Estimating microbial growth parameters from non-isothermal data: A case study with Clostridium perfringens

International Journal of Food Microbiology 118 (2007) 294 – 303 www.elsevier.com/locate/ijfoodmicro

Estimating microbial growth parameters from non-isothermal data: A case study with Clostridium perfringens Sarah Smith-Simpson a , Maria G. Corradini b , Mark D. Normand b , Micha Peleg b , Donald W. Schaffner a,⁎ a

Food Risk Analysis Initiative, Food Science Department, School of Environmental and Biological Sciences, Rutgers University, New Brunswick, N.J. 08901, United States b Department of Food Science, Chenoweth Laboratory, University of Massachusetts Amherst, MA 01003, United States Received 16 March 2007; accepted 7 August 2007

Abstract Microbial growth parameters are usually calculated from the fit of a growth model to a set of isothermal growth data gathered at several temperatures. In principle at least, it is also possible to derive them from non-isothermal (‘dynamic’) growth data. This requires the numerical solution of a rate model whose coefficients are nested terms that include the temperature profile. The methodology is demonstrated with simulated non-isothermal growth data on which random noise had been superimposed to emulate the scatter found in experimental microbial counts. The procedure has been validated by successful retrieval of the known generation parameters from the simulated growth curves. The method was then applied to experimental non-isothermal growth data of C. perfringens cells in cooled ground beef. The growth data collected under one cooling regime were used to predict the organism's growth patterns under different temperature histories. The practicality of the method is currently limited because of the relatively large scatter found in experimental microbial growth data and the relatively low frequency at which they are collected. But if and when the scatter could be reduced and the counts taken at short time intervals, the method could be used to determine the growth model in situ thus enabling to translate the changing temperature during processing, transportation or storage into a corresponding growth curve of the organism in question. © 2007 Elsevier B.V. All rights reserved. Keywords: Growth curves; Kinetic models; Predictive microbiology; Clostridium perfringens; Logistic growth

1. Introduction Modeling microbial growth has been a fertile field of research in the last two decades (Mc Meekin et al., 1993; McKellar and Lu, 2004; Brul et al., 2007). The starting point, almost without exception, has been the mathematical description of sigmoid isothermal growth curves in a closed habitat. Most of the resulting models can be classified as either rate models, empirical models or population dynamics based models. Rate models are all derivatives of the logistic (Verhulst) equation and the most well known in food microbiology is the Baranyi–Roberts model (Baranyi and Roberts, 1994). Other modifications of the logistic equation include an added power ⁎ Corresponding author. Tel.: +1 732 932 9611. E-mail address: [email protected] (D.W. Schaffner). 0168-1605/$ - see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.ijfoodmicro.2007.08.005

term (Fujikawa et al., 2004), “if” statements, (Oscar, 2005) and replacement of the cells' number by a logarithmic growth ratio (Peleg et al., 2007). Empirical models include the well known and widely used Gompertz model (McKellar and Lu, 2004) in its several varieties. Others include a three-step log linear discrete model (Buchanan et al., 1997), the solution of the logistic equation modified by an “if” statement (Augustin and Carlier, 2000; Oscar, 2005), a shifted logistic model (see below) and a model based on a combination of power terms (Corradini and Peleg, 2005; Peleg, 2006). Population dynamics models are based on the notion that the number of cells in the population, after any given time interval, is determined by the momentary sum of those added by cell division and those who only grew in size, from which the number of cells that have died has been subtracted (Taub et al., 2003; Doona et al., 2005). Unlike most of the models, which

