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

Analyses of survival data of a mixture of Salmonella spp. atfixed temperatures between 55°C (131°F) and 71.1°C (160°F)in ground beef matrices containing concentrations of salt between0 and 4.5%, concentrations of sodium pyrophosphate (SPP) between0 and 0.5%, and concentrations of sodium lactate (NaL) between0 and 4.5% indicated that heat resistance of Salmonella increaseswith increasing levels of SPP and salt, except that, for salt,for larger lethalities close to 6.5, the effect of salt wasevident only at low temperatures (<64°C). NaL did notseem to affect the heat resistance of Salmonella as much asthe effects induced by the other variables studied. An omnibusmodel for predicting the lethality for given times and temperaturesfor ground beef matrices within the range studied was developedthat reflects the convex survival curves that were observed. However,the standard errors of the predicted lethalities from this modelsare large, so consequently, a model, specific for predictingthe times needed to obtained a lethality of 6.5 log₁₀, was developed,using estimated results of times derived from the individual survivalcurves. For the latter model, the coefficient of variation (CV)of predicted times range from about 6 to 25%. For example, at60°C, when increasing the concentration of salt from 0 to 4.5%,and assuming that the concentration of SPP is 0%, the time toreach a 6.5-log₁₀ relative reduction is predicted to increasefrom 20 min (CV = 11%) to 48 min (CV = 15%), a 2.4 factor (CV= 19%). At 71.1°C (160°F) the model predicts that morethan 0.5 min is needed to achieve a 6.5-log₁₀ relative reduction.

INTRODUCTION

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Introduction

Salmonella enterica has long been recognized as an importantfood-borne pathogen, and it continues to be a leading cause offood-borne disease outbreaks associated with consumption ofmeat and poultry. The annual incidence of salmonellosis in theUS is 1.4 million cases, causing as many as 550 deaths (12),and the incidence of salmonellosis has increased during thepast 30 years. This reality stems from the ubiquitous occurrenceof Salmonella in the environment, its prominence in varioussectors of the agriculture industry, and the escalating movementof food and food ingredients worldwide. Important contributingfactors which lead to outbreaks of food-borne illness, includingsalmonellosis, are inadequate time and/or temperature exposureduring initial thermal processing (or cooking) and inadequatereheating to kill pathogens in retail food service establishmentsor homes (2, 15). Inadequate cooking was cited as a contributingfactor in 67% of the outbreaks in which Salmonella was an etiologicalagent (2). These outbreaks have implicated a variety of foods,including meat and poultry, milk, ice cream, cheese, eggs andegg products, chocolate, and spices, as vehicles of transmission (4).Current performance standards require that thermal processingschedules must achieve a 6.5-log₁₀ reduction of Salmonella forcooked beef, ready-to-eat roast beef, and cooked corned beefproducts (17).

An effective thermal process is necessary to control the potentialhazard of Salmonella in cooked meat products. A key to optimizationof the heating step is defining the target pathogen's heat resistance.While overestimating the heat resistance negatively impacts onproduct quality, underestimating increases the likelihood thatthe contaminating pathogen persists after heat treatment orcooking. Accordingly, teams of investigators have conductedthermal inactivation studies of different Salmonella serotypesin aqueous media and foods (4). Various factors affecting theheat resistance have been documented, including growth temperature,stage of growth, initial population, bacterial strains, compositionand pH of the heating menstruum, heat shock, and methodologyused for detection of survivors (16). In a study by Juneja etal. (7), when the heat resistance of Salmonella serotypes wasquantified 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 characterizationof the inactivation kinetics, there is a lack of informationon the effects of increasing concentrations of sodium pyrophosphate(SPP) and sodium lactate (NaL) in combination with various saltlevels on the heat resistance of the organism. Accordingly,the present study was carried out to quantitatively assess therelative effects and interactions of SPP, NaL, NaCl, and temperatureon the inactivation kinetics of Salmonella serotypes.

MATERIALS AND METHODS

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Materials and METHODS

Organisms.
A cocktail consisting of eight strains of different serotypesof Salmonella representing isolates from beef, pork, chicken,or turkey or from human clinical cases was used in this study.The information about these strains is given in Table 1. Thesestrains were preserved by freezing the cultures at -70°Cin vials containing tryptic soy broth (Difco laboratories, Inc.,Detroit, Mich.) supplemented with 10% (vol/vol) glycerol (SigmaChemical Co., St. Louis, Mo.).

