shelf life estimation from accelerated storage data

Trends in Food Science & Technology 18 (2007) 37e47

Viewpoint

Shelf-life estimation

from accelerated

storage data

Maria G. Corradini andMicha Peleg*

Department of Food Science, Chenoweth Laboratory,

University of Massachusetts, Amherst, MA 01003, USA

(Tel.: D1 413 545 5852; fax: D1 413 545 1262;

e-mail: [email protected])

Traditionally, deterioration rates at elevated temperatures are

extrapolated to lower ones assuming that their temperature de-

pendence obeys the Arrhenius equation, or a similar model.

For such methods to succeed, the spoilage kinetics must be

known in advance and be completely defined by a single

rate constant that must be independent of the food’s thermal

history. These assumptions may hold for certain foods and

model systems, but they cannot be universally applicable. In

a proposed alternative approach, no kinetics is assumed at

all and it is taken for granted that the food’s thermal history

can and does affect the deterioration pattern. Consequently,

the isothermal chemical degradation or microbial growth

data are described in terms of empirical models that have

the needed number of temperature dependent parameters,

usually two or three. The temperature dependence of these

is also described by ad hoc empirical models that are used

for the extrapolation to lower temperatures. This enables the

estimation of not only the ‘rate constant’ but also the whole

deterioration curve. The concept is demonstrated with the deg-

radation of vitamin C in frozen spinach and the growth of

a bacterium and yeast at different temperatures. An example

of lowering the salt concentration as a way of accelerating

bacterial growth is also given.

* Corresponding author.

0924-2244/$ - see front matter � 2006 Elsevier Ltd. All rights reserved.doi:10.1016/j.tifs.2006.07.011

The quality of the predictions largely depends on the

quality of the data gathered under the accelerating conditions

and the proximity of these to the conditions of normal

handling and storage. At least theoretically, the proposed

method predictions can be extended to non-isothermal storage

conditions even when the rate is a function not only of the

momentary temperature but also of the food’s previous

thermal history. In principle, the same methodology can be

used to predict quality improvement in products undergoing

aging.

IntroductionAccelerated storage has been a widely used method to

assess the shelf-life of foods, pharmaceuticals, cosmeticsand many other industrial products of limited durability.Since such products are especially designed to undergominimal changes during their normal distribution and stor-age, accelerated storage is perhaps the only practical wayto predict when such products will have to be withdrawnfrom the market, if not already consumed. Acceleratedstorage can also be helpful in predicting the shelf-life ofperishable commodities, such as refrigerated dairy ormeat products.

The fundamentals of accelerated storage have been dis-cussed in numerous publications. An excellent summary ofstate of the art in the field can be found in the work ofMizrahi (2004). In this book chapter, Mizrahi lucidlyexplains the principles underlying the method, presentsseveral mathematical models to estimate the deteriorationrate of stored foods and alerts the reader to potential sour-ces of error in the use of accelerated storage data. He alsostresses that although accelerated storage is almost alwaysassociated with elevated temperature, in principle at least,other accelerating factors can be used too, alone or incombination with high temperature. Mizrahi’s comprehen-sive summary is based on a large number of works ondeterioration kinetics, published by several research groupsover at least two decades. Almost invariably, the startingpoint of the deterioration kinetics models has been thatthe reaction’s order is known or can be determined fromexperimental data, and that the corresponding rate con-stant’s temperature dependence obeys the Arrhenius equa-tion. These two assumptions are supported by a large body

38 M.G. Corradini, M. Peleg / Trends in Food Science & Technology 18 (2007) 37e47

of experimental data and they are taken for granted inmuch of the literature on food spoilage, be it the result ofchemical processes or microbial growth, see also Labuza’swebpage (2000).

Yet, there is reason to suspect that while these twoassumptions have been appropriate and sufficient for pre-dicting the deterioration rate of many foods and possiblypharmaceutical and cosmetic products, there are spoilageprocesses that do not follow this assumed pattern (Peleg,Corradini, & Normand, 2004). The assumption that thedeterioration process can be characterized by a singlerate constant is limited to reactions that have a fixedkinetic order, i.e., that their progress rate or logarithmicrate, is not a function of time. Also, the use of the Arrhe-nius equation requires that under unchanged conditions,e.g., constant pH, moisture content, the rate constant,however defined, is only a function of temperature andhence totally unaffected by the food’s thermal history.But consider a hypothetical deterioration reaction assumedto be following the first order kinetics. According to theArrhenius model, if the food is kept at 45 �C fora week, say, and then cooled to and kept at 25 �C, itsexponential deterioration rate from that point onwardmust be exactly the same if it was initially kept at 5 �Cfor a week and then heated to and kept at 25 �C. Obvi-ously, this would be a very unlikely scenario in microbialdeterioration but one can also envision biochemical pro-cesses that, too, will not behave in such a manner. Al-though rarely questioned in the food literature, thenotion that all biochemical processes, regardless of theirmechanistic complexity and interactive pathways, musthave a single energy of activation is by no means self-evident. On the contrary, one can legitimately ask howall the intermediate steps could be coordinated in sucha way that they will always produce a single temperatureindependent energy of activation. If there is such a univer-sal mechanism, which is highly doubtful, its existence hasyet to be demonstrated and explained. Until this is done,one must consider systems that truly follow the Arrheniusmodel as an exception rather than the rule. No doubt theArrhenius equation can sometimes provide an acceptableapproximation of the temperature effect on biochemicalreactions, especially when they take place in a model sys-tem. But it will be difficult to envision how this modelapplies when the chemical environment drasticallychanges during the reaction. This is especially the casewhenever microbial growth or decline is involved andthe same can be said about processes involving enzymes.