S. Smith-Simpson et al. / International Journal of Food Microbiology 118 (2007) 294–303

295

Nomenclature Y(t) momentary logarithmic growth ratio, Log[N(t) / 0] N0 initial number of cells N(t) momentary number of cells at time t YA asymptotic logarithmic growth ratio. k a measure of the growth curve's steepness around its inflection point tc the time that marks the growth curve's inflection point k0, ck, tc0, ctc coefficients of the secondary models y[t] the numerical solution of the rate equation, representing the growth ratio vs. time relationship ΔY chosen scattered range Rn(t) random number with a uniform distribution between zero and one belong to rate or empirical groups, the models proposed by Taub et al. and Doona et al. can account for peak growth followed by mortality, the inevitable outcome of the habitat's resource depletion and its pollution by released toxic metabolites. It is true, however, that most foods become inedible or unsafe to eat long before the microbial population has reached the stationary phase, let alone the mortality stage, so models that only describe the sigmoid part of the growth curve are generally quite adequate for food systems. As such we will only deal with the part of the growth curve commonly identified as having a short or long ‘lag phase’ followed by ‘exponential’ growth that ends in a ‘stationary’ phase. It can be and has been shown that sigmoid isothermal microbial curves can be described by several mathematical models that have a very similar degree of fit as judged by statistical criteria (McKellar and Lu, 2004; Peleg, 2006). Since uniqueness, especially of the first and second kind of models, cannot be guaranteed, a ‘primary’ mathematical growth model should be chosen in light of considerations such as its theoretical strength, conformity with the principle of parsimony (Ockham's razor), i.e., maintaining the minimum number of adjustable parameters, convenience and the possibility to assign an intuitive meaning to its parameters. The test of a growth model's adequacy should not be its fit to experimental data but the ability to predict correctly the outcome of experiments not used in its derivation (Peleg, 2006). The strongest of these is its ability to predict non isothermal growth patterns from isothermal data. Examples can be found in the works of Koutsoumanis (2001), Fujikawa et al. (2004), Corradini and Peleg (2005) and Corradini et al. (2006). In general, that is for an arbitrary temperature history, predictions are achieved by developing a set of ‘secondary models’ that describe the temperature dependence of the primary models' parameters. When the non-isothermal temperature profile is known and can be expressed algebraically, these can be converted into functions of time, which in turn can be incorporated into the non-isothermal rate model's equation (see below) also known as a ‘tertiary model’. The resulting differential equation can then be solved, in most cases only numerically, to produce the non-isothermal growth curve, which could subsequently be compared with experimental growth data obtained under the same temperature history. It is important to note that the algebraic expression of the temperature profile is not a prerequisite and the growth curve

can be constructed for digitized sets of time-temperature data (Corradini and Peleg, 2005). Since the analysis that follows is based only on temperature profiles that can be described by continuous algebraic functions, this approach is not necessary here. A question that arises is whether the process can be reversed (at least theoretically) so that one could derive the isothermal growth parameters from non-isothermal data. If this is possible, then these isothermal growth parameters could be used to calculate the growth patterns not only under isothermal conditions at various temperatures but also under any chosen non-isothermal temperature history. Also, if the option of determining the growth parameters and their temperature dependence from non-isothermal data existed, then one could replace the set of experimental data at several constant temperatures, now the norm, with a single survival curve determined under a programmed temperature history. In practice, if the option became available, at least two such experiments with different temperature profiles would be required, for mutual verification (see below). The objectives of this work are to (1) explore the theoretical aspects of deriving microbial growth parameters from nonisothermal data using computer simulations, (2) test the concept with experimental growth data of Clostridium perfringens collection under dynamic (cooling) conditions and (3) evaluate the methodology's potential uses in light of practical considerations. 2. Theoretical background It has been shown in two previous studies that microbial survival parameters can be retrieved from non-isothermal inactivation data using numerical techniques (Peleg and Normand, 2004), and in certain simple situations analytically (Peleg et al., 2003). The starting point has been that the momentary (instantaneous) inactivation rate in a non-isothermal process is the isothermal rate at the momentary temperature at a time that corresponds to the momentary survival ratio. Thus, if the isothermal survival curves in the pertinent temperature range all follow a known model whose parameters' temperature dependence are also known, then the non-isothermal survival curve can be written in the form of a differential rate equation for any given temperature profile. In most cases, this equation cannot be solved analytically only numerically. A numerical

296

S. Smith-Simpson et al. / International Journal of Food Microbiology 118 (2007) 294–303

Fig. 1. Demonstration of the shifted logistic function's fit (Eq. (3)) to published isothermal growth data. The original experimental data of Candida sake are from Tyrer et al. (2004) and those of Pseudomonas from Koutsoumanis (2001).