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TABLE 1. Salmonella serotype cocktail sources

Products.
Raw 75% lean ground beef, used as the heating menstruum, wasobtained from a retail supermarket. The meat was separated intobatches for different treatments and mixed thoroughly with theadditives to be tested, i.e., each batch received various variablecombinations 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 meatstested were determined using a combination electrode (Sensorex,semimicro; A.H. Thomas, Philadelphia, Pa.) attached to a pHmeter (model 310; Orion, Boston, Mass.). The meat was placedinto 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, andirradiated (42 kGy) to eliminate indigenous microflora. Randomsamples were tested to verify elimination or inactivation ofmicroflora by diluting in 0.1% (wt/vol) peptone water (PW) toobtain1:1 meat slurry, followed by direct surface plating thesuspension (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 roomtemperature and 1.0 ml of the thawed culture was transferredto 10 ml of brain heart infusion (BHI) (Difco) broth in 50-mltubes and incubated for 24 h at 37°C. This culture was notused in heating tests, due to the presence of freeze-damagedcells. The inocula for use in heating tests were prepared bytransferring 0.1 ml of each culture to tubes of BHI broth (10ml) and incubating aerobically for 24 h at 37°C. These cultureswere then maintained in BHI for 2 weeks at 4°C. A new seriesof cultures was initiated from the frozen stock on a biweeklybasis.

A day before the experiment, the inocula for conducting theheating studies were prepared by transferring 0.1 ml of eachculture to 50 ml of BHI in 250-ml flasks, and incubating aerobicallyfor 18 h at 37°C to provide late-stationary-phase cells.On the day of the experiment, each culture was centrifuged (5,000x 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 log₁₀CFU/ml. The population densities in each cell suspension wereenumerated by spiral plating (model D; Spiral Biotech, Bethesda,Md.) appropriate dilutions (in 0.1% PW), in duplicate, ontoTSA plates. Approximately equal volumes of each culture were combinedin a sterile conical vial to obtain an eight-strain mixtureof Salmonella (8 log₁₀ CFU/ml) prior to inoculation of meat.

Experimental design.
A fractional factorial design was used to assess the effectsand 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 werestudied. Table 2 gives the 45 design points tested along withsome other information as explained below. For each experimentalcombination at least two replicates were obtained, and in totalthere were 110 survivor curves, two per experimental combination,for a total of 55 combinations, some of these the same, to give45 distinct combinations.

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TABLE 2. Estimated natural logarithm of time needed to obtain 6.5 lethality^a

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 wasblended with a Seward laboratory stomacher 400 for 5 min toensure even distribution of the organisms in the meat sample.Duplicate 3-g ground-meat samples were then weighed asepticallyinto sterile filtered stomacher bags (30 by 19 cm; Spiral Biotech).Negative controls consisted of bags containing meat samplesinoculated with 0.1 ml of 0.1% (wt/vol) PW with no bacterialcells. Thereafter, the bags were compressed into a thin layer(approximately 0.5 to 1 mm thick) by pressing them against aflat surface, excluding most of the air, and then were heatsealed using a Multivac (model A300/16; Multivac Inc., KansasCity, Mo.) packaging machine.

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

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

Statistical methods.
(i) Primary model.
Graphical examination of the observed survival curves revealedthat almost all the curves had a convex shape. Some of the curvesalso displayed "shoulders," suggesting a possible lag effect.The dependent variable used in the regressions is the observed log₁₀of N(t)/N(0), where N(t) is the number of cells at time t. The negativeof this quantity is referred to as the lethality at time t.The following equation,

(1)
where Eis the expected value operator and a and b (>0) are constants,has been used for fitting survival curves with the above describedproperties by various researchers (1, 10). This function providesthe flexibility to fit a variety of survival curves that have asymptoticconvex behavior. As t approaches infinity the derivative of theright side of equation 1 approaches 0. To allow for the possibilitythat, asymptotically, the survival curves approach a straightline with nonzero slope, we considered a model that involved addinganother term to the exponent:

(2)
where cis $>=$ 0. The asymptotic D value for survival curveof equation 2 thus is ln(10)/c. The derivative of the rightside of equation 2 approaches -e^abt^b ^-
1, as t approaches 0from the right, so that, if b is >1, then the slope at zerois zero, and if b is <1 then the limiting slope is minusinfinity. When b is >1, the survival curve has a "shoulder"and the point (time) of inflection (where the curve becomesconvex) is [(b - 1)/e^a]^1/b. Thus, for a given value of b of>1 (and c), smaller values of a provide curves with morepronounced shoulders and larger points of inflections.

(ii) Secondary model.
An omnibus model for predicting survival curves for any specifiedvalues of temperature, salt, SPP, and NaL, was determined byconsidering the parameters that are identified in equations 1or 2 to be at most quadratic polynomials of the independentvariables described in the Results section. Using higher orderpolynomials might result in a response surface with more thanone local maximum or minimum, which would be contrary to oura priori expectations, and, given the number of design points,a result contrary to this expectation probably could not be supportedand thus would not be believed but rather assumed to be a consequenceof experimental error. The desire is to determine a model thatincludes only statistically significant terms since including insignificantterms increases the standard error of predictions possibly withoutany corresponding reduction of bias (an example of Occam's maxim).Thus, the selection of terms in a model does not preclude othervariables that are not included in the model from being importantfor predicting lethality. Initially, stepwise regressions wereused to identify statistically significant variables from a quadraticresponse surface for inclusion in the model. The natural logarithmof the temperature was included among the variables consideredin the regression. Influential observations were determined byexamining studentized residuals (computed excluding the observation)and Cook's D statistic.

