**DOI:**10.1128/AEM.69.9.5138-5156.2003

## ABSTRACT

Analyses
of survival data of a mixture of *Salmonella* spp. at fixed
temperatures between 55°C (131°F) and 71.1°C
(160°F) in ground beef matrices containing concentrations of
salt between 0 and 4.5%, concentrations of sodium pyrophosphate
(SPP) between 0 and 0.5%, and concentrations of sodium lactate
(NaL) between 0 and 4.5% indicated that heat resistance of
*Salmonella* increases with increasing levels of SPP and salt,
except that, for salt, for larger lethalities close to 6.5, the effect
of salt was evident only at low temperatures (<64°C).
NaL did not seem to affect the heat resistance of *Salmonella*
as much as the effects induced by the other variables studied. An
omnibus model for predicting the lethality for given times and
temperatures for ground beef matrices within the range studied was
developed that reflects the convex survival curves that were observed.
However, the standard errors of the predicted lethalities from this
models are large, so consequently, a model, specific for predicting the
times needed to obtained a lethality of 6.5 log_{10}, was
developed, using estimated results of times derived from the individual
survival curves. For the latter model, the coefficient of variation
(CV) of predicted times range from about 6 to 25%. For example,
at 60°C, when increasing the concentration of salt from 0 to
4.5%, and assuming that the concentration of SPP is 0%,
the time to reach a 6.5-log_{10} relative reduction is
predicted to increase from 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 more than 0.5
min is needed to achieve a 6.5-log_{10} relative
reduction.

*Salmonella enterica* has long been recognized as an
important food-borne pathogen, and it continues to be a leading cause
of food-borne disease outbreaks associated with consumption of meat and
poultry. The annual incidence of salmonellosis in the US is 1.4 million
cases, causing as many as 550 deaths
(12), and the incidence
of salmonellosis has increased during the past 30 years. This reality
stems from the ubiquitous occurrence of *Salmonella* in the
environment, its prominence in various sectors of the agriculture
industry, and the escalating movement of food and food ingredients
worldwide. Important contributing factors which lead to outbreaks of
food-borne illness, including salmonellosis, are inadequate time
and/or temperature exposure during initial thermal
processing (or cooking) and inadequate reheating to kill pathogens in
retail food service establishments or homes
(2,
15). Inadequate cooking
was cited as a contributing factor in 67% of the outbreaks in
which *Salmonella* was an etiological agent
(2). These outbreaks have
implicated a variety of foods, including meat and poultry, milk, ice
cream, cheese, eggs and egg products, chocolate, and spices, as
vehicles of transmission
(4). Current performance
standards require that thermal processing schedules must achieve a
6.5-log_{10} reduction of *Salmonella* for cooked beef,
ready-to-eat roast beef, and cooked corned beef products
(17).

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.

## MATERIALS AND METHODS

Organisms.A cocktail consisting of eight
strains of different serotypes of *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. These strains were preserved by freezing the cultures at−
70°C in vials containing tryptic soy broth (Difco
laboratories, Inc., Detroit, Mich.) supplemented with 10%
(vol/vol) glycerol (Sigma Chemical Co., St. Louis,
Mo.).

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 × *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
log_{10} 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 log_{10} 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.

(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
log_{10} 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,
$$mathtex$$\[E{\{}\mathrm{log}_{10}[N(t)/N(0)]{\}}{=}{-}\mathrm{log}_{10}{\{}1{+}\mathrm{exp}[a{+}b\ \mathrm{ln}(t)]{\}}\]$$mathtex$$(1)
where
*E* is the expected value operator and *a* and *b*
(>0) are constants, has been used for fitting survival curves
with the above described properties by various researchers
(1,
10). This function
provides the flexibility to fit a variety of survival curves that have
asymptotic convex behavior. As t approaches infinity the derivative of
the right side of equation
1 approaches 0. To allow
for the possibility that, asymptotically, the survival curves approach
a straight line with nonzero slope, we considered a model that involved
adding another term to the exponent:
$$mathtex$$\[E{\{}\mathrm{log}_{10}[N(t)/N(0)]{\}}{=}{-}\mathrm{log}{\{}1{+}\mathrm{exp}[a{+}b\ \mathrm{ln}(t){+}ct]{\}}\]$$mathtex$$(2)
where
*c* is ≥0. The asymptotic *D* value for survival
curve of equation 2 thus is
ln(10)/*c*. The derivative of the right side of equation
2 approaches−
*e ^{a}*

*bt*, as

^{b}^{−1}*t*approaches 0 from the right, so that, if

*b*is >1, then the slope at zero is zero, and if

*b*is <1 then the limiting slope is minus infinity. When

*b*is >1, the survival curve has a“ shoulder” and the point (time) of inflection (where the curve becomes convex) 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 more pronounced shoulders and larger points of inflections.

(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,
$$mathtex$$\[f(x){=}\mathrm{ln}(10_{{-}x}{-}1)\]$$mathtex$$(3)
where
*x* = log_{10}(*rt*), is linear in the
unknown parameters, *a*, *b*, and *c*, so that
linear mixed effects model can be used for estimating the model
parameters as linear functions of the variables studied. Using linear
mixed effects models accounts, in a simple way, for the correlations
that exist among the observations. Using the variables identified from
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
considered as random effects. That is to say, if f(x) is assumed to be
a linear combination of *a* + *b*ln(*t*),
then, for example, it can be assumed that *a* is actually a
random variable that can be expressed as: *a* =μ
+ ε_{s} +ε
_{e(s)} + ε_{r}, whereμ
is the expected value of *a*, ε_{s} is
the error associated with factor *s*; ε_{e(s)}
is the error associated with the factor *e* nested within
*s*; ε_{r} is a residual error (nested within
*s* and *e*); and the error terms are independent, have
zero expected values and specified variances. The same type of
assumption is made for *b*, so that, in addition to the
variances, there are possible nonzero covariances between corresponding
errors at the same structural level associated with *a* and
*b*. The expected value of *a* and *b*,
themselves, are assumed to be linear combinations of the independent
variables with unknown coefficients that can be assumed to be random
variables. When including all possible variances and covariances, such
mixed effect models can have an enormous number of parameters for which
convergent solutions with estimable variances (nonsingular Hessian
matrix) sometimes are not readily attainable. Consequently simplifying
assumptions are made in order to reduce the number of parameters to“
manageable” levels. In this case, it is assumed that
only the constant or intercept terms—ones that are not
coefficients of an independent variable of temperature, salt, SPP or
NaL—are associated with random variables in the sense described
above. For details of using these models, the book by Pinheiro and
Bates (14) can be
consulted; the approach given in that book was followed here. In this
study, design combinations, and the replicates within these are
considered as factors. In considering whether to include terms in the
model, likelihood ratio tests based on the statistic *L*=
−2 ln(likelihood ratio), 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 terms improves the goodness-of-fit was made by
comparing the difference of the statistics, −2 ln(likelihood),
that are given in the PROC MIXED output, with the 95th percentile of a
chi-square distribution with q degrees of freedom. With each model
considered, the plots of the residuals versus the predicted values were
examined. Predictions of *x* = log_{10}
[*r*(*t*)], as a function of the selected
independent variables, were obtained by using the inverse of the
function of equation 3, and
the standard errors of these predicted values were obtained using the
linear approximation (first term of the Taylor series) of the inverse
function, and the asymptotic covariance matrix of the estimated values
of the parameters.

Of particular importance is the times needed
to obtain a 6.5-log_{10} 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-log_{10} 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:
$$mathtex$$\[t_{6.5}{=}\mathrm{exp}[\frac{f({-}6.5){-}a}{b}]\]$$mathtex$$(4)
where
*f* is given in equation
3. (If equation
2 were used, then direct
numerical procedures would be needed to solve for
*t*_{6.5}.)

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.

## RESULTS

Preliminary analysis.Equation
2 was fit for each growth
curve, with the restriction that *b* be >0 and
*c* be ≥0, where it was also assumed that *N*(0)
was a parameter with an unknown value. