Identification of the Kinetic Parameters of Thermal Micro-Organisms Inactivation

Abstract

A mathematical model for estimating the characteristics of the process of thermal inactivation of vegetative bacterial cells and their spores is presented. The model relates the change rate of the number of living cells as a nonlinear kinetic dependence of the p-th order, and the temperature constant of their inactivation rate is the Arrhenius function. A method for solving the inverse kinetic problem of identifying the parameters of this model from experimental data is proposed. The method is implemented through the minimization of the original functional, which reduces the number of variable parameters. The solution results of inverse problems for determining the kinetic model parameters based on the experimental data of thermal inactivation of bacterial spores B. subtilis and B. anthracis are presented. The obtained parameters are used to solve the direct problems of the dynamics of micro-organism inactivation. The calculation results represent the dependence on the time of the change number of inactivated micro-organisms, and the thermal exposure time for 99% of their deaths at different temperatures. A comparison of the results with other authors’ calculations and experimental data confirms the adequacy of the model, the high accuracy of the new solution method and the algorithm for its implementation. The developed model of thermal sterilization can be used for the selective deactivation of pathogens in the food products.

1. Introduction

The processes of thermal sterilization of biotechnological equipment, food products, medical materials, life support systems, etc., are based on both vegetative bacterial cells and their spore deaths at high temperatures. The high-temperature extent of exposure depends on the type of bacterial culture, concentration, physical condition and other factors. In particular, as concerns food products, heat treatment of milk is a mandatory requirement nowadays.

The death of the vegetative cells of micro-organisms when heated at a temperature range of 65–95 °C is called pasteurization. Thermal exposure at temperatures above 120 °C is called ultra-pasteurization (destruction of spore cultures [1]). The problem in both cases is to minimize the loss of food nutritional value [2,3]. Some types of bacteria are pathogens or spoilage agents and form spores with increased resistance. Their inactivation is an important factor in the development of food-processing technologies [4,5].

As a mathematical basis for describing the nonlinear dynamics of the number of micro-organisms contained in food products [5,6], the approaches of chemical kinetics (transition state theory) are successfully used. Models of chemical reactions occurring in gas–liquid and heterogeneous media adequately describe the biochemistry of many processes. This makes it possible to use well-established methods for the solutions to the physical chemistry problems of embryogenesis [7,8] and analysis of the life cycle of micro-organisms [9,10,11].

Many applications of the original and modified forms of the Arrhenius equation have been reported. These mathematical forms of the Arrhenius equation are used in various models: the growth rate of bacterial cultures; the respiration of plant leaves; the temperature dependence of the heart rate of amphibians; the kinetics of food spoilage, etc., [12]. As an example, the modeling of the behavior dynamics of living organisms is described by the kinetics of molecular reactions [13], and biological systems can be used to obtain certain values of the rate constants and activation energy.

To solve applied problems for quantitative estimation of the efficiency of thermal sterilization modes and predication of thermal inactivation, mathematical models are widely used: most important models consider the relevant features of these processes. The physiological, biochemical and physical properties of micro-organisms regulated by temperature, as well as effective modes of food processing, have been discussed [14]. Since the appearance of the original (1889) Arrhenius equation, many models have arisen to describe the population dynamics of organisms, the rate of enzymatic reactions and other temperature-dependent processes. Empirical approaches have been improved, and models based on the thermodynamic principles of biomolecular processes have become more complicated.

In particular, the criteria for evaluating the applicability of various approaches to studying the mechanisms of micro-organism inactivation are given [15]. New mathematical models were obtained by modifying the Arrhenius function, applying the Eyring–Polanyi transition state theory considering the diffusion and heat transfer equations [16,17,18]. The multifactorial model of extended quasi-chemical kinetics describes the continuous dynamics of some micro-organisms of bacterial cultures from the beginning of growth to the moment of complete inactivation [19]. These models enables to predict the growth or inactivation of micro-organisms as a function of environmental factors such as temperature, pH and water activity [20]. Structural models have been proposed, to describe microbial inactivation during heat treatment [21]. We note that in publications on similar subject matter, aspects related to the description of the model and the practical implementation of computational procedures are not considered together.

