Petros Taoukis of the National Technical University of Athens and Maria Giannakourou of the Technological Educational Institute of Athens review current approaches to food quality modelling and practical implications.


The quality of processed foods depends not only on the initial status of the raw materials but also on the changes occurring during processing and subsequent storage that may cause losses and decreased bioavailability.

Food quality, defined as ‘the assemblage of properties which differentiate individual units and influence the degree of acceptability of the food by the consumer or user[1]’, in general declines post harvesting or processing (with the notable exceptions of maturation and ageing).

Therefore, for each food product, there is a finite length of time after production for which it will retain an acceptable level of quality, defined as its ‘shelf-life’, under specific conditions of handling and storage. Although this term is frequently used due to its practical applicability, there is not a uniformly established definition of shelf life. The exact definition and the criteria for estimating it depend on specific commodities and on the definition’s intended use (i.e. for regulatory vs. marketing purposes). According to Regulation (EU) No 1169/2011, ‘date of minimum durability of a food means the date until which the food retains its specific properties when properly stored’.

The inconsistency of the nomenclature and current open dating practices worldwide can cause confusion and lead to excessive food waste[2].

The problem that arises is two-fold: on one hand, ‘good’ food is wasted due to an over conservative shelf life estimation and on the other hand, unacceptable products can remain in the food chain if the stated expiration date is based on inadequate quality modelling and/or unrealistic assumptions about the ability to impose the conditions of the food chain.

However, ‘use-by’ labelling should not be viewed as a guarantee of food safety, as this would imply absolute control and no temperature deviations throughout the food supply chain[2]. The ultimate goal of all parties involved in food supply is to optimally and sustainably balance consumer protection and satisfaction with the reduction of global food waste based on a quantitative knowledge of food quality, factoring in the uncertainties of the food chain.

It is crucial to have efficient tools to systematically measure and describe food quality degradation as a function of intrinsic and extrinsic factors in order to predict food status at any point in its processing or its distribution to the end user. This could be accomplished by appropriate mathematical models that can quantify food quality and allow optimisation of food processing and/or storage conditions to maximise quality retention. Such tools would provide the food industry with a scientific means to connect physical product properties, processing and distribution conditions with end-product quality and stability, as well as final consumer acceptance.

Since deteriorative reactions are bound to occur after harvesting, in processing and during storage and distribution, delivering quality food products depends on being able either to modify the instabilities of major constituents or to choose optimal storage conditions that minimise the kinetics of the associated biochemical or physical reactions.

Even if microbial action is controlled via appropriate combinatory treatments, quality is bound to decrease due to biochemical and physical reactions. Accordingly, in order to retard such deterioration and to extend the expected shelf life, it is crucial to understand and quantify, through appropriate kinetic equations, the effect of the main factors, intrinsic and/ or extrinsic to the food, that control component degradation.

This article reviews current use of food quality modelling to indicate directions for the use of methodologies in this field and to consider alternative approaches, by integrating knowledge on food properties, kinetics, and statistics; the ultimate scope is to provide the possibility to predict and control food quality attributes using reliable mathematical models.

It is crucial to have efficient tools to systematically measure and describe food quality degradation.

Food quality kinetics: fundamentals

Quality of foods mainly refers to microbiological, chemical, physical and enzymatic changes during processing and storage (including post-harvest changes). Foods, due to their nature, are unstable from a thermodynamic point of view, but they may be kinetically stable, meaning that it may take a long time before equilibrium is reached[3].

In order to make quality more tangible, it would be helpful to discriminate between the so-called intrinsic quality attributes, i.e. inherent to the product itself (chemical composition, physical structure, biochemical changes, microbial and chemical condition, nutritional value, etc.) and extrinsic attributes, linked to the product but not a property of the food itself, such as temperature, relative humidity, total pressure and partial pressure of different gases, light and mechanical stress, etc.

