Applied and Environmental Microbiology, September 2003, p . 5138-5156, Vol . 69, No . 9

Predictive Thermal Inactivation Model for Effects of Temperature, Sodium Lactate, NaCl, and Sodium Pyrophosphate on Salmonella Serotypes in Ground Beef

Vijay K . Juneja,^1* Harry M . Marks,² and Tim Mohr³

Eastern Regional Research Center, Agricultural Research Service, U.S . Department of Agriculture, Wyndmoor, Pennsylvania 19038,¹ Food Safety Inspection Service, U.S . Department of Agriculture, Washington, D.C . 20250,² Technical Service Center, Food Safety Inspection Service, U.S . Department of Agriculture, Omaha, Nebraska 68102³

Received 11 December 2002/ Accepted 19 June 2003

An effective thermal process is necessary to control the potential hazard of Salmonella in cooked meat products . A key to optimization of the heating step is defining the target pathogen's heat resistance . While overestimating the heat resistance negatively impacts on product quality, underestimating increases the likelihood that the contaminating pathogen persists after heat treatment or cooking. Accordingly, teams of investigators have conducted thermal inactivation studies of different Salmonella serotypes in aqueous media and foods (4) . Various factors affecting the heat resistance have been documented, including growth temperature, stage of growth, initial population, bacterial strains, composition and pH of the heating menstruum, heat shock, and methodology used for detection of survivors (16) . In a study by Juneja et al . (7), when the heat resistance of Salmonella serotypes was quantified in beef of different fat levels, asymptotic D values (D values for large times) increased with increasing fat levels . While the study by Juneja et al . (7) provided some characterization of the inactivation kinetics, there is a lack of information on the effects of increasing concentrations of sodium pyrophosphate (SPP) and sodium lactate (NaL) in combination with various salt levels on the heat resistance of the organism. Accordingly, the present study was carried out to quantitatively assess the relative effects and interactions of SPP, NaL, NaCl, and temperature on the inactivation kinetics of Salmonella serotypes .

 
Products.
Raw 75% lean ground beef, used as the heating menstruum, was obtained from a retail supermarket . The meat was separated into batches for different treatments and mixed thoroughly with the additives to be tested, i.e., each batch received various variable combinations of SPP (0.0 to 0.3%, wt/wt), NaL (0.0 to 4.8%, wt/wt), and/or NaCl (0.0 to 4.5%, wt/wt). The pH of the meats tested were determined using a combination electrode (Sensorex, semimicro; A.H . Thomas, Philadelphia, Pa.) attached to a pH meter (model 310; Orion, Boston, Mass.) . The meat was placed into stomacher 400 polyethylene bags (50 g/bag) and vacuum sealed . Thereafter, five of these bags were vacuum sealed in barrier pouches (Bell Fibre Products, Columbus, Ga.), frozen at -40°C, and irradiated (42 kGy) to eliminate indigenous microflora . Random samples were tested to verify elimination or inactivation of microflora by diluting in 0.1% (wt/vol) peptone water (PW) to obtain1:1 meat slurry, followed by direct surface plating the suspension (0.1 and 1.0 ml) on tryptic soy agar (TSA) (Difco) and incubating aerobically at 30°C for 48 h .

Preparation of test cultures.
To propagate the cultures, vials were partially thawed at room temperature and 1.0 ml of the thawed culture was transferred to 10 ml of brain heart infusion (BHI) (Difco) broth in 50-ml tubes and incubated for 24 h at 37°C . This culture was not used in heating tests, due to the presence of freeze-damaged cells . The inocula for use in heating tests were prepared by transferring 0.1 ml of each culture to tubes of BHI broth (10 ml) and incubating aerobically for 24 h at 37°C . These cultures were then maintained in BHI for 2 weeks at 4°C . A new series of cultures was initiated from the frozen stock on a biweekly basis .

A day before the experiment, the inocula for conducting the heating studies were prepared by transferring 0.1 ml of each culture to 50 ml of BHI in 250-ml flasks, and incubating aerobically for 18 h at 37°C to provide late-stationary-phase cells . On the day of the experiment, each culture was centrifuged (5,000 x g, 15 min, 4°C), the pellet was washed twice in 0.1% (wt/vol) PW and finally suspended in PW to a target level of 8 to 9 log₁₀ CFU/ml . The population densities in each cell suspension were enumerated by spiral plating (model D; Spiral Biotech, Bethesda, Md.) appropriate dilutions (in 0.1% PW), in duplicate, onto TSA plates . Approximately equal volumes of each culture were combined in a sterile conical vial to obtain an eight-strain mixture of Salmonella (8 log₁₀ CFU/ml) prior to inoculation of meat .

