mgGA-160mgGA-320mg40   3.2.4 Vitamin C retention  With reference to the above reported results, gallic acid was found to exhibit protective effect on both ACY and color retention during storage. Since fortification with vitamin C plays a role in ACY degradation, vitamin C retention was measured to gain better understanding on antioxidant mechanisms of gallic acid in cranberry juice. The results showed that gallic acid significantly (p < 0.05) increased vitamin C retention as compared to control juice (Figure 3.9). In order to increase vitamin C retention, ascorbate radicals possibly take protons from gallic acid, and then could convert back to L-ascorbic acid. However, in contrast to ACY and color retention, the level of gallic acid (80 to 320 mg/100 mL) showed no significant effect on vitamin C retention. It could be explained that gallic acid at 80 mg/100 mL possibly reached its maximum capacity to inhibit vitamin C oxidation. Moreover, vitamin C is a powerful antioxidant, which is susceptible to donate protons, and thus the oxidation reaction might occur more rapidly than the ability of gallic acid to inhibit the reaction of vitamin C (Yen and others 2002). This findings supported the hypothesis that the ability of gallic acid to stabilize reactive oxygen species  was predominant over the ability to decrease vitamin C oxidation.    41    Figure 3.9 Effect of gallic acid (0-320 mg/100 mL) on the vitamin C content of cranberry juice, fortified with 40-80 mg/100 mL vitamin C, during 16-day storage at 23 °C.  Treatments sharing same letters (a, b, c) and (A, B, C) are not significantly different from each HSD test,  = 0.05).  The results further showed that without vitamin C fortification, endogenous vitamin C in the juice stayed unchanged during 16 days of storage (7.1 mg/100 mL) (APPENDIX A, Table A2), and so did anthocyanin content (39.6 mg/L) and red color intensity (2 AU) (shown as a first bar in Figure 3.3 and Figure 3.5). Vitamin C fortification from 40 to 80 mg/100mL significantly accelerated color and ACY degradation. It is known that high doses of vitamin C can act as a pro-oxidant, which induces more free radical formation (Paolini and others 1999). At higher doses of vitamin C, another pathway of reactive oxygen species (ROS) formation could occur by chelating of ascorbate and metal ions in juice, such as Cu3+, Fe3+ (Rietjens and others 2002). The pro-oxidant effect may explain why fortified vitamin C caused color degradation, while endogenous vitamin C did not.  CbBbAbCaBaAaCaBaAaCaBaAa010203040506040 mg60 mg80 mgVitamin C (mg/100mL)GA-0mgGA-80mgGA-160mgGA-320mg42   3.3 Conclusions Gallic acid showed significant protective effect on ACY and color retention in cranberry juice. The higher the concentration of gallic acid, the higher ACY content and color retention were found. Generally, ACY and red color intensity of cranberry juice were effectively preserved at 320 mg/100 mL gallic acid, as it was the maximum gallic acid level in this study. Although gallic acid did not prevent vitamin C from oxidation, byproducts from the oxidation were neutralized by gallic acid, and hence gallic acid increased ACY and color retention. However, the addition of gallic acid could not completely overcome ACY and color degradation due to the effect from fortified vitamin C was very powerful. Therefore, concentration of fortified vitamin C needs to be taken into account to avoid or minimize a pro-oxidant effect.  The results demonstrated that gallic acid is a potential natural antioxidant compound that could be used in commercial cranberry juice. The high gallic acid concentration used in this study (320 mg/100 mL) would not develop astringent taste, but rather induce sweetness, as discussed in Chapter 2. These findings are not limited to cranberry juice but could be applied to other pigment-rich juices in order to preserve their endogenous ACY and natural color. Gallic acid is feasible to use in commercial juice production with regard to water solubility, and price. Cost of food-grade gallic acid is about $30/kg; adding gallic acid at 320 mg/100 mL would cost approximately $0.03 per 240 mL serving size. The use of gallic acid also offers a number of advantages for the juice industry in that it has no adverse effects, is colorless, and has the ability to induce non-caloric sweetness.    43   3.4 Limitations of the study Referring to ACY, color, and vitamin C retention, we believe that major pathway of ACY We also believe that gallic acid preserved ACY by deactivating the ROS because ACY retention increased with gallic acid concentration. The understanding of mechanisms was interpreted from retention of vitamin C and ACY. However, chemical analysis for H2O2 and ROS formation in juice was not performed in this study due to rapid decomposition of H2O2 and highly unstable ROS.                44        CHAPTER 4 OBJECTIVE TWO Estimating kinetic parameters of anthocyanins and color degradation during storage by using an inverse method of ordinary least squares            45   4.1 Materials and Methods 4.1.1 Mathematical Modeling 4.1.1.1 Estimation of the kinetic parameters in the primary model Data from all four replicates of ACY and color measurements during storage were used to estimate parameters k, C0, and n in an nth-order kinetic model which will be referred to as the primary model, Eq. (4.1) by the ordinary least squares inverse method in MATLAB (codes are shown in APPENDIX B). Parameter C0 is initial red color intensity (AU), or initial ACY content (mg/L), at time = day 1. The k is degradation rate constant (conc(1-n) day1). The n is reaction order, dimensionless.  Table 4.1 shows three parameters that were estimated for all 12 treatments of both color and ACY. Significance of parameters was determined by noting if 95% confidence intervals (CIs) did not contain zero. The R-matrix (correlation) was used to determine correlation between parameters, correlation value greater than 0.99 refers to high correlation. Relative error indicated accuracy of estimates.  The ease of estimating parameters was evaluated via the size of scaled sensitivity coefficients (SSC) (Dolan and Mishra 2013). The best model was determined by corrected Akaike Information Criteria (AICc). AICc (Eq. 4.2) is calculated based on the change of SS in regard to an increase or decrease in number of estimated parameters. Adding a parameter would always decrease SS; however, if the decrease is insufficient to justify the addition of a parameter, then AICc will be higher. Therefore, lower AICc indicates a better model.   (4.1)   46     (4.2)  where, N is number of data, p is number of parameters, K = p+1, SS is sum square of errors.  Table 4.1 Data analysis design for parameter estimation (k, C0, n) for color and anthocyanins at different concentration of vitamin C and gallic acid (12 treatments*).   Vitamin C   40 mg/100mL 60 mg/100mL 80 mg/100mL Color Gallic acid (0 mg/100 mL) k, C0, n (Trt-1) k, C0, n (Trt-5) k, C0, n (Trt-9) Gallic acid (80 mg/100 mL) k, C0, n (Trt-2) k, C0, n (Trt-6) k, C0, n (Trt-10) Gallic acid (160 mg/100 mL) k, C0, n (Trt-3) k, C0, n (Trt-7) k, C0, n (Trt-11) Gallic acid (320 mg/100 mL) k, C0, n (Trt-4) k, C0, n (Trt-8) k, C0, n (Trt-12) Anthocyanins Gallic acid (0 mg/100 mL) k, C0, n (Trt-1) k, C0, n (Trt-5) k, C0, n (Trt-9) Gallic acid (80 mg/100 mL) k, C0, n (Trt-2) k, C0, n (Trt-6) k, C0, n (Trt-10) Gallic acid (160 mg/100 mL) k, C0, n (Trt-3) k, C0, n (Trt-7) k, C0, n (Trt-11) Gallic acid (320 mg/100 mL) k, C0, n (Trt-4) k, C0, n (Trt-8) k, C0, n (Trt-12) *4 replications per treatment 4.1.1.2 Parameter correlation  If parameters were highly correlated, they could not be estimated simultaneously, and therefore, one parameter was fixed as a constant. Other researchers frequently fixed parameter n at 1 for both color and ACY degradation during storage; otherwise, n was assumed to be 0.5, or an integer (0, 1, 2) (Polydera and others 2003; Harbourne and others 2008; Wibowo and others 2015; Remini and others 2015). However, this study determined actual nth-order by varying n from 0.05 to 5 in 0.05 increments. The best fitted n-value for each treatment was the n that gave the smallest root mean square error (RMSE). Parameters k and C0 were then estimated simultaneously by 2-parameter estimation, holding the actual n constant. A similar procedure was performed by 47   Buckow and others (2010) to determine the n-value for ACY degradation in blueberry juice during 40 to 121 °C pasteurization at pressures from 0.1 to 700 MPa.      Upon determining the actual n-value of each treatment, parameter n from all 12 treatments of ACY and color was determined whether the n values were significant different (p < 0.05) as changing concentration of vitamin C and gallic acid concentration. The secondary equations for nACY and ncolor were modeled by multiple linear regression. The significant difference and modeling were done using JMP software, version 9.0.2 (SAS Institute, Inc., Cary, NC, USA).  