The current literature shows that a variety of approaches have been applied to describing food quality kinetics. Given the complexity of food systems, often empirical models have to be applied to describe the phenomena and the most frequent methodology used is based on the application of apparent kinetics, focusing on the predominant effect of temperature conditions (either isothermal or dynamic profiles). For practically and efficiently predicting quality status under real conditions, empirical models can adequately be used in lieu of mechanistic models, based on more complex thermodynamic and physical principles and in-depth study of the actual mechanisms that are involved in quality loss, with complicated reaction paths, leading to a variety of chemical products. Despite the conceptual gap that exists between these approaches, it is possible to apply the first procedure, attempting however to interpret the results using also the appropriate theoretical background.

In the established and more frequently employed two-step approach to food quality kinetics, the most representative quality indices are selected, and their changes are measured as a function of processing or storage time at constant temperature conditions. Then, an appropriate mathematical equation (primary model) is applied to describe the rate of these changes at each temperature. The second step is to select another equation (secondary model) that best describes the effect of temperature, or any other appropriate kinetic parameter dictated by the primary model, on the rate of changes.

Alternatively, the model parameters can be determined in a single step considering all isothermal datasets and performing a non-linear regression through appropriate mathematical equations, which are developed by incorporating the secondary into the primary model. Another approach involves the simultaneous determination of all kinetic parameters through a single experiment at dynamic non-isothermal conditions.

Despite giving reliable results and reducing significantly the experimental time, there are some practical disadvantages, such as the need for a more sophisticated optimisation method than the isothermal methodologies, the careful experimental design needed regarding measuring times and, most importantly, the risk of miscalculating the variability of kinetic parameters, if not enough degrees of freedom are used[4]. The developed primary and secondary models should be validated by comparing the calculated quality values with measured ones via additional independent experiments. Validated, mathematical models can be a useful tool to quantitatively predict quality at any stage and set of conditions in the food chain.

Basic principles

The ‘primary model’ describing either the loss of one or more quality indices, symbolised by A (e.g. a nutrient or characteristic flavour) or the formation of an undesirable product B (e.g. an off-flavour or discolouration) is in general expressed by Equation 1:

The quality factors [A] and [B] are quantifiable chemical, physical, microbiological or sensory parameters, identified and selected to representatively describe the quality deterioration of the particular food system. The constants k and k´ are the apparent reaction rate constants and n and n´ are the apparent orders of the respective reactions. The use of the term ‘apparent’ indicates that Equation 1 does not necessarily describe the mechanism of the measured phenomenon.

The reaction orders and constants are determined by fitting the change with time of the experimentally measured values of [A] or [B] to Eq. 1 by Differential or Integral Methods[5]. Integrating Eq.1 leads to a general expression for quality:

where Q can be defined as the quality function of the food and k, the apparent reaction rate constant, is a function of composition factors Ci, such as concentration of reactive compounds, inorganic catalysts, enzymes, reaction inhibitors, pH, water activity, as well as microbial populations, and of environmental factors, Ej, such as temperature, relative humidity, total pressure and partial pressure of different gases, light and mechanical stresses. The analytical form of Q(A) depends on the reaction order, and is shown in Table 1. For example, many food quality related actions are described as 1st order phenomena e.g. microbial death, thermal denaturation of proteins, vitamin loss in frozen, canned and dry food, oxidative colour loss, whereas other actions e.g. related to non-enzymatic browning, texture loss, overall sensory deterioration, as n=0 order.

Table 1 Quality function for different reaction order reactions

It is crucial to design the experimental procedure properly so as to include enough measurements of concentration change of the quality index to correctly define the appropriate apparent order and quality function. If the experiment is concluded too soon(<40%conversion), different reaction orders might seem equally appropriate from a goodness of fit point of view[4,5]. For example, as shown in Fig 1, both zero and first order (as well as different reaction orders, e.g. n=2 or 0.4) might seem adequate to predict time for up to 40% loss. However, if a 1st order reaction is wrongly represented by a 0 or 2nd order and the limit of acceptance (defining end of shelf life) is at 60%, then the shelf life would be calculated at 60 or 92 respectively instead of the correct 70 days.