Some approaches have used the Arrhenius function to study the thermal effect on organic compounds that are part of food products (vitamins, proteins, amino acids, etc., [22]). According to such approaches, the degree of thermal exposure in specific elements of the environment (media) can be estimated from the value of the inactivation energy $E_d$ during thermal destruction: for bacterial spores, the order of value is 200–400 kJ mol⁻¹, for proteins it is 160–170 kJ mol⁻¹ and for amino acids there are two times less [22]. Therefore, control of the mode of thermal exposure (temperature and duration) makes it possible to regulate the direction and selectivity of the sterilization process, as well as to evaluate the component composition of the product after the sterilization process. However, the strategy of temperature regulation of biotechnological processes has not yet been sufficiently developed [14].

The main aim of this study is to offer a kinetic model for the thermal inactivation of vegetative bacterial cells and their spores. This paper presents a solution method and a simple algorithm for the numerical implementation of the proposed approach to calculating the model parameters. To solve the inverse problem of kinetics according to experimental results, the inactivation energy $E_d$ and the frequency factor $A$ of the Arrhenius function are calculated to determine the constant $k_d\left(T\right)=A·\exp \left(-E_d/RT\right)$ of the inactivation rate of the vegetative bacterial cells and their spores. To achieve this, the dependence of the change rate $dF/dt$ of the dimensionless (relative to the initial quantity) number $F$ of living cells (spores) on the temperature function $k_d\left(T\right)$ and the order $p$ of the inactivation ($dF/dt\propto F^p$) can be used. It should be noted that in several publications, stepwise inactivation was studied, the order $p$ of the reaction mechanism being considered from 0 to 2 [18,19,22,23]. The problem of identifying the values ($p, A$ and $E_d$) is set as an optimization problem, to minimize the spread of one of the determined required parameters of the mathematical model—one of which is inherently constant [24,25,26]. The implementation of the algorithm is presented by a few Excel-formulas in the form of a short program code, which can be easily reproduced and verified.

2. Basics of the Method

In a closed system, the death of bacterial cells and their spores is considered the only irreversible process, and the rate of cell death in general is considered proportional to the actual number of living micro-organisms. Furthermore, it should be noted that the first-order kinetics of death does not always occur, especially for bacterial spores immediately after thermal exposure [2,10,18,19,22,23]. To enhance the generality of the proposed approach, when the equation of the model is formulated, the power law of this dependence $dF/dt\propto F^p$ is assumed.

A mathematical model is used to describe the process of death of the micro-organisms. The model relates the rate of change $F$ of the dimensionless (relative to the initial quantity of living cells) as a kinetic dependence of the $p$-th order where the inactivation rate constant is determined through the absolute temperature $T$ in the form of an Arrhenius function

$$\frac{dF}{dt}=-k_d{F}^p=-F^p·A·\exp \left(-\frac{E_d}{RT}\right),\hspace{1em} F\left(t=0\right)=F_0=1, R=8.314 \frac{J}{K·mol},$$

(1)

where: $t\ge$ 0 is time; [$t$] = min or s;

$A$ is pre-exponential (frequency factor); [$A$] = [$t$]; and

$E_d$ is the inactivation energy of the micro-organisms, determined by the relationship between the rate constant of death $k_d$ against temperature $T$ which is equivalent to the term activation energy of a chemical reaction, [$E_d$] = J·mol⁻¹.

The initial experimental data are given in table $\left\{t_{ij},F_{ij}\right\}$ ($i$ = 0, …, $N_j$) as a dependence of the dimensionless number of living bacterial cells or spores in different time moments $t_{ij}$ at a constant temperature $T_j$ ($j$ = 1, …, $M$) of biological culture. According to the physical meaning, the analyzed data should begin with the values $F_{0j}$ = 1.

