# Holistic Approach to the Uncertainty in Shelf Life Prediction of Frozen Foods at Dynamic Cold Chain Conditions

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{a}, of the Arrhenius equation, means that the confidence interval of one parameter depends on the value of the other parameter [30]. Joint confidence contours can be used to account for this statistical interdependence. Such plots provide information on the combinations of model parameter values, that are encompassed by the joint confidence ellipsoid. This would allow for the exemption of combinations that fall outside the ellipsoid that denotes the joint confidence boundary [31,32,33]. In recent literature, there are scarce studies that have estimated and plotted the joint confidence intervals of kinetic parameters [25,31,34,35,36,37,38]. Such approach would be particularly appropriate in order to account for the real uncertainty of model parameters, and thus proceed to realistic and reliable shelf life estimations, using Monte Carlo simulation. Although Monte Carlo techniques have been applied in recent literature for the probabilistic assessment of stochastic variability and uncertainty associated with various quality attributes of different food systems [16,18,26,27,28,29], the statistical interrelation of the values applied in the iterative simulation process has not been considered.

## 2. Materials and Methods

#### 2.1. Basic Principles

_{i}, E

_{j}), with C

_{i}describing factors related to food composition and E

_{j}representing different environmental factors [41,42].

_{ref}is the rate constant at the reference temperature T

_{ref}(K), R: the universal gas constant and E

_{a}: the activation energy (J/mol or cal/mol).

_{t}) quality function at time t, defined by Equation (1) in the case of isothermal conditions, is calculated, with T(t) describing the change of temperature as a function of time:

_{eff}is the value of the rate of the quality loss reaction at the effective temperature T

_{eff}. The T

_{eff}term represents the constant temperature that results in the same quality value as the variable temperature function over the same time period, which equals t

_{tot}.

_{i}of constant temperature T

_{i}(with ${{\displaystyle \sum}}^{\text{}}{\mathrm{t}}_{\mathrm{i}}={\text{}\mathrm{t}}_{\mathrm{tot}}$), and applying the Arrhenius equation as the secondary model, then Equation (3) can be alternatively written as Equation (4):

_{eff}can be estimated and subsequently, from the Arrhenius model, the effective temperature T

_{eff}can be calculated.

#### 2.2. Model Development and Determination of the Uncertainty of Kinetic Parameters

_{0}the initial Vitamin C concentration (mg/100 g of food), k

_{VitC}is the reaction rate of the Vitamin C oxidation at a fixed temperature and at Equation (2) T

_{ref}for frozen foods equals to −20 °C.

_{a}(in kJ/mol) and k

_{ref}(at −20 °C, in days

^{−1}).

^{®}was used and appropriate code was written, in order to derive the asymptotic standard errors (SE), calculated by Equation (7) based on the t-parameter for a confidence level (±SE

_{∙t(1–0.5a),n)}([1,51]).

_{a}and k

_{ref}can be described by a normal distribution curve, rather than a single value. Implementing this assumption, the variability calculated by the model of Equation (6) is incorporated within calculations of the shelf life of frozen green peas.

#### 2.3. Determination of the Variability of Temperature Conditions in the Cold Chain

#### 2.4. Shelf Life Assessment and Uncertainty Determination

_{a}and k

_{ref}variability are effectively represented by a normal distribution [14,25,57]. At each iteration, a random number is generated through an appropriate FORTRAN routine function, and a value is assigned to E

_{a}and k

_{ref}(independently the one from the other). The exact parameter value assigned is based on the discretization of the corresponding normal distribution curve (Figure 2a), and thus the corresponding value frequency. The construction of such Gaussian distributions is based on the estimate of the mean value, and the ±95% C.I. of the kinetic parameters of Equation (6).

_{a}and k

_{ref}), which means that the confidence interval of one parameter depends on the value of the other parameter. Therefore, joint confidence regions were derived using MATLAB and Equation (9); this information was used in order to exclude some of the pairs of values of E

_{a}-k

_{ref,}obtained by the random algorithm of Monte Carlo.

_{a}and k

_{ref}based on the discretization of the corresponding normal distribution curves (2) pairs of E

_{a}and k

_{ref}that do not fall within the estimated joint confidence intervals are excluded from further analysis and (3) a nested Monte Carlo algorithm is applied, where temperatures at each of the three stages are randomly selected based on the discretization of the distribution curves of Figure 1. With all parameter value assigned, Vitamin C retention is then calculated, based on Equation (6), and the shelf life can be accordingly estimated.