Figure 1 Loss of food quality as a function of time, showing difference between reactions of different apparent order.

Modelling temperature dependence of reactions in foods

Secondary mathematical models describe the dependence of rates, calculated from experiments in set constant conditions, on external factors, such as temperature, pH, aw etc. The paramount effect of temperature on food related reaction rates has long been at the centre of research since it strongly affects product quality and shelf life.

The first practical approach of a temperature secondary model is the Q10 value i.e. the ratio of the reaction rate constants (or equivalently the inverse ratio of shelf life, ts),when the food is stored at a temperature higher by 10°C. Most food literature reports end-point data rather than complete kinetic modelling of quality loss. The Q10 approach in essence introduces a temperature dependence equation of the form (Eq.3):

which implies that if ln k is plotted vs. temperature, a straight line is obtained. Equivalently, lnts can be plotted vs. temperature. Such plots are often called shelf life plots, where b is the slope of the shelf life plot and ko is the intercept. The shelf life plots are true straight lines only for narrow temperature ranges of 10 to 20°C.

However, the classical Arrhenius model (eq. 4) is most frequently used for modelling temperature dependence of quality changes.

where kref is the rate constant at the reference temperature Tref (K), R: the universal gas constant and Ea: the activation energy (J/mol or cal/mol). The Arrhenius equation implies that a straight line is obtained when the rate constant values are plotted vs 1/T. Other secondary models have also been used (such as the Williams-Landel-Ferry, the Eyring-Polanyi equations etc)[1,6,7].

Most of these equations, including the Arrhenius, although developed based on principles of chemical kinetics and thermodynamics, are empirically applied for complex food systems and do not imply a true mechanism. Hence the value of Ea is not really an ‘activation energy’, as defined in thermodynamics, but rather a measure of temperature dependence of the studied reaction.

Since the Arrhenius equation is widely employed (often in the same empirical sense) for a variety of phenomena in physics, biology and engineering, the Ea value offers a universal measure of comparison of temperature ‘sensitivity’. Food quality related reactions, chemical, biological and microbiological values of Ea are in the range of 50 to 120 kJ/mol, corresponding roughly to Q10 values of 2 to 5.

For microbiological reactions (growth and/or inactivation), more elaborated equations have been proposed (Belehradek, square root, log-logistic, probability model, γ-concept, etc)[4].

Practical implications: prediction of remaining shelf life

The application of the primary and secondary models would allow for both a reliable prediction of the quality loss of the product in question at a range of time-temperature conditions that differ from the experimental ones[1] and as a basic tool to implement a thorough shelf life kinetic study, based on the principles of Accelerated Shelf Life Testing (ASLT) methodology.

Given the fluctuations occurring in the real food chain, deviating from the ideal storage and/ or transport conditions, it is crucial to fully record and describe the effect of temperature on the loss of quality parameters. Based on numerous field surveys concerning handling practices for chilled and frozen foods[8], most foods will be exposed to a variable temperature environment, that not infrequently includes stages of abusive storage or transport/transfer conditions[9]. For example, 40% of the total time of the profile temperatures of frozen fish products is over the recommended temperature, varying between -16°C and -12°C[10]. Continuous monitoring and verification of the shelf life of food products is necessary and requires the development of practical systems that can monitor, record and translate the temperature effect of food quality from production to consumption.

Based on the two-step kinetic approach, once the respective equations have been selected and the shelf life criterion has been decided, it is possible to calculate the quality status at any point of the cold chain, under both isothermal and dynamic conditions.

The methodology frequently applied for such systematic shelf life studies is ASLT. This involves experimentation at higher testing temperatures in a thorough shelf life study and extrapolation of the kinetic results to normal, non-abusive storage conditions, taking into account the limitations of the Arrhenius law applicability. This procedure is used to significantly reduce the experimental time, through the acceleration of the quality deterioration reactions. ASLT steps can be summarised as follows[1,5]:

1 Determination of the main quality loss indices, which requires good knowledge of the system, previous experience and a thorough literature search.

2 Package selection for the shelf life test. Frozen, chilled and canned foods can be packaged in the actual product packaging. Dry products should be stored in sealed glass containers or impermeable pouches at the product’s specified moisture and aw.

3 Definition of the storage temperatures, with elevated temperatures being used to obtain data for prediction of shelf life at lower temperature or under variable time-temperature distributions.

4 Evaluation of the duration, type and frequency of testing, based on available information on the most likely Q10. If no information is available on the expected Q10 value, a minimum of three testing temperatures should be used. At each storage condition, at least six data points are required to minimise statistical errors, otherwise, the statistical confidence in the obtained shelf life value is significantly reduced.

5 Plot of the data as it is collected to determine the reaction order and to decide whether test frequency should be altered.

6 For each test storage condition, determination of the reaction order and prediction of the shelf life at the desired storage condition. In order to validate the model obtained, it is an effective practice to test the obtained predictive shelf life model by conducting an additional test at a controlled variable temperature.

For this purpose, the value of the quality function at time t, already defined by equation (2) in the case of isothermal conditions, is calculated by the following integral, where T(t) describes the change of temperature as a function of time:

where keff is the value of the rate of the quality loss reaction at the effective temperature. Teff is defined as the constant temperature that results in the same quality value as the variable temperature distribution over the same time period.

If the T(t) function can be described by a step sequence, or, equivalently, can be discretised in small time increments ti of constant temperature Ti (with Σti=ttot), then Equation 5 can be expressed by Equation 6, assuming the applicability of the Arrhenius equation:

From Equation 6, the value of keff can be estimated and subsequently, from the Arrhenius model, the effective temperature Teff can be calculated.

With an effective use of ASLT, shelf life modelling and determination that normally takes a year, can be completed in 2 to 3 months, if the testing temperature is raised by 20°C. The duration of the shelf life determination by ASLT depends on the temperature dependence of the quality deterioration phenomena as is shown in Table 2 for frozen foods.

Table 2, left, Time to complete an ASLT test for a frozen food of 1-year targeted shelf life at -18°C depending on the temperature sensitivity (Ea) of the shelf life determining reaction.

Continuous monitoring and verification of the shelf life of food products is necessary and requires the development of practical systems that can monitor, record and translate the temperature effect of food quality from production to consumption.

Case study: shelf life determination for frozen spinach

To study the shelf life of frozen spinach leaves, a literature review can assist in selecting the most representative quality indices. A systematic kinetic study was conducted at constant temperature, using as main indices Vitamin C and chlorophyll loss[11, 12]. A first-order reaction was found to describe the above chemical reactions, namely:

and the Arrhenius equation was applied to describe temperature effect:

where CVitC/chlor,0 is the initial value, kVitC/chlorref is the reaction rate at a reference temperature Tref (here equal to -18°C), Ea is the activation ener­gy of the chemical reactions and R is the universal gas constant. Based on available results, a shelf life experiment can be adequately designed (length and frequency of measurements), taking into account the range of Ea and the principles of ASLT meth­odology.

In Figure 2, indicative results and diagrams for Vitamin C loss vs. time, and the respective Arrhenius plot are illustrated[12]. Kinetic data used for this case study for the two quality indices are summarised in Table 3.

Figure 2 (a) Results for vitamin C loss vs time at five storage temperatures. Lines represent the first order fit (b) Arrhenius plot of the vitamin C loss rate for frozen spinach (with Tref = -18°C) and solid red lines are the 95% confidence bands.
Table 3 Estimated kinetic parameters for Vitamin C and total chlorophyll loss of frozen spinach leaves, based on isothermal data (-5, -8, -12, -18 °C)[11]

Based on these estimates, spinach total shelf life (ts) can be predicted at any constant temperature, based on a predetermined acceptability end criterion, e.g. 70% Vitamin C loss and 60% chlorophyll retention. For example, for Vitamin C loss, shelf life ts can be estimated, using the following equation and the results are given in Table 3:


Therefore, shelf life of the food product can be estimated at designated temperatures based on either quality criterion.

