Composite algorithms for predicting the rate of oxygen uptake and loss of shelf life of dry foods were developed based on basic mechanisms and kinetics of food.
Predicting Algorithms for Oxygen Uptake and Shelf Life of Dry Foods and the Application to Coffee.
C. Cardelli and T. P. Labuza.
C. Cardelli is with ConAgra
San Diego CA.
T. P. Labuza is at Department of Food Science and Nutrition, University of Minnesota, 1334 Eckles Ave., St. Paul, MN 55108.
Abstract Composite algorithms for predicting the rate of oxygen uptake and loss of shelf life of dry foods were developed based on basic mechanisms and kinetics of food deterioration as affected simultaneously by oxygen, aw and temperature. Temperature effect on reaction rate was modeled by the classical Arrhenius relationship but included the linear effect of aw increase on both the activation energy and pre-exponential factor. The rate of oxidation was modeled as a hyperbolic function of oxygen partial pressure. The three individual effects were fit into composite algorithms that fit very well to experimental data for roasted and ground coffee (r2= 0.98). Oxygen had the strongest impact on shelf life loss and oxygen uptake (Qo2= 1.1-10.5), followed by aw (Qa= 1.11.6) and temperature (Q10= 1.1-1.2). The proposed algorithms could be applied to other dry foods where shelf life is affected by oxygen, temperature and moisture simultaneously.
Much of the research on the understanding of basics mechanisms of deterioration in dehydrated foods was initiated by Prof. Marcus Karel in the sixties at MIT under NASA space food programs. There, Labuza and collaborators created a map for the rate of food deteriorating reactions as a function of water activity (aw)1. This map allows one to identify the main reactions of deterioration likely to occur in dehydrated foods, e.g. lipid oxidation and non-enzymatic browning (NEB). Further efforts were concentrated on modeling the kinetics of food deterioration as a function of oxygen partial pressure, temperature and aw. These are the controlling factors that may change during the storage and distribution of dehydrated foods stored in packaging semi-permeable to oxygen and water vapor 2. The approach for modeling has been to use well characterized equations derived from basic physical-chemical principles applying them to the study of deteriorating reactions in model systems and foods3,4. Labuza5 critically reviewed the basic mechanisms of lipid oxidation and proposed a simplified model for the rate of oxygen uptake and shelf life deterioration in foods as a function of oxygen partial pressure. This model has been used to study the oxidation of freeze dried shrimp3, freeze dried beef and chicken stew6 and potato chips4. The Arrhenius equation has been extensively used to model the effect of temperature on food deteriorating reactions2,7,8,9,10,11. More recently, the WLF equation has been proposed to model the effect of temperature on the rate of food deteriorating reactions in polymeric foods in the rubbery state12. Nelson and Labuza13 have shown that the Arrhenius equation is a better model over the narrow temperature range of storage for most dehydrated foods. The effect of moisture content on the rate of reactions of food deterioration has been more difficult to model. An increase in moisture first dissolves more reactants but then promotes dilution effects in the liquid phase14. Higher moisture decreases local viscosity, increases mobility and plasticizes bio-polymers promoting glass to rubber transitions. These effects complicate the kinetics and forced scientists to use pure curve fitting techniques to predict the rate of deterioration as a function of a w2. There has been intense debate about whether a linear a w model or the glass transition model
is more appropriate to describe the effect of moisture content on physical chemical changes in foods. However, both of these theories can be used in conjunction when modeling food deterioration15. Only a few papers have been published with algorithms for shelf life prediction of dry foods as a function of more than one environmental factor. Quast et al. 4 used basic kinetics equations and curve fitting techniques to create an algorithm for the rate of oxygen uptake of potato chips as a function of headspace O2 concentration, %RH and the extent of oxidation. Quast and Karel16 combined this work with that of Simon et al.3 and predicted the amount of rancidity and moisture gain in potato chips packaged in plastic pouches, as a function of headspace O2 concentration, %RH and extent of oxidation. Villota et al.17 used curve fitting and physical-chemical principles to build an algorithm for the prediction of the shelf life of dehydrated vegetables as a function of temperature and moisture. Labuza et al.18,19 described the basic physical-chemical principles to model moisture gain in packaged foods as a function of %RH. Cardoso and Labuza20 applied these principles to the modeling of moisture gain in packaged pasta as a function of variable external %RH and temperature. Kamman and Labuza21 used these data to predict thiamin loss in the same pasta during storage under variable time/ temperature/ humidity conditions. The objective of this paper was to develop algorithms for the prediction of the rate of oxygen uptake and the shelf life of dry foods as a function of oxygen partial pressure, temperature and aw based on basic physical-chemical principles. Roasted and ground (R/G) coffee was used as a representative food because of its high sensitivity to atmospheric oxygen22,23,24,25,26. Experimental data for R/G coffee was used to test the algorithms.