4.1.1.3 Error of parameters Error of estimates was determined by percent relative error, which was calculated from standard error of each estimate. Although parameter n was fixed as a constant in order to estimate parameters k and C0, errors of nACY and ncolor were determined by doing the reverse procedure, i.e., fixing different k values as previously estimated from the 2-parameter estimation shown in Section 4.2.4.2. Upon fixing k values, 2-parameter estimation was performed to estimate parameters n and C0, and the asymptotic parameter standard error was computed as the square root of the diagonal of the parameter variance-covariance matrix cov(a) (Dolan 2003):   (4.3) where; X is the sensitivity matrix, and MSE is the mean square error = SS/(N-p), N is number of data, p is number of parameters,    48   4.1.2 Developing secondary models for parameters as a function of vitamin C and gallic acid Degradation rate constants (kACY, kcolor) were functions of vitamin C and gallic acid (Eq. (4.4)). Empirical linear relationships between k and independent variables were determined by plotting k versus vitamin C, and k versus gallic acid, and conducting multiple linear regression. Because the variance of k values was not constant over vitamin C and gallic acid ranges, the k values were logarithmically transformed. In Eq. (4.4)1 6 were determined to be significant if the 95% CI (confidence interval) did not contain zero.    (4.4) AICc also indicated whether an additional parameter was necessary in the model. Residual plots were used to determine goodness of fit of the models with experimental values, by the following standard statistical assumptions: constant variance, additive errors, zero mean, uncorrelated errors (Dolan and Mishra 2013). Multiple linear regression was also used to model polynomial equations for Co and n, which are a function of vitamin C and/or gallic acid.   (4.5)    (4.6)    49   4.2 Results and Discussion 4.2.1 Parameter estimation According to a kinetic model, shown in Eq. (4.1), parameters k, C0, n were initially estimated by three-parameter estimation. Among 12 treatments, the estimates at vitamin C 60 mg/100 mL with gallic acid 80 mg/100 mL (Trt-6 according to Table 4.1) was chosen to present the results in Table 4.2.  From three-parameter estimation, it was found that confidence interval (CI) of the degradation rate constant of ACY (kACY) contained zero, showing that the kACY was not significant from zero (p > 0.05), and could be removed from the model. However, ACY were shown to decrease significantly (p < 0.05) during 16-day storage, which demonstrated that the kACY cannot be zero. According to R-matrix in Table 4.2, parameters kACY and nACY were highly correlated (correlation coefficient = k,n 0.9986), and therefore, should not be estimated simultaneously. The estimates of kACY and nACY were found to have large error of 118.42% and 24.23%, respectively. Therefore, three-parameter estimation was not appropriate to estimate kACY and nACY. For color, although kcolor did not contain zero, and itsk,n 0.9322 was low enough to allow simultaneous estimation; still, kcolor and ncolor were estimated separately to obtain better parameter accuracy and lower AICc (discussed later).       50   Table 4.2 Three parameter (k, C0, n) estimation for anthocyanins and color in cranberry juice fortified with vitamin C (60 mg/100 mL) and gallic acid (80 mg/100 mL), during 16-day storage at 23 °C. Kinetics Parameters CIs R-matrix Error (%) AICc Anthocyanins Co = 21.72 20.42 - 23.03  0.4546    2.94 34.27   k = 0.0077 0.011 - 0.026 118.52   n = 1.84 0.93 - 2.74 24.23 Color Co = 1.88 1.82 - 1.93  0.2608   0.4589   0.9322    1.0000 1.36 200.25   k = 0.034 0.027 - 0.040 9.56   n = 2.80 2.17 - 3.44 11.13  4.2.1.1 Determination of parameter correlation The scaled sensitivity coefficient (SSC) plots are helpful in determining which parameters can be estimated most accurately. As shown in Figure 4.1, SSC of all parameters k, C0, n were large enough to be estimated. SSC plots of C0 for both ACY and color were found to be larger than the size of n and k, which indicated that parameter C0 can be estimated more accurately and easier than n and k. In addition, SSC plots can also illustrate parameter correlation. In Figure 4.1A, shape of SSC plots for kACY and nACY were nearly identical, which indicated the high correlation between those two parameters; corresponded to k,n 0.9986 as shown in Table 4.2. In contrast to SSCACY, SSCcolor (Figure 4.1B) showed that all three parameters were not highly correlated and could be estimated simultaneously. Owing to the high correlation (0.9986), kACY and nACY could not be estimated simultaneously. One parameter, either k or n, needed to be fixed as a constant. The parameter n is commonly fixed rather than k. However, prior to fixing n, this study estimated n by using an innovative statistical procedure to assure that it was the best n that represented degradation reactions. 51        Figure 4.1 Representative SSC plots from three-parameter estimation of anthocyanins (A) and color (B) in cranberry juice fortified with vitamin C (60 mg/100 mL) and gallic acid (80 mg/100 mL), during 16-day storage at 23 °C.  52   Instead of fixing n as an integer (i.e., 0, 1, 2), this study let MATLAB vary n from 0.05 to 5 with 0.05 increments; total 100 n-values. Among 100 n-values, the n with the smallest RMSE was assigned as the best n, which would be later fixed in order to estimate for k and C0. This study found that n-value of reaction was not constant for all 12 treatments. The n significantly decreased with an increase in vitamin C concentration in fortification. Figure 4.2 shows representative plots of the RMSE versus n (reaction order) when C0 and k were estimated simultaneously. These plots are for 4080 mg/100 mL vitamin C fortification with no gallic acid addition, while other 3 pairs of plot (gallic acid 80320 mg/100 mL) were in Appendix B, Figure B1B3. As presented in Figure 4.2A, the smallest RMSE, nACY was 3.65, 1.75, and 1.45 at 40, 60, and 80 mg/100 mL vitamin C fortification, respectively. In contrast, Figure 4.2B shows that ncolor was 4.25, 2.55, and 2.35 at 40, 60, and 80 mg/100 mL vitamin C fortification, respectively.  From all the 12 treatments, this study found that although n-order altered with respect to vitamin C concentration, it did not change (p > 0.05) with gallic acid concentration, as presented in Figure 4.3. Therefore, this implied that n-value was a function of vitamin C only. Linear regression was performed to model predictive equations for nACY and ncolor, and resulted in Eq. (4.7) and Eq. (4.8), respectively. The empirical polynomial models to predict nACY and ncolor showed an RMSE of 0.386 and 0.247, while total range of nACY and ncolor were 3.1 and 2.5, respectively. According to Eq. (4.7) and Eq. (4.8), nACY and ncolor at vitamin C fortification of 40, 60, and 80 mg/100 mL were 3.3, 2.2, and 1.2 for ACY, and 4.4, 2.9, and 2.2 for color, respectively.  This is the first study reporting that nth-order for ACY and color degradation were different from 0 or 1, and changed with vitamin C concentration in fortification. Previous studies mostly estimated kACY by assuming first-order reaction (n = 1), without determining whether it was the 53   actual n for the degradation reaction (Bosch and others 2013; Ozkan and others 2005; Wang and Xu 2007). In addition, some studies have evaluated the best fitted n by comparing RMSE of various n at 0, 0.5, 1, 2, whereas the n greater than 2 was never even considered (Remini and others 2015; Buckow and others 2010; Wibowo and others 2015). None of the previous studies had reported that reaction order of ACY and color degradation could vary in such a wide range, i.e., from 1.2 to 4.4, as found in this study. In the present study, n-values were not assumed, but rather allowed to change freely with experimental measurements. Therefore, the n-values were the best fitted n of each treatment.          (4.7)      (4.8)    54     Figure 4.2 Representative plots of RMSE versus n (reaction order) of anthocyanins (A) and color (B) in cranberry juice fortified with vitamin C (40, 60, 80 mg/100 mL) and gallic acid (0 mg/100 mL), during 16-day storage at 23 °C.   55     Figure 4.3 Effect of gallic acid (0-320 mg/100 mL) on order of degradation reaction of anthocyanins (nACY) and color (ncolor) in cranberry juice, fortified with 40-80 mg/100 mL vitamin C, during 16-day storage at 23 °C.  Treatments sharing the same letters (a, b, c) and (A, B, C) are not significantly different from each HSD test,  0.05). AaBaCaAaBaCaAaBaCaAaBaCb0.01.02.03.04.040 mg Vit C60 mg Vit C80 mg Vit CnACYGA-0mgGA-80mgGA-160mgGA-320mgAaBbCaAaBbCaAaBbCaAaBaCa01234540 mg Vit C60 mg Vit C80 mg Vit CncolorGA-0mgGA-80mgGA-160mgGA-320mg56   4.2.1.2 Determining error of parameters The actual nth-orders were fixed accordingly with amount of fortified vitamin C, so that k and C0 were estimated by two-parameter estimation. Table 4.3 showed an example of the estimated parameters in cranberry juice fortified with vitamin C (60 mg/100 mL) and gallic acid (80 mg/100 mL), results for other treatments were in APPENDIX B, Table B3B4. Parameters k and C0 for both ACY and color were significantly different from zero, as the CIs did not contain zero. Correlation coefficients were lower than 0.95 (Co,k= 0.530 for ACY, Co,k = 0.509 for color), which indicated no correlation between parameters. Moreover, errors of parameters k and C0 from two-parameter estimation were lower than the three-parameter estimation, i.e., errors of kcolor and Cocolor decreased from 9.56 to 3.45% and from 1.36 to 1.20%, respectively. Also, AICc slightly decreased from 202.67 (color). Therefore, the two-parameter estimation was appropriate for both ACY and color models.  