Approaches for Modeling Thermal Inactivation of Foodborne Pathogens

Chapter 16

Approaches for Modeling Thermal Inactivation of Foodborne Pathogens 1

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Vijay K . Juneja , Lihan Huang , and Harry Marks

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Downloaded by CORNELL UNIV on July 7, 2012 | http://pubs.acs.org Publication Date: April 6, 2006 | doi: 10.1021/bk-2006-0931.ch016

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Eastern Regional Research Center, Agricultural Research Service, U.S. Department of Agriculture, 600 East Mermaid Lane, Wyndmoor, P A 19038 Regulations and Directives Development Staff (RDDS), Food Safety and Inspection Service, U.S. Department of Agriculture, Cotton Annex Building, 300 12 Street, S W , Washington, D C 20250 th

This chapter deals with some of the fundamental concepts concerning thermal processing of foods to eliminate foodborne pathogens. For most foodborne pathogens in a food matrix under isothermal conditions, the cell population generally decreases exponentially with heating time and, therefore, can be described by 1 -order kinetics. The resistance of microorganisms to heat is conventionally characterized by D and z values. For mixed cultures, if both D and z values follow 1 -order kinetics, a mixed-culture model can be used. With the latter model, the more heat-sensitive microorganisms will be preferentially inactivated, followed by the heat-resistant ones, explaining the "tail" effect observed in some survival curves. For more complex survival curves, a general "Weibull"-type model can be used. This model offers more flexibility in describing either convex, concave, or linear survival curves. This model is generally more accurate than the transitional linear model when used to describe convex and concave curves. st

st

© 2006 American Chemical Society

In Advances in Microbial Food Safety; Juneja, V., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 2006.

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Kinetic Analysis of Bacterial Inactivation by Heat - Chemists' Approach Like most chemical reactions, inactivation of bacteria is highly dependent on the time and temperature in the environment to which the bacteria are exposed. When a homogeneous population of a pure strain of bacteria is subjected to an environment maintained at a constant temperature, the reduction of bacterial counts generally follows l -order reaction kinetics (7, 2), which can be mathematically described by:


st

(1)

In this equation, C usually represents the count of bacteria per unit mass or volume, t is the total heating time, and k is the rate constant. Under isothermal conditions, this equation can be integrated, resulting in: (2) In eq 2, C is the initial counts of bacteria. This equation reveals a log-linear relationship between the bacterial count and the heating time under an isothermal condition, indicating that the logarithm of bacterial counts decreases linearly with time as heating progresses. A plot of the logarithm of bacterial counts versus time is referred to as an inactivation or survival curve. Equations 1 and 2 describe the change in the bacterial counts under an isothermal condition. However, the inactivation of microorganisms is highly dependent upon the heating temperature. The effect of temperature is usually manifested in the rate of inactivation during an isothermal process. Conventionally, the Arrhenius equation is used to described the relationship between the heating temperature and the rate of inactivation, such that: 0

k = Ae

(3)

In this equation, k is the rate of inactivation, A is a constant, also known as a frequency factor, E is the activation energy (kJ), R is the gas constant 8.314 χ 10" kJ/mol Κ and T is the absolute temperature (K). The Arrhenius equation was originally developed by a Swedish chemist, Svante Arrhenius. He developed a collision theory to describe the reactions among molecules. He hypothesized that a chemical reaction between reacting a

3

K


237 chemical species was a result of their molecules colliding with each other. The activation energy, E , was the minimum energy required for the chemical reaction to occur. Increasing the reaction temperature would increase the kinetic energy of reacting molecules and, thus, increase the rate of reaction. Equation 3 can be rewritten into a simpler form shown in eq 4. The new equation depicts a log-linear relationship between k and the inverse of T. If the logarithm of the rate is plotted against 1/T, a linear relationship should be observed. From the slope of the linear curve, the activation energy (E ) can be determined. The frequency factor can be calculated from the intercept: a

a


(4)

Thermal inactivation is generally believed to be caused by the irreversible denaturation of enzymes critical to support the biological activities of microorganisms. These enzymes are mostly complex large macromolecules distributed in the cells of microorganisms. At sufficiently high temperatures, these large protein molecules may be denatured and lose biological activity. Although eq 3 does not make any physical sense when it is applied to explain a physical process in a biological system such as inactivating microorganisms in foods, the l -order kinetics and the Arrhenius equation serve very well when used to describe the process of thermal inactivation of microorganisms in foods. st

