We are usually convinced more easily by reasons we have found ourselves than by those which have occurred to others.
Blaise Pascal Traditionally, and regardless of the assumed kinetics (see previous chap-ters), the database for calculating microbial survival has been a set of isothermal inactivation curves from which the microorganism’s or spore’s survival parameters have been obtained. This requires that
• A sample of food or other medium that contains a known number of the cells or spores being studied be heated instantaneously to the desired lethal temperature
• The sample be held at this temperature for a specified time
• The sample be cooled instantaneously to a temperature low enough to stop the destruction and also to disallow resumed growth of the survivors before they are counted
An instantaneous isobaric increase or decrease of a real object’s temper-ature is physically impossible, of course, because of heat transfer consid-erations. (It can be almost accomplished by very rapid pressurization or pressure release; see Chapter 6. However, the temperatures reached due to the mere application of ultrahigh pressure are usually still sublethal if the initial temperature is ambient and the high pressure’s own effect needs to be considered in such a case.) By external heating and cooling, it is only possible to approximate a “step temperature increase” to be followed by a “step temperature decrease” by shortening the come-up and cooling times of the sample as much as possible relative to the holding time. In contrast, to maintain a small specimen of cells’ or spores’ suspension at a constant temperature is usually fairly easy. This can be accomplished by placing the small specimen in a large volume of hot oil or water already at the desired temperature. Cooling is done by placing the small specimen
in ice water, which is also at a constant but low temperature. Because even a very small volume of liquid (much less than even 1 ml) can still hold a huge number of microbial cells or spores, reducing the treated specimen’s size is not considered a problem in most cases. To accomplish the rapid heating and cooling, the sample containing the cells or spores must be held in or passed through a capillary, to assure the shortest heat penetration distance and largest surface area to volume ratio.
Such an experimental procedure, however, cannot be used if the culture medium is too viscous and/or contains suspended solid particulates. This forces the experimenter to choose between two unattractive alternatives:
to use a surrogate medium that can be contained in or forced through a narrow tube or to pack the sample in a thin plastic bag or a capsule prior to the treatment. If the first option is chosen, relevant information regard-ing the medium’s effect on the survival pattern will be obviously lost. If the second option is exercised, the temperature distribution within the sample will be largely unknown and thus the overall reliability of the results might be hampered. Regardless of how the samples are heated and cooled, the previously mentioned technical solutions might be totally ineffective whenever one deals with treatments of very high temperatures and very short times. This is because, in such cases, even a few seconds at a temperature only slightly lower than the target temperature can still result in significant sporal inactivation.
A different situation with similar consequences may exist when the organism becomes extremely heat labile within a very narrow tempera-ture range. For example, an organism like Salmonella, Shigella, or E. coli perishes very fast at temperatures higher than about 60°C. Thus, an error of a single degree in the neighborhood of this temperature may be man-ifested in significant survival or a reduction of the cells’ number to an undetectable level.
Many of the preceding problems could be eliminated, at least in prin-ciple, if one could develop a procedure to determine the cells’ or spores’
survival parameter from nonisothermal inactivation data produced under slow heating regimes and, if necessary, slow cooling as well. Although the experimental procedures for such analyses are yet to be developed, one can envision that they will be much simpler and possibly more reliable than at least some of the “isothermal” treatments currently in use. The question that remains is whether the development of such procedures is theoretically possible — i.e., whether an organism’s or spore’s survival parameters can be retrieved from inactivation data obtained under nonisothermal conditions. What follows will demonstrate not only that this is possible, but also that the concept’s validity has already been confirmed for Salmonella cells and that there is evidence that it might be applicable to bacterial spores as well.