Table 4.3 Two parameter (k, C0) estimation for anthocyanins and color in cranberry juice fortified with vitamin C (60 mg/100 mL) and gallic acid (80 mg/100 mL), during 16-day storage at 23 °C. Kinetics Parameters CIs R-matrix Error (%) AICc Anthocyanins Co = 21.97 20.79 - 23.15 1.0000   0.5296 0.5296   1.0000 2.65 32.31   k = 0.0029 0.0025 - 0.0033 6.61   n = 2.2 (fixed)  - Color Co = 1.88 1.83 - 1.93 1.0000   0.5086     0.5086   1.0000     1.20 202.67   k = 0.033 0.031 - 0.035 3.45   n = 2.9 (fixed)  -  57   Errors of parameters nACY and ncolor were determined by fixing previously estimated kACY and kcolor (in Table 4.3) as a constant. Table 4.4 indicated that nth-order for both ACY and color were significantly different from zero and they were uncorrelated with parameter C0, (Co,n = 0.549 for ACY,  = 0.609 for color). Errors of nACY and ncolor from all 12 treatments were lower than 10%, whereas the maximum errors of nACY and ncolor were 3% and 6%, respectively.   Table 4.4 Two parameter (n, C0) estimation for anthocyanins and color in cranberry juice fortified with vitamin C (60 mg/100 mL) and gallic acid (80 mg/100 mL), during 16-day storage at 23 °C. Kinetics Parameters CIs R-matrix Error (%) Anthocyanins C0 = 21.96 20.76 - 23.16 1.0000     0.5485      0.5485     1.0000    2.69   k = 0.0029 (fixed)     n = 2.2 2.1 - 2.3 1.12 Color C0 = 1.88 1.83 - 1.93 1.0000      0.6086    0.6086      1.0000 1.30   k = 0.033 (fixed)     n = 2.9 2.6 - 3.1 3.87  4.2.2 Developing secondary models Secondary equations were modeled using parameters from all 12 treatments, while the parameters were estimated by two-parameter estimation. Since initial content of ACY and color differed due to vitamin C and gallic acid addition, empirical polynomial models for CoACY and Cocolor were proposed as Eq. (4.9) and Eq. (4.10). Error of the models were 1.18 of 13% of the range (CoACY), and 5.26% of the range (Cocolor). These equations were selected due to the lowest AICc (APPENDIX B, Table B5). 58    1 = 28.242 ×10-23 = 1.06×10-24 = 8.08×10-5   (4.9)   1 = 1.962 = 2.49×10-33 = 8.91×10-44 ×10-6              (4.10)  As illustrated in Figure 4.4, kACY and kcolor changed regarding vitamin C fortification with added gallic acid. Vitamin C fortification significantly (p < 0.05) increased kACY and kcolor, whereas gallic acid had significant positive impact on ACY and color, as exhibited by lower degradation rate constants. The increasing levels of gallic acid showed an increasing protective effect of ACY and color. This corresponded to findings in Chapter 3 that ACY and characteristic red color intensity of cranberry juice were preserved by adding gallic acid as an antioxidant. The curvatures of kACY versus vitamin C are shown in Figure 4.4A.  Even though the curvatures was not visually seen in Figure 4.4B, the curvature affected error in modeling predictive equation (data not shown). Therefore, log transformation was applied to both kcolor and kACY in order to make the error variance more constant.    59     Figure 4.4 Degradation rate constant of anthocyanins (A) and color (B) in cranberry juice fortified with vitamin C (4080 mg/100 mL) and gallic acid (0320 mg/100 mL), during 16-day storage at 23 °C.    0.000.020.030.050.060.0820406080100kacy((mg/L)day)Vitamin C foritification (mg/100 mL)(A)GA-0mgGA-80mgGA-160mgGA-320mg0.000.020.030.050.060.080.0920406080100kcolor(AUday)Vitamin C fortification (mg/100 mL)(B)GA-0mgGA-80mgGA-160mgGA-320mg60   Thus, predictive equations of degradation rate constant (kACY and kcolor) were modeled by multiple linear regression method.  As a result, logkACY and logkcolor could be calculated from Eq. (4.11) and Eq. (4.12), respectively.   1 = 2 = 7.2875×10-23 = 1.2934×10-34 = 1.8544×10-55 = 3.7809×10-4   (4.11)   1 = 2 = 2.0902×10-23 = 1.6303×10-34 = 1.6511×10-55 = -3.9769×10-46 = 2.3897×10-6                      (4.12)  The resulting models were the best models with regard to the lowest AICc, and all parameters1 to 6 were significantly different from zero (p < 0.05). Errors of the models were 0.95% of total range (logkACY) and 2.24% of total range (logkcolor). The 3D plots in Figure 4.5 illustrated goodness of fit between experimental logk (dot) and surface of predicted logk from models.       61       Figure 4.5 The 3D plots of logkACY and logkcolor from experiment () and predictive surface calculated from Eq. 4.11 (R2adjusted = 0.9994) and Eq. 4.12 (R2adjusted = 0.9948), respectively.  62     Figure 4.6 Representative model fittings of anthocyanins (A) and color (B) retention in cranberry juice fortified with vitamin C (60 mg/100 mL) and gallic acid (0320 mg/100 mL), during 16-day storage at 23 °C, while GA and Pred refer to gallic acid, and predicted values, respectively.  63   Model fitting results are presented in Figure 4.6 by plotting predictive lines with experimental measurement data. Predicted ACY and color retention (CACY and Ccolor) were calculated from differential equation, Eq. (4.1), by using ode45 in MATLAB, whereas n-values were calculated from Eq. (4.5) and Eq. (4.6), C0-values were calculated from Eq. (4.7) and Eq. (4.8), and k-values were calculated from Eq. (4.9) and Eq. (4.10).   Residual plots are more useful than R2 in order to determine goodness of fit of the models, since R2 can be misleading especially with curvature data. Residual plots can tell not only how close predicted and experimental measurements are, but  also present characteristics of the errors, which help determine whether it follows standard statistical assumptions. The residual plots for Figure 4.6 are shown in Figure 4.7. Goodness of fit was determined from constant bandwidth of the plot, which indicates constant variance. Other assumptions, such as additive errors and zero mean, were also met, Normal distribution of residuals illustrated by histograms are shown in Figure 4.8. For all 12 treatments of both ACY and color, there were mixed trends of the normality, which intepreted that some treatments did not show strong normality of residuals. However, except the normality, residual plots met most of statistical assumptions (constant variance, additive error, zero mean, uncorrelation). Therefore, these models could be used to predict ACY and red color retention during storage at any concentration of vitamin C and gallic acid, within the range of 40-80 mg/100 mL, and 0-320 mg/100 mL, respectively.  64      Figure 4.7 Representative residual plots showing difference between observed and predicted values of anthocyanins (A) and color (B) in cranberry juice fortified with vitamin C (60 mg/100 mL) and  gallic acid (80 mg/100 mL), during 16-day storage at 23 °C.  65      Figure 4.8 Representative histograms, plotted by dfittool in MATLAB, showing normal distribution of residuals in prediction of anthocyanins (A) and color (B) retention in cranberry juice fortified  with vitamin C (60 mg/100 mL) and  gallic acid (80 mg/100 mL), during 16-day storage at 23 °C.  66   4.3 Conclusions Vitamin C fortification typically accerelates degradation of color and ACY in cranberry juice. Gallic acid was used as a natural antioxidant to mitigate the degradation of ACY and color. The nonlinear inverse method of ordinary least squares is an appropriate tool for estimating parameters k, C0, n to avoid errors regarding model transformation, and also obtain physical meanings of estimates, i.e., correlation and error of parameters. Even though parameters kACY and nACY  were found to be higly correlated, an innovative statistical technique was performed to obtain accurate estimates. The results showed that order of degradation reaction (nACY and ncolor) decreased with increasing vitamin C fortification level  and varied from 1.2 to 4.4, while previous studies assumed n = 1, a first order reaction. Degradation rate constants (kACY and kcolor) were a function of both vitamin C and gallic acid, as they increased with vitamin C, but decreased with gallic acid. Statistical results from parameter estimation supported that the estimates of k, C0, and n in this study were accurate with respect to small error, significance from zero, and uncorrelation. Predictive equations modeled from estimated parameters showed good fit with experimental data as residual plots met most of statistical assumptions. The outcomes from this study have practical significance in that they not only introduced innovative statistical techniques to overcome highly correlation of parameters, but also proposed predictive empirical linear models for n-order (Eq. (4.7) and Eq. (4.8)), and for k (Eq. (4.11) and Eq. (4.12)) with comprehensive statistical information that could benefit to the juice industry for processing design with gallic acid implementation.      67   4.4 Limitations of the study The models proposed in this study are applicable for certain ranges of vitamin C and gallic acid concentration, which are 4080, and 0320 mg/100 mL, respectively. The models might not give an accurate predicted values if vitamin C and gallic acid concentration are beyond the ranges used in this study. In addition, cranberry juice in this study was prepared from only one source of commercial cranberry juice concentrate. Anthocyanin content and color intensity might differ according to different sources of cranberry juice, and some variations might be expected regarding application of proposed models in this study. However, these predictive models should give an idea of percent retention of ACY and color during storage, in which could be a guideline and thus reduce work in setting up new experiments.            68         CHAPTER 5 OBJECTIVE THREE Demonstrating practical application of results for the use in food industry, especially juice processor           69   5.1 Introduction Research findings would be very useful if they are applicable in real practice. The study in Chapter 3 revealed that gallic acid was an effective compound to mitigate color and anthocyanin degradation in cranberry juice fortified with vitamin C, and the models proposed in Chapter 4 are useful for predicting retention of anthocyanins and color in cranberry juice during storage. This Chapter provides examples of model application to illustrate the implementing of predictive models in process design. Efficacy of processing is commonly determined by the increase/decrease of percent retention. Therefore, in this study, protective ability of gallic acid over detrimental effect of vitamin C fortification is determined through an increase in retention of red color intensity and anthocyanin content in the cranberry juice. Retention ratio  of ACY and color in the juice can be calculated using Eq. (5.2), which is derived from differential equation of kinetic model (Eq. (5.1)). The explicit-model derivation is as follows:    Multiply to both sides:   (5.1) 70    Where t was total days of storage, t0 was initial storage day, C0, n, and k were estimated from predictive models proposed in Chapter 4, which were: 1. Anthocyanin predictive models: 1.1 Initial ACY concentration (CoACY)  1 = 28.242 = 8.71×10-23 = 1.06×10-24 = 8.08×10-5     (5.3)  1.2 Order of degradation reaction (nACY)         (5.4)  1.3 Degradation rate constant of ACY (kACY)  1 = 2 = 7.2875×10-23 = 1.2934×10-34 = 1.8544×10-55 = 3.7809×10-4   (5.5)  2. Red color intensity predictive models: 2.1 Initial color intensity (Cocolor)  1 = 1.962 = 2.49×10-33 = 8.91×10-44 = 1.96×10-6                (5.6)  2.2 Order of degradation reaction (ncolor)       (5.7) (5.2) 71   2.3 Degradation rate constant of color (kcolor)  1 = 2 = 2.0902×10-23 = 1.6303×10-34 = 1.6511×10-55 = 3.9769×10-46 = 2.3897×10-6                      (5.8)  Referring to experimental treatments in this study, ranges of vitamin C and gallic acid concentration used in the models were suggested to be between 4080, and 0320 mg/100 mL, respectively. Meanwhile, product shelf-life in this study (t) was 16 days and t0 was 1.   5.2 Example case study#1 in cranberry juice product. This product was made from cranberry juice concentrate, while juice in each bottle of 8 oz (240 mL) was fortified with 132 mg vitamin C. How much gallic acid would be recommended to maintain at least 50% color retention by the end of shelf-life? Solution procedures mg/100 mLC fortification at 132 mg/240 mL equals to 55 mg/100 mL.     72   A. Forward problem First of all, minimum and maximum color retention of the juice needed to be determined, to test whether 50% retention could be achieved by adding gallic acid. The gallic acid at 0 and 320 mg/100 mL are for minimum and maximum retention, respectively.  1A). At vitamin C = 55 mg/100 mL and gallic acid = 0 mg/100 mL, Cocolor (Eq. (5.6)), ncolor (Eq. (5.7)), and logkcolor (Eq. (5.8)) are: Cocolor =  1.78 AU ncolor =   3.2 logkcolor = 1.4140, hence kcolor = 0.0385 (AU)(1-n) day1 2A). Minimum color retention (Eq. (5.2)) = 0.4589, or 45.89% 3A). At vitamin C = 55 mg/100 mL and gallic acid = 320 mg/100 mL, repeat calculation in 1A. and 2A.  Cocolor =  2.04 AU ncolor =   3.2  logkcolor = 1.9316, hence kcolor = 0.0117 (AU)(1-n) day1 Maximum color retention = 0.6199, or 62.0% Therefore, 45.8962% is a range of color retention in cranberry juice with 55 mg/100 mL vitamin C fortification. According to the range, 50% color retention is within the limits.  The next step is to calculate what gallic acid concentration that would give 50% color retention. Equations (5.2), (5.6), (5.7), and (5.8) are needed in calculation. For Eq. (5.6) and (5.8), gallic acid is presented in many terms, which make it impossible to solve explicitly for gallic acid. 73   Therefore, this is a roots problem, and the solution to solve roots problem is giving an initial guess of gallic acid and iteratively solving until the guess value is close to actual value, such that function of value; f(x) ~ 0. B. Procedures for roots problem 1B). Use function fzero in MATLAB to solve roots problem, which is:  [x, fx] = fzero(function, x0)  X is location of the root, which gives an answer of gallic acid concentration (mg/100 mL)   fx is function evaluated at that root, the function for roots problem was set as follows:  Where, limit is the required color retention, retention is the value from iterative guessing of gallic acid, and f(x) = 0, meaning the gallic acid that gives difference between retention and limit close to zero. function is a function handle to the fx function, consists of Eq. (5.6), (5.7), and (5.8) for calculating retention (Eq. (5.2)). MATLAB syntax are shown in APPENDIX C. x0 is an initial guess of gallic acid concentration. 2B). The retention , Cocolor, kcolor, and ncolor are simultaneously calculated by guessing x0= 70 mg/100 mL.  3B). The results are: Gallic = 66.43 fx =   5.5511×10-17 74   In order to maintain 50% color retention in cranberry juice, gallic acid at 66.43 mg/100 mL needs to be added in the juice, which is fortified with vitamin C at 55 mg/100 mL. Therefore, each juice bottle (240 mL), which contains 132 mg vitamin C, needs gallic acid = 66.43 x 2.4  = 159.43 mg. The juice industry might be more interested in color retention than anthocyanin retention proposed ACY predictive models, which help predict ACY retention and the information could be useful for advertising and/or educating consumers regarding the health benefit components. The ACY retention in the juice adding gallic acid at 66.43 mg/100 mL can be calculated using Eq. (5.2), while CoACY, nACY, and kACY were calculated using Eq. (5.3), Eq. (5.4), and Eq. (5.5), respectively.  CoACY = 23.72  mg/L nACY = 2.49  logkACY = 2.8686, hence kACY = 0.0014 (mg/L)(1-n) day1 Therefore, there is 37.16% ACY retention in the cranberry juice (240 mL), which contains 132 mg fortified vitamin C and 159.43 (66.43 x 2.4) mg gallic acid.  5.3 Example case study#2 Since anthocyanins in cranberry juice help lower risk of human-diseases, a cranberry juice company would like to know how much anthocyanins have been lost in the current product (refer to the same juice product in Case study#1), and wondering if anthocyanin retention in the juice 75   could be increased to 50%. If yes, how much gallic acid will need to be added? (Note that juice was fortified with 132 mg vitamin C in each 240-mL bottle.) Solution procedures A. Forward problem Determine minimum and maximum ACY retention at gallic acid = 0 and 320 mg/100 mL, respectively. 1A). At vitamin C = 55 mg/100 mL, gallic acid = 0 mg/100 mL, CoACY, nACY and logkACY are calculated using Eq. (5.3), Eq. (5.4), and Eq. (5.5), respectively.  CoACY  =  21.8658 mg/L nACY =   2.49 logkACY = 2.7765, hence kACY = 0.0017 (mg/L)(1-n) day1 2A). Minimum ACY retention (Eq. 5.2)) = 35.43% Without gallic acid addition, the cranberry juice has lost approximately 65% of anthocyanins. 3A). At vitamin C = 55 mg/100 mL, gallic acid = 320 mg/100 mL, maximum ACY retention is calculated by repeating procedure 1A. and 2A. CoACY =  24.2236 mg/L nACY =   2.49 logkACY = 3.2201, hence kACY = 6.0241e-04 (mg/L)(1-n) day1 Maximum ACY retention = 0.5338, or 53.38% Thus, the target 50% ACY retention does not exceed the maximum limit. 76   B. Procedures for roots problem Similar to Case study#1, a roots problem is solved for gallic acid concentration that would give 50% ACY retention, whereas Eq (5.2), (5.3), (5.4), and (5.5) are used to solve for gallic acid. 1B). Use function fzero in MATLAB to solve roots problem, which is:  [x, fx] = fzero(function, x0)  X is location of the root, which gives an answer of gallic acid concentration (mg/100 mL)   fx is function evaluated at that root, the function for roots problem was set as follows:  Where, limit is the required ACY retention, retention is the value from iterative guessing of gallic acid, and f(x) = 0, meaning the gallic acid that gives difference between retention and limit close to zero. function is a function handle to the fx function, consists of Eq. (5.3), (5.4), and (5.5) for calculating retention (Eq. (5.2)). MATLAB syntax are shown in APPENDIX C. x0 is an initial guess of gallic acid concentration. 2B). The retention , CoACY, kACY, and nACY are simultaneously calculated by guessing x0= 70 mg/100 mL.  3B). The results are: Gallic = 282.3422 fx =   1.1102×10-16 77   Therefore, in order to maintain 50% ACY after storage, gallic acid at 282.34 × 2.4 = 677.62 mg will need to be added in the juice, which is fortified with 132 mg vitamin C. The above two case studies explained step-by-step calculation to address questions. However, overall trends of color and anthocyanin retention in cranberry juice were illustrated as 3D plots in Figure 5.1. Two main benefits of the 3D plots are: - To estimate maximum retention of color and ACY at any fortified vitamin C level in the juice. For instance, at 80 mg fortified vitamin C, maximum color retention is 49.19% (Figure 5.1A), whereas maximum ACY retention is 40.57% (Figure 5.1B). Therefore, both color and ACY retention in the juice with the 80 mg vitamin C would never reach 50%, even maximum gallic acid (320 mg) is added. - To identify vitamin C and gallic acid concentration that would give target retention of color and ACY.   5.4 Conclusions In summary, the model application provides predictive information, which would be useful for process design. The guideline obtained from models could save resources on doing experiments, especially money and time, and hence decisions or conclusions could be made rapidly. 78      Figure 5.1 Retention of color (A) and anthocyanin (B) in pasteurized cranberry juice after storage at 23 °C, 16 days, with addition of vitamin C (4080 mg/100 mL) and gallic acid (0320 mg/100 mL). 79         CHAPTER 6 OVERALL CONCLUSIONS AND FUTURE DIRECTIONS                    80   6.1 Overall conclusions The novel contributions of this study were: 1. It showed potential protective effect of gallic acid against detrimental effect of fortified vitamin C on color and anthocyanins in cranberry juice. Therefore, gallic acid is a potent natural antioxidant in addressing consumer complaints regarding color change in the juice.   2. It showed the feasibility of applying gallic acid in the juice industry with regard to its high water solubility, and desired lack of effect on color upon dissolving.   3. It explained that the most likely protective mechanism of gallic acid is neutralization of s.  4. It showed that parameters k and n of the kinetic model, which are degradation rate constant and order of degradation reaction, respectively, were highly correlated and cannot be estimated simultaneously.  5. It showed an innovative statistical method to estimate n-values at the smallest root-mean-square-error.  6. It is the first study to report that n-order for both anthocyanins and color varied in a wide range from 1.2 to 4.4, and decreased with increasing fortified vitamin C level.  7. It reported that n-order was a function of only vitamin C, but not gallic acid. In contrast, k and C0 were a function of both vitamin C and gallic acid.  81   8. It showed that proposed predictive models for k, n, and C0 accurately predicted color and anthocyanin retention, as evaluated from residual structure in residual plots and histogram.  9. It provided examples in model application, using roots problem solution technique, to calculate concentration of gallic acid that meets the goal of maintaining a minimum required color and/or anthocyanin retention in the juice.  6.2 Future directions The following topics are recommended for future study: 1. gallic acid addition.  2. Validation of secondary models with independent experiments on cranberry juice production.  3. This study was conducted with only cranberry juice, and thus all kinetic parameters (k, n, C0) estimated in the present study might not apply to other kinds of juice. Therefore, estimating kinetic parameters for other pigment-rich juices is recommended for  juices different from cranberry juice. The parameter estimation could be done by following procedures as used in this study, which are: 2.1 Estimate parameters k, n, C0 using ordinary least square (OLS) inverse method a. Check parameter correlation b. Check significance of parameters (CIs do not contain zero) c. Check error of parameters 82   2.2 Model predictive equations for k, n, C0 using multiple linear regression method a. Check error and AICc (corrected Akaike Information Criteria) of predictive models b. Check goodness of fit of the models via residual plots and histogram                83                APPENDICES   84   APPENDIX A  Experimental measurements for red color intensity, anthocyanins, and L-ascorbic acid content                         85   Table A1 Data of red color intensity, anthocyanins, and vitamin C in cranberry juice at day 0 after pasteurizing at 85 °C, the juice was not fortified with vitamin C and no antioxidant addition. Replication no. Color intensity (AU) Anthocyanins (mg/L) Vitamin C (mg/100 mL) 1 2.66 47.00 9.29 2 2.45 46.24 6.67 3 2.40 47.79 7.23 4 2.65 47.17 9.55 5 2.24 45.00 6.03 6 2.15 45.93 7.77 Avg ± SD 2.42 ± 0.21 46.52 ± 1.00 7.76 ± 1.41  Table A2 Data of red color intensity, anthocyanins, and vitamin C in cranberry juice pasteurizing at 85 °C, the juice was not fortified with vitamin C and no antioxidant addition. Storage (days) Color intensity (AU) Anthocyanins (mg/L) Vitamin C (mg/100 mL) 1 2.45 46.18 9.55  2.34 45.73 9.29 2 2.23 43.60 8.71  2.2 43.67 8.62 4 2.08 43.34 8.33  2.19 42.65 7.77 6 2.02 40.04 7.23  2.14 39.66 6.85 8 2.07 39.32 6.61  2.02 38.59 6.31 10 2.10 38.66 6.03  1.94 37.83 6.67 12 1.89 36.59 6.50  1.79 35.71 5.90 14 1.71 35.87 5.96  1.70 35.34 5.72 16 1.69 34.82 5.63  1.60 34.45 5.55 Avg ± SD 2.01 ± 0.24 39.6 ± 3.82 7.07 ± 1.22       86   Table A3 Data of red color intensity in cranberry juice fortified with vitamin C (4080 mg/100 mL) and gallic acid addition (0320 mg/100 mL). Vitamin C 40 mg/100 mL Storage (days) Color intensity (AU) GA-0mg GA-80mg GA-160mg GA-320mg 1 1.829 1.896 2.005 2.104  1.735 1.861 1.885 2.056  1.845 1.963 2.005 2.101  1.827 1.951 1.971 2.096 2 1.711 1.813 1.891 2.027  1.585 1.73 1.776 1.977  1.693 1.865 1.909 2.045  1.687 1.842 1.829 2.027 4 1.504 1.617 1.739 1.891  1.377 1.553 1.636 1.842  1.5 1.671 1.706 1.898  1.46 1.657 1.676 1.879 6 1.367 1.514 1.615 1.812  1.291 1.435 1.51 1.761  1.341 1.53 1.592 1.735  1.331 1.552 1.545 1.764 8 1.274 1.366 1.539 1.718  1.18 1.3014 1.426 1.663  1.215 1.432 1.502 1.688  1.21 1.433 1.446 1.665 10 1.231 1.34 1.468 1.681  1.13 1.266 1.387 1.628  1.172 1.333 1.414 1.604  1.147 1.402 1.359 1.606 12 1.161 1.273 1.427 1.611  1.083 1.246 1.327 1.552  1.111 1.303 1.355 1.574  1.136 1.363 1.328 1.571 14 1.109 1.251 1.378 1.571  1.053 1.194 1.292 1.528  1.073 1.269 1.296 1.5  1.093 1.305 1.3 1.49 16 1.085 1.206 1.308 1.493  1.007 1.162 1.245 1.466 87    Vitamin C 40 mg/100 mL Storage (days) Color intensity (AU) GA-0mg GA-80mg GA-160mg GA-320mg 16 1.022 1.209 1.263 1.498  1.041 1.251 1.21 1.446 Vitamin C 60 mg/100 mL 1 1.735 1.915 1.869 2.044  1.727 1.79 1.945 2.14  1.737 1.873 1.962 2.136  1.779 1.89 1.936 1.915 2 1.567 1.765 1.745 1.932  1.582 1.623 1.796 2.002  1.572 1.71 1.857 2.037  1.593 1.755 1.835 1.798 4 1.311 1.541 1.516 1.769  1.327 1.435 1.596 1.636  1.291 1.439 1.634 1.821  1.334 1.519 1.578 1.614 6 1.128 1.367 1.409 1.649  1.171 1.218 1.462 1.705  1.1 1.27 1.46 1.674  1.13 1.302 1.389 1.463 8 1.031 1.23 1.264 1.507  1.036 1.119 1.318 1.608  0.977 1.118 1.325 1.547  0.983 1.174 1.253 1.341 10 0.894 1.174 1.162 1.45  0.913 1.026 1.209 1.51  0.86 1.024 1.141 1.432  0.86 1.074 1.187 1.252 12 0.852 1.098 1.122 1.354  0.888 0.948 1.156 1.39  0.782 0.944 1.133 1.32  0.807 1.002 1.068 1.195 14 0.801 1.028 1.041 1.268  0.831 0.893 1.086 1.371  0.772 0.87 1.059 1.314  0.742 0.974 1.006 1.156 88    Vitamin C 60 mg/100 mL Storage (days) Color intensity (AU) GA-0mg GA-80mg GA-160mg GA-320mg 16 0.783 0.965 1.038 1.306  0.818 0.889 1.057 1.322  0.726 0.883 1.041 1.23  0.687 0.91 0.97 1.082 Vitamin C 80 mg/100 mL 1 1.681 1.746 1.896 1.975  1.735 1.862 2.04 1.997  1.747 1.761 1.799 1.886  1.678 1.833 1.851 1.921 2 1.481 1.615 1.766 1.871  1.555 1.718 1.891 1.878  1.533 1.661 1.782 1.745  1.43 1.655 1.755 1.779 4 1.183 1.325 1.504 1.68  1.226 1.399 1.611 1.642  1.222 1.362 1.452 1.511  1.115 1.356 1.489 1.549 6 1.039 1.105 1.349 1.558  1.026 1.227 1.418 1.49  0.993 1.093 1.264 1.319  0.926 1.168 1.31 1.359 8 0.911 1.007 1.18 1.401  0.881 1.054 1.233 1.338  0.867 0.92 1.162 1.178  0.821 0.95 1.208 1.262 10 0.812 0.843 1.031 1.296  0.751 0.955 1.142 1.242  0.753 0.793 1.052 1.057  0.705 0.888 1.049 1.143 12 0.748 0.809 0.99 1.152  0.68 0.835 1.042 1.175  0.667 0.749 0.979 0.996  0.644 0.821 1.02 1.037   89    Vitamin C 80 mg/100 mL Storage (days) Color intensity (AU) GA-0mg GA-80mg GA-160mg GA-320mg 14 0.668 0.714 0.898 1.132  0.631 0.782 0.95 1.095  0.639 0.694 0.874 0.936  0.573 0.705 0.923 0.971 16 0.662 0.734 0.876 1.085  0.586 0.726 0.918 1.037  0.625 0.638 0.883 0.837  0.552 0.695 0.886 0.834  Table A4 Data of anthocyanin content in cranberry juice fortified with vitamin C (4080 mg/100 mL) and gallic acid addition (0320 mg/100 mL). Vitamin C 40 mg/100 mL Storage (days) Anthocyanin content (mg/L) GA-0mg GA-80mg GA-160mg GA-320mg 1 22.94655 24.22136 29.76849 23.78493  23.46336 28.08023 26.63315 24.11799  26.70206 22.63646 32.14584 22.94655  22.77428 24.15245 26.49534 24.29026 2 20.74147 24.80708 21.15492 22.602  19.84566 22.25746 22.18855 21.67174  26.28861 21.3961 23.91127 22.91209  21.01711 21.98183 24.84153 26.08189 4 16.40023 18.74313 27.14997 22.73982  15.26324 18.39858 30.62985 20.01793  16.60696 17.7095 19.39776 19.94903  17.88177 20.46584 18.05404 20.39693 6 14.22962 17.53723 18.05404 20.43139  13.88507 15.36661 16.88259 21.36165  16.81369 18.57085 20.19021 23.98018  14.95316 17.77841 17.19268 20.29357 8 12.8859 15.6767 15.46997 16.98596  13.16153 13.98844 15.64224 17.15823  14.64307 16.1246 19.08767 22.1541  14.16071 14.95316 15.78006 17.05487 90    Vitamin C 40 mg/100 mL Storage (days) Anthocyanin content (mg/L) GA-0mg GA-80mg GA-160mg GA-320mg 10 13.54053 14.19516 15.81451 16.81369  10.95646 13.05817 12.61026 16.71032  12.67917 16.43469 15.33215 17.26159  14.67752 13.57498 15.19434 17.53723 12 10.43965 13.05817 15.78006 15.53888  10.64637 15.36661 13.09262 17.43386  14.36743 14.81534 14.67752 18.08849  14.95316 12.64472 15.09097 18.88094 14 10.4741 11.85227 12.43799 15.81451  10.4741 12.61026 13.54053 18.01959  11.64554 12.57581 13.91953 15.88342  14.98761 14.4708 14.95316 18.26077 16 8.78584 10.95646 12.43799 13.78171  10.26737 12.19681 15.40106 17.81286  10.40519 11.92118 15.26324 16.88259  12.40354 12.1279 13.60944 14.22962 Vitamin C 60 mg/100 mL 1 23.36 22.87764 24.08354 23.49782  22.18855 22.42973 23.15327 24.5659  18.39858 20.1213 23.53227 25.73734  18.36413 22.87764 26.32307 25.53062 2 19.39776 19.01876 20.53475 21.3961  18.43304 19.05321 27.08106 18.12295  22.63646 20.15575 22.56755 23.98018  20.70702 19.01876 21.01711 20.98265 4 14.26407 13.91953 17.43386 18.70867  14.81534 16.53805 17.3305 16.60696  16.88259 17.50277 17.15823 19.98348  14.12625 16.71032 17.84731 19.88012 6 13.05817 13.81616 14.50525 18.74313  11.23209 14.60861 15.46997 16.15906  12.81699 15.26324 15.46997 18.74313  12.33463 15.05652 16.9515 22.22301   91    Vitamin C 60 mg/100 mL Storage (days) Anthocyanin content (mg/L) GA-0mg GA-80mg GA-160mg GA-320mg 8 10.95646 11.85227 14.9187 14.5397  9.716106 11.74891 13.60944 14.43634  9.99174 13.12708 15.46997 20.22466  11.05982 11.301 14.57416 16.81369 10 7.063126 8.78584 12.95481 14.22962  11.57664 14.4708 12.26572 14.22962  8.820294 12.71363 13.54053 15.09097  9.612743 11.92118 14.71198 14.5397 12 6.477404 8.682477 12.36908 14.5397  7.855575 7.993392 10.99091 10.74973  4.789144 10.74973 13.95398 15.81451  10.8531 12.36908 15.88342 16.77923 14 7.614395 6.787492 8.854749 12.36908  6.959764 7.304306 8.613569 10.50855  7.476578 11.57664 11.16319 14.36743  6.718584 8.579114 11.02537 15.22879 16 7.338761 7.269852 8.820294 11.64554  6.132861 7.338761 8.30348 9.50938  7.924483 9.922831 9.647197 12.71363  7.166489 7.269852 10.8531 13.85062 Vitamin C 80 mg/100 mL 1 19.7423 17.26159 20.87929 21.46501  16.29687 22.63646 24.35917 21.3272  19.22549 24.22136 22.01628 22.36082  22.25746 26.80543 27.32224 28.42478 2 13.57498 17.02041 20.25911 19.98348  12.81699 16.91705 19.29439 18.50195  18.43304 18.57085 19.63894 20.25911  21.36165 23.77345 22.1541 23.98018 4 13.7128 13.50608 14.05734 18.26077  11.88672 12.40354 14.43634 17.88177  12.71363 18.26077 21.01711 19.81121  14.81534 20.70702 19.60448 18.77758   92    Vitamin C 80 mg/100 mL Storage (days) Anthocyanin content (mg/L) GA-0mg GA-80mg GA-160mg GA-320mg 6 10.40519 11.301 13.09262 21.43056  8.613569 11.68 15.02206 15.22879  11.74891 12.40354 16.88259 21.12047  11.23209 15.7456 16.60696 17.02041 8 7.097581 7.614395 12.71363 16.91705  8.992566 8.923657 12.43799 15.22879  4.789144 10.74973 13.95398 15.81451  10.8531 12.36908 15.88342 16.77923 10 5.891681 6.132861 8.648023 8.751386  7.993392 8.682477 10.0951 11.95563  7.132035 7.924483 12.1279 14.33298  7.82112 9.406017 14.40189 15.7456 12 3.686607 11.92118 8.337935 6.718584  3.824425 4.479056 9.785014 7.373215  7.683303 7.442123 10.33628 12.71363  6.615221 9.681651 10.30183 12.47245 14 3.996696 4.892507 8.0623 7.855575  3.204248 5.374867 9.061474 7.476578  5.030324 5.857227 8.78584 10.16401  6.20177 8.648023 9.543834 11.43882 16 4.823599 4.168967 7.132035 4.926961  3.54879 4.375693 8.0623 12.61026  4.341239 5.926135 7.132035 10.92201  5.443775 6.339587 9.233746 10.26737        93   Table A5 Data of L-ascorbic acid content in cranberry juice fortified with vitamin C (4080 mg/100 mL) and gallic acid addition (0320 mg/100 mL). Vitamin C 40 mg/100 mL Storage (days) Ascorbic acid content (mg/100 mL) GA-0mg GA-80mg GA-160mg GA-320mg 1 36.3229 41.13183 35.60204 42.47882  36.92952 32.80256 36.32256 43.47239  38.21195 33.11314 32.21028 33.45421  36.11892 34.45634 31.83397 33.2663 2 33.36274 29.72246 28.18709 32.68222  33.6729 22.03697 29.30677 28.75299  31.94378 32.51333 29.72681 29.19796  32.83359 31.77962 26.28412 28.85488 4 25.54045 25.27282 22.58175 24.10159  24.01687 15.88206 22.08526 23.60849  23.21415 22.70071 21.81643 21.54328  23.68709 23.38758 20.6925 20.28221 6 18.61035 15.57333 15.01102 15.93928  16.7043 10.16741 18.1156 14.69378  17.23206 14.32243 9.688326 12.04274  15.02702 14.23217 11.5714 11.70279 8 6.525108 6.295542 5.320056 4.666379  10.97347 4.491095 6.744318 6.443753  11.4693 11.61936 8.805205 9.745885  10.43647 10.08143 8.891652 6.821836 10 11.83682 12.82278 12.59733 14.31421  10.65812 12.50615 12.48349 12.25449  10.77112 7.370475 9.783123 6.898716  8.503306 6.812773 9.652785 7.053342 12 12.19921 12.01053 12.06532 12.1009  11.91086 11.94662 11.95495 11.66625  10.10084 8.699271 9.403349 8.565427  9.412753 8.597139 9.315885 8.963072 14 12.20504 11.61474 11.47486 11.34447  14.24295 12.3434 11.44224 12.7462  11.08961 9.401822 9.132461 8.717103  9.249405 8.162906 8.324431 8.715841   94    Vitamin C 40 mg/100 mL Storage (days) Ascorbic acid content (mg/100 mL) GA-0mg GA-80mg GA-160mg GA-320mg 16 5.01376 4.222359 4.257198 4.266516  4.391232 4.236779 3.872985 3.540066  10.22933 9.380121 9.620822 9.814859  10.16884 8.778902 9.284462 12.79641 Vitamin C 60 mg/100 mL 1 53.26456 55.19244 51.05578 60.3403  53.06407 59.16169 51.79537 48.74638  54.15472 52.54554 49.92 49.05714  52.32032 52.60621 50.03539 50.23886 2 48.49548 56.45525 48.07266 49.02292  52.54339 52.87737 50.3761 45.68101  48.98143 47.65177 44.11731 46.28749  47.36521 50.58611 48.67238 44.53013 4 42.09105 38.93365 40.41004 37.02235  45.35363 41.05715 40.66498 32.52461  41.81605 42.72694 36.14204 35.10553  40.71428 43.04518 37.94117 34.80431 6 32.32259 32.73227 29.01463 25.2124  32.77727 29.1968 30.18211 22.51106  31.91104 31.78803 30.22759 32.6144  29.20099 31.76333 33.57818 32.06954 8 30.42813 23.15081 20.12191 24.36153  26.2057 26.94568 20.44394 25.05665  23.88631 22.44871 25.23229 17.66713  23.49116 27.13346 25.76044 21.95771 10 17.75864 9.876488 14.16126 9.293497  15.65142 13.08254 10.28377 7.57621  19.93542 16.00908 14.56295 15.64509  20.84028 18.39063 17.16811 16.93603 12 17.41633 14.91641 14.1669 14.3175  14.6204 14.90733 18.25647 12.9838  15.85584 10.04533 11.34136 12.26397  12.39989 17.17749 11.8934 10.58831   95    Vitamin C 60 mg/100 mL Storage (days) Ascorbic acid content (mg/100 mL) GA-0mg GA-80mg GA-160mg GA-320mg 14 12.19908 12.6203 12.10529 12.2634  14.82886 12.17416 12.36419 12.59301  13.32676 10.36762 9.617971 10.43298  11.21497 10.79469 10.08815 9.326656 16 14.4353 12.41017 11.39981 10.69018  12.91019 12.29309 11.16125 10.27953  10.08432 9.064032 9.549488 13.0199  9.411888 9.460272 10.37045 12.44799 Vitamin C 80 mg/100 mL 1 63.74724 67.61069 87.24824 76.7644  62.3397 66.3471 61.30447 74.28407  68.21294 70.82319 66.41275 60.44126  67.15324 70.67804 67.83211 61.30195 2 61.2389 65.77362 69.7705 56.1496  61.06649 64.83855 64.93191 55.05276  63.12511 70.48266 63.69341 58.04704  65.1682 67.99291 63.92321 56.2545 4 54.61005 53.70349 31.75137 49.18951  52.95725 56.46677 27.88105 48.68375  55.63515 59.63047 56.40083 49.73936  58.69792 56.47696 56.21045 51.67425 6 50.52604 44.92376 50.63154 40.62398  49.42737 55.51184 43.00429 40.59164  52.32296 51.79776 50.01584 44.20368  49.4995 48.55869 48.03141 41.50566 8 46.56134 36.86035 34.62147 41.2536  47.21523 37.44363 36.69912 43.61777  41.69312 47.97312 37.63306 36.99927  46.92883 44.66159 35.23364 32.67509 10 33.15531 28.6903 33.01338 31.57013  33.2358 28.05638 27.30168 36.15037  34.56136 36.35589 32.3095 28.02707  41.24942 37.65857 25.78195 29.51492   96    Vitamin C 80 mg/100 mL Storage (days) Ascorbic acid content (mg/100 mL) GA-0mg GA-80mg GA-160mg GA-320mg 12 21.87883 16.14706 20.32687 13.57536  26.23138 23.31493 17.73734 13.74155  22.62652 19.05891 21.1077 22.13557  23.51296 22.13555 25.42336 17.84243 14 19.43074 22.04559 19.82995 16.10366  19.81042 21.96311 13.47741 20.61134  21.7281 20.71388 19.31578 18.11516  16.95148 21.92914 17.26456 21.54035 16 13.45866 17.88653 12.99896 12.53311  18.4545 16.82041 12.8103 14.01642  16.13667 10.89922 9.441339 16.74058  18.78586 12.9201 10.22739 15.8649                 97   APPENDIX B  Example MATLAB syntax and additional results for Chapter 4                 98   1. A syntax to estimate parameter n with smallest rmse  Example syntax to estimate parameter n for color intensity at one treatment (vitamin C 80, Gallic acid 320)  First function syntax: file name :  function y = forderdiff2Pnn(beta,t) %   called by inv_prob global len   tspan=t(1:len,1); n=t(1,2);%n comes from 2nd column [t,y]=ode45(@ff,tspan,beta(1));       function dy=ff(t,y) %function that computes the dydt                  dy(1)=-beta(2)*y(1).^n;     end t=[t;t;t;t]; y=[y;y;y;y]; end   Following syntax are to paste in MATLAB as a script file, while the previous function file is open:  clear  format compact %% Read in data global len  data =xlsread('data_4rep.xlsx'); x1=data(:,1); yobs1=data(:,2); yobs2=data(:,3); yobs3=data(:,4); yobs4=data(:,5); len=length(x1);  x=[x1;x1;x1;x1]; yobs=[yobs1;yobs2;yobs3;yobs4];   %% Initial parameter guesses   yo=2.0; k=0.0000005;%make smaller to be able to reach lowest rmse   beta0(1)=yo; %initial guess yo beta0(2)=k; %initial guess kr beta=beta0;%set beta=to the initial guesses   %% nlinfit returns parameters, residuals, Jacobian (sensitivity %coefficient matrix), %covariance matrix, and mean square error.  ode45 is solved many times iteratively  xn=x; %copy x into xn count =1; for n= 0.05:.05:5 xn(:,2)=n; % beta0(2)=beta0(2)*.8; [beta,resids,J,COVB,mse] = nlinfit(xn,yobs,@forderdiff2Pnn,beta0); beta rmse(count)=sqrt(mse) nx(count)=n; count = count +1; 99   end %% plot rmse versus n figure plot(nx,rmse,'-o') xlabel('n') ylabel('rmse') title('VitC80Gallic320')   2. Two-parameters (k and C0) estimation using Ordinary Least Square method, fixing n from the n with smallest rmse. Example syntax for estimating k and C0 for color at treatment with vitamin C 60 and Gallic acid 320  There are 3 function files: 2.1 Function forderdiff2P_for_SSC:  function y = forderdiff2P_for_SSC(beta,t) %first-order model, differential form %   called by inv_prob %global len   %tspan=t(1:len); tspan=t; [t,y]=ode45(@ff,tspan,beta(1));       function dy=ff(t,y) %function that computes the dydt         dy(1)=-beta(2)*y(1).^(2.9); %n from estimate n with lowest rmse         % avg for ACY n=0.35, 3.3, 2.2, 1.2 for vit 0,40,60,80, respectively         % avg for COLOR n=0.65,4.4, 2.9,2.2  for vit 0,40,60,80, respectively     end %t=[t;t;t;t]; y=[y;y;y;y]; end     2.2 Function  function y = forderdiff2P( beta,t ) %first-order model, differential form %   called by inv_prob global len   tspan=t(1:len);   [t,y]=ode45(@ff,tspan,beta(1));       function dy=ff(t,y) %function that computes the dydt                dy(1)=-beta(2)*y(1).^(2.9); % n from estimate n with lowest rmse     % avg n for ACY=0.35, 3.3, 2.2, 1.2 for vit 0,40,60,80, respectively         % avg n for COLOR=0.65, 4.4, 2.9,2.2 for vit 0,40,60,80, respectively     end t=[t;t;t;t]; y=[y;y;y;y]; 100   end  2.3 Function  function Xp = SSC_2P( beta, x, func ) %computes scaled sensitivity coefficients % uses forward-difference approximation % beta are the parameters % x is the independent variable % func is the model d=0.001; ypred=func(beta,x); figure for i = 1:length(beta)  %scaled sens coeff for forward problem             betain = beta; %reset beta             betain(i) = beta(i)*(1+d);             yhat{i} = func(betain,x);%function with only one perturbed parameter             Xp{i} = (yhat{i}-ypred)/d;%scaled sens coeff for ith parameter             ysensf=Xp{i}; hold on             h2(i) = plot(x,ysensf,'-b','LineWidth',2); end ysensf1=Xp{1}; ysensf2=Xp{2}; %extract data from cell array into vectors hold on YLine = [0 0];  XLine = [0 max(x)]; set(gca, 'fontsize',14,'fontweight','bold'); h2(1) = plot(x,ysensf1,'-b','LineWidth',2); h2(2) = plot(x,ysensf2,'-r','LineWidth',2); %h2(3) = plot(x,ysensf3,'-y','LineWidth',2);   legend(h2,'X''_{Co}','X''_k') xlabel('time (days)','fontsize',16,'fontweight','bold') ylabel('scaled sensitivity coefficient','fontsize',16,'fontweight','bold' ) plot (XLine, YLine,'k'); %plot a straight black line at zero grid on end    Syntax in script file:  %% example of nlinfit using file name = inv_soln_first_2P.m %This program can be used as a base for most nonlinear regression OLS %using nlinfit %% Housekeeping % clear all; % Clear the workspace. close all; % Close all figures. format compact %% Read in data global len   data =xlsread('data_4rep.xlsx'); x1=data(:,1); yobs1=data(:,2); yobs2=data(:,3); yobs3=data(:,4); yobs4=data(:,5); len=length(x1); 101     x=[x1;x1;x1;x1]; yobs=[yobs1;yobs2;yobs3;yobs4];   %% Initial parameter guesses yo=2; k=0.00005; % if error happen, make initial k very small i.e. 0.00005  beta0(1)=yo; %initial guess yo beta0(2)=k; %initial guess kr %beta0(2)=n; beta=beta0;%set beta=to the initial guesses   %% X' = scaled sensitivity coefficients using forward-difference % This is a forward problem with known approximate parameters xs=linspace(min(x1),max(x1),500); Xp=SSC_2P(beta, xs, @forderdiff2P_for_SSC); title('scaled sensitivity coefficients using initial guesses')   %% nlinfit returns parameters, residuals, Jacobian (sensitivity %coefficient matrix), %covariance matrix, and mean square error.  ode45 is solved many times iteratively   [beta,resids,J,COVB,mse] = nlinfit(x,yobs,@forderdiff2P,beta0); beta rmse=sqrt(mse) condX=cond(J) detXTX=det(J'*J) %type forderdiff2P.m  %uncomment if you wish to print the code of the function %% confidence intervals for parameters ci=nlparci(beta, resids,J)   %% R is the correlation matrix for the parameters, sigma is the standard error vector [R,sigma]=corrcov(COVB); R sigma relerr=sigma./beta' %relative error for each parameter   %% Confidence and prediction intervals for the dependent variable   %nonlinear regression confidence intervals-- 'on' means simultaneous %bounds; 'off' is for nonsimultaneous bounds; must use 'curve' for %regression line, 'observation' for prediction interval [ypred, delta] = nlpredci('forderdiff2P',x,beta,resids,J,0.05,'on','curve'); %confidence band for regression line [ypred, deltaob] =nlpredci('forderdiff2P',x,beta,resids,J,0.05,'on','observation');%prediction band for individual points   yspan=range(ypred)% total span of ypred relrmse=rmse/yspan % ratio of rmse vs. yspan   %simultaneous confidence bands for regression line CBu=ypred+delta; CBl=ypred-delta;   %simultaneous prediction bands for regression line PBu=ypred+deltaob; PBl=ypred-deltaob;   %% Output--ypred and yobs vs. t 102     figure  hold on h1(1)=plot(x(1:len),ypred(1:len),'-','linewidth',3); %predicted y values h1(2)=plot(x,yobs,'square', 'Markerfacecolor', 'r'); xlabel('time (days)','fontsize',16,'fontweight','bold') ylabel('Color (AU)','fontsize',16,'fontweight','bold') grid on   %% Output --CIs and PIs   %plot Cobs, Cpred line, confidence band for regression line h1(3) = plot(x(1:len),CBu(1:len),'--g','LineWidth',2); plot(x(1:len),CBl(1:len),'--g','LineWidth',2);   %plot prediction band for regression line h1(4) = plot(x(1:len),PBu(1:len),'-.','LineWidth',2); plot(x(1:len),PBl(1:len),'-.','LineWidth',2); legend(h1,'ypred','yobs','CB','PB')   %% residual scatter plot   figure hold on plot(x, resids, 'square','Markerfacecolor', 'b'); YLine = [0 0];  XLine = [0 max(x)]; plot (XLine, YLine,'R'); %plot a straight red line at zero ylabel('Observed y - Predicted y','fontsize',16,'fontweight','bold') xlabel('time (days)','fontsize',16,'fontweight','bold') grid on   %% number of runs = number of times moving from one residual to the next crosses zero n=length(yobs) rescross=resids(2:n).*resids(1:n-1);%multiply each pair of residuals res_sign=sign(rescross);%get the sign of each multiplied pair count=0; for i=1:n-1     if res_sign(i)<0 %if product of pair is < 0, that's a run         count=count+1;     end end fprintf('number of runs = %5.2f\n',count); minrun=(n+1)/2; %count should be >=minrun fprintf('Minimum required number of runs = %5.2f\n',minrun); %% residuals histogram--same as dfittool, but no curve fit here   [n1, xout] = hist(resids,10); %10 is the number of bins  figure  hold on  set(gca, 'fontsize',14,'fontweight','bold');  bar(xout, n1) % plots the histogram  xlabel('Y_{observed} - Y_{predicted}','fontsize',16,'fontweight','bold')  ylabel('Frequency','fontsize',16,'fontweight','bold')    Mean_of_error=mean(resids) % the closer to zero, the better-- prove assumption2 (no measurement error) %% scaled sensitivity coefficients using final estimated parameters % This is a double-check to make sure X' has not changed much   Xp=SSC_2P(beta,xs,@forderdiff2P_for_SSC); title('scaled sensitivity coefficients using beta estimates')   103   %% AIC analysis N=length(x); P=length(beta); K=P+1; SS=resids'*resids; AIC=N*log(SS/N)+(2*K) AIC_correct=AIC+(((2*K)*(K+1))/(N-K-1))    3. 3D plot for logkcolor predictive model (Figure 4.5B) Syntax in script file: %multiple linear regression clear all close all data=xlsread('K_color.xlsx','logK_color_n'); vitC=data(:,1); gallic=data(:,2); logk_color=data(:,3); vitCadj=data(:,4); gallicAdj=data(:,5); size(vitC); %X=[ones(size(logvitC)) logvitC gallic]; %X=[ones(size(vitC)) vitC gallic vitCadj.*gallicAdj]; %X=[ones(size(vitC)) vitC gallic vitCadj.*gallicAdj gallicAdj.*gallicAdj]; %X=[ones(size(vitC)) vitC gallic vitCadj.*gallicAdj vitCadj.*vitCadj]; X=[ones(size(vitC)) vitC gallic vitCadj.*gallicAdj vitCadj.*vitCadj gallicAdj.*gallicAdj]; [b, bint,r, rint, stats] = regress(logk_color,X);  format short e format compact b bint format short mse=stats(4); rmse=sqrt(mse) yspan=max(logk_color)-min(logk_color) Rsqr=stats(1)   %correlation of parameters covb=mse.*((X.'*X))^(-1); [R,sigma]=corrcov(covb)   %% AIC analysis N=length(X); P=length(b); K=P+1; resids=r; SS=resids'*resids; AIC=N*log(SS/N)+(2*K) AIC_correct=AIC+(((2*K)*(K+1))/(N-K-1))   %Adjusted Rsqaure SS_error=SS; y=logk_color; y_bar=mean(y); SS_total=sum((y-y_bar).^2); Rsq_adj = 1 - (N-1)./(N-P).*(SS_error./SS_total) %% scatter 3D plot figure scatter3(vitC,gallic,logk_color,'filled') hold on x1fit = min(vitC):2:max(vitC); x2fit = min(gallic):2:max(gallic); 104   [X1FIT,X2FIT] = meshgrid(x1fit,x2fit); % y = b(1) + b(2)*x1 + b(3)*x2 + b(4)*x1*x2 standard statistical model %YFIT = b(1) + b(2)*X1FIT + b(3)*X2FIT; %YFIT = b(1) + b(2)*X1FIT + b(3)*X2FIT + b(4)*(X1FIT-60).*(X2FIT-140); %YFIT = b(1) + b(2)*X1FIT + b(3)*X2FIT + b(4)*(X1FIT-60).*(X2FIT-140)+ b(5)*((X2FIT-140).^2); %YFIT = b(1) + b(2)*X1FIT + b(3)*X2FIT + b(4)*(X1FIT-60).*(X2FIT-140)+ b(5)*((X1FIT-60).^2); YFIT = b(1) + b(2)*X1FIT + b(3)*X2FIT + b(4)*(X1FIT-60).*(X2FIT-140)+ b(5)*((X1FIT-60).^2)+ b(6)*((X2FIT-140).^2); mesh(X1FIT,X2FIT,YFIT); xlabel('Vitamin C (mg/100mL)') ylabel('Gallic acid (mg/100mL)') zlabel('logk_{color} (AU^{(1-n)} day^{-1}) (n=4.4,2.9,2.2)') %title('3D Plotting') view(50,10) hold off   4. Model fitting for color retention (Figure 4.6B) Syntax in script file: clear  close all   %% Model fitting Vit C 40,Gallic 0 to 320 data=xlsread('colorData.xlsx','colorV40'); storage=data(:,1); v40G0=data(:,2); v40G80=data(:,3); v40G160=data(:,4); v40G320=data(:,5); figure hold on plot(storage,v40G0,'mo'); plot(storage,v40G80,'r*'); plot(storage,v40G160,'x','MarkerSize',10); plot(storage,v40G320,'^','Markerfacecolor','yellow'); legend('GA, 0 mg','GA, 80 mg','GA, 160 mg','GA, 320 mg','location','best') xlabel('storage (days)','fontsize',12,'fontweight','bold') ylabel('Red color intensity (AU)','fontsize',12,'fontweight','bold') title('Vit C 40'); axis([0 22 0.8 2.2]); text(17.5, 1.1,'Pred (GA, 0 mg)'); text(17.5, 1.3,'Pred (GA, 80 mg)'); text(17.5, 1.5,'Pred (GA, 160 mg)'); text(17.5, 1.6,'Pred (GA, 320 mg)');   % Predicted ACY vitC 40   C0color=xlsread('Co_color.xlsx');   for i=1:4 [t, c]=ode45(@color_pred, [1 17],[C0color(i)],[],i,1); hold on plot(t,c,'-','linewidth',1); clear t c end     %% Model fitting Vit C 60, Gallic 0 to 320 clear data=xlsread('colorData.xlsx','colorV60'); storage=data(:,1); 105   v60G0=data(:,2); v60G80=data(:,3); v60G160=data(:,4); v60G320=data(:,5); figure hold on plot(storage,v60G0,'mo'); plot(storage,v60G80,'r*'); plot(storage,v60G160,'x','MarkerSize',10); plot(storage,v60G320,'^','Markerfacecolor','yellow'); legend('GA, 0 mg','GA, 80 mg','GA, 160 mg','GA, 320 mg','location','best') xlabel('storage (days)','fontsize',12,'fontweight','bold') ylabel('Red color intensity (AU)','fontsize',12,'fontweight','bold') title('Vit C 60'); axis([0 22 0.5 2.2]); text(17.5, 0.8,'Pred (GA, 0 mg)'); text(17.5, 1,'Pred (GA, 80 mg)'); text(17.5, 1.2,'Pred (GA, 160 mg)'); text(17.5, 1.4,'Pred (GA, 320 mg)');   % Predicted ACY vitC 60   C0color=xlsread('Co_color.xlsx');   for i=5:8 [t, c]=ode45(@color_pred, [1 17],[C0color(i)],[],i,2); hold on plot(t,c,'-','linewidth',1); clear t c end   %% Model fitting Vit C 80, Gallic 0 to 320 clear data=xlsread('colorData.xlsx','colorV80'); storage=data(:,1); v80G0=data(:,2); v80G80=data(:,3); v80G160=data(:,4); v80G320=data(:,5); figure hold on plot(storage,v80G0,'mo'); plot(storage,v80G80,'r*'); plot(storage,v80G160,'x','MarkerSize',10); plot(storage,v80G320,'^','Markerfacecolor','yellow'); legend('GA, 0 mg','GA, 80 mg','GA, 160 mg','GA, 320 mg','location','best') xlabel('storage (days)','fontsize',12,'fontweight','bold') ylabel('Red color intensity(AU)','fontsize',12,'fontweight','bold') title('Vit C 80'); axis([0 22 0.3 2.2]); text(17.5, 0.6,'Pred (GA, 0 mg)'); text(17.5, 0.8,'Pred (GA, 80 mg)'); text(17.5, 1,'Pred (GA, 160 mg)'); text(17.5, 1.2,'Pred (GA, 320 mg)');   % Predicted ACY vitC 80   C0color=xlsread('Co_color.xlsx');   for i=9:12 [t, c]=ode45(@color_pred, [1 17],[C0color(i)],[],i,3); hold on plot(t,c,'-','linewidth',1); clear t c 106   end  5. Residual plot (Figure 4.7B) Syntax in script file: %% vitc 60 clear all data=xlsread('colorData.xlsx','colorV60'); storage=data(:,1); v60G80=data(:,3); %% plot residuals (predicted-observed)V60  t=[1 2 4 6 8 10 12 14 16 1 2 4 6 8 10 12 14 16 1 2 4 6 8 10 12 14 16 1 2 4 6 8 10 12 14 16]; logkcolorV60G80=b1+(b2.