Kinetic Analyses of Bacterial Inactivation by Heat - Food Scientists' Approach Food scientists and food microbiologists have traditionally been using a different approach to quantify the presence of microorganisms in foods. Rather than using the natural log of bacterial counts, as shown in eq 2, they prefer using the base-10 logarithm of bacterial counts to quantify the number of microorganisms in foods. They also observe a log-linear relationship between the log counts of bacteria and the heating time under isothermal conditions (eq 5). Instead of directly using the slope of the linear curve, a new term, D , was introduced (2). The D-value of an isothermal process now possesses some biological meaning. It is defined as the time needed to achieve one log (a 10fold) reduction in the bacterial population within the test medium under a constant heating temperature. Mathematically, D is the inverse of k:


238

log(c)=log(C )-^xi 0

D

ln(lO) k

2.303 *

(5)

(6)

k


Since D is directly related to the rate constant (k) under isothermal conditions, it can be used to correlate the effect of temperature on thermal inactivation of microorganisms in food systems. Historically, this correlation is also expressed as a log-linear relationship between D and T:

(7) ζ In this equation, ζ also has some biological meaning to food scientists and food microbiologists. This term (z) is defined as the increase in temperature needed to reduce the D value by one-log cycle (a 90% reduction in D value) within the test medium. It is an indicator for the effect of temperature on thermal inactivation of microorganisms under isothermal conditions. Determination of the kinetics data (k, E , D , and z) of microorganisms in foods is usually accomplished by graphical methods. Figures 1 and 2 illustrate the concept of using graphical methods to determine the thermal inactivation kinetics of microorganisms using the same data set and both approaches discussed in this section. These two figures clearly demonstrate that both approaches are equally suitable for kinetic analysis of thermal inactivation. However, the second approach (D and ζ values) are predominately used and historically accepted by food scientists and microbiologists. Therefore, the latter approach will be discussed in more detail in the following section. a

Using the Linear Kinetic Model The D and ζ values provide critical information about the heat resistance of the target microorganism for a thermal process. If the food to be pasteurized/sterilized can be instantaneously heated to and held at a constant process temperature, and i f the target organism is uniformly distributed within the food, then the D and ζ values can be directly used to accurately estimate and calculate the extent of bacterial destruction.



239

st

Figure 1. Graphical representation of the 1 -order inactivation kinetics under isothermal conditions and the dependence of inactivation rate upon heating temperature.

However, most foods cannot be instantaneously heated to a process temperature because heat transfer is a dynamic and transient process. If the food is solid, then heat must be transferred by conduction, which means that the outer layer of the food is heated first, and the center of the geometry still remains cold. This point is normally known as the "Cold Spot", or the coldest point in the food. To kill bacteria potentially present in the cold spot, heat must penetrate to this point. To design an effective thermal process, it is necessary to identify the potential cold spot in a food, and to increase the temperature of this point sufficiently high enough to kill the microorganisms. To ensure microbial safety, it is also necessary to identify the most heat-resistant target organism of concern. If this particular organism is inactivated, then it is fairly confident to declare that the food is safe with respect to this organism.


240


mm

Figure 2. Linear reduction of bacterial counts under isothermal conditions and the dependence ofD values on temperatures.

Since the temperature at the cold spot is not constant during heating, the D and ζ values cannot be directly used to calculate the extent of bacterial destruction. The total kill of bacteria is the integration of the entire transient process of heating. In particular, a parameter, called the F-value, which is the accumulated or integrated lethality, expressed as equivalent minutes at a specific reference temperature, T , is used for designing thermal processes (2). The value of F can be derived from the l -order kinetics (eq 1), as follows: Separating by terms, eq 1 can be re-written as: Ref

st

(8)


241 This equation can be integrated, resulting in

(9)

This equation can be further written as


(10)

B y changing the base of logarithm, the destruction of bacteria can then be expressed with terms more commonly used by food scientists and food microbiologists:

dt

D(T)

(ii)

The far left hand-side of eq 11 represents the total log reduction with respect to the initial counts of bacteria. The term D(T) in the far right side of equation represents the rate of bacterial destruction as a function of transient temperature. Since D(T) is a function of temperature (eq 7), the total log reduction of bacteria in a transient process can be expressed as:

(12)

where D and ζ are defined in eq 7. Food scientists and microbiologists compare dynamic processes with a process held under a known reference temperature, T f. For this temperature D = D f. Thus, for any thermal process, regardless isothermal or dynamic, the estimated total log reduction during a thermal process can be normalized to a 'process equivalent' total kill under the reference temperature (eq 13): 0

Re

0

Re

(13)


242 Since D

R e f

is the time required to achieve a 1-log reduction, the result of the

right side of eq 13 is multiplied by D f to estimate F

- the total heating time

T

Re

Re/

required to achieve desirable log reductions at the reference temperature ( T ) . With the concept expressed by the F- value, the time of a thermal process can be designed as multiples (n) of the D value ( D ) at the reference temperature for determining the obtained lethality. If the total heating time of a thermal process is equivalent to n x D f at the reference temperature, then the process is conventionally known as a nD process. With this concept, it is now possible to compare different time-temperature histories of any thermal process. However, we caution that before using the above equations for integrated lethalities in designing thermal processes experiments are needed to validate these equations. Ref

Ref


Re

Nonlinear Thermal Inactivation Curves Some experimental observations indicate that thermal inactivation curves are not always linear (3). One of the simplest cases where this would arise is for a culture containing two different strains of microorganisms with two distinct characteristics of thermal resistance. If each individual strain follows l -order kinetics, then models describing the inactivation kinetics for the mixed culture can be derived. Let's assume that the total initial concentration of a two-strain (strain 1 and 2) mixture of organisms is C , f is the fraction of Strain 1 in the initial inoculum, and C i and C are the concentrations of the two strains of microorganisms at any time of thermal inactivation. Since both strains follow the l -order inactivation kinetics, the concentrations of both strains of organisms during heating can be expressed as a function of time: st

0

2

st

2.303

C =fC e {

0

(14)

(15) Then total log counts of microorganisms at any given time of thermal inactivation are the sum of both strains. Therefore:


243 2.303


log(c) = log(C,

2.303

+C )=log(C )+log 2

(16)

0

Figure 3 shows the examples of hypothetical thermal inactivation curves of a two-strain mixture as affected by the relative ratio of the two strains with two different D values. A s defined previously, f is the initial fraction of the strain with a smaller D value in the two-strain mixture. If f = 0, the initial inoculum only contains the stain with a higher D value. And i f f = 1, then the initial inoculum consists of the first strain with a small D value. With a two-mixture inoculum, the isothermal inactivation curves show different degrees of upward concavity. Figure 3 ( A and B) clearly illustrates that the strain with a larger D value should be the target organism for thermal processing when the two strains are significantly different in thermal resistance. In these curves, the D value o f the first strain is only 25 or 50% of the other strain in the mixture. If both strains are in the food system at the same time, the strain with the lower D value would be preferentially inactivated. Even though the majority (90-99.9%) of the mixture is the first strain, the process is clearly determined by the inactivation of the second strain which is significantly higher in D value. If both strains are similar in D values, as illustrated in Figure 3 (C and D), the thermal inactivation is still primarily affected by the strain with the higher D value, but the impact of the relative fraction (f) of the mixture is less significant. The curves do not show significant non-linearity, and can be modeled by 1 order inactivation kinetics. Figure 4 shows an extreme example of hypothetical thermal inactivation curves with dramatically different heat resistance. The D value of the less heat resistant strain (Strain 1) is only 5% of the more resistant strain (Strain 2). There are 2-4 logs of Strain 2 organism in the total population of a 9 l o g mixture. The inactivation curves are linear during the initial stage of heating. However, the curves all level off, showing the "tailing effect", when the more heat resistant strain is present in the mixture. In most cases, the "tailing effect" usually indicates that a small fraction of organisms cannot be inactivated within the time frame of the study. Figure 4 illustrates that the "tailing effect" may be caused by the contamination with a small fraction of very heat resistant organism. Because the contaminating strain is extremely heat resistant when compared with the target organism, they can survive even though all the less heat resistant target is eliminated from the mixture. Since the heat inactivation study is usually conducted against the target organism, the sampling time and experimental plan are developed based on its heat resistance which is very low in D value when compared with the contaminating strain. The sampling time is too small for the contaminating organism. If the same sampling scheme developed for the target organism is used to sample the contaminating organism, heating time may not be st

10


244


sufficient to cause a significant change in the population of the contaminating organism. As a result, although the heating time may seem extremely long for the target organism, the contaminating organism will survive the heating, giving a "tailing effect".

Figure 3. Effect of mixed culture with different thermal resistance on thermal inactivation curves.

When observing a "tailing effect", the model expressed in eq 16 can be used to show the potential existence of the more heat resistant contaminating organism. It may be necessary to conduct a differential heating study to further identify the existence of the more heat resistant strain. The differential heating study can be conducted in two phases. In the first phase of the study, heating is used to eliminate the less heat resistant strain. Then, in the second phase the remaining organism may be sub-cultured, isolated, propagated, and further studied to examine i f the contaminating strain is different from the target organism. Another set of heat inactivation studies may be conducted to determine the thermal inactivation kinetics of the isolated contaminating organism.



245

Figure 4. Effect of mixed culture with distinctively different thermal resistance and the manifestation of "tailphase".

The Shoulder Effect The shoulder effect (Figure 5) is another category of the thermal inactivation curves commonly reported in the literature. The shoulder effect is usually manifested in the initial stage of heating where the microorganisms are not inactivated by the heating. Therefore the thermal inactivation curve is "flat" during the initial stage of heating. But after passing a threshold heating time (to), the inactivation curve becomes linear, as illustrated in Figure 5. If heat is instantaneously delivered to the food, and its temperature is immediately raised to the heating temperature, there may be some biological mechanisms responsible for the "shoulder effect" for some organisms. There may exist an initial energy barrier similar to the "activation energy" in chemistry. Sufficient thermal energy must be accumulated before showing the lethal effect. For these kinds of thermal inactivation curves, the kinetics can be modeled by:



246

log(c)=log(C ) 0

at tt

0

However, the "shoulder effect" may be caused by a simpler physical factor. Heating studies are usually conducted in some type of container. The amount of sample, thickness of containers, or existence of air pockets in the samples may affect the initial heat transfer process. The sample temperature cannot be instantaneously increased to the heating temperature used to kill the target organism. Most physical systems, such as food samples in containers, possess certain physical characteristics that affect the transfer of heat into the samples. One of the physical characteristics is the system's response time (τ), which, in this case, is defined as the time required for the temperature of the sample to reach 63.2% of the difference between the heating temperature and the initial temperature of the sample. The total time required for the food sample to reach the temperature of the heating medium is usually 3 times the response time.


247


Since the kinetic data is gathered under isothermal conditions, and the temperature of the sample does not increase instantaneously to the final heating temperature, the initial stage of heating should not be included in the kinetic data analyses. Therefore, it is necessary to measure the response time of the food sample subjected to isothermal heating. The initial sampling time must be sufficiently long (> 3τ), particularly at the higher temperature range where the D value is relatively low. The effect of response time, however, usually does not significantly affect the measurement of D values at lower heating temperature where the D values are substantially longer than the system's response time.

A General Model In the past few years, interest has increased to understand the response of a cocktail of different strains of an organism relative to thermal inactivation kinetics. In theory, the mixed culture model (eq 16) can be extended to understand the fraction and the D value of each strain in the cocktail. For organisms with similar heat resistance, the thermal inactivation curves should not deviate significantly from the linear kinetics. Therefore, the thermal inactivation curve should still show a linear trend. As illustrated in Figure 3 (C and D), the shape of thermal inactivation curves could, for practical purposes, be linear. Therefore the l -order kinetics (eqs 5 and 7) can be used to describe the response of the cocktail of microorganisms to heat under isothermal conditions. The D values measured under isothermal conditions would represent the overall heat resistance of the cocktail of microorganisms. Although the I -order kinetics can be applied to many microorganisms, it cannot be universally applied to all categories of microorganisms. Some organisms may not follow the l -order kinetics and, therefore, the thermal inactivation curves may show substantial deviation from the log-linear curves. Such curves may be either convex or concave (Figure 6). Since the rate of inactivation changes continuously with time, it is not accurate to use a D value to describe the kinetics of organisms showing obvious nonlinear characteristics during thermal inactivation. A simple generic mathematic model (4, 5) has been developed to describe the nonlinear behavior of thermal inactivation. With an assumption that the rate of inactivation may change with time under a constant temperature (eq 18), the new model is capable of describing convex, concave, and linear thermal inactivation curves (eq 19): 8t

s

st



248

t

Figure 6. A thermal inactivation may be linear, convex, or concave.

a

— = -kCt dt

log(C)=log(C )+5i

(18)

a

0

(19)

In eq 19, α is a constant that determines the shape of thermal inactivation or survival curves. If α = 1, then the equation is reduced to the linear form. If α > 1, the semi-logarithmic inactivation curve shows downward concavity (concave). And i f α < 1, then the inactivation curve is upwardly concaved (convex), sometimes referred to as "tailing." According to Huang and Juneja (4) this model is more accurate in describing nonlinear survival curves. Figure 7 shows examples of hypothetical nonlinear curves shown in Figure 3 (A and Β with f = 0.999) and described by this model. As shown in this figure, eq 19 provides a smooth and accurate description of convex-shaped thermal inactivation curves in the region shown. Figure 8 shows examples of concave curves fitted by eq 19. 9



249

Figure 7. Examples of convex curves (Figure 3A and 3B) fitted with the Weibulltype general model (eq 19).

Figure 8. Examples of concave curvesfittedwith the Weibull-typè general model (eq 19).


250

Table I: Some Survival Curves, ln(p(t)) versus t, where p(t) is the Probability that a Specific Cell W i l l be Viable at Time t. The Function h Represents the Derivative of the Survival Curve; h(0) is the Derivative at time = 0. asym. D - value Represents the Approximate D - Value for Large Times (asymptotic D - value). The Names are Those Sometimes Used to Describe the Model Model : ln(P(t)) =


-kt™

k

Parameter restrictions k>0,b>-l.

k, w, >0.

Weibull

Convex (b 0), no asym. D-value Concave, asym D value = ln(10)k" , h(0) = 0

Two stage

1

-fo+ln(l+--(l-0) w -kt+ln(l+bt)

Model Name

Properties

Concave, asym D-value = Ιη(ΙΟ)^ , h(0) = b-k h o < 0 , 0 < α < 1 , Convex, asym D-value = ln(10) b>0. [ho(l-a)] • ,h(0) = h Convex, asym D α, β > 0 value = 0 Convex, b,c>0 asym D-value = ln(10)c" Sigmoidal, asym D c>0 value = ln(10)c k,b>0.

Adjusted two-stage

1

ho[(l-a)t-ab(exp(-t/b) -D]

Modified Poisson

1

0

-aln(l+pt) -ln(l+exp(a+bln(t) + ct))

Gamma Log-Logistic (b>0)

1

Logistic (b =0) References: Weibull: Huang and Juneja (4\ Peleg and Cole (5); Two-stage: Juneja et al. (7); Modified Poisson: Hans (8) and Sharpe and Bektash (9); Gamma: Bazin and Prosser (10); and log-Logistic: Juneja et al. (//). -ln(l+exp(a+ct)) + ln(l+exp(a))

_1

The above model is sometimes referred to as the Weibull model, because of its connection to a Weibull distribution of survival times (5). In fact, researchers over the years have developed many models, in addition to the ones described above, that describe survival curves. Examples of some curves and their properties are given in Table I. To determine the "best" model for describing survival curves, statistical analyses are needed. A good textbook to consult for modeling inactivation data is Mixed-effects Models in S and S-Plus, written by Pinheiro and Bates (6).


251 Much research is needed for gaining an understanding of the mechanisms that lead to different types of nonlinear curves. It is possible that prior treatment - what has happened to the cells before being subjected to a thermal treatment could significantly affect the shape of the survival curves. Even slight differences in the environment and preparation of cultures might have a significant effect on survival curves, particularly for small and large times, because such effects might affect the physiology of the cells. This effect could explain, in part, the variations seen in survival curves during experiments.


References 1.

Rahn, O. In Biodynamica Monograph No. 3; Luyet, B . J., Ed.; Biodynamica, Normandy, M O , 1945; pp 9-41. 2. Stumbo, C. R. Thermobacteriology in Food Processing. Academic Press, Inc.: Orlando, F L , 1973. 3. Moats, W . A. J. Bacteriol. 1971, 105, 165-171. 4. Huang, L.; Juneja, V . K . J. Food Prot. 2001, 64, 2078-2082. 5. Peleg, M.; Cole, M. B . Crit. Rev. Food Sci. 1998, 38, 353-380. 6. Pinheiro, J.; Bates, D. Mixed-Effect Models in S and S-Plus. Springer¬ -Verlag, New York, N Y , 1998. Juneja, V . K.; Eblen, B . S.; Marks, H . M. Int. J. Food Microbiol. 2001, 70, 37-51. 8. Hans, Y. W . Can. J. Microbiol. 1975, 21, 1464-1467. 9. Sharpe, K . ; Bektash, R. M. Can. J. Microbiol, 1977, 23, 1501-1507. 10. Bazin, M. J.; Prosser, J. I. Physiological Models in Microbiology, Vol. 1. C R C Press, Inc.: Boca Raton, F L , 1988. 11. Juneja, V . K . ; Marks, H . M.; Mohr, T. Appl. Environ. Microbiol, 2003, 69, 5138-5156. 7.


Approaches for Modeling Thermal Inactivation of Foodborne Pathogens

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