The concept that kinetic parameters can be retrieved from nonisother-mal data is not new, of course (see Mizrahi et al., 1978, for example). The proposed departure from the previous approach is the explicit assertion that the momentary rate of change, whether it is inactivation (see Chapter 1 and Chapter 2) or growth, depends not only on the momentary tem-perature (and/or other conditions that may apply), but also on the sys-tem’s momentary state. Or in other words, the momentary rate depends on the process’s history as well as on the momentary conditions. This means that the validity of neither the first order kinetics model or that of the Arrhenius equation can be taken for granted. For nonisothermal microbial inactivation, this means that, in general, the inactivation rate is a function of the momentary temperature and momentary survival ratio. Organisms or spores whose logarithmic inactivation rate is a function of temperature only are those that follow the first-order mortality kinetics. However, as has been demonstrated, they are only a special case and should be con-sidered the exception rather than the rule (see van Boekel, 2002, for example).
The Linear Case
Linear Survival at Constant Rate Heating
Consider an organism whose isothermal survival curves in the lethal temperature range and on the pertinent time scale are log linear for all practical purposes. The thermal inactivation of Listeria monocytogenes as reported by Stephens et al. (1994) can serve as an example. The log linear isothermal survival equation, as has been already mentioned, can be viewed as a special case of the Weibullian–power law model with a shape factor power of unity (n = 1):
log10S(t) = b(T)t (4.1)
or
(4.2)
If the temperature dependence of b(T) can be described by the log logistic model (Peleg et al., 2003; see Figure 4.1):
b(T) = loge{1 + exp[k(T – Tc)]} (4.3) Then, under nonisothermal temperature profile, the rate equation will be:
d S t
dt b T
T
log ( )
10 ( )
=
=
const
(4.4)
In the case in which the temperature increases linearly:
T(t) = T0 + vt (4.5)
where T0 is the initial temperature and the heating rate (in deg.nin–1), in this particular case, the momentary nonisothermal survival rate can be written as:
(4.6)
and the survival curve equation will be:
(4.7)
FIGURE 4.1
The b(T) vs. T relationship of Listeria monocytogenes fitted with the log logistic model (Equa-tion 4.3) and the discontinuous model (Equa(Equa-tion 4.9). The original survival data are from Stephens, P.J. et al., 2004, J. Appl. Microbiol., 77, 702–710. (Adapted from Peleg, M. et al., 2003, Bull. Math. Biol., 65, 219–234. With permission, courtesy of Elsevier Ltd.)
5
4
3
b(T)
2
1
0
50 55
Temperature(°C)
60 65
d S t
dt e k T t Tc
log ( )
log [ exp{ [ ( ) ]}]
10 = − 1+ −
d S t
dt e k T vt Tc
log ( )
log [ exp[ ( ) ]]
10
1 0
= − + + −
log10 ( ) log { exp[ ( 0 )]}
0
1
S t e k T vt Tc dt
t
= −
∫
+ + −The integral at the right side of the equation has an analytical solution (see the Mathematica® book by Wolfram Research). Thus, the nonisother-mal survival curve can be described explicitly by the expression:
(4.8)
PolyLog[n,x] (also known as the Jonquire’s function) is a standard func-tion in Mathematica® and can be used as an ordinary function in all relevant mathematical operations, including nonlinear regression. Thus, if it is known a priori that the isothermal survival curves of a given organism at the given medium are all log linear and if a linear heating profile can be programmed — i.e., T0 and the hating rate, v, are known — then one can calculate that organism’s survival parameters, k and Tc, by nonlinear regression using Equation 4.8 as a model. (Notice that only when all the isothermal survival curves of the organism in question are log linear will there be only two survival parameters, as in the traditional model. If, however, even some of the isothermal survival curves are curvilinear, then at least three survival parameters will be required to account for the organism’s heat resistance — see below.)