It is worth mentioning that, according to the form of the dependence (1), it is similar to the Wigner–Polanyi equation for thermal desorption of molecules adsorbed on the solid-state surface [25,27].

At constant temperature, the rate coefficient $k_d$ = const, in such a way that, after separating the variables and integrating the right- and left-hand sides of Equation (1),

$$\underset{F_0}{\overset{F}{{\displaystyle \int }}}\frac{dF}{F^p}=-\underset{0}{\overset{t}{{\displaystyle \int }}}{k}_{d}dt=k_{d}t,$$

(2)

the time dependence for the dimensionless number of living micro-organisms is as follows:

$$F\left(t\right)=\left\{\begin{array}{c}\exp \left(-k_{d}t\right), \mathrm{if} p=1,\\ {\left[1-\left(1-p\right)k_{d}t\right]}^{1/\left(1-p\right)}, \mathrm{if} p\ne 1.\end{array}\right.$$

(3)

From Equation (3) can be expressed the value of the pre-exponential factor

$$A=\frac{1}{t}·\exp \left(\frac{E_d}{RT}\right)·\left\{\begin{array}{c}-\ln \left(F\right), \mathrm{if} p=1,\\ \left[1-F^{1-p}\right]/\left(1-p\right), \mathrm{if} p\ne 1.\end{array}\right.$$

(4)

In accordance with Equation (4), for each range of values $t_{0j}-t_{ij}$ of the initial data table $\left\{t_{ij},F_{ij}\right\}$ ($i$ = 0, …, $N_j$), $T_j$ ($j$ = 1, …, $M$), a set of constants $A_{ij}$ can be calculated:

$$\begin{gathered} A_{ij}=\frac{1}{t_{ij}}·\exp \left(\frac{E_d}{R{T}_j}\right)·\left\{\begin{array}{c}-\ln \left(F_{ij}\right), \mathrm{if} p=1,\\ \left[1-F_{ij}^{1-p}\right]/\left(1-p\right), \mathrm{if} p\ne 1,\end{array}\right. \\ i=1,\dots ,N_j j=1,\dots ,M. \end{gathered}$$

(5)

It is assumed that the “true” values $p,A,E_d$ of the triplet are reached when the set of calculated values $A_{ij}$ has a minimum spread, since the pre-exponential factor $A$ is a constant essentially. This assumption can be mathematically expressed using the functional $\mathcal{F}$, which is a statistical function of the coefficient of variation [24,25,28]:

$$\mathcal{F}\left(p,E_d\right)=\frac{1}{\overline{A}}{\left[\frac{1}{{{\displaystyle \sum }}_k^M{N}_k}{\displaystyle \sum }_{j=1}^M{\displaystyle \sum }_{i=1}^{N_j}{\left(A_{ij}-\overline{A}\right)}^2\right]}^{0.5} , \overline{A}=\frac{1}{{{\displaystyle \sum }}_k^M{N}_k}{\displaystyle \sum }_{j=1}^M{\displaystyle \sum }_{i=1}^{N_j}{A}_{ij}.$$

(6)

In other words, the unknown values of the parameters $p$ and $E_d$ are chosen in such a way that for a set of constants $A_{ij}$, calculated by Formula (6), total deviation should be minimally different from the average value $\overline{A}$. The obtained values $p,E_d$ and $A$ = $\overline{A}$, as a result of minimization, determine the optimal values of the kinetic triplet.

The undoubted advantage of the proposed approach is the search process of the optimal values $A,E_d,p$, where only two variables, $E_d,p$, vary. The functional $\mathcal{F}$ corresponds to the variation coefficient, which can be easily calculated using statistical Excel functions.

3. Results and Discussion

The foregoing approach was examined in two examples of the problem-solving of identification kinetic parameters according to the experimental data of the spore’s inactivation of two bacterial cultures, B. subtilis and B. anthracis. The first example concerned the identification of the kinetic triplet ($p,A,E_d$) for a given tabular dependence resulting from the quantitative analysis of the number of viable B. subtilis spores over time at different temperatures [10]. The solution problem was executed in MS Excel spreadsheets, which do not require programming experience in high-level languages. To find the optimal values of the kinetic triplet ($p,A,E_d$) the Solver was used. The algorithm for calculating the values of the kinetic triplet $p,A,E_d$, defined by the following steps, is illustrated in Figure 1.

1. The initial data represent the dependence of the relative number of living cells or spores $F_{ij}$ over time (min) at different temperatures (358, 363, 383, and 393 K), entered into the cell range C3:F13 (Figure 1).

2. Sufficiently arbitrary initial values of the parameters $E_d/R$ and $n$ were entered into cells N3 and N4 where, after the Solver execution, optimal values were formed. The parameter $n=100p$ (one hundredfold the order of the process) in the Solver Parameters dialog box Subject to the Constraints was defined as an integer. In this example, the values were set to 11,111 and 11, respectively. The Excel formulas in cells N16 and N7 determined the optimal values $E_d$ and $p$ directly.

3. To calculate $A_{ij}$ according to the above relationship (5), the formula in cell H5 was entered, which to down to row 13 and to the right to column K, and was copied by dragging the fill handle marker [28] (the bold arrows in Figure 1).

4. In Set Target Cell N10, where the Objective Function was optimized, Formula (6) for calculating the functional (coefficient of variation), using the Excel functions STDEV.P(·), was entered. According to the relationship (6) above, the average value of the pre-exponential factor $\overline{A}$ was calculated in cell N15 using the AVERAGE(·) function.

The Solver Parameters dialog box was set up to search the minimum of the Objective Function (radio button To Minimum [Min]) by the solving method GRG Non-Linear; the flag Make Unconstrained Variables Non-Negative was set. The constraints indicated that the number in cell N4 was defined as the integer $N$4 = integer. The click button Solve in the Solver Parameters dialog box is running optimization.

When the optimization was performed, the optimal values $p,E_d$ and $A$ were displayed in cells N10, N16 and N15; consequently, Equation (1) can be represented as

$$\frac{dF}{dt}=-3.04·{10}^{30}F·\exp \left(-\frac{222076}{RT}\right).$$

(7)

The quantitative adequacy of the methods was characterized by the determination coefficient $R^2$ by the proportion of the model error variance [24,28]:

$$R^2=1-\frac{{{\displaystyle \sum }}_{j=1}^M{{\displaystyle \sum }}_{i=1}^{N_j}{\left(F_{ij}-{\widehat{F}}_{ij}\right)}^2}{{{\displaystyle \sum }}_{j=1}^M{{\displaystyle \sum }}_{i=1}^{N_j}{\left(F_{ij}-{\overline{F}}_{ij}\right)}^2}, {\overline{F}}_{ij}=\frac{{{\displaystyle \sum }}_{j=1}^M{{\displaystyle \sum }}_{i=1}^{N_j}{F}_{ij}}{{{\displaystyle \sum }}_{j=1}^M{N}_j},$$

(8)

where $F_{ij}$ and ${\widehat{F}}_{ij}$ were the initial experimental and calculated values according to the above model (7), using the above relationship (3).

The determination coefficient was calculated using the Excel functions SUMXMY2(·), VAR.P(·) and COUNT(·) (cells C15:F15 and J15 in Figure 1).

For the obtained parameters of the biokinetic model, the characteristics of the death of B. subtilis spores under thermal exposure were calculated.

The content percent $F^d\left(t\right)$ of inactivated spores, depending on the time $t$ at temperature $T_{*}$ was determined by the following relationship, which was obtained from Formula (3) at $p$ = 1, considering Equation (7):

$$F^d\left(t\right)=1-F\left(t\right)=\left\{1-\exp \left[-3.04·{10}^{30}·\exp \left(-\frac{222076}{R{T}_{*}}\right)t\right]\right\}·100\%, T_{*}=\mathrm{const}.$$

(9)

The time-dependent curves required for the death of 99% of the bacterial spores at different exposure temperatures, and the corresponding curves of the percent of dead spores versus exposure time, are shown in Figure 2 and Figure 3 respectively.

The curves were essentially exponential; a small increase of temperature had a significant effect on the death of spores.

The comparison of the time required to kill 99% of the spores, between the calculated value according to Equation (3) and the given value [10], was less than 4.9%. The calculated value of the inactivation energy $E_d$ = 222076 J·mol⁻¹ differed from the given value [10] by 1.2%.

For mathematical models that include the temperature Arrhenius function, analysis of linear correlation between the results of the direct problem solving and experimental data is a powerful tool for examining these models [1,29]. The results obtained at certain values of the kinetic parameters of the Arrhenius function must always be compared with the original experimental data to confirm the actual application based on the accepted assumptions. When the proposed approach was used, the attained value $R^2$ > 0.999 indicated the quantitative adequacy, and the high level of the approximation accuracy for the identification of the kinetic parameters of the model.

To analyze the mortality level of the micro-organisms, the parameter $D$—the tenfold decrease time (or decimal reduction time) in number for certain values at constant temperatures of the content of a bacterial culture was used [15,19,23,30]. To approbate the proposed approach, a second example relating to the identification of kinetic parameters for $D$-type experimental data could be used in the following form:

$$D\left(T\right)=\ln \left(10\right)\exp \left(E_d/RT\right)/A,\hspace{1em}\hspace{1em}\left[D\right]=s,$$

(10)

which was derived from Equation (3) at $p$ = 1 and $F$ = 0.9.

The identification of the kinetic parameters $E_d$ and $A$ of model (1) or Equation (10) was considered by the example of the data of spore mortality B. anthracis in liquid media [30]. The experimental temperatures and the corresponding $D$ values for $N$ = 8 data pairs $\left\{T_i,D_i\right\}$ were given in the cell range B3:C10 (Figure 4). The algorithm of the solution was similar to (Figure 1) for B. subtilis, cited above.

A sufficiently arbitrary initial value of $E_d/R$ was entered into cell H3—for example, 22,222. To calculate the pre-exponential factor $A_i=\ln \left(10\right)\exp \left(E_d/R{T}_i\right)/D_i$ for the first pair of initial data ($T_1,D_1$), Formula (10) was entered into cell E3 and copied down to row 10 by a fill handle marker [28]. On the basis of the obtained values of a set of constants $A_i$, $i$ = 1, …, $N$ (E3:E10), in cell H9 the average value $\overline{A}$ was calculated. Similar to Equation (6), the functional $\mathcal{F}\left(E_d\right)$ was represented by the variation coefficient determined in cell H5.

When the Solver was being set up to minimize the Objective Function in cell H5 with the Changing Cell H3 of variables, the unknown values $A$ and $E_d$ were displayed in cells H9 and H10 correspondingly (Figure 4).

The estimation of the determination coefficient according to the square of the Pearson correlation coefficient [28], by comparison of the initial data with the calculated results (Equation (10)), through the Excel-function RSQ(·), was $R^2$ > 0.999, which indicated a high adequacy of the obtained values of the kinetic parameters of the model. The calculated value of the inactivation energy $E_d$ = 293,640 J·mol⁻¹ differed from the value obtained by the authors [30] for the ANR-1 strain by 0.8%. A good correlation between the calculated results (solid line) and experimental measurements of the $D$-value of the spore reduction time for the three indicated strains of B. anthracis is illustrated in Figure 5.

In real devices for thermal pasteurization, when the liquid containing B. anthracis culture is heated from the initial $T_0$ (20 °C) to a certain temperature, the spores die off gradually. In such a case, the temperature induction period [18] can have a significant effect on the time of the process. The calculation results for estimating the temperature change of a liquid medium containing spores with mass $m$ from a heater at a temperature $T_h$ = 373 K (100 °C) on a surface $S$ are shown in Figure 6. The estimate was calculated for a simple model which described the process of the temperature change of a liquid medium with heat capacity c under heat transfer forced convection in accordance with the Newton–Richmann law, where α is the heat transfer coefficient [18,31]:

$$cm\frac{dT}{dt}=\mathsf{\alpha }S\left(T_h-T\right), T\left(t=0\right)=T_0.$$

(11)

The temperature change of the liquid medium depending on time was determined by the analytical solution of Equation (11), which took the exponential form (Figure 6), when $T\left(t\to \infty \right)\to T_h$:

$$T\left(t\right)=T_h-\left(T_h-T_0\right)\exp \left(-t/t_{*}\right), t_{*}=cm/\left(\mathsf{\alpha }S\right),$$

(12)

where the time to reach the relaxation temperature $T_{rel}\approx$ 71 °C ($T_{rel}=T_h-\left(T_h-T_0\right)/\mathrm{e}$, $\mathrm{e}$ is the Euler number) was ≈4 s, which was estimated according to the given thermophysical characteristics of the liquid [32].

Using the dependence (11) of the temperature change $T\left(t\right)$, when the liquid medium containing B. anthracis was heated, the mortality $F^d\left(t\right)$ could be estimated by integrating Equation (1) over time:

$$F^d\left(t\right)=1-F\left(t\right)=\left\{1-\exp \left[-8.225·{10}^{39}\underset{0}{\overset{t}{{\displaystyle \int }}}\exp \left(-\frac{293640}{RT\left(t\right)}\right)dt\right]\right\}·100\%.$$

(13)

The technique of integrating the Arrhenius function in one Excel cell, by using CSE array formula [28], is described in [33].

The time-dependent curves of the relative number of dead spores (%) in the liquid medium for two modes of temperature exposure, at instantaneous heating to 100 °C and gradual heating with the induction period, are shown in Figure 7. The latter curve demonstrates a sigmoid character, where the inflection point is close to the moment when the temperature of the liquid medium reaches the temperature of the heater with an accuracy of $\approx$1%. For $F^d\left(t\right)$ = 0.5, and the lag length $\mathsf{\Delta }$ is about 14 s.

4. Conclusions

1. A model to describe the death of micro-organisms (cells or spores) as a kinetic function of a certain order, when the temperature dependence of the inactivation rate is determined by the Arrhenius function, is proposed. To solve the inverse problem of identification of the kinetic parameters of the model, an original statistical functional is used, which makes it possible to reduce the number of variable parameters.

2. For the proposed approach, an algorithm of implementation in the most common computing environment, MS Excel spreadsheets, is presented, which is distinguished by the high accuracy, simplicity and clarity of a short program code. The step-by-step representation of the detailed algorithm, and screenshots of the final worksheets in MS Excel, provide easy reproduction and verification.

3. The obtained parameters of the biokinetic models are used to solve direct problems of the population dynamics of micro-organisms under thermal effects. For different types of temperature exposures, the calculation results are presented both as forms of time dependent on the percent change of the inactivated micro-organisms, and as the necessary time required to reach the death of 99% of the bacterial spores. In addition, the time-dependence of the relative percent of dead spores for the instantaneous heating to 100 °C of the model medium, and for the gradual temperature increase that occurs in practice, are obtained.

4. Comparison with the calculations of other authors, high-determination coefficients of the results of direct problem solving with initial values, and comparison with experimental data confirm the adequacy of the model, the proposed new solution method and the algorithm of implementation.