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

(3)
where x = log₁₀(rt),is linear in the unknown parameters, a, b, and c, so that linearmixed effects model can be used for estimating the model parametersas linear functions of the variables studied. Using linear mixedeffects models accounts, in a simple way, for the correlations thatexist among the observations. Using the variables identifiedfrom the stepwise regressions, a mixed effects model was fit (14).Nested error structures were assumed, where experimental condition,s, and replicate within experimental condition, e(s), were consideredas random effects. That is to say, if f(x) is assumed to be alinear combination of a + bln(t), then, for example, it canbe assumed that a is actually a random variable that can beexpressed as: a = µ + ${varepsilon}$ _s + ${varepsilon}$ _e(s) + ${varepsilon}$ _r, where µ is theexpected value of a, ${varepsilon}$ _s is the error associated with factor s; ${varepsilon}$ _e(s) is the error associated with the factor e nested within s; ${varepsilon}$ _r is a residual error (nested within s and e); and the errorterms are independent, have zero expected values and specifiedvariances. The same type of assumption is made for b, so that,in addition to the variances, there are possible nonzero covariancesbetween corresponding errors at the same structural level associatedwith a and b. The expected value of a and b, themselves, areassumed to be linear combinations of the independent variableswith unknown coefficients that can be assumed to be random variables.When including all possible variances and covariances, such mixedeffect models can have an enormous number of parameters forwhich convergent solutions with estimable variances (nonsingularHessian matrix) sometimes are not readily attainable. Consequentlysimplifying assumptions are made in order to reduce the numberof parameters to "manageable" levels. In this case, it is assumedthat only the constant or intercept terms—ones that arenot coefficients of an independent variable of temperature,salt, SPP or NaL—are associated with random variablesin the sense described above. For details of using these models,the book by Pinheiro and Bates (14) can be consulted; the approachgiven in that book was followed here. In this study, designcombinations, and the replicates within these are consideredas factors. In considering whether to include terms in the model,likelihood ratio tests based on the statistic L = -2 ln(likelihoodratio), compared to the 95th percentile of a chi-square distribution(0.05 significance level) with appropriate degrees of freedom,was used. That is, evaluating whether the addition of q termsimproves the goodness-of-fit was made by comparing the differenceof the statistics, -2 ln(likelihood), that are given in thePROC MIXED output, with the 95th percentile of a chi-squaredistribution with q degrees of freedom. With each model considered,the plots of the residuals versus the predicted values were examined.Predictions of x = log₁₀ [r(t)], as a function of the selected independentvariables, were obtained by using the inverse of the functionof equation 3, and the standard errors of these predicted valueswere obtained using the linear approximation (first term ofthe Taylor series) of the inverse function, and the asymptoticcovariance matrix of the estimated values of the parameters.

Of particular importance is the times needed to obtain a 6.5-log₁₀relative reduction. The above model could be used for estimatingthese times, however, a more direct approach was used: for eachindividual experiment, using the estimated survival curve, anestimate of the time for a predicted 6.5-log₁₀ relative reduction,t_6.5, was derived, and the natural logarithm of this estimatedtime was used as the dependent variable in a mixed effects regressionanalysis, as described above. From equation 1, the predictedtime, t_6.5, to obtain a 6.5 lethality is obtained as follows:

(4)
where f is given in equation 3. (If equation 2were used, then direct numerical procedures would be neededto solve for t_6.5.)

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

RESULTS

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Results

Preliminary analysis.
Equation 2 was fit for each growth curve, with the restrictionthat b be >0 and c be $>=$ 0, where it was alsoassumed that N(0) was a parameter with an unknown value. Ofthe 98 estimated curves for which the estimate of b was >0,26 of them had a c of >0, and of these 6 had estimated valueof c significantly greater than zero at the one-sided 0.10 level andonly 2 at a significance better than 0.05. The pooled root mean squareerror (RMSE) for fitting equation 2 is 0.548 compared to 0.500for equation 1. Thus, it appears that, for individual survivalcurves, equation 2 does not generally provide a significantlybetter fit than does equation 1. Hence, for this analysis, equation1 is used.

Furthermore there were 18 values for which nondetection was recorded.For these, when it was assumed that there was 1 cell so that thelog₁₀ value would equal 0, using equation 1, the average predicted log₁₀value was 0.37 and only 3 of the 18 data values had positiveresiduals. 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 betweenincluding these 18 values and assuming a log₁₀ value of 0, anddeleting these 18 values are small. For example, the model presentedin this paper (deleting the 18 results) predicts that, at 71.1°Cand with salt, SPP, and NaL = 0%, the time needed to obtaina 6.5 lethality is 0.60 min with an error CV of 18.3%, whilewhen the 18 data points are included, the estimated time is0.54 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 estimatedtime is 20.1 min with a CV of 10.8%; with the data points, theestimate is 19.1 min with a CV of 11.5%. Insofar as the lowlevels 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 paperare not affected whether or not they are included, it was decidednot to include these data. With these points deleted, an examinationof the residuals of the regressions using equation 1 revealedthat for smallest positive times (3 s), the predicted modelunderestimated, on the average, the observed lethalities. Thepossibility exists that these values could be affected by thetemperature 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 thefitted curves for 60 and 71.1°C, respectively. The headingsinclude the order number of the design point, followed by theconcentrations of salt, SPP, and NaL. For each design point,there were two replicate experiments; in the figures, the datapoints labeled by the same symbol are from the same experiment.These graphs show the fit of equation 1 to the observed data; similarpatterns exist for the other temperatures. Figures showing the observeddata and fitted curves derived from the omnibus model are givenlater. For the 110 fitted survival curves using equation 1,the pooled RMSE was 0.480 log₁₀, and the average R-square valueswas 0.971; however, there was a fitted survival curve that hadan exceptionally low R-square value of 0.75 and which had only fivemeasured values where the difference between the lowest and highestvalues was 2.64 log₁₀. In the appendix is a table that givesthe estimated parameter values of a and b, and the estimatedtimes to obtain a 6.5 lethality for each survival curve.

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FIG. 1. Observed and fitted survival curves for the different combinations of values of salt, SPP, and NaL, for temperature = 60°C. The first number in the heading for the graphs is a design number designator; the following three values represent the salt, SPP, and NaL, values, respectively. For each design point there were two experiments; data points labeled with the same symbol are from the same experiment.

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FIG. 2. Observed and fitted survival curves for the different combinations of values of salt, SPP, and NaL, for temperature = 71.1°C. The first number in the heading for the graphs is a design number designator; the following three values represent the salt, SPP, and NaL, values, respectively. For each design point there were two experiments; data points labeled with the same symbol are from the same experiment.

Stepwise regressions of the estimated parameters, a and b, andthe estimated natural logarithm of the times needed to obtaina 6.5 log₁₀ lethality, ln(t_6.5), obtained from equation 4, wereperformed, where the independent variables consisted of allpossible terms of a quadratic polynomial in temperature, ln(temperature)salt, SPP, and NaL. Two observations were found to have largestudentized residuals (greater than 3 in absolute value) forpredicting ln(t_6.5). These two observations and the one with 0.75R-squared value are identified on Figures 3 to 5, which provideplots of ln(t_6.5) versus levels of salt, SPP and NaL, respectively,with linear regression lines, by temperature. For each of thesepoints, it can be seen that, relative to other points with thesame x-axis value, the value of ln(t_6.5) is "separated" fromthe other values of ln(t_6.5) in at least one of the figures.For example, in Fig. 3, for salt, two points are identifiedwith a dark squares representing observations at 65°C, butcorresponding predicted values are quite apart from the regionwhere the other predicted values are for that temperature andwithin the region of the predicted value for 60°C. As a resultof this analysis, these three points, representing 3 survival curvesand the data associated with them, were deleted, leaving data from107 survival curves in the analysis.

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FIG. 3. Scatter plot of estimated natural log of times (m) needed to obtain a 6.5 lethality, ln(t_6.5), from individual nonlinear regression (equation 1) versus levels of salt (%), with linear regression lines by temperature. Points excluded from analyses are indicated with asterisks superimposed on symbols.

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FIG. 5. Scatter plot of estimated natural log of times (minutes) needed to obtain a 6.5 lethality, ln(t_6.5), from individual nonlinear regression (equation 1) versus levels of NaL, with linear regression lines by temperature. Points excluded from analyses are indicated with asterisks superimposed on symbols.

An examination of the influence statistics of the stepwise regressionsrevealed one data point that was highly influential for predictingvalues of the parameter a, which can be seen on Fig. 5, at 55°Cand 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 valuefor its replicate data point, and by the location of the data pointat the most extreme boundary of the region or range of valuesof the independent variables. Because of the high relative degreeof influence, there is a strong argument to delete this datapoint, particularly so if its inclusion would actually changeany general conclusion. However, this did not happen; the effectof including this point was to increase by slight amounts theRMSE 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 leavethe point in the analysis.

Using the results from the 107 survival curves in the stepwiseregression, 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 variableto 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 variableb, the only variable was the square of salt.

The fact that a function of salt entered the stepwise regressionfor b and a function of temperature did not needs further investigation.Figure 6 is scatter plot of the estimated values of b versussalt levels, with quadratic regression lines for each temperature.All but one value of b are greater than 1, the exception for65°C. It is not clear that the values of b are not dependenton temperature; on the average, the highest values of b arefor 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 of4.9; and then 65°C, with an average of 3.8. The analysisof variance indicated a temperature effect, and when ln(b) isthe dependent variable, ln(temperature) entered the stepwiseregression first, followed by the square of temperature, interactionof salt and temperature, temperature, salt, and last the squareof salt, which had a significance levels of 0.08. Consequently,for the omnibus model, it is not to be assumed that b, the coefficientof ln(time) in equation 1, is not dependent upon temperature.Furthermore, the figure shows an inconsistent dependency ofthe value of b on the salt level, where, for the three highesttemperatures, the values of b are on the average increasingwith salt level, with a convex shaped quadratic curve; however,for 55°C, the relationship is reversed (the quadratic curveis concave, where the maximum value is between 2 and 2.5% salt).However, this type of interaction: a concave relationship forone temperature and convex for the others, was not expected,and would, if representing a true relationship, imply a morecomplex model than anticipated. Rather, it was assumed thatthis pattern was a result of experimental variation.

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FIG. 6. Scatter plot of estimated values of parameter b in equation 1 from individual nonlinear regressions versus levels of salt, with quadratic regression lines by temperature.

Table 2 presents the means and standard deviations of the estimatedvalues of ln(t_6.5) from the derived 107 regressions using equation1 for the 45 design combinations of this study. Included in thistable are the mean of the estimated ln(t_6.5), the standard deviationof 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 weightedvariances for the individual estimates ln(t_6.5), with weightequal to the degrees of freedom of the regression. The correlationsbetween these standard deviations and errors with the meansof the ln(t_6.5) were not significantly different from zero;thus, pooling variances over the row entries of the table providesa rough summary of the goodness-of-the fit of equation 1 forindividual survival curves. The last row of the table thus includesthe pooled standard deviations, weighted by the number observationsminus 1. The entry 0.641 in the last row of Table 2 in the columnheaded by between experiment standard deviation of ln(t_6.5) canbe used to compute, roughly, a probability range of estimatedtimes from single experiments, by adding and subtracting anappropriate quantile of the standard normal distribution times0.641 to the estimated ln(t_6.5) and taking the exponential ofthe resultants. Thus, a 99% percent probability range would bealmost factor of 30; for example, the range associated withan estimated time of 10 min would be 1.9 to 52 min. A good portionof this range is due to the error of the regression: the rangefor 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 lastcolumn.

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

The above results and a close examination of Table 2 revealsthat the NaL does not have consistent relationship with t_6.5 (noteparticularly the results for 55°C, rows 6 and 8). This canbe seen from Table 3, where, assuming that the other three variablesare constant, the number of times that the geometric mean oft_6.5, for a larger value of the 4th independent variable, isgreater than that for a smaller value of the same independentvariable is given for each temperature. For each temperatureother than 65°C, there are 4 such comparisons for each variable;and for 65°C, there were 10 such comparisons. The same statisticis 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 increasingrelationship holds for NaL is near 50%, whereas for salt andto a lesser extent SPP, the percentages are larger, with the notableexception for t_6.5 for salt at 60 and 65°C. The resultsindicate that while a relationship seems to exist for lethalitieswith salt for low lethalities, and SPP, there does not appearto be as strong or as clear relationship of lethalities andNaL.

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TABLE 3. Number of times there is an increasing relationship of the estimated times needed to achieve t_3.0 and t_6.5 for a given variable, holding three of the other variables constant

However, in the stepwise regressions, the interaction term ofSPP and NaL was selected. The examination of the influence statisticsthat are part of the output did not reveal any particular observationthat would be having an inordinate amount of influence on thisinteraction 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 thatthree of the four linear regression lines—for temperatures55 to 65°C—are decreasing and nearly mutually parallel,while the fourth line at 71.1°C is nearly horizontal. Forthe variable b there is a similar pattern. Thus, it is seenwhy the interaction term enters the regression, possibly reflectingsome sort of synergistic effect of the two compounds on the lethality.

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FIG. 7. Scatter plot of estimated values of parameter a in equation 1 from individual nonlinear regressions versus levels the product of SPP and NaL, with linear regression lines by temperature.

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 the107 experiments, excluding the 3 experiments identified above,were included in this analysis. Models were considered by addingor 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 designcombination (n = 55), replicate (2) within design combination,considered as random factors. All variances and correlationswere significantly different from zero, and when any of theseparameters were eliminated from the model the likelihood ratio statisticwas significant, at better than the 0.05 level. Three models werefound to provide basically the same likelihood: (i) the model givenin Table 4 that presents the estimated fixed parameter valuesof the model; (ii) the same model except the interaction ofsalt 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 averagestandard error of prediction increases only slightly, from 0.4232to 0.4236]; and (iii) instead of salt and the interaction ofSPP and NaL, the terms SPP and NaL are used [for this model, thevalue of -2 ln(likelihood function) is slightly less, by about1, than that of the model of Table 4, and the average standarderror of prediction decreases slightly, from 0.4232 to 0.4223].The model of Table 4 is chosen for further analysis insofaras it does include the salt variable, whereas the third modeldoes not, and choosing this model over the second model was justa matter of choice, based on the stepwise regression for the variablea where salt rather than interaction of salt with temperaturewas selected. Furthermore, adding other terms, such as salt,SPP, or the square of SPP did not decrease the likelihood functionsignificantly. Only adding a term involving the square of salt andthe interaction of the square of salt and temperature for bdid the model goodness-of-fit criterion improve significantly(P = 0.03), even though the average of the standard errors ofpredictions increased to 0.4635. However, as discussed abovein connection with Fig. 6, the interaction term was not considered;and when deleting that term, the significance of the salt squaredterm vanished. Models including the parameter c of equation2, suggesting that there would be a asymptotic nonzero D value,were also considered. These models also met our 0.05 significancelevel 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 themodel of Table 4 is chosen for further analysis, though it beingselected does not imply that terms not in the model do not havean effect on the lethality—in particular, that there isnot an interaction between salt and temperature or there isnot, asymptotically, a nonzero D value.

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TABLE 4. Estimates of parameter values used for predicting lethality

For the model of Table 4, the standard errors of the predictedlog₁₀ of the relative reduction range up to 1 log₁₀, the highervalues for the higher and lower temperatures of 55 and 71.1°C.Figure 8 is the plot of the residuals of the predicted log₁₀of the relative reduction versus the predicted values. As isseen, there is a trend in the residuals; this type of trendexisted for all the models discussed above, and is a resultof the incompleteness of the design, the transformation, andthe existence of the variance components. When examining theresiduals of the predicted logit, treating the random effectparameters as fixed, the correlation is 0.08(P = 0.04). Forthe model of Table 4, the estimated mean values of b, with standarderrors, 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). Fromthe mixed effects model, the between-experiment standard deviationof b is estimated to be approximately 1.41, so that, for example,at 65°C, the 95% probability interval is estimated to be3.53 ± 2.82 = (0.71, 6.35), ignoring the uncertaintyof the estimated values. Thus, while it is expected that survivalcurves will have shoulders, the model predicts that some experimentswill not have shoulders, a consequence of experimental variation.

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FIG. 8. Scatter plot of residuals versus predicted log₁₀ relative reductions obtained from omnibus model (Table 4) with linear regression line.

Figure 9 presents the fitted survival curves for the 45 distinctdesign combinations studied, together with the observed log₁₀relative reductions. The x axis represents a measure of timethat is normalized by dividing the actual time by the estimatedtime to achieve a 6.5 lethality derived from the omnibus model.The graphs also include curves representing estimated 90% upperand lower probability bounds for the obtain lethalities depictingthe expected probability range of survival curves for singleexperiments, derived using the estimated variance componentsof the mixed effect model. With a few exceptions, the curvesderived from the omnibus model provide a reasonable fit or coverageof the observed data points.

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FIG. 9. Observedand fitted survival curves derived from the omnibus model (Table 4) for the 45 combinations of values of temperature (in degrees Celsius) and salt, SPP, and NaL concentrations (percentages) studied (Table 2). The middle line represents the predicted survival curve, and the two outer lines represent 90% upper and lower probability bounds, depicting the expected probability range of survival curves for single experiments. The time axis has been normalized by dividing the actual time by the predicted time to obtain a 6.5 lethality derived from the omnibus model. Each panel (a to e) shows graphs for nine combinations.

As mentioned in the previous sections, of particular importanceis the prediction of the times, t_6.5, needed for a 6.5 log₁₀relative reduction, using equation 4. The estimated times forobtaining a 6.5 lethality derived from the omnibus model comparereasonably well (with a few exceptions) with those obtaineddirectly from the individual nonlinear regressions of equation1 given in Table 2. Figure 10 is a plot of the difference betweenthe mean estimated natural logarithm of the times, reportedin Table 2, and the corresponding estimate derived from theomnibus regression, versus the average of the two estimates.However, the standard errors of predictions from the omnibusmodel are large.

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FIG. 10. Scatter plot of difference between mean of estimated natural logarithm of time needed to obtain a 6.5 lethality, ln(t_6.5), obtained from individual regressions (Table 2) and omnibus model (Table 4) versus average of the two estimates.

Thus, instead of using the omnibus model for estimating thetimes needed to obtain a 6.5 lethality, a regression with dependentvariable ln(t_6.5) obtained from the 107 individual regressionsof equation 1 can be used directly. With few exceptions, thereplicate standard deviations are small compared to the magnitudeof the residuals; thus, a simple linear regression was usedwith the maximum likelihood estimation method, where the dependentvariable was the mean of the estimated ln(t_6.5) for each designcombination (55 in total). The selected model is given in Table5. 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 ofSPP increased L by 0.5; adding the square of ln(temperature)decreased L by 1.4; adding a linear term for SPP decreased Lby 0.1; adding NaL and the interaction of SPP and NaL decreasedL by 1.7; using the interaction of SPP and ln(temperature) insteadof the interaction of salt and ln(temperature) increased L by14.7. The linear regression line of the residuals versus thepredicted values of ln(t_6.5) is virtually flat (Fig. 11). Thestandard errors of the predicted ln(t_6.5) for the data rangefrom 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 canbe approximated as 100% times the standard error of the estimatedln(times). An estimated CV of 20% implies that a 99% confidenceinterval of the estimate of the expected times needed to obtaina 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 ofabout 5; thus, an estimated time of 10 min would have 99% confidenceinterval of 4.5 to 22 min. Table 6 gives the estimated timesto obtain a 6.5 lethality obtained from the individual nonlinearregressions using equation 1 (Table 2); the linear mixed effectsregression using these estimated times (actually the natural logof them); and the omnibus model, given in Table 4. Includedare the estimated CV's obtained from the latter two regressions,estimated by multiplying the standard error of the estimatedln(time).