The application of the model is demonstrated in Figure 4.2. The figure shows a simulated heating curve where the temperature increases at constant rate (top) and corresponding survival data generated for Listeria with Equation 4.8 as a model, to which random noise was added in order to make them realistic (bottom). Also shown in the figure is the fitted survival curve calculated by nonlinear regression again using Equation 4.8 as a model (solid line). To test the procedure to estimate the survival
FIGURE 4.2
Simulated linear heating curve (left) and corresponding generated survival data of Listeria monocytogenes (right). The open circles are the generated data (with superimposed noise) and solid line the fit of Equation 4.8. For the agreement between the generated and retrieved parameters see text. (From Peleg, M. et al., 2003, Bull. Math. Biol., 65, 219–234. With permis-sion, courtesy of Elsevier Ltd.)
Temperature(°C)
Time(min)
Log S(t)
Time(min)
0 10 20 30 40
0 10 20 30 40
70 60 50 40 30 20
0
−2
−4
−6
log ( ) { , exp[ ( )]}
eS t = −PolyLog2 − k T0+ −vt Tc −PolyLLog k T T kv
{ , exp[ (2 − 0− c)]}
parameters from nonisothermal data, the generation and regression sur-vival parameters k and Tc were compared. They were (Peleg et al., 2003):
As one would expect (see following), the degree of agreement largely depends on the survival data’s scatter. More importantly, for the method to work at all, enough data at the lethal temperature region are needed and the more the better. As can be seen from Figure 4.2, Tc can be estimated by visual inspection of the curve — that is, by locating the temperature where the survival curve takes a dip (see figure). In contrast, information regarding the magnitude of k is primarily obtained from the steepness of the survival ratio’s drop at temperatures beyond Tc, i.e., from data gathered at the lethal regime.
It is already evident from the preceding example that the survival curve of a heat-sensitive organism like Listeria beyond its Tc can be too steep to allow meaningful determination of k, especially if the experimental sur-vival data have a considerable scatter. Therefore, the slower the heating rate is, the more moderate will be the survival curves’ dive (more on this issue later). This will make it easier to calculate the magnitude of k and the resulting value will be more reliable. Obviously, the heating rate should not be so slow that it will allow the organism to increase its heat resistance through physiological adaptation; in this case, the method could not be used (see Hassani et al., 2005). This constraint applies to all that follows in this chapter and to the prediction of survival patterns in general as already stated in Chapter 2.
Linear Survival at Varying Heating Rate
In light of heat transfer considerations, creating a linear temperature pro-file (constant rate heating) requires a programmed and tightly controlled heater. Such a piece of equipment might not be available in many labo-ratories; thus, it would be advantageous if the survival parameters, k and Tc, could be estimated from survival data obtained under natural temper-ature profiles of the kind shown in Figure 4.3. Such profiles can be obtained by immersing a well-stirred sample in a hot water or oil bath — an inexpensive standard piece of equipment that exists in almost any microbiology laboratory. (The survival data would be obtained by with-drawing small specimens periodically, cooling them with ice, and then proceeding with the count.)
Survival parameter Generation parameter Regression parameter n 1.0 (by definition) 1.0 (by definition)
k 1.01˚C–1 0.96˚C–1
Tc 60.3˚C 60.4˚C
Source: Peleg et al., 2003.
Unfortunately, when the temperature profile is curvilinear, the rate model’s equation (Equation 4.4) cannot be solved analytically. To over-come this problem, one can replace the log logistic component of the model (b(T) in Equation 4.3) by the discontinuous approximation:
b(T) = if [T ≤ Tc, 0, k (T – Tc)] (4.9) This says that, if T ≤ Tc, b(T) = 0; otherwise, it will be k(T – Tc).
The fit of this discrete model to the Listeria’s survival rate data is shown in Figure 4.1. (Because it does not account for the curvature of b(T) around Tc, its use will provide an underestimate of the inactivation rate and thus a more conservative assessment of the process’s lethality, at least theoret-ically.) The expression of the temperature dependence of b(T) as a discon-tinuous function will render the survival model:
FIGURE 4.3
Simulated temperature profiles using Equation 4.11 through Equation 4.13 as models. Notice that when inserted into Equation 4.9, the survival model’s equation (Equation 4.10) will have an analytical solution. (From Peleg, M. et al., 2003, Bull. Math. Biol., 65, 219–234. With permission, courtesy of Elsevier Ltd.)