Models for the Effect of O2 on the Kinetics of Food Deterioration Foods susceptible to lipid oxidation are usually stored in packages with low permeability to oxygen. Under these conditions, oxygen level in the headspace is the limiting factor controlling the rate of lipid oxidation. The uptake of oxygen (Xt) during the storage time (t) can be modeled following apparent-zero order kinetics (Eq. 1). The rate of oxygen uptake (Ro2) is a function of the partial pressure of oxygen (po2) in the head space and can be modeled by the hyperbola in Eq. (2)5. This hyperbola begins at the
origin and becomes asymptotic to k1, as in Eqns. (3) and (4) respectively. The parameters k1 and k2 are proportional to the rates of lipid initiation and chain propagation, respectively. These parameters are found from Eq. (2) by non-linear regression or by converting this hyperbola into the straight line form shown in Eq. (5) and applying linear regression27. Xt = R o . t
(1)
2
R O 2 = k1
R O2
min
R O2
max
1 Ro2
=
=
p O2
(2)
k2 + p O 2
lim R p O2 ♦ 0
=
=0
lim R = k1 pO 2 ♦ ×
1 k2 1 + k1 k p O2 1
(3)
(4)
(5)
When dry foods are stored at a constant po2 until the end of the shelf life (θs), a critical amount of O2 (Xo) is taken up by the product according to Eq. (1). Thus, θs is normally inversely proportional to Ro2 (Eq. 6). For different levels of po2 , then θs follows Eq. (7). Usually Xo is unknown and it is incorporated as a derived constant into the model as in Eq. (8). The parameters k3 and k4 are proportional to the rate of lipid initiation and chain propagation, respectively, and are determined by linear regression. The θs tends to infinity at po2 = 0 if other reactions of deterioration are not present, and becomes asymptotic to a minimum value equal to 1/k3 at increasing po2 (Eqns. 9 and 10, respectively).
There is only one reference in the literature, published by Heiss et al.25, who reported an Xo = 120µg O2/g for roasted and ground (R/G) coffee stored at room temperature in air (un-stated moisture). No experimental data were shown in this paper.
θs =
θs =
θs =
Xo RO2
(6)
1 k 1 = Xo . + 2 k1 pO2 k1 k1 k + p 2 O2 Xo pO2
1 pO2
k 3 k +p 4 O2
=
1 k4 1 + k3 p k 3 O2
θ s max = lim θ s = ∞ pO2 → 0 1 X θ s min = lim θ s = o = k3 pO2 → ∞ k1
(7)
(8)
(9)
(10)
Models for the Effect of Temperature on the Kinetics of Food Deterioration The Arrhenius model in Eq. (11) was empirically derived in 1889 for the hydrolysis of sugars, confirmed later by thermodynamic and statistical mechanical principles and extensively applied to model the temperature dependence of deteriorating reactions in food systems2,7,8,9,10,11.
kT = ko . e
E − a R.T
(11)
According to the Arrhenius equation, both ko (pre-exponential factor) and Ea (activation energy) are constant and independent of temperature7. Although deviations may occur at very high temperatures, they are not important in the temperature range at which dry foods are stored8. Nelson and Labuza13 critically reviewed the literature concerning the applicability of the Arrhenius model in products in the rubbery state. They concluded that this model may be useful for describing the temperature dependence of reaction rates over the narrow temperature range most foods are tested or stored at, e.g Ð20 to 0¡C for frozen products and 4 to 45¡C for refrigerated, dry or shelf stable products. Deviations may be observed at higher temperatures when a large temperature range is examined, then the use of the WLF equation is probably more applicable.
Models for the Effect of aw on the Kinetics of Food Deterioration A number of prediction models based on pure curve fitting or combinations of basic kinetics and curve fitting have been proposed in the literature for the effect of aw on the rates of deteriorative reactions in foods 4,8,9,11,16,17,29,30,31,32,33,34,35 and for the effect of aw in the deterioration of drugs36. Among these models, those proposed by Mizrahi et al.30; Villota et al.17; and Singh et al.34 deserve attention. These models can be transformed into simple equations as shown in Eqns. (12) to (14), by using a linear approximation for the water vapor sorption isotherm, assuming that θs follows apparent-
zero order kinetics and the effect of temperature on the rate of deterioration follows the Arrhenius equation. Thus, the simplified Mizrahi, Villota and Singh equations predict a reduction in Ea at increasing aw. This has been observed for reactions of food deterioration and has been attributed by Labuza28 to an enthalpy / entropy compensation phenomena. Mizrahi et al.30: Ea = co .
e-c1 . m
lnEa = ln E* - c . a w
(12)
Ea = E * − b . a w
(13)
Villota et al.17: lnθ s = c o +
Ea 1 . + c1 . (m − BET ) R T
♦
Singh et al.34: Ea = E* − b . (aw − c) R
Ea = E * − b . a w
(14)
The linear model in Eq. (14) has been successfully fitted to literature data by Cardelli (Ph.D thesis, University of Minnesota, St. Paul, MN, USA, 1997) for the reaction of non-enzymatic browning in dehydrated cabbage29,30 and apples34, and to the loss of ascorbic acid in tomato juice37, infant cereal38 and dehydrated apples34. The increase in moisture content can influence not only the Ea but also ko. Labuza28 speculated from pure gas and chemical kinetics that the value of ko can be related to the probability or frequency of breakdown of an activated complex in a food deteriorating reaction. Thus, ko is inversely proportional to the viscosity of the aqueous phase. If the local viscosity decreases with increasing aw, i.e. greater molecular mobility, then one should expect ko and thus the overall rate constant to increase. The increase in ko at increasing aws has been verified for the rate degradation of betanine and vulgaxanthin-I in beet slices 9. For these systems the influence of aw on ko could be modeled by the exponential equation (15). * + ln ko = ln ko c . aw ♦
ko = k*o .
e(c . aw )
(15)
Building Algorithms for Predicting the Rate of O2 Uptake and Shelf Life of Dry Foods as a Function of Oxygen Partial Pressure, aw and Temperature Predicting algorithms for the Ro2 and θs deterioration of R/G coffee as a function of po2 , aw and temperature were derived using literature data. The effect of po 2 on Ro2 was modeled by Eq. (2). The Arrhenius relationship was incorporated into the model for the effect of temperature on the reaction rate. The linear effect of aw on the energy of activation (Eq. 14) and the exponential effect of aw on ko (Eq. 15) were also built into the model. It should be noted that if the parameter ÒcÓ is temperature dependent, it is impossible to separate the effect of aw on ko from the effect of aw on the Ea. For this situation, a simple model called Algorithm-1 was developed to predict Ro2 as a function of po2, aw and temperature; with ko=ko*= constant and Ea = E*-b aw. If Algorithm-1 did not fit well, a more complex Algorithm-2 was tested by incorporating separately the effect of aw into ko and Ea. Algorithms 1 and 2 are shown in Eqns. (21) and (22), respectively. Following similar reasoning, Algorithms-1Õ and 2Õ for the prediction of θs of R/G coffee were built, assuming a pseudo-zero order kinetics for shelf life loss as shown in Eqns. (23) and (24), respectively. The rates ko1 and ko3 were called Òapparent-initiationÓ and the rates and ko2 and ko4 were identified as Òapparent-propagationÓ oxidation. They are the result of a simplification of the kinetics model for the rate of oxygen uptake in dry foods at low levels of oxygen and are proportional to the rates of initiation and propagation oxidation.
k = k *o . e
Since
c.a w .e
−
Ea* − d aw R T
ex . ey = e(x+ y)
(16)
(17)
Ea − d . aw c . aw − R . T * k =k .e
(18)
o
If c is temperature dependent, c =
'
c , then RT
c' .a w − Ea + d .aw R . T R . T R . T k = k * . e
(19)
o
k = k *o . e
−
Ea* − c' aw − d aw RT
= k *o .e
−
Ea* − b aw R T
(20)
Algorithm 1:
RO2 =
* o1
k .
ko*2
e
e
−
−
Ea* 1 − b1 aw
RT
E − b2 a
w
* a2
RT
+
. pO2
(21)
pO2
Algorithm 2:
RO2 =
[k [k
* o2
* o1
e(
e(
c1 a w )
c2 aw )
]
]
e
. .e
− E1* + d1 aw RT
− E2* + d2 aw RT
+
. pO2
pO2
(22)
Algorithm 1':
θs =
ko* 4 .e
− E4* + b4 aw RT
ko* 3 .e
− E3* + b3 aw RT
+
.
pO2
(23)
pO2
Algorithm 2':
θs
[k =
* o4
[k
* o3
e(
c4 aw )
e(
]
c3 a w )
.e
]
.
− E4* + d4 aw RT
e
− E3* + d3 aw RT
+
pO2
(24)
. pO2
Testing the Algorithms The algorithms above were tested by fitting them to experimental Ro2 and _s data reported by Cardelli (Ph.D. thesis, University of Minnesota, St. Paul, MN, USA, 1997), for R/G coffee stored at constant levels of oxygen, aw and temperature. The parameters for the algorithms were determined by non-linear regression using the Quasi-Newton method and the computer program SYSTAT (Systat Inc., Evanston, Il) as described by Wilkinson39. The program generated plots for the residuals and calculated the r2 values, which were used to compare the goodness of the fit. Algorithms 1 and 1Õ fitted well to the rates of oxygen uptake and shelf life data with r2 > 0.98 and results in Tables 1 and 2, respectively. The domain for Algorithm-1 is 0.5 to 21 %O2, aw 0.106 to 0.408 and 4 to 35oC. Domain for Algorithm-1Õ is 1 to 21%, aw 0.106 to 0.408 and 4 to 35oC. Figures 1 to 4 show tri-dimensional plots for Ro2 and _s predicted with the models above for R/G coffee at 10 and 30°C, and various combinations of %O2 and aw within the experimental range of storage. As expected, O2 increase had the most important effect on the rate of deterioration, followed by the rise in temperature and aw.
Table 1: Parameters for Ro2 Predicting Algorithm-1.
Parameter ko1
Magnitude
Units µg/g ds/week
370146
E*1
3880
cal/mole
b1
-20.0
cal/mole
ko2
398
%O2
2220
cal/mole
-1259
cal/mole
E*2
b2 2
r
0.988
Table 2: Parameters for Shelf Life Predicting Algorithm-1Õ
Parameter
Magnitude
Units
ko3
8356
week-1
E*3
6055
cal/mole
b3
2255
cal/mole
ko4
7841
%O2
3352
cal/mole
b4
-682
cal/mole
r2
0.983
E *4
µ g/ g ds/ week)
400
300 200
2
Rate O Uptake (
500
100
0 0
2
4
6
8
10
0.3
12
14
16
18
20
% Oxygen
0.4
aw
0.2
21, 0.1
Uptake ( Rate O µ g/ g ds/ week) 2
Figure 1. Rate of O2 Uptake in R/G Coffee at 10°C from Algorithm-1.
500 400 300 200 100 0 0
0.4
2
4
6
8
10
% Oxygen
12
14
16
0.2
18
20
aw
21, 0.1
Figure 2. Rate of O2 Uptake in R/G Coffee at 30°C from Algorithm-1.
80
Shelf Life (weeks)
70 60 50 40 30 20 10 0 1
2
4
6
8
0.3 10
12
14
16
0.2
18
20
% Oxygen
0.4
aw
21, 0.1
Figure 3. Shelf Life of R/G Coffee at 10°C from Algorithm-1Õ.
80 70 Shelf Life (weeks)
60 50 40 30 20 10 0 1
2
4
6
8
10
12
14
16
18
% Oxygen
Figure 4. Shelf Life of R/G Coffee at 30°C Algorithm-1Õ.
0.2 20
21, 0.1
0.4 0.3
aw
Figure 5 shows the effect of aw increase on the Ea for the rates of apparentinitiation oxidation (k1 and k3) and apparent-propagation oxidation (k2 and k4) for Ro2 and _s loss. The plot was built using Eq. (14) and the parameters in Tables 1 and 2 valid for the whole domain of the algorithms. Figure 5 shows also the effect of aw increase on the Ea for Ro2 and _s loss for a typical oxygen level of 3%, modeled by Eq. (25) and (26) respectively. As expected Ea decreased at increasing aw for most of rates with the exception of a few cases, which remained constant (k1) or showed a slight increase (k2 and k4). The increase in Ea at higher aw translates into lower rates of apparent-propagation oxidation (k2 and k4), which could be a consequence of reduced free radical activity due to low solubility in water and dilution effects. The magnitude of the Ea ranged between 2 to 6 Kcal/ mole, which indicates that the deterioration is controlled by diffusion phenomena. This could be a consequence of the fact that coffee is in the glassy state for the range conditions at which coffee was stored (Cardelli, Ph.D. thesis, University of Minnesota, St. Paul, MN, USA, 1997). In this state, the limiting factor is that oxygen needs to diffuse through the matrix before it can react with the coffee lipids. Even though lipid oxidation has high temperature dependence, diffusion controls the rate at which O2 can diffuse to the lipid. Thus, when temperature is increased, the amount of oxygen available for reaction is what limits the reaction rate.
Ro2 ≅ ko . e
θs ≅
−
( E * − b . aw ) R.T
1 = kθ s ko .e
1 − E * + b aw RT
(25)
(26)
8.0 Ro2 k1 7.0
k2 k S.LIFE
Energy of Activation (kcal/mole)
k3 k4
6.0
5.0
4.0
3.0
2.0
1.0
0.0 0
0.1
0.2
0.3
0.4
0.5
Water Activity
Figure 5 Water Activity Effect on the Energy of Activation for Rates of Shelf Life Loss and O2 Uptake.
Effect of %O2 on the Rate of O2 Uptake and Sensory Shelf Life of R/G Coffee There is no simple index in the literature to describe the effect of oxygen increase on the rates of deterioration of shelf life of foods affected by lipid oxidation. To study this problem we defined the Qo2 index as shown in Eqns. (27) and (28) for the rate of oxygen uptake and shelf life, respectively. The Qo2 index represents the relative acceleration of the rate of deterioration when the level of oxygen is increased by 1%. Results are shown in Figure 6 for typical storage conditions. The acceleration in the deterioration is extremely sensitive to %O2. The Qo2 index is high at low oxygen levels (Qo2 = 10.5) but decreases to 1.1 when the level of oxygen increases from 0.1 to 5%,
and remains almost constant at Qo2 =1.1 within the range from 5 to 21% oxygen. In other words, there is approximately a 1000% acceleration in the deterioration when oxygen level is changes from 0.1 to 1.1% versus only 10% acceleration when oxygen concentration increases from 5 to 6%. This explains why coffee shelf life is prolonged dramatically by vacuum packing coffee in high barrier containers 23,24,25.
QO =
R %O2 + 1%
2
=
(27)
R %O2 θ s %O
2
(28)
θ s %O2 + 1%
12 S.Life 10
Ro2
6
Qo
2
8
4
2
0 0
5
10
15
20
% Oxygen
Figure 6. Qo2 for the Shelf life and Rate of O2 Uptake of R/G Coffee (22°C and aw= 0.270).
Effect of aw on the Rate of O2 Uptake and Sensory Shelf Life of R/G Coffee The sensitivity to changes in aw was determined by calculating the Qa index40. In practice, Qa represents the relative increase in the rate of deterioration per 0.1 rise in aw. Table 3 shows Qa values for representative storage conditions. There is 60% acceleration in shelf life loss per 0.1 aw increase versus only 10% acceleration for the rate of oxygen uptake. This explains why packages with high barrier to water vapor promote longer coffee shelf life. The different sensitivity to aw increase observed for the rates of shelf life loss and the rate of oxygen uptake indicates that reactions other than lipid oxidation, e.g. the Maillard reaction, may participate in shelf life deterioration. This could be related also to changes in the activation energy for each step of the free radical reactions.
Table 3: Qa for the Shelf Life and Rate of O2 Uptake of R/G Coffee (10% O2, 22°°C).
aw
Qa Shelf Life
Qa Ro2
0.14
1.6
1.1
0.27
1.6
1.1
0.42
1.6
1.1
Effect of Temperature on the Rate of O2 Uptake and Shelf Life Table 4 shows Q10 changes for R/G stored from 4 to 35oC and typical storage conditions. Temperature effect was approximately constant, with 20% acceleration in the deterioration per each 10oC increase. This explains why coffee shelf life shelf life can be slightly extended by storing the product under refrigeration. It must be noticed that %O2 is still the main factor controlling shelf life, as previously seen in Figures 3 and 4. Thus, R/G coffee stored at a w 0.1 in packages with
1% residual O2 will last for 55 weeks at 30oC versus only 5 weeks for product stored in air. If temperature is reduced to 10oC, coffee stored at 1%O2 will be extended to 72 weeks versus only 7 weeks for product in air.
Table 4: Q10 for the Shelf life and Rate of O2 Uptake of R/G Coffee (10% O2, a w = 0.27).
T (°°C)
Q10 Shelf Life
Q10 Ro2
4
1.2
1.2
22
1.2
1.2
35
1.1
1.2
Conclusions and Future Trends Algorithms for the prediction of the rate of oxygen uptake and shelf life were derived from basic kinetics of food deterioration. The proposed algorithms showed good fit to kinetics data for roasted and ground coffee with r 2 > 0.98. Oxygen had the most important effect on shelf life loss and rates of oxygen uptake with 10 fold acceleration when increasing level from 0.1 to 1.1% O2. Water activity had the second most important effect, with 60 and 10 % acceleration per 0.1 a w increase for the rate of shelf life loss and oxygen uptake, respectively. Finally, temperature increase in 10 oC promoted no more than 20% acceleration for shelf life deterioration and oxygen uptake. Energy of activation for rates of reaction was < 6 kcal/mole, which suggest that the deterioration is controlled by diffusion phenomena. Further investigation is necessary to unveil the following: •
The fundamental role of the Maillard reaction to R/G coffee deterioration.
•
The applicability of the proposed algorithms to other dry foods affected by lipid oxidation and stored under constant levels of oxygen, a w and temperature.
•
The development of a shelf life predicting procedure for dry foods packaged in plastic pouches by combining the algorithms above and basic mechanisms of oxygen and water vapor transfer through the packaging material.
In order to test the suitability of the algorithms oxygen uptake needs to be measured, usually by gas chromatography, which is very cumbersome. The development of novel methods to determine the consumption of oxygen in dry foods would greatly contribute to extend the application of the proposed algorithms. Sensory shelf life data is the second piece of information needed to develop predicting procedures for packaged foods. Sensory shelf life is usually determined by means of a trained panel or consumer testing. Alternative techniques using staggered tasting by un-trained consumers deserve attention. They are more representative than using trained tasters and cheaper than consumer testing.
Symbols aw:
water activity.
BET: monolayer moisture from BET isotherm (g water/ 100g drysolids). c, b, d:
constants for the linear effect of aw on the energy of activation (kcal/mole).
Ea:
energy of activation (Kcal/mole).
E*a :
energy of activation for the rate ki at a reference aw (Kcal/mole).
k:
rate of reaction.
k o:
pre-exponential factor.
k*o :
pre-exponential factor for the rate ki at a reference aw.
k 1:
rate of apparent-initiation oxidation for Ro2 (µg O2/g dry solids/week).
k 2:
rate of apparent-propagation oxidation for Ro2 (atm/100).
k 3:
rate of apparent-initiation oxidation for θs (week-1).
k 4:
rate of apparent-propagation oxidation for θs (atm/100).
m:
moisture (g water / 100g dry solids).
i
i
po2 : oxygen partial pressure (atm/100). Qa:
relative increase on the rate of deterioration per 0.1 aw increase.
Qo2: relative increase on the rate of deterioration per 1% O2 increase. Q10:
relative increase on the rate of deterioration per 10oC increase.
Ro2: rate of oxygen uptake (µg O2/g dry solids/ week). R:
gas constant.
t:
time (weeks).
T:
absolute temperature (K).
Xo :
cumulative amount of oxygen taken up at the end of shelf life (µg O2/g dry solids).
Xt :
cumulative amount of oxygen taken up at time t (µg O2/g dry solids).
Greek Letters: θs:
shelf life (weeks).
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12. Slade, L. and Levine, H. (1985) ÔIntermediate moisture systemsÕ in Proceedings of the Discussion Conference on Concept of Water Activity. Girton College. Cambridge. U.K. (Faraday Div., Ind. Phys. Chem. Group, eds.), Royal Soc. of Chem. 13. Nelson, K. A. and Labuza, T.P. (1994) ÔWater activity and food polymer science: Implications of state on Arrhenius and WLF models in predicting shelf lifeÕ in J. Food Eng. 22, 271-289. 14. Labuza, T.P. (1975) ÔInterpretation of sorption data in relation to the state of constituent waterÕ in Water Relations of Foods. (Duckworth, R. B., ed.), pp. 155172, Academic Press. 15. Tauokis, P. S.; Labuza, T.P. and Saguy, S. (1997) ÔKinetics of food deterioration and shelf-life predictionÕ in Handbook of Food Engineering Practice (Valentas, K, J.; Rotstein, E.; Singh, P.R., eds.), pp. 361-403, CRC Press. 16. Quast, D. G. and Karel, M. (1972) ÔComputer simulation of storage life of foods undergoing spoilage by two interacting mechanismsÕ in J. Food Sci. 37, 679-683. 17. Villota, R.; Saguy, I. and Karel, M. (1980) ÔAn equation correlating shelf life of dehydrated vegetable products with storage conditionsÕ in J. Food Sci. 45, 398-401. 18. Labuza, T.P., Mizrahi, S. and Karel, M. (1972) ÔMathematical models for optimization of flexible film packaging of foods for storageÕ in Transactions of the ASAE. pp:150-155, ASAE. 19. Labuza, T.P. and Contreras-Medellin, R. (1981) ÔPrediction of moisture protection requirements for foodsÕ in Cereal Foods World. 7(26), 335-343. 20. Cardoso, G. and Labuza, T.P. (1983) ÔPrediction of moisture gain and loss for packaged pasta subjected to a sine wave temperature/ humidity environmentÕ in J. Food Technol. 18, 587-606. 21. Kamman, J.F. and Labuza, T. P. (1983) ÔReaction Kinetics and Accelerated Tests Simulation as a Function of TemperatureÕ in Computer Aided Techniques in Food Technology (Saguy, I., ed.), pp. 71-115, Marcel Dekker. 22. Elder, L.W. (1940) ÔStaling vs. rancidity in roasted coffeeÕ in Ind. and Eng. Chemistry 32(6), 798-801.