The above questions concerning the fundamentals of thetraditional theories of deterioration kinetics suggest that itwould be worthwhile to at least explore the merits of alter-native modeling approaches. What follows will describea way to interpret accelerated storage data without any pre-conceived kinetic model and to provide demonstrations ofhow the methodology could be used in shelf-life predic-tions from accelerated storage data.

Theoretical backgroundAccelerated spoilage, be it the result of chemical reac-

tions or microbial growth, can be accomplished in morethan one way. Storage at an elevated temperature is themost common, of course, followed by exposure to highrelative humidity. But in principle at least, the spoilagerate can also be increased by the removal of an inhibitoryfactor. For example, suppose that the microbial shelf-lifeof a given meat product at ambient temperature is deter-mined by the amount of salt that it contains. Then theproduct’s spoilage would be accelerated if the salt con-centration is intentionally lowered. Obviously, the twomethods, i.e., reducing the salt and elevating the tem-perature can be combined (see Mizrahi, 2004). The math-ematical treatment of the results, however, might be muchmore difficult e see below. But let us first concentrate onthe first two options. The main problem is how to devisea reliable method, or methods, that will allow the extrap-olation of the accelerated storage data in a manner thatwill guarantee a correct prediction of what will happenunder normal storage and handling conditions. Had thedeterioration kinetics been completely known, e.g., thereaction order could be established unambiguously andso the temperature or salt concentration dependence ofthe pertinent rate of the process, one could use the corre-sponding traditional mathematical model to calculate therate at any temperature or salt concentration of interest.This holds good provided that there is evidence or goodreason to believe that the deterioration follows the samekinetics under both the normal and accelerated storageconditions.

There are situations, however, where the deteriorationkinetics is not known a priori and there is insufficient evi-dence to conclude that it follows any of the traditionalmodels. Or, worse still, there might be systems where thereis reason to suspect that any model based on a fixed reac-tion kinetic order, and consequently the Arrhenius equation,must be inapplicable. Indeed, certain processes followmodels that have two or more temperature dependentparameters, in which case the rate cannot be expressedby a single constant, as the Arrhenius equation demands.Microbial growth, as already mentioned, is a perfect example.Many isothermal growth curves have a sigmoid shapewhose mathematical description requires no less than twoconstants and in many cases at least three. Thus, a ‘maxi-mum specific rate’ vs. temperature relationship alone willbe insufficient to account for the phenomenon known as‘lag’, which can vary independently with temperature too.The same can be said about the ‘asymptotic growth level’,although it usually shows only weak temperaturedependence.

Degradation reactions or microbial inactivation that fol-lows the Weibullian model (see Corradini & Peleg, 2004and Eq. (1) below), would be another example if the shapefactor or power, n, happens to be temperature dependent.But even when the power n could be fixed as a constant,

39M.G. Corradini, M. Peleg / Trends in Food Science & Technology 18 (2007) 37e47

the meaning of any ‘energy of activation’ would remainobscure. A Weibullian decay model, like many growth oraccumulation models, describes situations where the rate,however defined, is not only a function of temperaturebut of time too. How can such models be used for extra-polation? To answer the question, we will consider two kindsof scenarios: one a chemical degradation process havinga non-linear kinetics and the other ‘logistic’ (sigmoid)microbial growth.

Non-linear degradation at elevated temperaturesTypical decay patterns at different temperatures are

shown in Fig. 1 e top left. The purpose of the acceleratedstorage is to predict the curve shown at the top right of thefigure (i.e., at a lower temperature) from those shown atthe top left (i.e., at high temperatures). Fitting of isothermaldecay data can be done with more than one model, using anyof the many commercial software packages available. TheWeibullian model is one example (Corradini & Peleg, 2006):

Vitamin C in Spinach

0 50 100 150 200

0

0.2

0.4

0.6

0.8

1

Prediction & Observation

-20ºC

Time (days)

0 25 50 75 100

0

0.2

0.4

0.6

0.8

1

Isothermal Degradation

-12ºC

-8ºC

-3ºC

Time (days)

C(t)/C

0

-24 -20 -16 -12 -8 -4 0

0

0.02

0.04

0.06

0.08

0.1

b(T

) (d

ays

-n)

b(T) = 0.077*exp[0.081 T]

Extrapolated

0.5

0.75

1

1.25

1.5

-24 -20 -16 -12 -8 -4 0

n(T) = 1.44*exp[0.04 T]

n(T

)

Extrapolated

0

25

50

75

100

-24 -20 -16 -12 -8 -4 0

Temperature (ºC)

k1(T

) (d

ays)

Extrapolated

k1(T) = 8.5*exp[-0.11 T]

0

0.25

0.5

0.75

1

1.25

1.5

-24 -20 -16 -12 -8 -4 0

Temperature (ºC)

Extrapolated

k2(T) = 0.15-0.05T

k2(T

)

Fig. 1. Prediction of vitamin C loss in frozen spinach using Eqs. (1) and (2) as models (solid and gray curves, respectively). The original data are fromGiannakourou and Taoukis (2003).


YðtÞ ¼ CðtÞC0

¼ expð � btnÞ ð1Þ

The one below is another:

YðtÞ ¼ CðtÞC0

¼ 1� t

k1þ k2tð2Þ

In both models C(t) and C0 are, respectively, themomentary and initial concentrations of the monitoredcompound, be it a vitamin, pigment, etc. The constants,b and n, or k1 and k2 are temperature dependent parameters.The first order kinetic decay is just a special case of Eq. (1)where n¼ 1. The zero order kinetics is just a special case ofEq. (2) where k2¼ 0. Thus, all that follows will also applyto the zero and first kinetics if and when they are encoun-tered but not vice versa.

Similar models can be written for growth or ‘accumula-tion’. In Eq. (1), this can be done by reversing the sign ofthe exponential argument and in Eq. (2) by changing thesign of the second term on the right. Notice that bothmodels have two temperature dependent coefficientsb and n or k1 and k2. Although these two models oftenhave a very similar degree of fit, their continuation beyondthe ‘experimental range’ diverges. Yet, there can be a sub-stantial time span for which they can be used inter-changeably (Peleg, 2006). Now, suppose we have enoughisothermal data to plot the temperature dependence of thesemodels’ coefficients and express them algebraically (aftercurve fitting), the resulting empirical expressions can beused to calculate the coefficients at any desired tempera-ture. And if the two models hold over a certain pertinenttemperature range, one can construct the whole C(t)/C0

vs. time relationship at any specified temperature withinit. If the above method indeed works, then the predictionsobtained by using Eqs. (1) and (2) as models should beclose (Peleg et al., 2004). The same applies to any modelthat fits the experimental data well, provided that it too isused within the ‘overlap range’ only. The agreement be-tween the predictions of different models by itself is nota proof that they are all correct, but it strongly suggeststhat they might be. In reality, especially when the databasehas only few and scattered entries, the divergence of thecurves will start close to the edge of the experimentalrange, in which case only one or none of the models wouldprovide reasonable predictions. In any case, the decisivetest would be a comparison between the predicted curvesand those recorded experimentally. But, once the underly-ing model or models are affirmed in this way, then it orthey could be used to predict the deterioration pattern notonly under isothermal conditions, but also under almostany non-isothermal temperature regime e see below.

Deterioration as a result of microbial growthMost of the reported isothermal microbial growth curves

have a sigmoid shape of the kind shown in Figs. 2e4 e top

right. Many of these can be described, with a comparabledegree of fit, by several three-parameter empirical models.Among them is the shifted logistic function (Corradini &Peleg, 2005, 2006):

YðtÞ ¼ log

�NðtÞN0

�¼ a

1þ exp½kðtc � tÞ� �a

1þ expðktcÞð3Þ

The ‘‘power model’’ is an alternative:

YðtÞ ¼ log

�NðtÞN0

�¼ a0mtm

bmþ tmð4Þ

N(t) and N0 are the momentary and initial numbers, respec-tively, and a, k and tc or a0, m and b temperature dependentcoefficients.

According to these two models, at t¼ 0, whenN(t)¼N0, Y(0)¼ 0. As t / N, N(t) approaches a{1�1/[1þ exp(ktc)} in the first model and a0m in the second.[Three is the minimal number of adjustable parametersthat is required to describe the sigmoid growth curve ofmany bacteria in a ‘closed habitat’. This is the main rea-son for choosing Eqs. (3) and (4) as models and not thebetter known four-parameter Gompertz and Baranyi andRoberts models.]

If in the accelerated storage experiment the growthcurve is determined at various temperatures, then and asbefore, the temperature dependence of the coefficients ofEq. (3) or (4) can be plotted and expressed algebraicallyby ad hoc empirical models. Here again, if the samemodel holds over the entire pertinent temperature range,the terms a(T ), k(T ) and tc(T ) or a0(T ), m(T ) and b(T ),could be used to generate the complete growth curve atany desired constant temperature within this range. Likein chemical degradation, the model’s coefficients can beconverted into functions of time, i.e., a(t)¼ a[T(t)],k(t)¼ k[T(t)], which in turn could be used to generatenon-isothermal growth curves (Corradini & Peleg, 2005;Peleg, 2000).

Growth after removal of an inhibitory factorThe principles outlined above are also applicable to

accelerated storage achieved by lowering the level of aninhibitory factor (Fig. 4). Consider microbial growth incured meats or cheeses as an example, be it of the product’s‘‘natural microflora’’ or of a pathogen. The presence ofsalts usually suppresses microbial growth. In cured meats,the salts are sodium chloride and sodium nitrite e part ofit added directly and part produced by reduction of thealso added sodium nitrate. In cheeses, the salt is primarilysodium chloride sometimes supplemented by a chemicalpreservative. In both products, lowered water activity alsocontributes significantly to their relative stability. In princi-ple, if one could maintain a constant water activity and tem-perature, then reducing the salt content would acceleratethe growth by removal of the inhibitory agent. As far as


0 50 100 150 200 250

0

2

4

6

8

10

Pseudomonas fluorescens in REM

Isothermal Growth

Time (h)

Y(t) =

L

og

[N

(t)/N

0]

10ºC

15ºC

20ºC

0 100 200 300 400

0

2

4

6

8

10


Time (h)

5ºC

k(T) = 0.017*exp(0.08 T)

0 5 10 15 20 25

0

0.025

0.05

0.075

0.1

0.125

k(T

) (h

-1)

Extrapolated

a(T) = 10 (fixed)

0 5 10 15 20 25

0

25

50

75

100

125

tc(T

) (h

)

tc(T) = 118.*exp(-0.09 T)

Extrapolated

0 5 10 15 20 25

0

50

100

150

Temperature (ºC)

b(T

) (h

)

b(T) = 139.4*exp(-0.09 T)

a’(T) = 2.5 (fixed)

0 5 10 15 20 25

1

1.5

2

2.5

3

3.5

Temperature (ºC)

m(T

)

m(T) = 2.3+0.004 T

Extrapolated

Extrapolated

Fig. 2. Prediction of Pseudomonas fluorescens growth in a ready-to-eat meal (REM) using Eqs. (3) and (4) as models with a fixed a and a0, respectively.The original data are from Tyrer, Ainsworth, Ibanoglu, and Bozkurt (2004).

the methodology is concerned, the procedure should bevery similar. The first step is to record the growth curvesof the organism of interest, or of the total microbial popu-lation, at different salt levels. Then the data should be fittedwith the appropriate model, Eq. (3) or (4), for example.This would be followed by expressing these equations’coefficients as a function of the salt concentration, i.e.,

a(Csalt), k(Csalt) and tc(Csalt) or a0(Csalt), m(Csalt) andb(Csalt). Once these terms are determined, they could beused to predict the growth patterns at any elevated salt con-centration, provided that the chosen model indeed holdsover the entire pertinent salt concentration range. At leastin principle, the same can be achieved in other productsby elevating the pH or the amount of moisture, lowering


Candida sake in REM

0

1

2

3

4

5

6

7

Isothermal Growth

10ºC

15ºC

20ºC

0 50 100 150 200 250

Time (h)

Y(t) =

L

og

[N

(t)/N

0]

0

1

2

3

4

5

6

7


5ºC

0 100 200 300 400

Time (h)

a(T) = 7 (fixed)

0

0.025

0.05

0.075

0.1

k(T

) (h

-1)

k(T) = 0.02*exp(0.06 T)

Extrapolated

0 5 10 15 20 25

0

25

50

75

100

125

150

Extrapolated

tc(T

) (h

)

tc(T) = 146.9*exp(-0.07 T)

0 5 10 15 20 25

a’(T) = 2.35 (fixed)

0

25

50

75

100

125

150

Extrapolated

0 5 10 15 20 25

Temperature (ºC)

b(T

) (h

)

b(T) = 159.*exp(-0.07 T)

1.5

2

2.5

3

m(T

)

Extrapolated

0 5 10 15 20 25

Temperature (ºC)

m(T) = 2.3+0.003 T

Fig. 3. Prediction of the growth of Candida sake in a ready-to-eat meal (REM) using Eqs. (3) and (4) as models with a fixed a and a0, respectively. Theoriginal data are from Tyrer et al. (2004).

the sugar(s) or a chemical preservative concentration, etc.Such modifications, however, can influence the inhibitoryeffect of synergistic co-factors. But if the collateral changesare all manifested in the magnitude of a single inhibitoryfactor, like the salt concentration, i.e., a(Csalt), k(Csalt),then these could be used as such to predict the microbialgrowth patterns. Otherwise, all the factors and their

interactions would have to be accounted for individually,which would complicate the mathematical analysis consid-erably and require an expanded database.