Time(min) 0 0
10 20 30 40 50 60 70 80
5 10 15 20
Temperature (°C)
(4.10)
where tc is the time to reach Tc, i.e., T(tc) = Tc.
Notice that if the function that describes the temperature profile, T(t), can be integrated analytically, so would the term k[T(t) – Tc]. Consequently, for any such thermal history, the survival curve, log10S(t) vs. t, can be calculated analytically using Equation 4.10 as a model. Examples of inte-grable expressions that can fit many experimental heating curves are:
• Monotonic temperature increase with an asymptote T0 + 1/a2:
(4.11)
• Monotonic temperature increase at a progressively decreasing rate:
T(t) = T0 + a3tm (4.12)
• Logistic increase with an asymptotic temperature Ttarget:
(4.13)
where
T0 is the initial temperature a is a constant
m, Ttarget, and tch are constants
In fact, the three simulated heating curves shown in Figure 4.3 were produced with Equation 4.9 through Equation 4.11 as models. Most likely, one of the preceding expressions could be used as a regression model for real experimental heating curves. However, if none of the three provides adequate description of the experimental heating curve(s), then one can try an alternative model that can be integrated analytically or use a sum of any number of the same terms (Equation 4.11 through Equation 4.13) with different parameters or any of their combinations (a mixed model).
This is because the sum of integrable expressions is also integrable.
log10S t( ) If T t[ ( ) Tc, ,0 k T t[ ( ) T dtc] ]
t t
c
= ≤ −
∫
−T t T t
a a t ( )= +
+
0
1 2
T t T T T
a tch t ( )= + exp[ (− )]
+ −
0
0
1 4 target
Modern statistical software allows expressions with an ‘if statement’ to serve as regression models, as already mentioned. Therefore, once T(t) has been defined in the preceding manner and Equation 4.10 is solved, the solution in the form of log10S(t) = f(k, Tc, t) can be used to estimate the survival parameters k and Tc by nonlinear regression. This is demonstrated with a simulated heating curve using Equation 4.12 as a model (Figure 4.4, left). The corresponding survival data, generated with superimposed noise to make them appear realistic, are also shown in Figure 4.4 (right).
The solid line is the fitted survival curve using the solution of Equation 4.10 as the regression model when Equation 4.12 represented the temper-ature profile, T(t). The regression yielded the following results:
The regression parameters demonstrate that although Tc could be esti-mated fairly accurately using the discontinuous model, the magnitude of k was about 20% lower than the correct value. Whether such a discrepancy is significant in the practical sense can be determined by comparing the model’s predictions with survival ratios obtained experimentally, as will be shown below.
FIGURE 4.4
A simulated temperature profile using Equation 4.12 as a model (left) and corresponding generated survival data of Listeria monocytogenes (right). The open circles are the generated data (with superimposed noise) and solid line the fit of Equation 4.10. For the agreement between the generated and retrieved parameters, see text. (From Peleg, M. et al., 2003, Bull.
Math. Biol., 65, 219–234. With permission, courtesy of Elsevier Ltd.)
Survival parameter Generation parameter Regression parameter n 1.0 (by definition) 1.0 (by definition)
k 1.01˚C–1 0.81˚C–1
Tc 60.3˚C 59.8˚C
Source: Peleg et al., 2003
Temperature (°C) Log S(t)
Time(min)
−80
−6
−4
−2 0
30 40 50 60
10 20 30 40 50
0 10 20 30 40 50
Time(min)
The Nonlinear Case
Weibullian–Power Law Inactivation at Arbitrary Heating Rate History The previous discussion addressed the special and probably rare case of cells or spores whose isothermal survival curves are all log linear. In reality, one would expect that most isothermal survival curves would not be log linear, but rather follow the Weibullian–power law model. In one of its simplest forms (see Chapter 1 and Chapter 2), the shape factor, or power, n, is fixed and the temperature dependence of the rate parameter, b(T), can be described by the log logistic model (Equation 4.3) or any alternative two parameters secondary model. For any given temperature profile, T(t), the momentary survival rate is the differential equation:
(4.14)
As previously shown, once the organism’s survival parameters, namely, n, k, and Tc, are all known and T(t) can be written as a continuous or discrete algebraic expression, this equation can be solved numerically by Mathematica® to produce the corresponding curve, log10S(t) vs. t.