23. Clinton, W. P. (1980) ÔConsumer and expert evaluations of stored coffee productÕ in Proceedings of the 9th ASIC Meeting. (International Association of Coffee Scientists, ed.), pp: 273-285, ASIC. 24. Radtke, R. (1979) ÔZur kenntnis des sauerstoffverbrauchs von rostkaffee und seiner auswirkung auf die sensorisch ermittelte qualitat des kaffeegetranksÕ in Chem. Mikrobiol. Technol. Lebensm. 6, 36-42. 25. Heiss, R., Radtke, R. and Robinson, L. (1977) ÔPackaging and marketing of roasted coffeeÕ in Proceedings of 8 th. ASIC Colloquium on Coffee (International Association of Coffee Scientists, eds.), pp. 163-174, ASIC. 26. Clarke, R.J. (1993) ÔThe shelf-life of coffeeÕ in Shelf Life Studies of Food and Beverages (Charalambous, G., ed.), pp. 801-819, Elsevier. 27. Karel, M. and Labuza, T.P. (1969) ÔOptimization of protective packaging of space foodsÕ, Final Report Contract No. [F 41-609-68-c-0015], NASA. 28. Labuza, T.P. (1980) ÔThe effect of water activity on reaction kinetics of food deteriorationÕ in Food Technol. 34(4), 36-41, 59. 29. Mizrahi, S., Labuza, T.P. and Karel, M. (1970) ÔComputer - aided predictions of extent of browning in dehydrated cabbageÕ in J. Food Sci. 35(6), 799-803. 30. Mizrahi, S., Labuza, T.P. and Karel, M. (1970) ÔFeasibility of accelerated tests for browning in dehydrated cabbageÕ in J. Food Sci. 35(6), 804-807. 31. Aguilera, J. M., Chirife, J., Flink, J. M. and Karel, M. (1975) ÔComputer simulation of non-enzymatic browning during potato dehydrationÕ in Lebensm. Wiss. und Technol. 8(3), 128-133. 32. Mizrahi, S. and Karel, M. (1978) ÔEvaluation of Kinetic Model for Reactions in Moisture -Sensitive Products Using Dynamic Storage ConditionsÕ in J. Food Science. 43, 750-753. 33. Dennison, D. B. and Kirk, J. R. (1978). ÔOxygen effect on the dehydration of ascorbic acid in a dehydrated food systemÕ in J. Food Sci. 43, 609-612, 618. 34. Singh R. K., Lund D. B. and Buelow F.H. (1980) ÔStorage stability of intermediate moisture apples: Kinetics of quality changeÕ in J. Food Sci. 48, 939-944.
35. Saguy, I. and Karel, M. (1987) ÔIndex of deterioration and simulation of quality lossesÕ in Objective methods in food quality assessment (Kapsalis, J., ed.), pp. 233260, CRC Press Inc. 36. Nakabayashi, K., Shimamoto, T. and Mima, H. (1980) ÔStability of packaged solid dosage forms II shelf life prediction for packaged sugar-coated tablets liable to moisture and heat damageÕ in Chem. Pharm. Bull. 28(4), 1099-1106. 37. Riemer J. and Karel M. (1977) ÔShelf life studies of vitamin C during food storage: prediction of ascorbic acid retention in dehydrated tomato juiceÕ in J. Food Proc. Preserv. 1(4), 293-312. 38. Vojnovich C. and Pfeiffer, V.P. (1970) ÔStability of ascorbic acid in blends of wheat flour, CSM and Infant CerealsÕ in Cereal Sci. Today. 15(9), 317-318. 39. Wilkinson, L. (1989) ÔStatisticsÕ in SYSTAT: The System for Statistics, pp. 446-471, SISTAT Inc. 40. Loncin, M., Bimbenet, J.J. and Lenges, J. (1968) ÔInfluence of the activity of water on the spoilage of foodsÕ in J. Food Technol. (Br.). 3:131.