solution of a differential equation, in contrast with an algebraic expression, cannot be used as a model for standard regression procedures. However, it can serve as a basis for similar error minimization methods (see below) that would yield the sought model parameters. The actual minimization procedure requires that the differential rate equation be solved numerically at every iteration. Still, in at least some cases, a solution can be reached within very reasonable time, from several seconds to a few minutes with the current generation of personal computers, depending on the closeness of the initial parameter estimates. The methodology has been validated with computer simulations and actual inactivation data of Salmonella in a growth medium and ground chicken breasts, assuming a Weibullian isothermal survival pattern with a constant exponent (Peleg and Normand, 2004). It has also been applied successfully to extract the survival parameters of E. coli in water disinfected with a dissipating chemical agent (Corradini and Peleg, 2003). The concept can be extended to non-isothermal growth if we accept the notion that the momentary growth rate is the isothermal growth rate at the momentary temperature, at a time that corresponds to the momentary growth ratio — an

assumption that can be verified experimentally. The growth ratio itself, Y(t), can be defined as either:   N ðtÞ Y ðtÞ ¼ log10 ð1Þ N0 or Y ðtÞ ¼

N ðtÞ  N0 N0

ð2Þ

where N0 is the cells' initial number and N(t) the momentary number at time t. Notice that in either case, the initial growth ratio is zero by definition, i.e., Y(0) = 0. Many sigmoid isothermal survival curves can be described by the shifted logistic model (Corradini and Peleg, 2005; Corradini et al., 2006; Peleg, 2006) that has the form:   1 1 Y ðtÞ ¼ YA  ð3Þ 1 þ exp½kðtc  tÞ 1 þ expðk tc Þ where YA, k and tc are temperature dependent coefficients. The fit of this model to published growth data (Candida sake from Tyrer

Fig. 2. Schematic view of the assumed isothermal growth parameters' temperature dependence.

S. Smith-Simpson et al. / International Journal of Food Microbiology 118 (2007) 294–303

et al. (2004) and Pseudomonas from Koutsoumanis (2001) is demonstrated in Fig. 1. Although the model shown in Eq. (3) has a complicated appearance, it has quite a few advantages over other empirical growth models. It has only three adjustable parameters, namely YA, k and tc, and these have a clear intuitive meaning. According to this model, the asymptotic growth ratio (the growth level at the ‘stationary phase’) is YA{1 − 1 / [1 + exp(ktc)]}. However since in many cases 1 / [1 + exp(ktc)] is much smaller than one,

297

YA is a rough measure of the asymptotic growth ratio. The parameter k is a measure of the growth curve's steepness around its inflection point whose location is marked by tc. In growth simulations, what is known as the ‘lag time’ can be extended or shortened by increasing or decreasing the magnitude of tc. Eq. (3) with Y(t) defined as either the linear or logarithmic growth ratio is a flexible model and it can describe experimental growth patterns characterized by either a short or long ‘lag time’, a feature that

Fig. 3. Simulated heating and cooling curves, their corresponding (scattered) growth ratios and the ‘fit’ of the non-isothermal rate model constructed with Eqs. (6)–(9). The random scatter was produced with Eq. (11). The agreement between the generated and retrieved growth parameters for several runs is shown in Table 1.

298

S. Smith-Simpson et al. / International Journal of Food Microbiology 118 (2007) 294–303

would be primarily reflected in the magnitude of tc, as already stated. Eq. (3) also has an analytical inverse, i.e. the time to reach any given growth ratio, t⁎, at a given temperature, T, can be expressed algebraically, i.e.,

chemical inhibitors (e.g. sodium chloride, nitrite). In any case, acceptable approximations would be in the form of: YA ðT Þ ¼ constantðassessed from the data Þ