Simultaneous alteration of two or more factorsSuppose one wants to accelerate a product’s chemical

deterioration by simultaneously elevating its temperature


0 200 400 600 800 10000 200 400 600 800

0

1

2

3

4

5

6

7

0

1

2

3

4

5

6

7

Yersinia enterocolitica (% NaCl, 7ºC, pH = 7)

Iso-concentration Growth Prediction & Observation

1%

5%

Time (h) Time (h)

2%

3%

4%

6%

Y(t) =

L

og

[N

(t)/N

0]

0 1 2 3 4 5 6 7

0

0.02

0.04

0.06

0.08

% NaCl

k(T) = 0.07/{1+exp[0.8(T-2.6)]}

a(T) = 6.5 (fixed)

k(T

) (h

-1)

Extrapolated

0

100

200

300

400

500

600

700

0 1 2 3 4 5 6 7

% NaCl

Extrapolated

tc(T

) (h

)

tc(T) = 26.7*exp(0.5 T)

Fig. 4. Prediction of the salt suppressed growth of Yersinia enterocolitica using Eq. (3) as a model with a fixed a. The original data are from the FoodStandards Agency (undated).

and water activity or accelerate its microbial spoilage byraising the temperature and lowering the salt content. Inthe traditional kinetic models, at least in microbiology,the rate constant’s temperature dependence is usually de-scribed by either the Arrhenius equation or the squareroot model originally proposed by Ratkowski (McMeekin,Olley, Ross, & Ratkowsky, 1993). When two or more in-fluential factors are known to affect the growth or inacti-vation rate, pH and/or water activity, say, then theArrhenius model has been frequently ‘‘amended’’ by theaddition of a term or terms that are functions of the pH,aw, etc. Such models can be used for curve fitting butthey have a serious conceptual problem. There is simplyneither evidence nor theoretical reason for the effects oftemperature, pH, aw, or any other factor for that matter,

to be additive or multiplicative as such models imply(Peleg, 2006). More probably, any change in these factors,affects the magnitude of the process rate equation’s coef-ficients, in a manner that can only be determined experi-mentally. For example, suppose that a certain deteriorationprocess follows the model expressed by Eq. (1) and thetemperature dependence of its rate parameter b(T ) isgiven by the log logistic model (Corradini & Peleg,2005, 2006):

bðTÞ ¼ loge

�1þ exp½kðT� TcÞ�

�ð5Þ

where k and Tc are coefficients. These degradation coef-ficients, however, are functions of pH, salt concentration,aw, and other factors, i.e., k¼ k(pH, Csalt, aw, .) and


Tc¼ Tc(pH, Csalt, aw, .). The exact nature of these func-tions is usually unknown and hence they too ought to bedetermined from actual data. The same applies to thecoefficients of all the other models (Eqs. (3)e(5)), where,k1¼ k1(T, pH, aw, .), k2¼ k2(T, pH, aw, .), tc¼ tc(T,pH, aw, .), etc. These functions, as already mentioned,must be determined experimentally and there is no rea-son to assume that any of them follow a universal rule.This means that determining these functions and theirown coefficients will require a considerable experimentaleffort, and expense, making the whole attempt unattrac-tive for most food systems. Perhaps there are reactionsor growth patterns, where the effects of factors likepH, aw, the presence of salts and the like can be ex-pressed by simple algebraic terms, but they are yet tobe identified.

Demonstration of the concept with publishedexperimental chemical degradation and microbialgrowth dataVitamin C loss in frozen spinach

Published degradation curves of vitamin C in frozenspinach stored at three temperatures, �3, �8 and �12 �C,are shown in Fig. 1 (top left). Superimposed on the re-ported data is the fit of two different empirical models(Eqs. (1) and (2)). The temperature dependence of these,fitted with yet another set of ad hoc models, is alsoshown in the figures. If these ‘secondary models’ couldbe extrapolated to lower temperatures, as shown in themiddle and bottom plots, then one could predict the deg-radation curve at �20 �C. The predicted curves areshown at the top right corner of the figure togetherwith the corresponding experimental data. Although thefirst model’s prediction was an underestimate and thatof the second an overestimate, see Fig. 1 (top right),both predictions were fairly close to actual observations,despite that the extrapolation was based on three temper-atures only. Alone, this demonstration would be insuffi-cient to support the proposed approach. Yet, the factthat two very different empirical models yielded a similarprediction suggests, at least, that the method might havemerit.

Pseudomonas growth in a ready-to-eat mealSigmoid growth curves of Pseudomonas fluorescens in

a ready-to-eat meal stored at 20, 15 and 10 �C are shownin Fig. 2 (top left). Also shown in the figure is the fit ofthe slightly modified logistic function (Eq. (3)) and thepower model (Eq. (4)). Eq. (3) has three adjustable param-eters, the minimum necessary to describe sigmoid curvesof the kind investigated. The parameter a, is a markerof the asymptotic growth level, specified by a{1�1/[1þ exp(�ktc)]}, where k is a marker of the curves’ steep-ness around its inflection point and tc a marker of the latter’slocation. For economy, and in order to avoid a large scatter

or unreasonable ‘secondary model’ parameters, the value ofa was fixed. The same was done with the alternative model(Eq. (4)) and for the same reasons. [Since the original pub-lished growth data have a considerable scatter, standard non-linear regression, which is solely based on minimizing themean squared error, can yield parameters that are physicallymeaningless.] The temperature dependence of the twomodels’ parameters is also shown in the figure. Asbefore, each was described by an ad hoc empirical model,which was subsequently used for extrapolation to thelower temperature. Comparison of the predicted growthcurves at 5 �C calculated by the two models with experi-mental observations is also shown in Fig. 2. Both models’predictions were higher than the experimental data, withthe discrepancy reaching about one log unit at the moreadvanced growth stage. But in light of the considerablescatter in the original ‘‘high temperature’’ data and thatthere were only three temperatures to work with, thematch is not unreasonable. That the two models, despitetheir different mathematical construction, gave almostidentical predictions suggests that the discrepancy wasdue primarily to the original data’s imperfections andnot to a methodological flaw in the procedure. Surprisingas it might sound, it is not easy to find high quality micro-bial growth data at several temperatures. Moreover, be-cause usually the published curves were only determinedonce, even a slightly shifted curve at one temperaturecould have a large effect on the secondary model’s para-meters and consequently on the model predictions’reliability.

Candida sake in a ready-to-eat mealThe method used for predicting the growth of P. fluo-

rescens at 5 �C was also used to predict the growth pat-tern of Candida sake (a yeast) at this temperature. Theoriginal growth data with the two growth models’ fit(Eqs. (3) and (4)) are shown in Fig. 3 (top left). The fig-ure also shows the temperature dependence of the growthparameters and the ‘secondary’ models that describethem e middle and bottom plots e together with the em-pirical expressions used in the extrapolation. The modi-fied logistic model’s prediction almost perfectlymatched the actual growth curve of the Candida (seefigure). The prediction of the ‘power’ model (Eq. (4)),whose fit was somewhat worse than that of the modifiedlogistic function at 10 �C (see figure), was slightly higherthan the actual observation. Still, the maximum discrep-ancy hardly exceeded a half log unit, i.e., it was onlyslightly larger than the scatter in the experimental growthdata. The better overall predictions than in the Pseudo-monas case are most probably due to the considerablysmaller spread of the Candida’s ‘‘high temperaturesdata’’ because here too there were only three growthcurves available to determine the model and its coeffi-cients’ temperature dependence.


The inhibitory effect of salt on Yersinia enterocolitica’sgrowth

The previous examples were based on extrapolationfrom high temperatures to low. However, the proposedmethodology can also be applicable when the accelerat-ing factor is a lower salt concentration. Published growthcurves of Y. enterocolitica (a food borne pathogen usu-ally transmitted through improperly cooked meats) at7 �C and pH 7 in the presence of 1, 2, 3 and 4%NaCl, are shown at the top left of Fig. 4. Superimposedon the experimental data is the fit of the modified versionof the logistic function (Eq. (3)). The temperature depen-dence of its parameters, also shown in the bottom plot ofthe figure, was fitted with ad hoc empirical models as be-fore. These were used to predict the growth curves at 5%and 6% salt concentrations as seen in the plots at the topright of the figure. Although the match between the pre-dictions and the reported experimental growth data is farfrom being perfect, it is certainly not unreasonable inthe 5% salt case, where the discrepancies were mostlyon the order of half to one log unit. As for the 6%salt, the experimental results themselves had a largespread, even by microbiological standards. Yet, the pre-dicted growth curve was not totally out of line, consider-ing the data scatter at the low salt concentrations,particularly when the storage exceeded 400 h. Asbefore, each of the individual predictions might not beconsidered as convincing evidence to the fact that themethodology has merits. But that the same methodgave predictions of similar quality in very different sys-tems, described by totally different mathematical models(Eqs. (1)e(4)), and even for different kinds of accelerat-ing factors, suggests that the concept is not totally unre-alistic. And once more, no prediction, be it of the growthof Yersinia in salty habitat, growth of Candida and Psue-domona at various temperatures or of vitamin loss incold storage, has been based on any special preconceivedkinetics. On the contrary, very different mathematicalmodels could produce similar results, demonstratingthat uniqueness is not a prerequisite. Therefore, thechoice of a model should be guided primarily by practi-cal considerations and not by any attempt to conform toa traditional kinetic theory.

Simulation of deterioration processes undernon-isothermal conditions

Once the deterioration model has been found and thetemperature dependence of its parameters expressed alge-braically, it can be converted into a differential rate equa-tion that can be solved numerically for almost anyconceivable thermal history. This is true for both degrada-tion processes, such as those that can be characterized byEq. (1) or (2), and for growth/accumulation processes likethose that can be characterized by Eq. (3) or (4) (Peleget al., 2004). The same can be said about alternative math-ematical models.

As an example, consider the vitamin C loss in frozenspinach (Fig. 1), the kinetics of which can be expressedby Eqs. (1) and (5). Under non-isothermal conditions, thismodel will become (Corradini & Peleg, 2006):

dCðtÞC0dt

¼�b½TðtÞ�exp�� b½TðtÞ�t�n½TðtÞ�

�n½TðtÞ�t�

n½TðtÞ��1

n½TðtÞ� ð6Þ

where T(t) is the temperature profile and t* is the inverse ofEq. (1), i.e., the time that corresponds to the momentaryratio C(t)/C0.

Eq. (6) is an ordinary differential equation (ODE) andcan be solved in Mathematica� with the ‘NDSolve’ com-mand for almost any conceivable T(t), including discontin-uous profiles whose expression requires ‘if statements’.Examples of simulated scenarios of vitamin C loss infrozen storage under fluctuating temperatures are shownin Fig. 5 e left. [Although the shown degradation curveswere produced by Mathematica�, they can also be pro-duced by other advanced mathematical programs. Theycan even be produced by general purpose software likeMS Excel�, after conversion of the model’s rate equationinto a difference equation. In this form, however, difficul-ties may arise if the temperature fluctuations are toosharp.]

All the above also pertains to the deterioration caused bymicrobial growth. Consider Candida’s growth in the ready-to-eat meal (Fig. 3), when described by Eq. (3) as the‘primary model’. The asymptotic growth level was fixed,i.e., a(T )¼ 7.0 and the temperature dependence of the pa-rameters k(T ) and tc(T ) was characterized by the algebraicexpressions that are shown in the figure. These, as well asthe temperature profile, T(t), can be incorporated into therate model’s equation, which becomes (Corradini & Peleg,2005):

dYðtÞdt¼

k½TðtÞ�a½TðtÞ�exp�k½TðtÞ�ftc½TðtÞ� � t�g

��1þ exp½k½TðtÞ�ftc½TðtÞ� � t�g�

�2ð7Þ

where, again, t* is the inverse of Eq. (3), i.e., the time thatcorresponds to the momentary growth ratio, Y(t).

Eq. (7) was used to generate the non-isothermal growthcurves that are shown in Fig. 5 e right, together with thecorresponding temperature histories. These plots demon-strate that the temperature history and the rate equation’scomplexities are not a hindrance to the rate equation’s so-lution. Despite the cumbersome appearance of the result-ing mathematical expression, the model is solved almostinstantaneously by Mathematica� and probably by othermathematical programs. In fact, the simulations can berepeated with random histories, thus allowing estimationof the probabilities of spoilage as a function of time underuncertain temperature conditions. This, however, is a to-tally separate topic that is discussed in detail elsewhere(Corradini, Normand, & Peleg, in press). What’s impor-tant here is the demonstrated possibility to develop


0 5 10 15 20 25 30

0

-5

-10

-15

-20

0 25 50 75 100 125 150

0

5

10

15

20

Vitamin C in Spinach

Tem

peratu

re (ºC

)

Candida sake in REM

TEMPERATURE PROFILE TEMPERATURE PROFILE

0 5 10 15 20 25 30

0

0.25

0.5

0.75

1

Time (days)

C(t)/C

0

DEGRADATION CURVES

0 25 50 75 100 125 150

0

1

2

3

4

5

6

7

Time (h)

GROWTH CURVES

Y(t) =

L

og

[N

(t)/N

0]

Fig. 5. Simulated non-isothermal vitamin C degradation and Candida sake growth curves generated with Eqs. (6) and (7) (left and right, respectively).Notice that the complexity of the temperature history has no noticeable effect on the solution of the model’s rate equations. Also, once the model’scoefficients have been determined from the isothermal data e the actual values are shown in Figs. 1 and 3 e they can be used to generate spoilage

curves under almost any conceivable temperature regime within the range of the model’s applicability.

a predictive deterioration model from accelerated storagedata without having to assume a kinetic order. If andwhen such a model can be derived for a given food,then, as shown in Fig. 5, it could be used to predict thequality consequences of an almost unlimited variety ofstorage conditions.

Concluding remarksShelf-life estimation from accelerated storage results,

in the form of chemical degradation or microbial growthdata, does not require that the process kinetics be as-sumed a priori. On the contrary, there is good reasonto believe that many, or perhaps even most of the dete-rioration processes in foods might not follow any of thestandard kinetic models. The assumption that the processor reaction must follow a particular kinetic is thereforeunnecessary. The same can be said about the use ofthe Arrhenius equation to describe the deteriorationrate’s temperature dependence. It too need not apply,

especially when the isothermal deterioration curve isnot log linear. Reasonable predictions of what happensat low temperatures, and in one case at high salt concen-tration, could be obtained by more than one model andwithout any prior knowledge of the process’s details. Itis even suggested that at least two different mathemati-cal models should be used for mutual verification, thusincreasing the predictions’ reliability. The describedmethod is consistent with the fact that the deteriorationrate, however defined, need not be a function of temper-ature only (and, other factors like pH, water activity,etc.), but also of time. Thus, in contrast with the tradi-tional Arrhenius equation, the role of the food’s thermalhistory is not only acknowledged but also becomes anintegral part of the deterioration model itself. The math-ematical models to describe the deterioration can bechosen solely on the basis of their fit and convenience;they do not have to conform to any theoretical require-ment. However, since they are used for extrapolation,their usefulness is limited to conditions that are not

