Growth of Clostridium perfringens from spore inocula in cooked cured beef: development of a predictive model

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1 Growth of Clostridium perfringens from spore inocula in cooked cured beef: development of a predictive model V.K. Juneja a,, J.S. Novak a, H.M. Marks b, D.E. Gombas c Abstract The objective of this study was to develop a model to predict the growth of C. perfringens from spores at temperatures applicable to the cooling of cooked cured meat products. C. perfringens growth from spores was not observed at a temperature of 12 C for up to 3 weeks. The two parameters: germination, outgrowth, and lag Ž GOL. time and exponential growth rate, EGR, were determined using a function derived from mechanistic and stochastic considerations and the observed relative growths at specified times. A general model to predict the amount of relative growth for arbitrary temperature was determined by fitting the exponential growth rates to a square root Ratkowsky function, and assuming a constant ratio of GOL and generation times. The predicted relative growth is sensitive to the value of this ratio. A closed form equation was developed that can be used to estimate the relative growth for a general cooling scenario and determine a standard error of the estimate. The equation depends upon microbiological assumptions of the effect of history of the GOL times for gradual changes in temperature. Applying multivariate statistical procedures, a confidence interval was computed on the prediction of the amount of growth for a given temperature. The model predicts, for example, a relative growth of 3.17 with an upper 95% confidence limit of 8.50 when cooling the product from 51 to 11 C in 8 h, assuming a log linear decline in temperature with time. Published by Elsevier Science Ltd. Keywords: Cooling; Clostridium perfringens; Cooked cured meat; Temperature; Spore growth 1. Introduction A human pathogen, enterotoxigenic Clostridium perfringens, is found in soil, water, air, intestinal tract, and a variety of raw and processed foods, particularly meat and poultry. This ubiquitous microorganism is a leading cause of food poisoning worldwide, and thus, a continuing concern to the food-service industry. Because of the ubiquitous distribution, the presence of spores of this pathogen must be assumed in various animal or plant products. If the food is to be safe, either thermal treatment should be sufficient to destroy the spores, or their germination, outgrowth and subsequent multiplication into a dangerously high number of vegetative cells must be prevented by the manipulation and control of one or more factors such as storage temperature, ph, and preservatives. Cooking temperatures, if designed to inactivate C. perfringens spores, may negatively impact the organoleptic attributes of foods. Accordingly, the mild heat treatment given to fresh tasting, high-quality, ready-to-eat, extended shelf-life refrigerated foods is aimed at the destruction of vegetative cells of spoilage and pathogenic bacteria; heat-

2 resistant spore-forming foodborne pathogens, including C. perfringens spores can survive the thermal process. Furthermore, the heat treatment could serve as an activation step if it were not lethal to the spores. In such minimally processed foods, if the rate and extent of cooling after cooking is not sufficient, heat-activated surviving spores pose a potential public health hazard due to the potential to germinate, outgrow and multiply. Also, if the products are not transported, distributed, stored and handled under refrigeration, there is a likelihood of C. perfringens spore germination and vegetative cell growth. Inadequate cooling practices in retail food operations have been cited as a major cause of food poisoning with C. perfringens ŽBryan, 1978; Bean & Griffin, In the United States, the organism has been implicated in an estimated yearly average Ž from of approximately 14 million cases of foodborne illnesses with an average of 41 hospitalizations and seven deaths per year ŽMead, Slutsker, Dietz, McCaig, Bresee, Shapiro, Griffin & Tauxe, In 1994, the total cost of illnesses due to C. perfringens was estimated at $123 million in the US ŽAnonymous, The estimated large number of illnesses due to C. perfringens clearly stresses the importance of cooling foods quickly after cooking. Published research suggests that C. perfringens can grow in media supplemented with various levels of curing salts. While growth was not inhibited by 4% wv NaCl, some strains do not grow in 56% NaCl and most strains failed to grow in 78% NaCl ŽRoberts & Derrick, Gough and Alford Ž reported that C. perfringens growth was not inhibited at 8000 ppm of sodium nitrite but was inhibited when the concentration was increased to ppm. It is worth mentioning that the inhibitory effect of sodium nitrite is enhanced when it is heated Ž Davidson & Juneja, Roberts and Derrick Ž reported that the inhibitory concentrations of sodium nitrite can be lowered if combined with other curing salts. In their study, the levels of sodium nitrite necessary to inhibit the strains tested dropped from 300 ppm to 25 ppm when the concentration of NaCl was increased from 3% to 6%. Juneja, Whiting, Marks and Snyder Ž present a model for predicting the relative growth of C. perfringens from spores, through lag, exponential and stationary phases of growth, at temperatures spanning the entire growth temperature range of approximately 1050 C. The growth medium used in the study by Juneja et al. Ž was trypticasepeptone glucoseyeast extract broth. Models pertaining to the behavior of surviving C. perfringens spores during cooling of cooked beef supplemented with preservatives, however, are not available. The purpose of this research was to develop a model which can be used to help determine the safety of cured products or those supplemented with low levels of preservatives, which have been subjected to continuous cooling temperatures. Because of the potential health hazard in cooling cooked foods, discussed above, the United States Department of Agriculture Ž USDA. requires that, during the cooling of certain meat and poultry products, the relative growth of C. perfringens should not exceed one log Ž USDA, Thus, the primary application of the model developed in this paper would be to predict small to moderate amounts of relative growth of C. perfringens from spores during cooling of cooked cured beef products. Many models used for predicting relative growth Ž van Gerwen & Zwietering, are based on empirical data fitting and cover the whole growth cycle, including lag, exponential and stationary phases of growth. These models are often difficult to interpret microbiologically and depend upon assumptions concerning the maximum densities or relative growth possible Ž Juneja et al., 1999; Buchanan, An implicit consequence when using these curves is that the relative growth rate depends upon the number of cells at a given time. Thus, for example, the predicted relative growth at a given time would be different for initial population densities of one cell, 10 cells, or 100 cells per unit of product and so forth. There exists evidence that, for certain situations, lag times can be a function of cell densities, through intercellular communication enhanced by substances produced by the cells Ž Kaprelyants & Kell, In such situations, average cell lag times for populations with densities Žone cell per gram. could be considerably larger than those for populations with larger densities; however, in a graph presented in the above cited paper ŽKaprelyants & Kell, a nearly average constant cell lag time seems to exist for populations with densities greater than approximately 10 cellsg. In the application of models for predicting relative growth of C. perfringens cell populations, the initial densities are assumed to be in the range of approximately cellsg, corresponding to the approximate range of initial levels observed in this study. In this range, we have not been able to find information supporting assumptions which imply that lag times would be significantly dependent upon initial population densities. Consequently, we believe, some researchers have searched for different and simpler models that do not depend upon the initial levels of the population ŽBuchanan, Whiting & Damert, 1997; Baranyi, 1998; Coleman & Marks, to be used for predicting relative growth of cell populations. For this paper, we have selected to use models that have been developed recently ŽBaranyi, 1998; Coleman & Marks, for predicting the small relative growth of a population of cells initially in lag phase. The predictions of these models apply only to a population of cells, numbering approximately 10 per gram or more, that are initially in lag, and remain in the exponential

3 phase of growth. Thus, the difficulty of interpretation, caused by assumptions on the maximum density or growth, is eliminated. 2. Materials and methods 2.1. Test organisms and spore production Three strains of Clostridium perfringens, NCTC 8238 Ž Hobbs serotype 2., NCTC 8239 Ž Hobbs serotype 3., and NCTC Ž Hobbs serotype 13. were used in this study. The spores were produced in a modified formulation of Duncan and Strong sporulation medium as previously described Ž Juneja, Call & Miller, After washing the spore crop of each strain twice and resuspending in sterile distilled water, the stock spore suspensions were stored at 4 C. A spore cocktail containing all three strains of C. perfringens was prepared immediately prior to experimentation by mixing equal numbers of spores from each suspension. This spore mixture was not heat-shocked prior to use Growth mediumproducts Ground beef was obtained from Hatfield Quality Meats, Inc. Ž Hatfield, PA, USA.. The proximate analysis of meats performed by the supplier indicated that the beef contained 15% fat, 66% moisture, 1% ash and 18% protein. Brine Ž 3.5%. was thoroughly mixed in beef before the meat was placed into stomacher 400 polyethylene bags Ž 100 gbag. and vacuum-sealed. Thereafter, five of these bags were vacuum-sealed in barrier pouches ŽBell Fibre products Corp., Columbus, GA, USA., frozen at 40 C and irradiated Ž 42 kgy. to eliminate indigenous microflora. Random samples were tested to verify elimination of microflora by diluting in 0.1% Ž wt.vol.. peptone water Ž PW., spiral plating Ž Spiral Biotech, Bethesda, MD, USA; Model D. onto dishes containing Tryptic soy agar Ž TSA; Difco. and then incubating, both aerobically and anaerobically, at 37 C for 48 h Sample preparation, and inoculation Sodium nitrite Ž 120 ppm. and the cocktail of three strains of C. perfringens were added Ž 1 ml. to 100 g of the thawed irradiated beef products. Thereafter, the inoculated meat was blended with a Seward laboratory stomacher 400 for 5 min to ensure an even distribution of sodium nitrite and the organisms in the meat sample. Duplicate 5-g ground meat samples were then weighed aseptically into 3019-cm sterile filtered stomacher bags Ž Spiral Biotech, Bethesda, MD, USA.. Negative controls included bags containing meat samples inoculated with 0.1 ml of 0.1% Ž wv. peptone water with no bacterial spores. The bags were compressed into a thin layer Ž approx mm thick. by pressing against a flat surface, excluding most of the air, and then heat sealed under vacuum Žnegative pressure of 1000 millibars.. One bag, randomly selected, was opened and heat-shocked at 75 C for 20 min. The samples were serial diluted in 0.1% peptone water Ž wt.vol., surface plated with a Spiral plater ŽModel D, Spiral Biotech, Bethesda, MD, USA. on Shahidi Ferguson perfringens agar as described previously Ž Juneja, Marmer & Call, The lower limit of detection by this procedure was 21 CFUml. The total C. perfringens population was determined after 48 h of anaerobic incubation. This was recorded as the initial inoculated numbers of bacterial spores before cooking Incubation temperatures, sampling times and bacterial enumeration Simulating the conditions that occur in the food industry, the vacuum-packaged bags containing the meat samples were immersed in a programmable water bath Ž Techne, ESRB, Cambridge, UK. at 10 C. The temperature of the water bath was programmed to increase linearly to 60 C in a period of 1 h. This process simulated the cooking of rare roast beef and heat-shocked the spores. Samples after cooking were enumerated and the data were recorded as spore numbers after cooking. Thereafter, all samples were incubated in a constant temperature water bath stabilized at 10.0, 12.0, 15.6, 19.0, 21.1, 23.9, 26.7, 29.4, 32.2, 35.0, 37.8, 40.6, 43.3, 46.1 or 48.9 C. The length of time Žin seconds. for the samples to reach an equilibrium temperature can be considered instantaneous, or negligible, in comparison to the lag phase duration times Žin hours., and thus would have virtually no effect on the observed growth curves. At frequent intervals, appropriate for each growth temperature, samples were taken out for C. perfringens count enumeration as described above. Two independent experiments were done at each temperature. For each replicate experiment, an average cfug of two platings of each sampling point was used to determine estimates of the growth kinetics. 3. Statistical methodology 3.1. Primary relatie growth model Models for predicting relative growth during the lag and exponential phases of growth ŽBaranyi, 1998; Coleman & Marks, 1998., have been developed by keeping track of the status of the original cells, O, and new cells, D, that are either a result of completing an acclimation or lag phase Ž Baranyi, 1998., or are a result

4 of cell division Ž Coleman & Marks, Baranyi Ž assumed that the random time of death of an original cell Ž thus becoming a new D cell. and the random time of a D cell birth ŽD cell division into two D cells. were distributed as exponential distributions. Coleman and Marks Ž developed a more general equation, assuming an arbitrary survival function for the time of death of the original cell Žbecoming two new cells. using generating functions of the stochastic distribution of the number of cells. Defining Ž. t as the infinitesimal cell death rate of the O cell, i.e. the O cell either becomes a D cell with probability q, 0q1, or actually dies, with probability 1q. Also, define Ž. t to be the infinitesimal cell birth rate of a D cell. Let m Ž. A t be the number of cells of type A Ž O or D. at time t. The following set of differential equations: O m Ž t. Ž t. m Ž t. Ž 1. m D Ž t. qž t. m O Ž t. Ž t. m D Ž t. O Ž. Ž. with boundary conditions, m 0 N and m 0 0, O 0 D are derived, reflecting the assumptions that the population of O-cells follows a first order kinetic decay and the D-cells increase due to two sources: Ž. 1 the deaths of O-cells; and Ž. 2 the births of D-cells. Eq. Ž. 1 corresponds to equations used by Baranyi Ž with the exception here that the transition rates, Ž. t and Ž. t, are time-dependent and there is an additional paramet t ter q. Defining Ž t. Hd and Ž t. Hd, 0 0 the solution to Eq. Ž. 1 is: Žt. m t N e 2 O 0 H Žt. ŽŽ.Ž.. D 0 0 m t N qe e d Assuming that Ž. t equals a constant,, so that the growth of the population of D cells is a simple exponential growth process, the expected relative growth, Ert Ž.is: H Žt. t t ŽŽ.. 0 EŽrŽ t.. e e q e d Ž 3. Fig. 1. Comparison of three relative growth curves Ž logarithm relative growth vs. time. where for large times the relative growths are nearly the same, but for small times the curvatures of the curves are different. The random times for original cells to become acclimated and divide as newly Ž a born cells is assumed to be distributed as an exponential distribution for one curve, a Weibull distribution, 1exp t b., where a1, b0, for another curve..., and for the third curve Ž., it is assumed that original cells actually die with non-zero probability, 1q. The unit associated with the time variable is arbitrary.

5 Eq. Ž. 3 accounts for the curvature that is seen at the beginning of typical growth curves. As an example, a convenient probability distribution for the time of an O cell transformation with density of form, ht Ž. Ž. Žt. Ž. te is the Weibull distribution, t t, where and are constants. The infinitesimal cell death rate or hazard function for the O cells, Ž. t,is Ž. 1 equal to t t. The expected time for an O cell to be in the germination, outgrowth, and lag phase is the expected value of a random variable with Weibull Ž. Ž 1 cumulative distribution, Ht, and equals 1.. The relative growth model, Eq. Ž. 3, then has five parameters:,, q, and N. When 1, Ht Ž. 0 describes the exponential cumulative distribution, and Eq. Ž. 3 becomes: Ž 1. t 1 t 1q e q e EŽrŽ t.. Ž 4. 1 Fig. 1 provides examples of Eqs. Ž. 3 and Ž. 4, assuming a pure linear birth process for the D cells. When q1, the growth curve is referred to as the improper exponential. The values of the parameters for the equations are chosen so that, for relatively large times, the expected relative growths depicted by the three curves are nearly the same, but that the expected relative growths at times near zero are different. Near times equal to zero, the log-relative growth curve based on the Weibull distribution is flatter than the curves based on the exponential distribution. The curve based on the exponential distribution where q is not assumed to be equal to 1 initially for times greater than zero is negative. A mathematical definition of lag time for a non-linear microbiological growth curve log Ž CFUunit. 10 vs. time has been defined as the intersection of the horizontal line, ylog Ž N. 10 0, and the tangent line of the curve with slope equal to the maximum derivative Ž McMeekin, Olley, Ross & Ratkowsky, For Eq. Ž. Ž. t 4 assuming 1 so that Ht1e, the mathematical lag time is defined as the intersection of the horizontal line, ylog Ž N. 10 0, and the asymptotic line of the curve as t approaches infinity. The mathematical lag time, lagtime, is thus: lnž 1. lnž q. lagtime Ž 5. The expected relative growth when tlagtime is a function of c when q1, and is equal to: c EŽrŽ lagtime.. 1 Ž 6. 11c Ž 1c. Hence, the expected relative growth at tlagtime is between 1 and 2, or in common logarithmic units, between 0 and 0.3. At tlagtime, the survival function of O cells, St1H Ž. Ž. t is: c 1 S lagtime 1c 7 1 Thus, e SŽ lagtime. 1, or that at time equal to the mathematical definition of lag less than 63.21% of the O cells have transformed to D cells Secondary models The above equations apply for a constant temperature, T. To derive equations that apply for arbitrary temperatures, the values for the derived parameters are considered to be functions of temperature, and statistical regression procedures are used to derive secondary models that express the parameter values as a function of temperature. The parameter measures the exponential growth rate in the natural log scale. Following customary procedures, the actual secondary model is performed with the exponential growth rate expressed in the common log Ž base 10. scale, egr lnž 10.. For generalizing egr, the Ratkowsky function Ž McMeekin et al., of the form: 12 bžtt. 12 min ma x egr Ž T. až TT.1e Ž 8. is used, where a, b, T min, and Tmax are parameter values determined from a regression analysis ŽJuneja et al., For TT min,ortt max, egr is defined as zero. For generalizing lag times, the ratios of the lag times to the generation times for D cells the time needed for doubling the populationlnž. 2 are analyzed ŽRoss, From Eq. Ž 5., this ratio for a given temperature is: lnž 1. lnž q. Rat Ž 9. lnž 2. The time it takes for an O cell to be come acclimated and transform to a D cell depends upon many factors related to the environment and history of the cells Ž Baranyi & Roberts, For a given temperature, the lag times, and thus the conditional expected value of the ratios of the lag time to the generation time for a given experiment are not assumed to be equal; rather they are considered to be random variables. An analysis of variance is performed to determine the betweenexperiment variance component of the ratios, computed from Eq. Ž. 9. Statistical analyses were performed using SAS for Ž windows, release 7.00 SAS, 1999.; figures were cre- Ž ated using Microsoft Excel 97 SR-2l Microsoft,

6 . Ž 1997 and S-Plus 2000 release 3 S-Plus, 2000., and some calculations were performed using Mathcad 7 Ž Professional Mathcad 7, Results Table 1 presents the estimated exponential growth rate, egr, and lagtime using Eqs. Ž. 4 and Ž. 5, assuming q1, the generation time, and common logarithm of the ratio of lagtime to the generation time. Fig. 2am depicts the measured levels log Ž CFUg. 10 and fitted curves using Eq. Ž. 4 with q1, for each temperature studied from 15.6 to 48.9 C and for each replicate. R 2 values were generally greater than approximately 0.95, being, on average, higher for the higher temperatures Ž 35 C.. An exception was for the second replicate for temperature 48.9 C, where R ; the low value was caused by a higher than expected variability in the measurements. However, the derived curve is similar to the one derived for the other replicate at this temperature and the observed results indicate that there was substantial growth; thus the derived kinetics for the curve are used in the secondary model. Among replicate growth curves, it can be seen by observing Table 1 and Fig. 2am that there is high variability in many cases in the initial or lag stages of growth. Observations that are shown and used in determining the values for the parameters are depicted in the figures; the observed points that are not depicted were those that appeared to be near stationary levels. Two temperature replicates Ž at 23.9 and 32.2 C. experiments were completely deleted because there was very little observed growth. Other observations that were deleted are indicated in the figures. Curves using Eq. Ž. 4 where q was an unknown value were also fitted. Most of the estimated values of q were very close to 1 Ža logit transform for q was used to avoid boundary conditions.. However, convergence was not achieved for some curves and for those where convergence was obtained, the standard errors of the estimates were very large. Consequently, the fitted curves assume q1. At temperatures of 10 and 12 C, there was no apparent growth in the 500 h for which observations were made. Fig. 3 is a scatterplot of the square root of the exponential growth rate, egr, vs. temperature, together with the fitted Ratkowsky function Eq. Ž. 8. For the highest temperature studied Ž 48.9 C. there was substantial growth, so that this high temperature does not provide an approximation to the Tmax parameter of Eq. Ž. 8. From a study of the relative growth of C. perfringens in broth Ž Juneja et al., an estimate of T max was 51 C. For determining the parameter values of the Ratkowsky curve, it thus was assumed that T 51 max Table 1 Summary statistics of fitted curves, egr, the exponential growth rate, lag time, generation time and the log generation time 10 of the ratio of lag time to the Temp EGR Lag time Generation Log10 ratio lag time C log Ž CFUu. h Ž h. time Ž h. to generation

7 C. The other parameter values Ž with standard errors. were estimated: a0.035 Ž ; b0.231 Ž ; and T C Ž 2.10 C. min. Fig. 4 is scatterplot of the 22log Ž 10 lagtimegeneration time. estimated values vs. temperatures together with the linear regression line, between temperatures 21.1 and 48.9 C. Between these temperatures, there is no significant relationship between the log ratios and temperatures. An analysis of variance of the log ratio values between these temperatures indicated statistically insignificant temperature or replication effects. The distribution of the log ratio values between these temperatures is approximately normal, with a mean value, m, of Žcorresponding geometric mean of Fig. 2. Ž am. Plots of observed common logarithmic levels of C. perfringens vs. time Ž h. and estimated growth curves, at a given temperature. Data points not used to determine the growth curves are designated by.

8 Fig. 2. Continued. approx. 8. and standard deviation, s, of The lower 90% confidence limit of the geometric mean is The standard deviation includes contributions of a measurement error component and an experimental error component that might rise from slightly Žunavoi- dable. different conditions of the experiments. A further analysis of variance, ignoring temperature and replication factors, was performed accounting for the variance within an experiment for a given temperature and replication. The within experiment measurement variance for a given experiment was estimated using the asymptotic covariance matrix of the estimated values of the parameters and of Eq. Ž. 4, and an approximation in and based on the linear terms of a Taylor series expansion of Eq. Ž. 9. The approximate variance estimates ranged between and 0.232, with one exception, at 48.9 C, where, for the second replicate, an estimate of 1.29 was obtained. Excluding

9 percentile of a predicted ratio value is used. This percentile is derived by considering the quantity: ym t Ž Ž s ns b. Fig. 2. Continued. this one value, the average within experimental variance, s, is Using this estimate, an estimate of the between experiment variance, s 2 b, of approximately Ž standard deviation, s, of approx b is derived. For modeling purposes, it is assumed that the population of log ratio values are normally distributed, independent of the temperature, with meanm. For the calculations presented in this paper, the lower x-th where y is a random log ratio value, to be distributed as a t-distribution with 21 d.f. The first term of the expression: s 2 ns 2 b, in the denominator, accounts for the uncertainty of the mean and the second accounts for the expected variability among experiments. Thus, for example, the lower 90th percentile of the ratio value, Rat, is derived to be approximately When tlagtime, from Eqs. Ž. 5, Ž. 6 and Ž. 8, the expected relative growth of the population can be calculated, as a function of Rat. Thus, for Rat equal to the geometric mean, 8.068, the expected relative growth is approximately 1.98, or common logarithms, 0.296; for Rat equal to the lower 90th percentile, the expected relative growth is approximately 1.88, or in common logarithms, 0.274log Discussion In the present study, the equations developed above apply for arbitrary, but fixed temperatures. For a given 12 Fig. 3. Plot of square root of the exponential growth rate log h,egr, vs. temperature, together with fitted Ratkowsky curve, egr až 10 T T.1expŽbTT. 12. min max

10 Fig. 4. Plot of common log of ratio of lag time to generation time and linear regression line for temperatures between 21.1 and 48.9 C. temperature, from Eqs. Ž. 8 and Ž. 9, values of and are derived, and then the relative growth curve, Eq. Ž. 4, can be determined. Of interest is using the above equations for determining relative growth when the temperatures are changing over time, described by the function, TgŽ. t, where T is the temperature at time t. The underlying assumptions in developing Eq. Ž. 2 is that processes, Ž. t and Ž. t, describing instantaneous kinetic rates of O-cells becoming D-cells and D-cells dividing, depend strictly upon the present time, t, and not on historical events or the rate of cooling. While it is assumed that the exponential growth rate Ž egr. values, and thus Ž. t, do not depend upon the historical experience of the parent cells, such an assumption with regard to the lag times of cells and thus the quantity Ž. t cannot be made; lag times appear to depend upon the historical experience of the cells ŽZwietering, De Wit, Cuppers & Van t Riet, 1994; Baranyi & Roberts, Research Ž Zwietering et al., shows that the lag times increase over that expected at a given temperature, after adjusting for the time spent in lag at a prior temperature. This is interpreted to mean that Ž. t, the infinitesimal cell death rate of original O cells and becoming D cells, decreases when the cells experience a temperature change. If this were so, then, when temperatures are changing, using Eq. Ž. 2 would provide estimates of the expected relative growth that would be expected to be higher than the true amount of relative growth. Mathematically, as the temperature, T, changes, the growth curve, EŽ T., changes. In order to maintain continuity in the estimate of the relative growth moving from one growth curve, EŽ T. 0, to the other curve, EŽ T., the change in EŽ T. 1 1 would be added to the value of the relative growth before the temperature change. The mathematical problem is where Žfor what value of time. to begin measuring changes in EŽ T. 1.To address this problem, it has been assumed ŽJuneja et al., that the effect of history on the lag times can be expressed as a function, fž. t, reflecting the accumulated ratios of length of times in lag spent up to a specified time. Assume that LtT Ž. log Ert Ž 10, for any fixed temperature, T, can be written as a function Gk Ž ŽzT t.,. where k, z and are functions of temperature, and zt is any pivot point on the time axis which can be expressed as a function of temperature. Let be the time such that g T where g is an arbitrary function of time. The function f is defined as the translation from the pivot point zgt Ž. along the time axis reflecting the accumulated ratios of length of times to pivot points spent on previous growth curves up to time, i.e. f zž gž t..ž1 t H z 1 ŽŽ.. g t d s. Note, this function can be negative 0 and when the temperature is constant fž. t zž T. t.

11 The expected relative growth over time, assuming Ž. changing temperature, Lt, can be expressed as: H t LŽ t. GŽkg Ž. fžzž.,ž g... d Ž where G is the derivative of G with respect to time evaluated at time zero and held at constant temperature. Estimates of relative growth obtained from Eq. Ž. 2 and Eq. Ž 11. are the same, and thus the function f provides a geometric interpretation of Eq. Ž. 2. As an example, Eq. Ž. 2 was calculated for a hypothetical cooling of product from 51 to 11 C int x hours where temperature, T, at time t, is given by: kt g t T T e T 12 1 a a where T1 is the initial temperature of the product, Ta is the ambient air temperature Žhere assumed equal to 0 C., and k is the exponential cooling rate. The derived Ratkowsky Eq. Ž. 8 is used and it is assumed to Ž m be g g 2 1, where m represents the ratio of the mathematical lag time, lagtime, to the generation time for the D cells Eq. Ž. 9. Table 2 provides the common logarithm of the expected relative growth from Eq. Ž. 2, for t x6, 710 h, for as- sumed values of m5.223, corresponding to the predicted lower 90th percentile value of the ratios of lagtime to the generations times; m6.857, corresponding to the lower 90% confidence limit of the geometric mean of the ratios; and m8.068, corresponding to the estimated geometric mean of the ratios. As can be seen from the table, the predictions of relative growth depend heavily on the assumed ratio of the lag and generation times. The standard error of the estimate for 8 h, assuming that m8.068, was determined using linear terms of a Taylor series approximation Ž Rao, 1973., expanding the estimator as a function of the four parameters: a, b, T min, and m, and assuming that the estimate of m is not correlated with the other estimates. In this expansion, the percentage of the variance due to the estimate of the parameter m, assuming m equals 8.068, is approximately 70%, indicating a high, relative sensitivity of the predicted estimates to the values of the variable, m. A model has been developed to estimate the relative growth of C. perfringens in beef broth ŽJuneja et al., Eq. Ž 11. was used where the function G was the logistic function: GtM, Ž B. 71expŽBŽ tm.. where B and M are constants, dependent upon temperature, and the coefficient, 7, represents the approximate average maximum relative growth of C. perfringens observed in the study. Using results reported by Juneja et al. Ž 1999., assuming the product is cooled log linearly from 51 to 11 C, the logistic function predicts a relative growth of C. perfringens of approximately 2.2 if the cooling takes 6 h; 4.6 if the cooling takes 8 h; and 16 if the cooling takes 10 h. These values are slightly higher than the predictions: for 6 h; for 8 h; and for 10 h, obtained from Table 2, when assuming the ratio of the mathematical lag time to the generation time for cells in exponential phase of growth is equal to 8.068, the estimated geometric mean. For purposes of investigating the effect of temperature change on the growth curve, a simple assumption that Ž. t is proportional to the exponential cooling rate, kž. t could be considered. If k were constant, then the temperature is given by Eq. Ž 12.. Thus, a possible model could assume that, for a given exponential cooling rate, k, the infinitesimal death rate of original cells is given by: Ž t. Ž t. Ž 13. k k where k is a decreasing function of k, such that 0 1 and 0. Fig. 5 presents hypothetical growth curves for values of 1, 0.75, 0.50 and 0.25, ask suming T 51 C, T 0 C, and k0.192, so that for 1 a t8 h, the temperature is 11 C. Growth curves could be determined for different constant exponential rates of temperature change to explore the relationship of k as a function of k and to evaluate if the above model Table 2 Predicted common logarithm of expected relative growth of C. perfringens in cured beef when cooled, log linearly, from 51 to 11 C over 6, 710 h, assuming different values of the ratio of the mathematical lag time to the generation time of cells in the exponential phase of growth Hours Ratio of mathematical lag time to generation time a b c a 90th lower percentile. b 90% lower confidence limit of geometric mean. c Geometric mean.

12 Fig. 5. Plot of common log of relative growth vs. time for a hypothetical scenario of cooling in 8 h from 51 to 11 C, assuming an ambient temperature of 0 C. The topmost curve is derived from data presented in this paper, assuming that the infinitesimal rate that original cells leave the lag phase, Ž. t, does not depend on history. Curves below the top one are derived assuming infinitesimal rates equal to Ž. t, for 0.75, 0.50 and 0.25, where the bottom curve corresponds to can provide approximate estimates of relative growth for constant exponential rates of temperature change. Research to do this is being planned. The present study has assessed the growth of C. perfringens from spores inocula, in beef supplemented with curing salts, when the beef was cooked slowly to 60 C and then cooled rapidly to a specified temperature. Under these conditions, the growth of C. perfringens from spores over a period of 21 days at 10 and 12 C was not observed. These observations are in agreement with previous studies; Goepfert and Kim Ž reported that C. perfringens growth does not begin in foods stored at 15 C or below, even after extended storage. However, this may not hold true when hot foods are cooled and the rate of cooling is not sufficiently fast. Solberg and Elkind Ž reported that C. perfringens vegetative cells increased by three log 10 cycles in 3 days at 15 C and 5 days at 12 C, but the growth was restricted at 10 and 5 C. It is possible that the growth kinetics at other temperatures would also be affected by the rate of cooling. In addition to the lack of history assumption used for deriving Eq. Ž. 2, the assumption of an exponential distribution, used in deriving Eq. Ž. 4 of the times for O cells to be in the GOL phase may not be innocuous. This assumption entails a constant hazard function; i.e. the instantaneous probability that a cell will exit the GOL phase, given that it has not yet done so, is constant over time. However, cells probably need to go through various development stages that may be influenced by the environment or changes in the environment. Thus, the effects of the changes in the environment might not be felt immediately by the cells, delaying, so to speak, the developmental events. In other words, it seems plausible to hypothesize that the hazard function would increase over time, as would be Ž a the case for a Weibull distribution, 1exp t b., with a 1. A paper ŽChea, Chen, Montville & Schaffner, concerning the time for germination of Clostridium botulinum cells that were examined microscopically reported that the Weibull distribution provided a best fit to data from among many distributions considered, though the exponential distribution provided an adequate fit. A further assumption used in deriving Eq. Ž. 2 is that once the cells leave the GOL phase, they have the same growth characteristics as new offspring cells. Indeed it might be that the developmental events leading to a cell leaving the GOL phase and the events needed for cell division are operating closer in parallel rather than in series or sequentially, as was assumed for the model used in this paper. The consequence, of course, is that the microbiological interpretation of the model s parameters is questioned, even though the model provides an adequate fit of observed data. The types of experiments needed to test these types of hypotheses would be more involved than the ones conducted for this study. Many observations near the estimated times where the cells would be leaving GOL phase as well as perhaps microscopic examination of the cells would be needed. In summary, this paper presents a model for predict-

13 ing small to moderate relative growth of C. perfringens during cooling and subsequent storage of certain cooked cured meat products, which is assumed not to be dependent upon the initial levels of the population when between 10 and 10 4 CFUg. The predictions are sensitive to an assumed ratio of the lag time to the generation time; more research is needed to obtain better understanding of the ratio. Also, as discussed above, research is needed to study the kinetics of growth as a function of the cooling rate in order to validate assumptions and equations presented in this paper. Thus, it may be premature to use the model for predicting C. perfringens growth in cured beef products. References Anonymous Ž Food poisoning an overview. International Poultry Produce, 4, Baranyi, J., & Roberts, T. A. Ž A dynamic approach to predicting bacterial growth in food. International Journal of Food Microbiology, 23, Baranyi, J. Ž Comparison of stochastic and deterministic concepts of bacterial lag. Journal of Theoretical Biology, 192, Bean, N. H., & Griffin, P. M. Ž Food borne disease outbreaks in the United States, : pathogens, vehicles, and trends. Journal of Food Protection, 53, Bryan, F. L. Ž Factors that contribute to outbreaks of foodborne disease. Journal of Food Protection, 41, 816. Buchanan, R. L. Ž Using spreadsheet software for predictive microbiology applications. Journal of Food Safety, 11, Buchanan, R. L., Whiting, R. C., & Damert, W. C. Ž When is simple good enough: a comparison of the Gompertz, Baranyi, and three-phase linear models for fitting bacterial growth curves. Food Microbiology, 14, Chea, F. P., Chen, Y., Montville, T. J. & Schaffner, D. W. Ž Modeling the germination kinetics of Clostridium botulinum 56A spores as affected by temperature, ph, and sodium chloride. Journal of Food Protection, 63Ž. 8, Coleman, M. E., & Marks, H. M. Ž Qualitative and quantitative risk assessment. Food Control, 10, Davidson, P. M., & Juneja, V. K. Ž Antimicrobial agents. In A. L. Branen, P. M. Davidson, S. Salminen, Food Addities Ž p New York: Marcel Dekker, Inc. van Gerwen, S. J. C., & Zwietering, M. H. Ž Growth and inactivation models to be used in quantitative risk assessments. Journal of Food Protection, 61Ž 11., Goepfert, J. M., & Kim, H. V. Ž Behavior of selected foodborne pathogens in raw ground beef. Journal of Milk and Food Technology, 38, Gough, B. J., & Alford, J. A. Ž Effect of curing agents on the growth and survival of food poisoning strains of Clostridium perfringens. Journal of Food Science, 30, Juneja, V. K., Whiting, R. C., Marks, H. M., & Snyder, O. P. Ž Predictive model for growth of Clostridium perfringens at temperatures applicable to cooling of cooked meat. Food Microbiology, 16, Juneja, V. K., Marmer, B. S., Call, J. E. Ž Influence of modified atmosphere packaging on growth of Clostridium perfringens in cooked turkey. Journal of Food Safety, 16, Juneja, V. K., Call, J. E., Miller, A. J. Ž Evaluation of methylxanthines and related compounds to enhance Clostridium perfringens sporulation using a modified Duncan and Strong medium. Journal of Rapid Methods of Automated Microbiology, 2, Kaprelyants, A. S., & Kell, D. B. Ž Do bacteria need to communicate with each other for growth? Trends in Microbiology, 4, Mathcad 7 Professional Ž Cambridge Mass., USA: MathSoft Inc. Mead, P. S., Slutsker, L., Dietz, V., McCaig, L. F., Bresee, J. S., Shapiro, C., Griffin, P. M., & Tauxe, R. V. Ž Food-related illness and death in the United States. Emerging and Infectious Diseases, 5, McMeekin, T. A., Olley, J. N., Ross, T., & Ratkowsky, D. A. Ž Predictie Microbiology: Theory and Application. New York: J. Wiley & Sons, Inc. Microsoft Exel 97 SR Microsoft Corporation. Rao, C. R. Ž Linear Statistical Inference and its Applications. NY: John Wiley. Roberts, T. A., & Derrick, C. M. Ž The effect of curing salts on the growth of Clostridium perfringens Ž welchii. in a laboratory medium. Journal of Food Technology, 13, Ross, T. Ž Predictie food microbiology models in the meat industry. Meat and Livestock, Australia. Solberg, M., & Elkind, B. Ž Effect of processing and storage conditions on the microflora of Clostridium perfringens-inoculated frankfurters. Journal of Food Science, 35, SAS for Windows Ž Cary NC, USA: SAS Institute, Inc. S-Plus 2000 Professional Release 3 Ž Cambridge Mass., USA: MathSoft Inc. US Department of Agriculture, Food Safety and Inspection Service Ž Performance standards for the production of certain meat and poultry products. Federal Register, 64Ž. 3, Zwietering, M. H., De Wit, J. C., Cuppers, H. G. A. M., & Van t Riet, K. Ž Modeling of bacterial growth with shifts in temperature. Applied and Enironmental Microbiology, 60Ž. 1,