ð7Þ

kðT Þ ¼ k0 expðck T Þ

ð8Þ

The isothermal momentary growth rate according to this model is:

tc ðT Þ ¼ tco expðctc T Þ

ð9Þ

dY ðtÞ kðT ÞYA ðT ÞexpfkðT Þ½tc ðT Þ  t ⁎ g ¼ 2 dt f1 þ expfkðT Þ½tc ðT Þ  t ⁎ gg

where k0, ck, tc0 and ctc are the coefficients of the secondary models. A schematic view of these relationships is given in Fig. 2. Thus, in our case, the growth rate model has four adjustable parameters, namely k0, ck, tc0 and ctc, and one fixed, namely YA(T). The rationale for fixing YA(T) is based on empirical observation. In principle, it can vary with temperature and in some systems probably does. Yet, in most systems YA is only a weak function of temperature because it is primarily determined by the habitat's carrying capacity. For any given temperature profile, T(t), the three secondary models can be readily converted into functions of time, i.e.,

t⁎ ¼

  1 exp½kðT Þtc ðT ÞfYA ðT Þ þ Y ðtÞf1 þ exp½kðT Þtc ðT Þgg Loge kðT Þ YA ðT Þexp½kðT Þtc ðT Þ  Y ðtÞf1 þ exp½kðT Þtc ðT Þg ð4Þ

ð5Þ

Under non-isothermal conditions, i.e., where the temperature profile is T(t), the momentary rate will be (Corradini and Peleg, 2005; Corradini et al., 2006): dY ðtÞ k½T ðtÞYA ½T ½texpfk½T ðtÞ½tc ½T ðtÞ  t ⁎ g ¼ 2 dt f1 þ expfk½T ðtÞ½tc ½T ðtÞ  t ⁎ gg

ð6Þ

where t⁎ is defined by Eq. (4), with T(t) replacing T. As shown in Corradini and Peleg (2005) and Corradini et al. (2006), although Eq. (6) appears cumbersome it can be solved by Mathematica® (Wolfram Research, Champaign IL, USA) or MS Excel® (see http://www-unix.oit.umass.edu/~aew2000/ MicrobeGrowthModelA.html). With few exceptions, the mathematical description of sigmoid isothermal growth curves requires a model with no less than three adjustable parameters [in our case YA(T), k(T) and tc(T)]. Modeling the temperature dependence of each of these three growth parameters would require a different secondary model, each having at least two parameters. This means, in general, that a non-isothermal growth curve described by Eq. (6) as a model will have at least six coefficients that would have to be determined by the minimization procedure. The changing temperature profile can be described by any convenient ad hoc empirical mathematical model. Hence determination of the profiles' parameters through standard regression is a straightforward exercise. But although one should always attempt to keep the number of adjustable parameters to a minimum, this need not be always a major consideration in the choice of a temperature profile model. Exactly how an organism's three growth parameters vary with temperature in a given medium is unknown a priori, therefore, the types of models required to describe their temperature dependence can only be assumed. In our case, hints on the general character of the YA(T), k(T) and tc(T) vs. T relationships came from previous work, in which they had been determined experimentally for C. perfringens in cooled ground ham (Corradini et al., 2006). It should be mentioned that although the growth patterns in ground ham and the medium used in this study (ground beef), are generally similar, they may differ slightly because ham (and other cured meats) contains

Table 1 Comparison of the growth parameters and their retrieved values from simulated non-isothermal data having two levels of scatter a Regime

Heating b

Maximum noise span (log10 units) 0.5

Heating b

1.0

Cooling d

0.5

Cooling d

1.0

Run 1 2 3 4 5 Mean c 1 2 3 4 5 Mean c 1 2 3 4 5 Mean c 1 2 3 4 5 Mean c

Generation parameters k0

Ck

tc0

Ctc

0.01725

0.125

150

0.100

160 153 153 141 151 152 125 190 137 136 175 153 141 145 164 157 134 148 175 131 150 163 130 150

0.102 0.099 0.105 0.095 0.108 0.102 0.088 0.112 0.087 0.093 0.119 0.10 0.099 0.099 0.104 0.102 0.094 0.10 0.110 0.095 0.102 0.099 0.095 0.10

Extracted parameters 0.014 0.138 0.013 0.141 0.022 0.111 0.020 0.123 0.032 0.099 0.020 0.122 0.017 0.128 0.012 0.151 0.001 0.249 0.014 0.126 0.030 0.099 0.015 0.151 0.013 0.146 0.018 0.125 0.015 0.133 0.017 0.124 0.022 0.110 0.017 0.128 0.013 0.140 0.018 0.122 0.012 0.142 0.007 0.185 0.019 0.115 0.014 0.141

MSE 0.013 0.020 0.015 0.017 0.016 0.070 0.055 0.063 0.066 0.064 0.010 0.011 0.010 0.015 0.014 0.040 0.061 0.055 0.037 0.054

a The growth parameters are defined by Eqs. (6)–(11), which serve as the rate model. b The heating profile equation is T(t) = 30 exp(−0.0325t). c Mean in this case represents the best estimate of parameter. d The cooling profile equation is T(t) = 10 exp(0.0325t).

S. Smith-Simpson et al. / International Journal of Food Microbiology 118 (2007) 294–303

YA(t) = YA[T(t)], k(T) = k[T(t)] and tc(t) = tc[T(t)]. These in turn can be inserted into Eq. (6) which will become the growth rate model for the particular thermal history. A numerical solution of a differential equation in Mathematica® is found by the function NDSolve, which yields an “interpolation function”, a dense set of the sought function's numerical values within a range specified by the user. Thus, if y′[t] is the time derivative of the sought growth curve defined by the right side of Eq. (6) that we will call f[t] for simplicity, then (in the syntax of Mathematica®) the solution in the form of y[t] vs. t will be rendered by the function: y½t ¼ NDSolve½fyV½t ¼¼ f ½t; y½0 ¼¼ 0g; y½t; ft; 0; tCycleg ð10Þ This means that the growth ratio vs. time relationship, Y(t), represented here by y[t], is the numerical solution of dy(t) / dt = f(t), with the boundary condition that at t = 0, y[0] = 0. The sought solution is the function y[t], i.e., Y(t), whose generated values would be between time zero and the specified final time tCycle.

299

Once created in this way, the interpolation function y[t] is essentially treated by Mathematica® as a regular function. With the growth curve's equation specified in this manner, one can calculate its coefficients by minimizing the mean squared distance (error) between a set of experimental growth data and the corresponding values of the interpolation function y[t]. Notice that y[t] is just a function defined by Eq. (10) and Mathematica® will calculate its value in the prescribed manner whenever called for. To determine the growth parameters, we have chosen Mathematica®'s ‘NMinimize’ function using the “Nelder-Mead” method option. We should note that this method has been tested in previous studies against two other algorithms and yielded practically identical results. The above analysis indicates that determination of the growth parameters from non-isothermal data is technically possible, provided that the secondary models' formats are known or can be assumed. In other words, the mathematical tools to extract the growth parameters from non-isothermal growth data already exist and are available as commercial software. This approach would also be valid had an alternative isothermal growth model

Fig. 4. Examples of non-monotonic temperature profiles and corresponding growth curves generated with Eq. (6) as the rate model. Notice that the temperature profile's complexity is no hindrance to the rate equation's numerical solution.

300

S. Smith-Simpson et al. / International Journal of Food Microbiology 118 (2007) 294–303

been selected, e.g., Y(t) = am tm / (bm + tm), where a, b and m are the three temperature dependent coefficients (Corradini and Peleg, 2005). The difference would only be in the rate model's formula and the mathematical expressions that would describe the growth parameters' temperature dependence. It is important to note, as in more ‘standard methods’ of nonlinear regression, the number of experimental data points should be large enough to be commensurate with the number of adjustable parameters. Also, as the number of adjustable parameters increases, the ability of the regression analysis to produce a unique solution decreases, a problem which is only aggravated when the experimental data have considerable scatter. Thus, there is a premium to reduce the number of adjustable parameters to the minimum that the system allows, in this case, four (see Eqs. (7)–(9)). Even if a sufficient number of data points can be assured and the number of adjustable parameters has been reduced, one decisive question remains: will the procedure yield correct or at least reasonable growth parameters when applied to scattered

experimental growth data? The answer to this question will come from two tests: the successful retrieval of the generation parameters from simulated non-isothermal growth curves that include random “noise” and the ability of a rate model derived from experimental non-isothermal growth data to predict isothermal or other non-isothermal growth curves recorded under different temperature histories. 3. Testing the model with simulated growth data Simulated temperature profiles and corresponding growth curves generated with Eq. (6) as the underlying model are shown in Fig. 3, and as shown in this figure the same methodology can be applied to cooling as well as heating. The profiles, of non linear monotonic temperature increase and decrease, left and right respectively, represent realistic scenarios of fish or poultry being taken out of refrigeration and the microbial proliferation that follows cooling of cooked but unsterilized foods like ham or roast beef. For comparison,