Assuming the following hypothetic distribution scenario (two initial isothermal steps, e.g. distribution to warehouse and retail display), it is possible to predict the remaining shelf life of the product, when stored in a domestic freezer of average temperature -12°C. At this temperature, the index that defines the end of shelf life (Table 3) is Vitamin C loss; in this context, using the 70% Vitamin C loss for end-criterion, the first order reaction and the Arrhenius results, values of kvitC, and the remaining concentration of Vitamin C (and thus the SLRrem) can be calculated after each stage (Figure 3).

Figure 3 Quality retention at each stage of a three-isothermal step time-temperature
scenario of frozen spinach distribution-SLrem calculation at domestic storage
assumed to be at -12°C

In Figure 3, shelf life has reached its end after 158 days of rotation, taking into account the equivalent isothermal steps of the assumed t-T profile.

However, temperature in each stock-rotation step is not actually a single value, but is better described by a distribution of values. Therefore, taking into account the actual distribution of the effective temperatures at each stage, SLRrem at the end of the 158d-cycle, is expressed by a distribution, showing also the uncertainty of the mean estimate, information that should not be overlooked.

In our case, by introducing the actual temperature distributions of Stages 1, 2 and domestic storage (Figure 4), the SLRrem after almost 160 days in the frozen chain shows that almost 20% of products are beyond their limit of acceptability (based on the nutritional limit of 70% Vitamin C loss).

The above equations with the appropriate parameters can be easily integrated in user friendly software, often referred to as ‘tertiary models’, that can serve as practical tools for examining alternative scenarios in the food chain leading to improved monitoring and optimisation and a more realistic, scientifically supported shelf life declaration.

There are no uniform, established theories for predicting food reaction kinetics.

Figure 4 SLrem distribution after 158 days in the stock rotation of frozen spinach,
taking into account the real temperature conditions of warehouse transport, retail
display and domestic storage.

Conclusions and further considerations

There are no uniform, established theories for predicting food reaction kinetics. Kinetic models and parameters based on experimental observations are empirical in nature and derive from a specific food matrix and the processing/storage conditions. Additionally, caution is necessary when making assumptions about the application of particular primary or secondary models.

Another important observation is that there has been little attention in the food science literature to the statistical quality of parameters, and especially their uncertainty. Therefore, in addition to the well-known biological variability of foods, it is important to study the significance of the uncertainty of parameters in more detail, certainly if the scope is to make real model predictions[13]. In order to alleviate this weakness, the use of techniques, such as Monte Carlo simulations, seem very promising.

A final issue relates to the dynamic conditions occurring in the distribution chain of perishable foods. Application of an optimised quality and safety assurance system for the chilled and frozen distribution chain requires continuous monitoring and control of storage conditions, from production to consumption, using smart or intelligent packaging to monitor shelf life in the non-isothermal conditions of the food chain[10].

Time Temperature Integrators (TTI) are smart labels that show an easily measurable, time-temperature dependent change that cumulatively reflects the time-temperature history of the food product. Based on reliable models of shelf life and the kinetics of the product and the TTI response, the effect of temperature can be monitored and quantitatively translated into food quality, from production to the point of consumption.

The selection and use of the optimum TTI for a particular product could lead to realistic control of the cold chain, while reliable estimation of the quality status and the remaining shelf life could be performed, allowing for better management and optimisation along the food chain[1].

Petros S. Taoukis Professor

National Technical University of Athens, School of Chemical Engineering, Laboratory of Food Chemistry and Technology, Division IV- Product and Process Development, Iroon Polytechniou 5, Zografou 15780 Athens, Greece

Telephone 30-210-7723171


Maria C. Giannakourou Associate Professor

Technological Educational Institute of Athens, Faculty of Food Technology and Nutrition, Department of Food Technology, Agiou Spiridonos, 12210, Egaleo, Athens, Greece

Telephone 30-210-5385511



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