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TABLE 5. Estimates of parameter values using linear model for predicting the natural logarithm of the (minimal) times needed to obtain a 6.5-log₁₀ relative reduction of Salmonella^a

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FIG. 11. Scatter plot of residuals versus predicted natural logarithm of the time needed to obtain a 6.5 lethality, ln(t_6.5), using the model in Table 5, where the dependent variable is the ln(t_6.5) derived from individual regressions of 107 observed survival curve.

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TABLE 6 Estimated times from individual regressions (), regression using these times (), and omnibus mixed effects regression ()

The predicted effect of SPP on the predicted times based onmodel using the individual estimates, ln(t_6.5), is clear: the higherthe level of SPP, the more time it takes to achieve a 6.5 log₁₀relative reduction. The interaction of temperature and saltis statistically significant (P = 0.0001). Example predictions:for a temperature = 60°C, salt = 0%, SPP = 0%, the predictedtime 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 predictedtime 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%. Increasingsalt from 0 to 4.5% at 60°C increases the time needed toobtain a 6.5 log₁₀ relative reduction by a 2.4 factor (CV =18.6%), significant at <0.0001; at 63.5% the factor is about1.5, significant at the 0.06 level, but at 71.1°C the differencein the times is not significant (P = 0.18).

However, for lower lethalities, the salt effect seems to bemore pronounced. A similar analysis as above was performed forthe 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 (coefficientof 1.90; P = 0.05). Interaction terms or terms involving NaL didnot improve the fit of the model with the data; however, as mentionedabove, this does not mean that such terms are not important. Thepoint of mentioning this analysis is to support the resultsfrom Table 3 that salt has an effect for low lethalities, whereasthe 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, asseen above for the result at the end of the last paragraph for71.1°C, the estimated effect of decreasing the heat resistancewas not statistically significant.

DISCUSSION

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Discussion

D values are used often in reporting kinetic results of inactivation studies,but many researchers (3, 6, 7) have reported nonlinear curvesfor Salmonella spp. Murphy et al. (13) fit nonlinear curves thathave initial lag times, but reported only D values. Even inpapers that report D values, it is often stated that the linearportion of the survival curve was used, implying that the actual survivaldata indicated some type of nonlinear shape. Thus, comparisons ofpredicted times needed to obtain specified lethalities using nonlinearmodels with predicted times based on reported D values may notbe appropriate; it might be that the shoulders and the tailsof the survival curves are affected by the concentrations ofthe factor being studied, for example, as seems to be the caseregarding the effect of fat concentrations on the heat resistanceof Salmonella in meat and poultry (7, 9). In this work, the survivalcurves were nonlinear and displayed tailing, and thus it was notpossible to report D values and use them for direct comparisonswith other published results. Hence, the comparisons that arepresented below need to be understood as only rough approximations.

There have been many studies showing that high concentrationsof 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 increasesthe heat resistance of Salmonella at lower temperatures. Inthe Mañus et al. (11) paper, the effects of salt concentrationson the heat resistance of Salmonella enterica serovar Typhimuriumwas studied in broth at 58°C. Results given in that paper(in terms of D values) indicated that increasing levels from0 to 4.5% would double the D value, implying that the time neededto obtain a fixed lethality would double. For the model givenin Table 5, to obtain a 6.5-log₁₀ reduction, at 58°C whenincreasing the salt from 0 to 4.5% and assuming that the concentrationof 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, whichis slightly higher than the estimated factor of 2 reported inthe work of Mañus et al. (11).