Fig. 5. Two cooling curves of cooked ground beef (A and B) and corresponding growth curves of C. perfringens cells. The solid lines are the ‘fitted’ curves using Eq. (6) as the rate model. The dotted lines are the predicted growth curves. The one on the left, for temperature profile A, was generated with the parameters derived from the growth curve under temperature profile B and the one on the right, for temperature profile B, was generated with the parameters derived from the growth curve under temperature profile A. Circles of the same shade represent experimental observations from the same experiment.

S. Smith-Simpson et al. / International Journal of Food Microbiology 118 (2007) 294–303

isothermal profiles at a low and high temperature are also shown (as dashed lines). The random “noise” in the non-isothermal growth data was produced by the algorithm (Peleg and Normand, 2004): Yscattered ðtÞ ¼ Ysmooth ðtÞ þ DY ½2Rn ðtÞ  1

ð11Þ

where ΔY is the chosen scattered range in terms of Y(t) as defined by Eq. (1) or (2) and Rn(t) is a random number with a uniform distribution between zero and one. This algorithm produces random values in the range of Y(t) ± ΔY, i.e., with a span of 2ΔY. The generation parameters and their retrieved values in five runs, using the minimization procedure and Eq. (6) as the nonisothermal growth model are listed in Table 1. Since Y(T) is a weak function of temperature at best, the adjusted parameter estimates were fairly robust against minor changes in its value, and similar estimates have been obtained when YA(T) varied between 4.7 to 5.3. The table shows that as long as the scatter is not too high, reasonable estimates of the parameters could be obtained even when four parameters must be estimated. This

301

analysis demonstrates that the method can work in principle, but not that it will work in every case. With fewer data points and more outliers, a solution either might not be reached at all (i.e., convergence does not occur) or the rendered parameters are unrealistic or biologically impossible (e.g., having a negative sign). As expected, the generated parameters (see Fig. 3 and Table 1), are close, but not identical to the original values. This was mostly due to the superimposed scatter and to the fact that an adjustment of the magnitude of one parameter in order to minimize the sum of squared errors inevitably results in a corresponding change in the magnitude of the other parameters. Deviations from the “true” values had only a minor effect on the predictive ability of the model created with the retrieved parameters, as demonstrated in Fig. 4. This figure shows three temperature profiles (top) and corresponding to growth data generated with the original parameters listed in Table 1 with superimposed noise (filled circles in bottom plots). It also shows the predictions of the rate model constructed with the retrieved parameters (the means of the first set of runs that are listed in Table 1). These examples demonstrate that in principle, growth data obtained under a monotonic heating or

Fig. 6. Examples of additional experimental non-isothermal growth curves of C. perfringens in cooling cooked ground beef and their predictions with the rate model whose parameters had been calculated from the growth data shown in Fig. 5 (Profiles A and B). Circles of the same shade represent experimental observations from the same experiment.