It need not concern us here that the model’s equation can also be solved by the incremental method (Chapter 3), even when T(t) is not expressed as an algebraic term. It is of concern that Equation 4.14 with n ≠ 1 can only be solved numerically and therefore it is not a proper model for standard or “canned” nonisothermal regression procedures of the kinds that come with statistical software packages.
To determine the survival parameters from a given set of experimental inactivation data, therefore, requires the programming of a special numer-ical procedure. One such procedure is based on the simplex or FindMin-imum algorithm. When either is employed, the rate equation is solved numerically with initial guesses and subsequently newly generated values of the survival parameters. The mean square error is calculated at each iteration, and the process is repeated until the survival parameters calcu-lated in this way yield a solution to the model’s equation within the specified accuracy. The result is a set of estimated survival parameters that can be used to test the model’s validity against fresh data.
Testing the Concept with Simulated Data
To test the preceding procedure, we have used a set of simulated survival data corresponding to three different temperature profiles (Figure 4.5),
d S t
with known survival parameters, n, k, and Tc, on which a random noise was superimposed. The temperature profiles’ equations and correspond-ing results are shown in Table 4.1 and Table 4.2, respectively. Examples of the simulated curves with two noise levels are shown in Figure 4.6.
In principle and especially with simulated data when one has control of the number of entries and the random deviations, one can determine all three survival parameters simultaneously. However, this might not be possible when the survival ratios that serve as the database are few and have a considerable scatter. The reason is that letting the power n be an adjustable parameter can result in a totally unrealistic estimate of the value of Tc. Remember that although Tc is a component of a secondary model determined by regression, it still carries a clear physical meaning — i.e., it is the marker of the temperature region in which lethality starts to accelerate. Thus, its calculated estimate must be congruent with the actual temperature where this happens, regardless of whether a different value will result in a better fit as judged by statistical criteria.
To avoid this problem, we have fixed the magnitude of n (at various values) and only determined the values of k and Tc by the previously mentioned literature procedure. That we got the same results with the FindMinimum and simplex algorithms provided mutual verifications (Peleg and Normand, 2004). (As will be seen later, justification of fixing
FIGURE 4.5
The simulated temperature profiles generated with the models listed in Table 4.1. (From Peleg, M. and Normand, M.G., 2004, Crit. Rev. Food Sci. Nutr., 44, 409–418. With permission, courtesy of CRC Press.)
0 2 4 6 8 10
0 20 40 60 80 100
Temperature(°C) c
b a Simulated temperature profiles
Time(min)
the magnitude of n comes from the model’s ability to predict the outcome of different processes correctly, regardless of the chosen value of n, as long as it is not too far away from the “correct” value.) The procedure was tested with simulated nonisothermal survival data of the kind depicted in Figure 4.6. The purpose was to demonstrate that the generation param-eters could be retrieved with reasonable accuracy even from scattered survival ratios. Table 4.2 shows that, with only five simulated “replicates,”
one can get fairly close estimates of the generated parameters k and Tc. As could be expected, the quality of the estimates declined as the scat-ter’s amplitude increased. Obviously, the replicates’ spread is not a param-eter that the experimenter can tightly control in microbial inactivation
one can get fairly close estimates of the generated parameters k and Tc. As could be expected, the quality of the estimates declined as the scat-ter’s amplitude increased. Obviously, the replicates’ spread is not a param-eter that the experimenter can tightly control in microbial inactivation