Blackburn et al. (3) also reported that higher levels of saltincreased heat resistance, but noted that beyond 3.5% the heatresistance stayed about the same. The highest value for ourstudy was 4.5% so that the linear effect for a given temperaturethat is used in the model developed within this paper is notinconsistent with the findings in Blackburn et al. (3). A predicted Dvalue for Salmonella serovar Enteritidis of 0.9 min at 60°Cfor beef with a salt level of 0.23% (wt/wt, aqueous phase) wasreported (3) (Table 6). Using this D value, to obtain a 6.5-log₁₀relative reduction of Salmonella, the estimated time neededwould be 5.85 min. However, in that same paper, a graph is presentedthat shows that the estimated times needed to obtain a 5-log₁₀relative reduction from the nonlinear model that was used inthat paper is about twice as large as those estimated usingthe predicted D values. We should point out that the model usedin the Blackburn paper had the property that the probabilityof viable cells did not approach 0 as t approaches infinity.It seems reasonable to assume that the Blackburn, et al. (3) modelreflected the behavior of their observed survival curves for relativelylarge times, thereby possibly being relatively "flat" with extensivetailing. Thus, it is quite possible that, for a 6.5 log₁₀ relativereduction, the ratio of the of the predicted times obtain fromthe nonlinear curves versus the estimated times obtained fromthe D values would be substantially larger than the estimatedvalue of 2 for the ratio for obtaining a 5 lethality; the ratiofor a 6.5 lethality could be more than 3. If the ratio were3, then the estimated time needed to obtain a 6.5 log₁₀ wouldbe 17.6 min. From the model given in Table 5 (assuming SPP = 0%,salt = 0.23%, and temperature = 60°C), the estimated timeis 21.1 m, with an error CV of 10.3%, and thus a lower 95% confidencebound of the needed time is about 17.8 min. Thus, the differentestimates may not be statistically different.

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

The survival curves were convex. An omnibus model, assumingnonlinear survival curves, was developed for predicting theobtained lethality when cooking beef for a fixed amount of timeat a fixed temperature between 55°C (131°F) and 71.1°C(160°F), where the beef matrices contain concentrationsof salt between 0 and 4.5%; SPP between 0 and 0.5%; and NaLbetween 0 and 4.5%. The selected model included terms involvingsalt and the interaction of SPP and NaL. While the former termis not surprising, the latter term, by itself, without the presenceof terms for SPP and NaL, presents difficulties in explainingthe model: is there is a synergistic effect of the two compoundson lethality, at least for relatively small times, or is thestatistical significance of this interaction term alone, withoutterms for SPP or NaL, just a fluke and that there really areSPP and NaL effects that were masked due to variability whichcould have only been detected in this study when the two compoundswere both present. Further research is needed to clarify this.

For the omnibus model, however, the standard errors of predictionare large. Since there is special interest of the times neededto obtain a 6.5 lethality, a model was developed for predicting thetimes needed to obtained a lethality of 6.5 log₁₀, using directlythe estimated times to achieve a 6.5 lethality obtained from regressionsof the individual survival curves. For the latter model, theCV of predicted times range from about 7 to 30%. The results indicatethat at 71.1°C, the times needed to obtain a 6.5-log₁₀ lethalitycould exceed 0.5 min.

The derived estimated times needed to obtain a 6.5-log₁₀ lethalityseem to be higher than predictions derived from reported D valuesin the published literature. A contributing reason for thiscould be due to the nonlinear survival curves. Predictions basedsolely on D values from the "linear" portion of the survivalcurves could be biased because of tailing of the survival curvesfor large times and because of the lag times—shouldersof the survival curves for small times before the linear kineticinactivation begins. However, many researchers have reportednonlinear survival curves for Salmonella, and thus predictedtimes for obtaining specified lethalities need to be based onmodels of these types of curves and not on D values.

In addition, the standard errors or CVs of the estimated lethalitiesfor a given time or predicted times needed to obtain a givenlethality seemed rather large, in some cases, exceeding 20%,giving rise to rather large confidence intervals of estimatedvalues. 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%, impliesa 99% confidence interval covering 5.8 to 17.1 min. The confidence intervalsdo not include the variability that may arise from slight misspecificationsof conditions, so that in actuality, to assure that processeswould be meeting lethality objectives, larger upper bounds mightbe needed. While no standards of predictions have been establishedby professional organization, we suggest that, for omnibus modelsthat need to satisfy multiple needs, the CV's of estimates ofthe expected values of times to achieve specific lethalitiesshould not be much larger than 10%, so that confidence intervalswould not be "too" wide. Consequently, for these types of studiesmore observations are needed, perhaps, more than two or three timesas many as in this study. More research is needed to clarifythe conditions that create nonlinear curves and to develop modelsfor them.

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FIG. 4. Scatter plot of estimated natural log of times (m) needed to obtain a 6.5 lethality, ln(t_6.5), from individual nonlinear regression (equation 1) versus levels of SPP, with linear regression lines by temperature. Points excluded from analyses are indicated with asterisks superimposed on symbols.

APPENDIX

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Appendix

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

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TABLE A1. Estimated parameters from individual regressions

ACKNOWLEDGMENTS

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

Mention of a brand or firm name does not constitute an endorsementby the U.S. Department of Agriculture over others of a similarnature not mentioned.

FOOTNOTES

* Corresponding author. Mailing address: U.S. Department of Agriculture, Agricultural Research Service, Eastern Regional Research Center, 600 E. Mermaid Ln., Wyndmoor, PA 19038. Phone: (215) 233-6500. Fax: (215) 233-6581. E-mail: vjuneja{at}arserrc.gov.

REFERENCES

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