302

S. Smith-Simpson et al. / International Journal of Food Microbiology 118 (2007) 294–303

cooling regime can be used to predict growth patterns under more elaborate thermal histories that may include temperature oscillations. It should be emphasized that any of these analyses pertain to the same organism in the same original growth state, cultured in the same medium and under time-temperature conditions covered by the experiment from which the model has been derived. In other words, the rate model should not be used for extrapolation to time–temperature histories that are outside the temperature range and time scale of the original data, let alone to a different habitat or organism. 4. Testing the model with experimental growth data of C. perfringens Examples of experimental cooling curves of ground beef and the corresponding growth data of C. perfringens cells are shown in Figs 5 and 6). These data are taken from published work by Smith-Simpson and Schaffner (2005) and experimental and methodological details can be found there. The only modification of the original data was the conversion of the cell counts into logarithmic growth ratios as defined by Eq. (1). It should be noted that the experimental growth data presented in Figs. 5 and 6 have been selected from a larger dataset and some of the data contained in Smith-Simpson and Schaffner (2005) could not be used for the analysis. In some data the experimental scatter was too large, the data points too few or there were outliers that dramatically skewed the growth pattern. Such imperfections in microbial datasets are typical and will be addressed in relation to the result's interpretation and the method's potential implementation below. Fig. 5 shows two experimental cooling curves and the corresponding (non-isothermal) growth data of C. perfringens cells. Both sets had a sufficient number of points and a scatter within a range of about one log unit. Thus, knowing the temperature profile equation, we could apply the same minimization procedure that had been used to recover the growth parameters of the simulated data based on Eq. (6) as the rate model. On the basis of the growth data themselves we have assumed that the asymptotic logarithmic growth ratio YA was ∼ 5, for the experimental time scale and temperature range under consideration. Since the growth parameters' “correct values” were unknown, the model's initial validation had to come from its ability to predict the growth curves under cooling regimes not used in the original model formulation. The first demonstration of this capability is given in Fig. 5. This figure shows the predictions of the growth curve under Profile A with the model when its parameters had been derived from Profile B and vice versa. The agreement between prediction and observation provided the ‘mutual verification’ of the model. Additional examples of such predictions and the corresponding experimental data (filled circles) are shown in Fig. 6. The predicted curves shown in both figures were in reasonable agreement with the experimental data and only in a few cases did the discrepancy between the estimated and observed growth ratio exceeded what could have been expected from the scatter of the experimental data themselves.

5. Conclusions The ability of the described rate model to predict nonisothermal growth patterns from data obtained under different non-isothermal conditions has strong support. Evidence for this statement has come from the simulated experiments, in which the expected outcome can be known, and from experiments where one set of data is used to predict another. But while the principle has been validated, the practicality of the method remains uncertain at this time. The first and foremost is that sigmoid microbial growth curves of the kind encountered in foods require a primary model having at least three temperature-dependent parameters. This means that the general non-isothermal rate model can have as many as six adjustable parameters; two each to account for the temperature dependence of the asymptotic growth ratio, YA(T), the rate parameter, k(T), and the time characteristic, tc(T) or their equivalents in alternative models. Regardless of the regression procedure's sophistication, to obtain meaningful (let alone unique) parameters from scattered data of the kind found in food microbiology is a difficult task. In this work we have been successful, to an extent, because we could assume that the asymptotic growth ratio was practically temperature independent, an assumption that is generally acceptable. This might not be always the situation, so we suggest that at least two nonisothermal temperature profiles should be examined simultaneously for mutual verification. Obviously, the smaller the scatter in the original experimental data, the higher is the probability that the predictions will match the observations, which is also true for models derived from isothermal data. Such isothermal models must also have at least six parameters and their determination will require tests at several constant temperatures. Provided that counting methods with a much reduced scatter are developed for both options, the choice between them would hinge on what's more practical; obtaining a set of isothermal data at various temperatures or under two programmed temperature regimes. In fact, the former will also have to include at least one non-isothermal experiment in order to validate the model. How easy or difficult it is to set up programmed temperature experiments is a minor technical issue, given the availability of programmable water baths. If the non-isothermal method is proven practical for a given system, then it would enable the determination of the growth parameters in situ by taking periodic counts in the food while its temperature is continuously monitored. Once these parameters are determined they will enable one to estimate the product's future fate by translating its time–temperature record into a growth curve. The principle can also be implemented in shelflife estimation, by predicting the whole growth curve rather than the growth rate alone (Corradini and Peleg, 2007). While additional development is needed, both the technology and mathematical methods to accomplish such a task already exist. Acknowledgements This work was supported in part by USDA CSREES National Integrated Food Safety Initiative, the New Jersey