*vitc2)+(b3.*Gallic2)+(b4.*(vitc2-60)*(Gallic2-140))+ (b5.*(vitc2-60)*(vitc2-60))+(b6.*(Gallic2-140)*(Gallic2-140)); kcolorV60G80 = 10.^logkcolorV60G80; predictedV60G80=((n-1).*kcolorV60G80.*(t-to) + Co2.^(1-n)).^(1/(1-n)); obsV60G80=v60G80.'; residV60G80=predictedV60G80-obsV60G80; figure; hold on plot(t,residV60G80,'*'); YLine= [0 0]; XLine= [0 max(t)]; plot(XLine, YLine,'R'); xlabel('storage (days)','fontsize',12,'fontweight','bold') ylabel('Color_{observed} - Color_{predicted}','fontsize',12,'fontweight','bold') title('Residual plot Vit60Gallic80') hold off    107     Figure B1 Plot of RMSE versus n (reaction order) of anthocyanins (A) and color (B) in cranberry juice with vitamin C (4060 mg/100 mL) and gallic acid (80 mg/100 mL), during 16-day storage at 23 ºC.  108     Figure B2 Plot of RMSE versus n (reaction order) of anthocyanins (A) and color (B) in cranberry juice with vitamin C (4060 mg/100 mL) and gallic acid (160 mg/100 mL), during 16-day storage at 23 ºC.  109     Figure B3 Plot of RMSE versus n (reaction order) of anthocyanins (A) and color (B) in cranberry juice with vitamin C (4060 mg/100 mL) and gallic acid (320 mg/100 mL), during 16-day storage at 23 ºC.   110   Table B1 Summary of n-values with smallest RMSE of all 12 treatments for color and anthocyanins. Vitamin C (mg/100mL) Gallic acid (mg/100mL) Color Anthocyanins n rmse n rmse 40 0 4.25 0.03949 3.65 1.7281   80 4.55 0.04826 3.25 1.2703   160 4.35 0.04384 3.35 2.6143   320 4.45 0.02641 2.95 1.6144 Average 4.4   3.3   60 0 2.55 0.03466 1.75 1.5841   80 2.85 0.05606 1.85 1.4558   160 2.75 0.03475 2.35 1.6091   320 3.45 0.09398 2.65 1.9869 Average 2.9   2.15   80 0 2.35 0.042204 1.45 1.9023   80 2.05 0.049306 1.45 2.4894   160 2.25 0.054702 1.55 1.8463   320 2.15 0.08425 0.55 2.5833 Average 2.2   1.25    Calculate nACY and ncolor at different vitamin C concentration using following equations;         (4.5)      (4.6)    111   Table B2 The nACY and ncolor calculated from Eq. (4.5) and Eq. (4.6), respectively. Vitamin C nACY   ncolor 40 3.3 4.4 60 2.2 2.9 80 1.2 2.2  Table B3 Two parameter (k, C0) estimation for anthocyanins in cranberry juice fortified with vitamin C (40-80 mg/100 mL) and gallic acid (0-320 mg/100 mL), during 16-day storage at 23 °C. Vitamin C (mg/100 mL) Gallic acid (mg/100 mL) n (fixed) CoACY ;  %Error  kACY ;     %Error 40 0 3.3    24.16;     3.10%    1.0410-4 ;    8.51%  80 3.3    24.89;     2.12%   7.4910-5;     6.22%  160 3.3    28.14;     3.93%   6.3210-5;    11.16%  320 3.3    24.04;     2.40%   3.0410-5;    10.60% 60 0 2.2    21.70;     3.11%   4.1010-3;     7.21%  80 2.2    21.97;     2.65%   2.9010-3;     6.61%  160 2.2    24.37;     2.58%   2.3010-3;     6.66%  320 2.2    23.77;     3.06%   1.5010-3;     9.53% 80 0 1.2    19.05;     3.76%   6.9610-2;     7.70%  80 1.2    22.14;     4.17%   6.1410-2;     8.74%  160 1.2    22.79;     2.87%   4.3810-2;     6.81%  320 1.2    23.31;     3.85%   3.6810-2;     9.97%     112   Table B4 Two parameter (k, C0) estimation for color in cranberry juice fortified with vitamin C (40-80 mg/100 mL) and gallic acid (0-320 mg/100 mL), during 16-day storage at 23 °C. Vitamin C (mg/100 mL) Gallic acid (mg/100 mL) n (fixed) Cocolor ;  %Error  kcolor ;     %Error 40 0 4.4    1.82;     0.92%    1.4710-2 ;    3.17%  80 4.4    1.93;     1.01%   8.4010-3;     3.91%  160 4.4    1.97;     0.88%   6.8010-3;     3.59%  320 4.4    2.09;     0.47%   3.6010-3;     2.34% 60 0 2.9    1.77;     0.88%   4.9210-2;     2.33%  80 2.9    1.88;     1.20%   3.2710-2;     3.45%  160 2.9    1.94;     0.78%   2.4810-2;     2.40%  320 2.9    2.05;     1.71%   1.5410-2;     6.25% 80 0 2.2    1.71;     1.02%   7.8510-2;     2.40%  80 2.2    1.83;     1.10%   6.1410-2;     2.68%  160 2.2    1.92;     1.08%   4.1010-2;     2.99%  320 2.2    1.95;     1.60%   3.3810-2;     4.75%           113   Developing secondary model for CoACY using multiple linear regression in JMP software An empirical polynomial equation is;  Table B5 Model comparison using p-value of parameters, and AICc as criteria. Term Parameters p-value,  = 0.05 AICC Model#1 Intercept vitC Gallic (vitC-60) x (Gallic-140) (vitC-60)2 (Gallic-140)2  1 = 27.83 2 -2 3 = 1.06 x 10-2 4 = 2.866 x 10-4 5 = 1.53 x 10-3 6 -5   < 0.0001 0.0033 0.0106 0.1157 (not sig diff) 0.3750 (not sig diff) 0.0227  68.76 Model#2 Intercept vitC Gallic (Gallic-140)2  1 = 28.24 2 -2 3 = 1.06 x 10-2 6  8.08 x 10-5   < 0.0001 0.0033 0.0121 0.0278  53.24 Model#3 Intercept vitC Gallic   1 = 27.69 2 -2 3 = 6.41 x 10-3   < 0.0001 0.0108 0.1217 (not sig diff)  54.66  Model#2 has the lowest AICC among those three models, and all parameters in the mmodel#2 are significant different from zero. Therefore, model#2 is the best model for CoACY.         114   APPENDIX C  MATLAB syntax for Chapter 5                 115    1. problem in Case study#1: function x = color_func( Gallic, limit ) limit = 0.5; % require 50% color retention  vitC = 55; % mg/100 mL fortified concentration in juice t= 16; %end of storage (day) to=1; %start storage at day 1 % Eq. (5.7) n-order as a function of vitamin C n=6.2-0.055*(vitC)+0.001*(vitC-60).^2;   b1=  -2.6121e+00; b2=   2.0902e-02; b3=  -1.6303e-03; b4=   1.6511e-05; b5=  -3.9769e-04; b6=   2.3897e-06; % Eq. (5.8) logkcolor as a function of vitamin C and gallic acid  logkcolor=b1+(b2.*vitC)+(b3.*Gallic)+(b4.*(vitC-60)*(Gallic-140))+ (b5.*(vitC-60)*(vitC-60))+(b6.*(Gallic-140)*(Gallic-140)); kcolor = 10.^logkcolor;   c1=  1.96;  c2= -2.49e-03; c3= 8.91e-04; c4= -1.96e-06;  % Eq. (5.6) Co(color) as a function of vitamin C and gallic acid Co= c1+(c2.*vitC)+(c3.*Gallic)+(c4.*(Gallic-140)*(Gallic-140));    %retention =C/Co; retention = [((kcolor*(n-1)*(t-to))/(Co^(1-n)))+1]^(1/(1-n)); function = limit - retention;   end The statement (script file) to solve the roots problem is: Xo = 70; % initial guess of gallic acid that gives 50% retention (C/Co=0.5) [Gallic, fx] = fzero(@(Gallic)color_func(Gallic), Xo)  2.  function x = acy_func( Gallic, limit ) limit = 0.5; % require 50% ACY retention  vitC = 55; % mg/100 mL fortified concentration in juice t= 16; %end of storage (day) to=1; %start storage at day 1 % Eq. (5.4) n-order as a function of vitamin C n=5.31-0.0513*(vitC);   b1=  -6.7882e+00; b2=   7.2875e-02; b3=  -1.2934e-03; 116   b4=   1.8544e-05; b5=  -3.7809e-04;  % Eq. (5.5) logkacy as a function of vitamin C and gallic acid logkacy=b1+(b2.*vitC)+(b3.*Gallic)+(b4.*(vitC-60)*(Gallic-140))+ (b5.*(vitC-60)*(vitC-60)); kacy = 10.^logkacy;   c1=  28.24;  c2= -8.71e-02; c3= 1.06e-02; c4= -8.08e-05; % Eq. (5.3) Co(acy) as a function of vitamin C and gallic acid Co= c1+(c2.*vitC)+(c3.*Gallic)+(c4.*(Gallic-140)*(Gallic-140));     %retention =C/Co; retention = [((kacy*(n-1)*(t-to))/(Co^(1-n)))+1]^(1/(1-n)); x = limit - retention; end The statement (script file) to solve roots problem is: Xo = 70; % initial guess of gallic acid that gives 50% retention (C/Co=0.5) [Gallic, fx] = fzero(@(Gallic)acy_func(Gallic), Xo)  3. 3D plot color retention (Figure 5.1A) Syntax in script file:  vitC = linspace(40,80,4)'; Gallic = linspace(0,320,4)';   t= 16; %end of storage (day) to=1; %start storage at day 1 n=6.2-0.055.*(vitC)+0.001.*(vitC-60).^2; % n-order as a function of vitamin C   b1=  -2.6121e+00; b2=   2.0902e-02; b3=  -1.6303e-03; b4=   1.6511e-05; b5=  -3.9769e-04; b6=   2.3897e-06;   c1=  1.96;  c2= -2.49e-03; c3=  8.91e-04; c4= -1.96e-06; Co= c1+(c2.*vitC)+(c3.*Gallic)+(c4.*(Gallic-140).*(Gallic-140)); % Co_color regression   figure x1fit = min(vitC):1:max(vitC); x2fit = min(Gallic):10:max(Gallic); [X1FIT,X2FIT] = meshgrid(x1fit,x2fit);   logkcolor=b1+(b2.*X1FIT)+(b3.*X2FIT)+(b4.*(X1FIT-60).*(X2FIT-140))+ (b5.*(X1FIT-60).*(X1FIT-60))+(b6.*(X2FIT-140).*(X2FIT-140)); kcolor = 10.^logkcolor; n=6.2-0.055.*(X1FIT)+0.001.*(X1FIT-60).^2; 117   Co= c1+(c2.*X1FIT)+(c3.*X2FIT)+(c4.*(X2FIT-140).*(X2FIT-140)); YFIT = 100.*[((kcolor.*(n-1).*(t-to))./(Co.^(1-n)))+1].^(1./(1-n)); % YFIT = retention mesh(X1FIT,X2FIT,YFIT); axis([40 80 0 320 20 80]) xlabel('Vitamin C (mg/100 mL)') ylabel('Gallic acid (mg/100 mL)') zlabel('Color retention (%)') %title('3D Plotting') view(50,10)                  118                REFERENCES               119   REFERENCES Alfa Laval, Inc. 2016. 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