Experimental design.
A fractional factorial design was used to assess the effects and interactions of heating temperature, SPP, NaL, and NaCl . Levels of the factors studied are as follows: heating temperature, 55, 60, 65, and 71.1°C; NaCl, 0.0, 0.75, 1.5, 2.5, 3.0, 3.75, and 4.5%; SPP, 0.0, 0.15, 0.30, 0.40, 0.45, and 0.50%; NaL, 0.0, 1.0, 1.5, 2.5, 3.0, 4.0, and 4.5% .

Forty-five different design points of the above factors were studied . Table 2 gives the 45 design points tested along with some other information as explained below . For each experimental combination at least two replicates were obtained, and in total there were 110 survivor curves, two per experimental combination, for a total of 55 combinations, some of these the same, to give 45 distinct combinations .

 
Sample preparation and inoculation.
The cocktail of eight strains of Salmonella was added (0.1 ml) to 50 g of thawed, irradiated ground meat . The inoculated meat was blended with a Seward laboratory stomacher 400 for 5 min to ensure even distribution of the organisms in the meat sample . Duplicate 3-g ground-meat samples were then weighed aseptically into sterile filtered stomacher bags (30 by 19 cm; Spiral Biotech) . Negative controls consisted of bags containing meat samples inoculated with 0.1 ml of 0.1% (wt/vol) PW with no bacterial cells . Thereafter, the bags were compressed into a thin layer (approximately 0.5 to 1 mm thick) by pressing them against a flat surface, excluding most of the air, and then were heat sealed using a Multivac (model A300/16; Multivac Inc., Kansas City, Mo.) packaging machine .

Thermal inactivation and bacterial enumeration.
The thermal inactivation studies were carried out in a temperature controlled circulating water bath (Techne, ESRB, Cambridge, United Kingdom) stabilized at 55, 60, 65, or 71.1°C according to the procedure as described by Juneja et al. (8) . Bags for each replicate were then removed at predetermined time intervals, placed into an ice-water bath and analyzed within 30 min . Surviving bacteria were enumerated by surface plating appropriate dilutions, in duplicate, on to TSA supplemented with 0.6% yeast extract and 1% sodium pyruvate, using a spiral plater .

Samples not inoculated with Salmonella cocktail were plated as controls . Also, 0.1- and 1.0-ml aliquots of undiluted suspension were surface plated, where necessary . All plates were incubated at 30°C for at least 48 h prior to counting colonies . For each replicate experiment, average numbers of CFU per gram of four platings of each sampling point were used to determine estimates of the lethality kinetics .

Statistical methods.
(i) Primary model.
Graphical examination of the observed survival curves revealed that almost all the curves had a convex shape . Some of the curves also displayed "shoulders," suggesting a possible lag effect . The dependent variable used in the regressions is the observed log₁₀ of N(t)/N(0), where N(t) is the number of cells at time t . The negative of this quantity is referred to as the lethality at time t . The following equation,

(ii) Secondary model.
An omnibus model for predicting survival curves for any specified values of temperature, salt, SPP, and NaL, was determined by considering the parameters that are identified in equations 1 or 2 to be at most quadratic polynomials of the independent variables described in the Results section . Using higher order polynomials might result in a response surface with more than one local maximum or minimum, which would be contrary to our a priori expectations, and, given the number of design points, a result contrary to this expectation probably could not be supported and thus would not be believed but rather assumed to be a consequence of experimental error . The desire is to determine a model that includes only statistically significant terms since including insignificant terms increases the standard error of predictions possibly without any corresponding reduction of bias (an example of Occam's maxim) . Thus, the selection of terms in a model does not preclude other variables that are not included in the model from being important for predicting lethality . Initially, stepwise regressions were used to identify statistically significant variables from a quadratic response surface for inclusion in the model . The natural logarithm of the temperature was included among the variables considered in the regression . Influential observations were determined by examining studentized residuals (computed excluding the observation) and Cook's D statistic .

One advantage of equation 1 is that the logit transformation on the quantity 1 - r(t), where r(t) = N(t)/N (0), or equivalently, the transformation,

Of particular importance is the times needed to obtain a 6.5-log₁₀ relative reduction . The above model could be used for estimating these times, however, a more direct approach was used: for each individual experiment, using the estimated survival curve, an estimate of the time for a predicted 6.5-log₁₀ relative reduction, t_6.5, was derived, and the natural logarithm of this estimated time was used as the dependent variable in a mixed effects regression analysis, as described above . From equation 1, the predicted time, t_6.5, to obtain a 6.5 lethality is obtained as follows:

Nonlinear regressions, stepwise regressions, and linear mixed effects models were computed using PC SAS, release 8, using the available default options, with the exception for the mixed effects models, where the maximum-likelihood method was used .

Furthermore there were 18 values for which nondetection was recorded . For these, when it was assumed that there was 1 cell so that the log₁₀ value would equal 0, using equation 1, the average predicted log₁₀ value was 0.37 and only 3 of the 18 data values had positive residuals . The measurements at these levels are relatively inaccurate, and the pooled RMSE decreased slightly when not including them . The differences in the models and predictions discussed below between including these 18 values and assuming a log₁₀ value of 0, and deleting these 18 values are small . For example, the model presented in this paper (deleting the 18 results) predicts that, at 71.1°C and with salt, SPP, and NaL = 0%, the time needed to obtain a 6.5 lethality is 0.60 min with an error CV of 18.3%, while when the 18 data points are included, the estimated time is 0.54 min with CV of 19.6%; thus, the difference is about 10% lower when including the points . For 60°C for the same circumstances, there is a 5% difference: without the 18 data points, the estimated time is 20.1 min with a CV of 10.8%; with the data points, the estimate is 19.1 min with a CV of 11.5% . Insofar as the low levels associated with these 18 samples are not measured accurately; including them increases the standard error of predictions; and the model structure and basic conclusions of this paper are not affected whether or not they are included, it was decided not to include these data . With these points deleted, an examination of the residuals of the regressions using equation 1 revealed that for smallest positive times (3 s), the predicted model underestimated, on the average, the observed lethalities . The possibility exists that these values could be affected by the temperature come-up times more substantially than other values, though it is considered that the come-up time is negligible . Consequentially, data for times equal 3 seconds were deleted .

Figures 1 and 2 contains plots of the observed data and the fitted curves for 60 and 71.1°C, respectively . The headings include the order number of the design point, followed by the concentrations of salt, SPP, and NaL . For each design point, there were two replicate experiments; in the figures, the data points labeled by the same symbol are from the same experiment . These graphs show the fit of equation 1 to the observed data; similar patterns exist for the other temperatures . Figures showing the observed data and fitted curves derived from the omnibus model are given later . For the 110 fitted survival curves using equation 1, the pooled RMSE was 0.480 log₁₀, and the average R-square values was 0.971; however, there was a fitted survival curve that had an exceptionally low R-square value of 0.75 and which had only five measured values where the difference between the lowest and highest values was 2.64 log₁₀ . In the appendix is a table that gives the estimated parameter values of a and b, and the estimated times to obtain a 6.5 lethality for each survival curve .

 
Stepwise regressions of the estimated parameters, a and b, and the estimated natural logarithm of the times needed to obtain a 6.5 log₁₀ lethality, ln(t_6.5), obtained from equation 4, were performed, where the independent variables consisted of all possible terms of a quadratic polynomial in temperature, ln(temperature) salt, SPP, and NaL . Two observations were found to have large studentized residuals (greater than 3 in absolute value) for predicting ln(t_6.5) . These two observations and the one with 0.75 R-squared value are identified on Figures 3 to 5, which provide plots of ln(t_6.5) versus levels of salt, SPP and NaL, respectively, with linear regression lines, by temperature . For each of these points, it can be seen that, relative to other points with the same x-axis value, the value of ln(t_6.5) is "separated" from the other values of ln(t_6.5) in at least one of the figures . For example, in Fig. 3, for salt, two points are identified with a dark squares representing observations at 65°C, but corresponding predicted values are quite apart from the region where the other predicted values are for that temperature and within the region of the predicted value for 60°C . As a result of this analysis, these three points, representing 3 survival curves and the data associated with them, were deleted, leaving data from 107 survival curves in the analysis .

 
An examination of the influence statistics of the stepwise regressions revealed one data point that was highly influential for predicting values of the parameter a, which can be seen on Fig. 5, at 55°C and an NaL concentration of 4.5% at about a value of ln(t_6.5) of 8 . The high degree of influence for this data point is "caused" by the large difference between its value of ln(t_6.5) and the value for its replicate data point, and by the location of the data point at the most extreme boundary of the region or range of values of the independent variables . Because of the high relative degree of influence, there is a strong argument to delete this data point, particularly so if its inclusion would actually change any general conclusion . However, this did not happen; the effect of including this point was to increase by slight amounts the RMSE and the standard error of prediction for the omnibus model . The standard deviation of the replicate values of ln(t_6.5), while among the largest of the replicate standard deviations, was not the largest, so consequently, it was decided to leave the point in the analysis .

Using the results from the 107 survival curves in the stepwise regression, for ln(t_6.5), the first variable selected was ln(temperature), followed by salt, the interaction of salt and ln(temperature), and the square of SPP . For the parameter a, the first variable to enter was ln(temperature), followed by the square of salt, the square of temperature, and the interaction of SPP and NaL, represented as the product of SPP and NaL. For the variable b, the only variable was the square of salt .

The fact that a function of salt entered the stepwise regression for b and a function of temperature did not needs further investigation . Figure 6 is scatter plot of the estimated values of b versus salt levels, with quadratic regression lines for each temperature . All but one value of b are greater than 1, the exception for 65°C . It is not clear that the values of b are not dependent on temperature; on the average, the highest values of b are for 55°C, with an average of 5.6; followed by 71.1°C, with an average of 5.0; then by 60°C, with an average of 4.9; and then 65°C, with an average of 3.8 . The analysis of variance indicated a temperature effect, and when ln(b) is the dependent variable, ln(temperature) entered the stepwise regression first, followed by the square of temperature, interaction of salt and temperature, temperature, salt, and last the square of salt, which had a significance levels of 0.08 . Consequently, for the omnibus model, it is not to be assumed that b, the coefficient of ln(time) in equation 1, is not dependent upon temperature . Furthermore, the figure shows an inconsistent dependency of the value of b on the salt level, where, for the three highest temperatures, the values of b are on the average increasing with salt level, with a convex shaped quadratic curve; however, for 55°C, the relationship is reversed (the quadratic curve is concave, where the maximum value is between 2 and 2.5% salt) . However, this type of interaction: a concave relationship for one temperature and convex for the others, was not expected, and would, if representing a true relationship, imply a more complex model than anticipated . Rather, it was assumed that this pattern was a result of experimental variation .

 
Table 2 presents the means and standard deviations of the estimated values of ln(t_6.5) from the derived 107 regressions using equation 1 for the 45 design combinations of this study . Included in this table are the mean of the estimated ln(t_6.5), the standard deviation of these among the replicate experiments for each design combination, and the pooled standard error of these estimates due to regression, obtained by taking the square root of the sum of the weighted variances for the individual estimates ln(t_6.5), with weight equal to the degrees of freedom of the regression . The correlations between these standard deviations and errors with the means of the ln(t_6.5) were not significantly different from zero; thus, pooling variances over the row entries of the table provides a rough summary of the goodness-of-the fit of equation 1 for individual survival curves . The last row of the table thus includes the pooled standard deviations, weighted by the number observations minus 1 . The entry 0.641 in the last row of Table 2 in the column headed by between experiment standard deviation of ln(t_6.5) can be used to compute, roughly, a probability range of estimated times from single experiments, by adding and subtracting an appropriate quantile of the standard normal distribution times 0.641 to the estimated ln(t_6.5) and taking the exponential of the resultants . Thus, a 99% percent probability range would be almost factor of 30; for example, the range associated with an estimated time of 10 min would be 1.9 to 52 min . A good portion of this range is due to the error of the regression: the range for an estimated 10 min due to regression would 3.6 to 28 m, obtained using 0.401, the entry of the last row in the last column .

The between-experiment variance can be thought of as a sum of variance components: between replicate, within design combination (n = 55), and between repeated-design combinations . The between-repeated-design combination variance components is based on five design combinations for which replicates were repeated (from Table 2, top to bottom): three, two, three, five, and two times . The within-design combination variance component depends on 52 replicates, since three results were deleted. The analysis of variance on ln(t_6.5) indicated a negative between repeated-design combination variance component; however, the highest five replicates accounted for 63% of the sum of the variances suggesting that the underlying distribution of results is not normal (P = 0.10, based on 10,000 simulations) . A similar analysis was performed for t_3.0, the estimated time needed to achieve a 3.0 lethality . Here, the intra-repeated-design combination correlation was 86% indicating, relatively, a very high variance between repeated-design combinations . Consequently for the models, it is assumed that there is a nonzero between repeated-design variance component .