S. Smith-Simpson et al. / International Journal of Food Microbiology 118 (2007) 294–303

Agricultural Experiment Station and the Massachusetts Agricultural Experiment Station of Amherst. References Augustin, J.C., Carlier, V., 2000. Mathematical modeling of the growth rate and lag time for Listeria monocytogenes. International Journal of Food Microbiology 56, 29–51. Baranyi, J., Roberts, T.A., 1994. A dynamic approach to predicting bacterial growth in food. International Journal of Food Microbiology 23, 277–294. Brul, S., Van Gerwen, S., Zwietering, M., 2007. Modelling Microorganisms in Food. Woodhead Publishing Limited, Cambrigde, England. Buchanan, R.L., Whiting, R.C., Damert, W.C., 1997. When is simple good enough: a comparison of the Gompertz, Baranyi and tree-phase linear models for fitting bacterial growth curves. Food Microbiology 14, 313–326. Corradini, M.G., Peleg, M., 2003. A model of microbial survival curves in water treated with a volatile disinfectant. Journal of Applied Microbiology 95, 1268–1276. Corradini, M.G., Peleg, M., 2005. Estimating non-isothermal bacterial growth in foods from isothermal experimental data. Journal of Applied Microbiology 99 (1), 187–200. Corradini, M.G., Peleg, M., 2007. Shelf-life estimation from accelerated storage data. Trends in Food Science and Technology 18, 37–47. Corradini, M.G., Amézquita, A., Normand, M.D., Peleg, M., 2006. Modeling and predicting non-isothermal microbial growth using general purpose software. International Journal of Food Microbiology 106 (2), 223–228. Doona, C.J., Feeherry, F.E., Ross, E.W., 2005. A quasi-chemical model for the growth and death of microorganisms in foods by non-thermal and highpressure processing. International Journal of Food Microbiology 100, 21–32. Fujikawa, H., Kai, A., Morozumi, S., 2004. A new logistic model for Escherichia coli growth at constant and dynamic temperatures. Food Microbiology 21, 501–509.

303

Koutsoumanis, K., 2001. Predictive modeling of the shelf life of fish under nonisothermal conditions. Applied and Environmental Microbiology 67, 1821–1829. McKellar, R., Lu, X. (Eds.), 2004. Modeling Microbial Responses on Foods. CRC Press, Boca Raton, FL. Mc Meekin, T.A., Olley, J.N., Ross, T., Ratkowsky, D.A., 1993. Predictive Microbiology: Theory and Application. John Wiley & Sons, New York, NY. Oscar, T.P., 2005. Development and validation of primary, secondary, and tertiary models for growth of Salmonella typhimurium on sterile chicken. Journal of Food Protection 68, 2606–2613. Peleg, M., 2006. Advanced Quantitative Microbiology for Food and Biosystems: Models for Predicting Growth and Inactivation. CRC Press, Boca Raton, FL. Peleg, M., Normand, M.D., 2004. Calculating microbial survival parameters and predicting survival curves from non-isothermal inactivation data. Critical Reviews in Food Science and Nutrition 44, 409–418. Peleg, M., Normand, M.D., Campanella, O.H., 2003. Estimating microbial inactivation parameters from survival curves obtained under varying conditions—the linear case. Bulletin of Mathematical Biology 65, 219–234. Peleg, M., Corradini, M.G., Normand, M.D., 2007. The logistic (Verhulst) model for sigmoid microbial growth curves revisited. Food Research International 40, 808–819. Smith-Simpson, S., Schaffner, D.W., 2005. Development of a model to predict growth of C. perfringens in cooked beef during cooling. Journal of Food Protection 68 (2), 336–341. Taub, I.A., Feeherry, F.E., Ross, E.W., Kustin, K., Doona, C.J., 2003. A quasichemical kinetics model for the growth and death of Staphylococcus aureus in intermediate moisture bread. Journal of Food Science 68, 2530–2537. Tyrer, H., Ainsworth, P., Ibanoglu, S., Bozkurt, H., 2004. Modelling the growth of Pseudomonas fluorescens and Candida sake in ready-to-eat meals. Journal of Food Engineering 65 (1), 137–143.