47od Science & Technology 18 (2007) 37e47

too different from those that exist in the acceleratedstorage experiments. For example, they cannot be ex-tended to low temperatures if freezing occurs or to ex-ceedingly high salt concentrations that might kill theorganisms rather than merely inhibit its growth. Withinthe applicable temperature range, the method could beused not only to predict isothermal deterioration butalso changes induced by regular and irregular tempera-ture fluctuations. The mathematical procedure to obtainan isothermal prediction does not require special pro-gramming, only access to common non-linear regressionsoftware. The prediction’s reliability will be primarilydetermined by the quality of the accelerated storageexperimental results and their reproducibility. It willalso depend on how many elevated temperature levelshave been employed, or the number of levels of any otheraccelerating factors. The more numerous the levels, andthe closer they are to the ‘normal storage conditions’, thehigher would be the predictions’ reliability. In practice,though, feasibility considerations will dictate the numberand range of the accelerating conditions and these will de-termine the prediction’s quality. Because foods rarely haveexactly the same composition, some discrepancies betweenthe predictions and reality would be inevitable, regardless ofthe method used to obtain them. Therefore, an occasionaldiscrepancy between reality and expectation does notnecessarily invalidate the concept. Theoretically, the meth-odology can be extended to situations where several factorsaffecting the deterioration are altered simultaneously. How-ever, the mathematical treatment of such accelerated storageexperimental results might be too elaborate and the requireddatabase too large for practical implementation.

In principle at least, the same methodology can beused ‘‘in reverse’’, i.e., to predict the beneficial effectsof aging, for example, from short term experiments. Butsince we have no data of this kind to work with, it ismerely a speculation at this point. Admittedly, the numberof successful predictions that we have presented in thisarticle is rather small. Yet, it is probably sufficient todemonstrate that the concept might work at least inselected systems.

M.G. Corradini, M. Peleg / Trends in Fo

AcknowledgmentContribution of the Massachusetts Agricultural Experi-

ment Station.

References

Corradini, M. G., Normand, M. D., & Peleg, M. Non-linear kinetics:principles and potential food applications. Food engineering:Integrated approaches. Springer Verlag (in press).

Corradini, M. G., & Peleg, M. (2004). A model of non-isothermaldegradation of nutrients, pigments and enzymes. Journal of theScience of Food and Agriculture, 84(3), 217e226.

Corradini, M. G., & Peleg, M. (2005). Estimating non-isothermalbacterial growth in foods from isothermal experimental data.Journal of Applied Microbiology, 99(1), 187e200.

Corradini, M. G., & Peleg, M. (2006). Prediction of vitamins loss duringnon-isothermal heat processes and storage with non-linear kineticmodels. Trends in Food Science & Technology, 17(1), 24e34.

Food Standards Agency (undated). Funded data generated at IFR.Records ID B002_209 to B002_214. Available fromhttp://www.combase.com (Combined Database for PredictiveMicrobiology).

Giannakourou, M. C., & Taoukis, P. S. (2003). Kinetic modeling ofvitamin C loss in frozen green vegetables under variable storageconditions. Food Chemistry, 83(1), 33e41.

Labuza, T. P. (2000). FScN 8334, Reaction Kinetics of Food Deterio-ration. Available from http://courses.che.umn.edu/00fscn8334e

1f/FScN8334_Reading.html.McMeekin, T. A., Olley, J. N., Ross, T., & Ratkowsky, D. A. (1993).

Predictive microbiology: Theory and application. New York, NY:John Wiley & Sons.

Mizrahi, S. (2004). Accelerated shelf life tests. In R. Steele (Ed.),Understanding and measuring the shelf-life of food. Cambridge,UK: Woodhead Publishing.

Peleg, M. (2000). Modeling and simulating microbial survival in foodssubjected to a combination of preservation methods. In G. V.Barbosa-Canovas, & G. W. Gould (Eds.), Innovations in foodprocessing. Lancaster, PA: Technomic.

Peleg, M. (2006). Advanced quantitative microbiology for food andbiosystems: Models for predicting growth and inactivation. BocaRaton, FL: CRC Press.

Peleg, M., Corradini, M. G., & Normand, M. D. (2004). Kinetic modelsof complex biochemical reactions and biological processes.Chemie Ingenieur Technik, 76(4), 413e423.

Tyrer, H., Ainsworth, P., Ibanoglu, S., & Bozkurt, H. (2004).Modelling the growth of Pseudomonas fluorescens and Candidasake in ready-to-eat meals. Journal of Food Engineering, 65(1),137e143.

http://www.combase.com

http://courses.che.umn.edu/00fscn8334%20131f/FScN8334_Reading.html

http://courses.che.umn.edu/00fscn8334%20131f/FScN8334_Reading.html

shelf life estimation from accelerated storage data

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