The above results and a close examination of Table 2 reveals that the NaL does not have consistent relationship with t_6.5 (note particularly the results for 55°C, rows 6 and 8) . This can be seen from Table 3, where, assuming that the other three variables are constant, the number of times that the geometric mean of t_6.5, for a larger value of the 4th independent variable, is greater than that for a smaller value of the same independent variable is given for each temperature . For each temperature other than 65°C, there are 4 such comparisons for each variable; and for 65°C, there were 10 such comparisons . The same statistic is given for t_3.0, the estimated time needed to achieve a 3.0 lethality . As is evident from this table, with the exception of 65°C, for both t_3.0 and t_6.5 the percentage of times that the above increasing relationship holds for NaL is near 50%, whereas for salt and to a lesser extent SPP, the percentages are larger, with the notable exception for t_6.5 for salt at 60 and 65°C . The results indicate that while a relationship seems to exist for lethalities with salt for low lethalities, and SPP, there does not appear to be as strong or as clear relationship of lethalities and NaL .

 
However, in the stepwise regressions, the interaction term of SPP and NaL was selected . The examination of the influence statistics that are part of the output did not reveal any particular observation that would be having an inordinate amount of influence on this interaction term . Furthermore, from a scatter plot (Fig. 7) of the estimated values of a versus the product of SPP and NaL, with linear regression lines by temperature, it is seen that three of the four linear regression lines—for temperatures 55 to 65°C—are decreasing and nearly mutually parallel, while the fourth line at 71.1°C is nearly horizontal . For the variable b there is a similar pattern . Thus, it is seen why the interaction term enters the regression, possibly reflecting some sort of synergistic effect of the two compounds on the lethality .

 
Secondary models.
For the mixed linear effects model for fitting the logit transformation, f(x) = ln(10^-x - 1), where x is the log₁₀ of the relative reduction, a nested error structure: temperature, combination within temperature, and replicate within experimental combination, was considered . Observations at time zero were deleted . Observations from the 107 experiments, excluding the 3 experiments identified above, were included in this analysis . Models were considered by adding or deleting parameters and were evaluated by considering the -2 ln(likelihood ratio), as described under "Statistical methods" section . The error structure assumed was nested, with design combination (n = 55), replicate (2) within design combination, considered as random factors . All variances and correlations were significantly different from zero, and when any of these parameters were eliminated from the model the likelihood ratio statistic was significant, at better than the 0.05 level . Three models were found to provide basically the same likelihood: (i) the model given in Table 4 that presents the estimated fixed parameter values of the model; (ii) the same model except the interaction of salt and ln(temperature) is used instead of salt [for this model, the value of -2 ln(likelihood function) is only slightly larger, by 0.05, than that of the model of Table 4, and the average standard error of prediction increases only slightly, from 0.4232 to 0.4236]; and (iii) instead of salt and the interaction of SPP and NaL, the terms SPP and NaL are used [for this model, the value of -2 ln(likelihood function) is slightly less, by about 1, than that of the model of Table 4, and the average standard error of prediction decreases slightly, from 0.4232 to 0.4223] . The model of Table 4 is chosen for further analysis insofar as it does include the salt variable, whereas the third model does not, and choosing this model over the second model was just a matter of choice, based on the stepwise regression for the variable a where salt rather than interaction of salt with temperature was selected . Furthermore, adding other terms, such as salt, SPP, or the square of SPP did not decrease the likelihood function significantly . Only adding a term involving the square of salt and the interaction of the square of salt and temperature for b did the model goodness-of-fit criterion improve significantly (P = 0.03), even though the average of the standard errors of predictions increased to 0.4635 . However, as discussed above in connection with Fig . 6, the interaction term was not considered; and when deleting that term, the significance of the salt squared term vanished . Models including the parameter c of equation 2, suggesting that there would be a asymptotic nonzero D value, were also considered . These models also met our 0.05 significance level criterion (e.g., 0.01) when compared to the model of Table 4, but provided, on the average, higher standard errors of predictions, and, more important, negative estimates of c for some conditions, contrary to the restriction that c > 0 . Consequently the model of Table 4 is chosen for further analysis, though it being selected does not imply that terms not in the model do not have an effect on the lethality—in particular, that there is not an interaction between salt and temperature or there is not, asymptotically, a nonzero D value .

 
For the model of Table 4, the standard errors of the predicted log₁₀ of the relative reduction range up to 1 log₁₀, the higher values for the higher and lower temperatures of 55 and 71.1°C . Figure 8 is the plot of the residuals of the predicted log₁₀ of the relative reduction versus the predicted values . As is seen, there is a trend in the residuals; this type of trend existed for all the models discussed above, and is a result of the incompleteness of the design, the transformation, and the existence of the variance components . When examining the residuals of the predicted logit, treating the random effect parameters as fixed, the correlation is 0.08(P = 0.04) . For the model of Table 4, the estimated mean values of b, with standard errors, are: at 55°C, 5.08 (0.43); at 60°C, 3.79 (0.28); at 65°C, 3.53 (0.26); and at 71.1°C, 4.33 (0.45) . From the mixed effects model, the between-experiment standard deviation of b is estimated to be approximately 1.41, so that, for example, at 65°C, the 95% probability interval is estimated to be 3.53 ± 2.82 = (0.71, 6.35), ignoring the uncertainty of the estimated values . Thus, while it is expected that survival curves will have shoulders, the model predicts that some experiments will not have shoulders, a consequence of experimental variation .

 
Figure 9 presents the fitted survival curves for the 45 distinct design combinations studied, together with the observed log₁₀ relative reductions . The x axis represents a measure of time that is normalized by dividing the actual time by the estimated time to achieve a 6.5 lethality derived from the omnibus model . The graphs also include curves representing estimated 90% upper and lower probability bounds for the obtain lethalities depicting the expected probability range of survival curves for single experiments, derived using the estimated variance components of the mixed effect model . With a few exceptions, the curves derived from the omnibus model provide a reasonable fit or coverage of the observed data points .

 
As mentioned in the previous sections, of particular importance is the prediction of the times, t_6.5, needed for a 6.5 log₁₀ relative reduction, using equation 4 . The estimated times for obtaining a 6.5 lethality derived from the omnibus model compare reasonably well (with a few exceptions) with those obtained directly from the individual nonlinear regressions of equation 1 given in Table 2 . Figure 10 is a plot of the difference between the mean estimated natural logarithm of the times, reported in Table 2, and the corresponding estimate derived from the omnibus regression, versus the average of the two estimates . However, the standard errors of predictions from the omnibus model are large .

 
Thus, instead of using the omnibus model for estimating the times needed to obtain a 6.5 lethality, a regression with dependent variable ln(t_6.5) obtained from the 107 individual regressions of equation 1 can be used directly . With few exceptions, the replicate standard deviations are small compared to the magnitude of the residuals; thus, a simple linear regression was used with the maximum likelihood estimation method, where the dependent variable was the mean of the estimated ln(t_6.5) for each design combination (55 in total) . The selected model is given in Table 5 . Adding variables did not significantly improve the model, when using the chi-square approximation to L = -2 ln(likelihood ratio) statistic as a criterion . Using SPP instead of the square of SPP increased L by 0.5; adding the square of ln(temperature) decreased L by 1.4; adding a linear term for SPP decreased L by 0.1; adding NaL and the interaction of SPP and NaL decreased L by 1.7; using the interaction of SPP and ln(temperature) instead of the interaction of salt and ln(temperature) increased L by 14.7 . The linear regression line of the residuals versus the predicted values of ln(t_6.5) is virtually flat (Fig. 11) . The standard errors of the predicted ln(t_6.5) for the data range from about 0.065 to 0.20, however, for product at 71.1°C, a salt concentration of 4.5%, and an SPP concentration of 0% the standard error is 0.25 . The CV of the estimated times can be approximated as 100% times the standard error of the estimated ln(times) . An estimated CV of 20% implies that a 99% confidence interval of the estimate of the expected times needed to obtain a 6.5 lethality ranges by a factor of about 3 (based on 50 df); thus, for example, an estimated time of 10 min would have 99% confidence interval of 5.8 to 17.1 m; an estimated CV of 30% implies that a 99% confidence interval ranges by a factor of about 5; thus, an estimated time of 10 min would have 99% confidence interval of 4.5 to 22 min . Table 6 gives the estimated times to obtain a 6.5 lethality obtained from the individual nonlinear regressions using equation 1 (Table 2); the linear mixed effects regression using these estimated times (actually the natural log of them); and the omnibus model, given in Table 4 . Included are the estimated CV's obtained from the latter two regressions, estimated by multiplying the standard error of the estimated ln(time) .

 
The predicted effect of SPP on the predicted times based on model using the individual estimates, ln(t_6.5), is clear: the higher the level of SPP, the more time it takes to achieve a 6.5 log₁₀ relative reduction . The interaction of temperature and salt is statistically significant (P = 0.0001). Example predictions: for a temperature = 60°C, salt = 0%, SPP = 0%, the predicted time is 20.2 m, with a CV of 10.8%; for 60°C, 4.5% salt, and 0% SPP, the predicted time is 48.2 min with CV of 15.2%; for 71.1°C (160°F), 0% salt, and 0% SPP, the predicted time is 0.60 m, with CV of 18.3%; for 71.1°C, 4.5% salt, and 0% SPP, the predicted time is 0.36 m, with CV of 24.7%. Increasing salt from 0 to 4.5% at 60°C increases the time needed to obtain a 6.5 log₁₀ relative reduction by a 2.4 factor (CV = 18.6%), significant at <0.0001; at 63.5% the factor is about 1.5, significant at the 0.06 level, but at 71.1°C the difference in the times is not significant (P = 0.18) .

However, for lower lethalities, the salt effect seems to be more pronounced . A similar analysis as above was performed for the estimated time to obtain a 3 lethality, t_3.0 . The "best" model included ln(temperature), the square of ln(temperature), salt (coefficient of 0.369, P = 0.002), and SPP (coefficient of 1.90; P = 0.05) . Interaction terms or terms involving NaL did not improve the fit of the model with the data; however, as mentioned above, this does not mean that such terms are not important. The point of mentioning this analysis is to support the results from Table 3 that salt has an effect for low lethalities, whereas the model of Table 5 for t_6.5 suggests that, at higher temperature, the salt effect seems to be less pronounced for larger lethalities, or, depending upon temperature, possibly reversed, though, as seen above for the result at the end of the last paragraph for 71.1°C, the estimated effect of decreasing the heat resistance was not statistically significant .

There have been many studies showing that high concentrations of salt increase heat resistance of Salmonella (5) . For example, results reported by Mañus et al. (11), seem to agree, in part, with our finding that increasing levels of salt increases the heat resistance of Salmonella at lower temperatures . In the Mañus et al . (11) paper, the effects of salt concentrations on the heat resistance of Salmonella enterica serovar Typhimurium was studied in broth at 58°C . Results given in that paper (in terms of D values) indicated that increasing levels from 0 to 4.5% would double the D value, implying that the time needed to obtain a fixed lethality would double . For the model given in Table 5, to obtain a 6.5-log₁₀ reduction, at 58°C when increasing the salt from 0 to 4.5% and assuming that the concentration of SPP is 0%, the estimated time needed increased 3.1-fold, with CV of 19.8%, so that a lower 95% confidence limit is 2.3-fold, which is slightly higher than the estimated factor of 2 reported in the work of Mañus et al. (11) .

Blackburn et al . (3) also reported that higher levels of salt increased heat resistance, but noted that beyond 3.5% the heat resistance stayed about the same . The highest value for our study was 4.5% so that the linear effect for a given temperature that is used in the model developed within this paper is not inconsistent with the findings in Blackburn et al. (3) . A predicted D value for Salmonella serovar Enteritidis of 0.9 min at 60°C for beef with a salt level of 0.23% (wt/wt, aqueous phase) was reported (3) (Table 6) . Using this D value, to obtain a 6.5-log₁₀ relative reduction of Salmonella, the estimated time needed would be 5.85 min. However, in that same paper, a graph is presented that shows that the estimated times needed to obtain a 5-log₁₀ relative reduction from the nonlinear model that was used in that paper is about twice as large as those estimated using the predicted D values . We should point out that the model used in the Blackburn paper had the property that the probability of viable cells did not approach 0 as t approaches infinity . It seems reasonable to assume that the Blackburn, et al . (3) model reflected the behavior of their observed survival curves for relatively large times, thereby possibly being relatively "flat" with extensive tailing . Thus, it is quite possible that, for a 6.5 log₁₀ relative reduction, the ratio of the of the predicted times obtain from the nonlinear curves versus the estimated times obtained from the D values would be substantially larger than the estimated value of 2 for the ratio for obtaining a 5 lethality; the ratio for a 6.5 lethality could be more than 3 . If the ratio were 3, then the estimated time needed to obtain a 6.5 log₁₀ would be 17.6 min . From the model given in Table 5 (assuming SPP = 0%, salt = 0.23%, and temperature = 60°C), the estimated time is 21.1 m, with an error CV of 10.3%, and thus a lower 95% confidence bound of the needed time is about 17.8 min . Thus, the different estimates may not be statistically different .

Conclusion.
The results of data analysis indicated that salt and sodium pyrophosphate (SPP) significantly affect the heat resistance of Salmonella spp . Increasing the level of SPP increases the heat resistance . Increasing the salt levels increases the heat resistance for lower temperatures (<63.5°C), but for higher temperatures and large lethalities, salt levels did not significantly affect the heat resistance . NaL did not seem to affect the heat resistance of Salmonella as much as the effects induced by the other variables studied .

The survival curves were convex . An omnibus model, assuming nonlinear survival curves, was developed for predicting the obtained lethality when cooking beef for a fixed amount of time at a fixed temperature between 55°C (131°F) and 71.1°C (160°F), where the beef matrices contain concentrations of salt between 0 and 4.5%; SPP between 0 and 0.5%; and NaL between 0 and 4.5% . The selected model included terms involving salt and the interaction of SPP and NaL . While the former term is not surprising, the latter term, by itself, without the presence of terms for SPP and NaL, presents difficulties in explaining the model: is there is a synergistic effect of the two compounds on lethality, at least for relatively small times, or is the statistical significance of this interaction term alone, without terms for SPP or NaL, just a fluke and that there really are SPP and NaL effects that were masked due to variability which could have only been detected in this study when the two compounds were both present . Further research is needed to clarify this .

For the omnibus model, however, the standard errors of prediction are large . Since there is special interest of the times needed to obtain a 6.5 lethality, a model was developed for predicting the times needed to obtained a lethality of 6.5 log₁₀, using directly the estimated times to achieve a 6.5 lethality obtained from regressions of the individual survival curves . For the latter model, the CV of predicted times range from about 7 to 30% . The results indicate that at 71.1°C, the times needed to obtain a 6.5-log₁₀ lethality could exceed 0.5 min .

The derived estimated times needed to obtain a 6.5-log₁₀ lethality seem to be higher than predictions derived from reported D values in the published literature . A contributing reason for this could be due to the nonlinear survival curves . Predictions based solely on D values from the "linear" portion of the survival curves could be biased because of tailing of the survival curves for large times and because of the lag times—shoulders of the survival curves for small times before the linear kinetic inactivation begins . However, many researchers have reported nonlinear survival curves for Salmonella, and thus predicted times for obtaining specified lethalities need to be based on models of these types of curves and not on D values .

In addition, the standard errors or CVs of the estimated lethalities for a given time or predicted times needed to obtain a given lethality seemed rather large, in some cases, exceeding 20%, giving rise to rather large confidence intervals of estimated values . For example, based on the 50 df of the model of Table 5, an estimate of an expected 10 m, with a CV error of 20%, implies a 99% confidence interval covering 5.8 to 17.1 min . The confidence intervals do not include the variability that may arise from slight misspecifications of conditions, so that in actuality, to assure that processes would be meeting lethality objectives, larger upper bounds might be needed . While no standards of predictions have been established by professional organization, we suggest that, for omnibus models that need to satisfy multiple needs, the CV's of estimates of the expected values of times to achieve specific lethalities should not be much larger than 10%, so that confidence intervals would not be "too" wide . Consequently, for these types of studies more observations are needed, perhaps, more than two or three times as many as in this study . More research is needed to clarify the conditions that create nonlinear curves and to develop models for them .

 
Shown in Table A1 are estimated parameters from individual regressions (equation 1) .

 
We thank John Phillips of the Agricultural Research Service, who developed the factorial design used in this study, read the manuscript, and made helpful comments, and Mark Cutrufelli of the Food Safety and Inspection Service, who consulted with us and provided helpful advice during the design phase of the study . We also thank anonymous reviewers for many helpful comments which led, we think, to substantial improvements in the paper .

Mention of a brand or firm name does not constitute an endorsement by the U.S . Department of Agriculture over others of a similar nature not mentioned .

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