## 3. Results

#### 3.1. Application of the Holistic Approach to Shelf Life Prediction in the Frozen Green Peas Cold Chain

_{a}= 102.31 ± 17.91 kJ/mol and k

_{ref}= 0.00196 ± 0.000795 days

^{−1}(as calculated out of a two step procedure and a linear regression analysis), the frozen green peas shelf life is estimated at −20 °C at 390 days (and 248 days at −18 °C). Results from a one step analysis, namely E

_{a}= 104.24 ± 11.34 kJ/mol and k

_{ref}= 0.00177 ± 0.000494 days

^{−1}, are slightly different from those estimated via the 2-step analysis. Using a one step analysis, frozen green peas shelf life is also estimated at −18 °C at approximately 250 days. It can be also observed that when applying the two-step approach, the 95% C.I., (calculated via regression analysis) are usually wider than those calculated with a global-one step approach [22].

_{a}and k

_{ref}variabilities are described by a normal distribution (Figure 2a). The Gaussian distributions illustrated in Figure 2a were constructed based on the estimate of the mean value, (for example E

_{a}= 104.24 kJ/mol) and its standard deviation, (in the case of E

_{a}, σ = 5.8 kJ/mol). The same procedure was followed to construct the corresponding distribution curve for k

_{ref}.

_{a}and k

_{ref}parameters, based on their normal distribution curve, in order to estimate the shelf life (Equation (6)) at an arbitrarily chosen temperature of −18 °C, using the 50% Vitamin C loss as the acceptability limit. Results for Shelf Life (SL) calculation, including its uncertainty, are depicted in Figure 2b, expressed as a frequency curve with a mean value (SL estimate) ± 95% C.I., equal to 254.2 ± 29.9 days, giving a more realistic prediction than the single value estimation of 250 days, based on Arrhenius parameters’ mean estimated values. The necessary number of Monte Carlo simulations for a specific kinetic model is not clearly defined in literature (ranging from some hundreds to few thousands), since there are other factors, such as the degrees of freedom, the range of inputs, possible parameter interactions, etc., that affect algorithm performance [58]. However, the number of 10

^{4}(applied in this study) seems to be a frequently used number of iterations [59,60] and an acceptable compromise between computing power/time and result accuracy.

_{a}and k

_{ref}parameters of the Arrhenius equation were estimated for degradation of cyanidins under dynamic conditions, by generating artificial data of the initial measurements, superimposing the experimental error. The use of a Gaussian distribution is a common practice in food engineering [18,61]; however, the same methodology can be implemented in case a different probability distribution describes better data variability [28,62,63,64,65,66].

_{a}-k

_{ref,}values obtained by the random algorithm of Monte Carlo, were excluded. Given these pairs of E

_{a}and k

_{ref}, the shelf life at −18 °C (arbitrarily chosen) is calculated, which is described by a distribution even narrower than the one of Figure 2b. (Figure 3b, blue line), deriving a shelf life of 248.5 ± 21.3 days (95% C.I.).

_{a}-k

_{ref}are considered, the remaining Shelf Life after 130 days of handling is estimated at 106 days (at a ‘reference’ temperature of −18 °C), (black line, Figure 4a).

^{4}different temperature cases (3 of the random scenarios are depicted in Figure 4a, using different colours) were ran and the remaining shelf life at the end of the 130 days cycle was calculated (Figure 4b). In this approach, it is important to bear in mind that kinetic parameters E

_{a}-k

_{ref}in Equation (6) were assumed constant, with fixed values equal to the mean estimates of regression analysis (Figure 2a).

_{a}-k

_{ref}is selected considering not merely the distributions depicted in Figure 2a, but also the intercorrelation between these parameters, as depicted in the Joint Confidence Intervals (Figure 3a). Having incorporated within the model both temperature variability and parameter uncertainty, remaining shelf life predictions become significantly broader, as it is clearly depicted in Figure 6, where results deriving from different approaches are comparatively depicted. When all possible sources of error are incorporated, remaining shelf life is estimated at 93.4 ± 110.5 (days), vs. 106.0 ± 71.3 (days) when only temperature variability is considered, vs. 112.4 ± 26.0 when temperatures are assumed constant during the three stages (and equal to the mean estimate of each distribution of Figure 1), accounting only for parameter uncertainty. All these predictions can also be compared to the mean value predicted (106 days) without considering any source of error.

#### 3.2. Effect of Parameter Uncertainty

_{a}-k

_{ref}, as obtained by the 1-step non linear regression, based on the Vitamin C measurements. If an artificial error is introduced in raw data, and the derived ±95% confidence intervals (using MATLAB) are assumed broader (as depicted in the joint confidence region within Figure 7a), it can be observed that the contribution of parameter uncertainty becomes more pronounced, than that depicted in Figure 6. Similarly, if initial measurements are improved so as to provide much narrower ±95% confidence intervals (as depicted in the joint confidence region within Figure 7b), one could conclude that parameter uncertainty introduces a relatively small error in comparison to the effect of the storage temperature variability that could practically be neglected.

#### 3.3. Effect of Temperature Variability

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**Temperature distribution during (

**a**) production/distribution warehouse, (

**b**) retail display and (

**c**) domestic storage for frozen foods handling (FRISBEE database).

**Figure 2.**(

**a**) Normal distribution of E

_{a}and k

_{ref}values, based on the mean value and the standard deviation estimated by the one-step non linear analysis and (

**b**) Shelf Life estimated at −18 °C.

**Figure 3.**(

**a**) Joint Confidence Intervals, depicting the correlation between parameters E

_{a}and k

_{ref}and (

**b**) Shelf Life estimated at −18 °C, taking into account the correlation of the two kinetic parameters (). Black line represents the SL distribution without considering E

_{a}-k

_{ref}correlation, blue line after excluding the E

_{a}-k

_{ref}pairs of Figure 3a.

**Figure 4.**(

**a**) Vitamin C degradation, for different temperature scenarios of the cold chain of frozen green peas (black line represent the C/Co value for the average temperatures of all stages) and (

**b**) Remaining Shelf Life estimated at a reference temperature of −18 °C, for the 2000 different distribution scenarios through Monte Carlo technique (kinetic parameters E

_{a}-k

_{ref}in Equation (6) assumed fixed).

**Figure 5.**Flow chart of the methodology used, showing the remaining shelf life of frozen green peas, after 130 days in the cold chain, when both temperature variability and parameter uncertainty were incorporated within model’s prediction algorithm.

**Figure 6.**Remaining shelf life of frozen green peas, after 130 days in the cold chain, when only temperature variability (dashed blue line), only parameter uncertainty (red line) or both temperature variability and parameter uncertainty (solid light blue line) were incorporated within model’s prediction algorithm.

**Figure 7.**Remaining shelf life of frozen green peas, after 130 days in the cold chain, when only temperature variability (dashed blue line), only parameter uncertainty (red line) or both temperature variability and parameter uncertainty (solid light blue line) were incorporated within model’s prediction algorithm. (

**a**) when assuming broader 95% C.I. (

**b**) when assuming narrower 95% C.I. Relative Joint Confidence Regions are depicted within plots.

**Figure 8.**Remaining shelf life of frozen green peas, after 130 days in the cold chain, when only temperature variability (dashed blue line), only parameter uncertainty (red line) or both temperature variability and parameter uncertainty (solid light blue line) were incorporated within model’s prediction algorithm. (

**a**) for slightly improved temperature conditions at the domestic level (

**b**) assuming ideal conditions at the domestic level. Temperature conditions of the third stage are depicted within plots.

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**MDPI and ACS Style**

Giannakourou, M.; Taoukis, P.
Holistic Approach to the Uncertainty in Shelf Life Prediction of Frozen Foods at Dynamic Cold Chain Conditions. *Foods* **2020**, *9*, 714.
https://doi.org/10.3390/foods9060714

**AMA Style**

Giannakourou M, Taoukis P.
Holistic Approach to the Uncertainty in Shelf Life Prediction of Frozen Foods at Dynamic Cold Chain Conditions. *Foods*. 2020; 9(6):714.
https://doi.org/10.3390/foods9060714

**Chicago/Turabian Style**

Giannakourou, Maria, and Petros Taoukis.
2020. "Holistic Approach to the Uncertainty in Shelf Life Prediction of Frozen Foods at Dynamic Cold Chain Conditions" *Foods* 9, no. 6: 714.
https://doi.org/10.3390/foods9060714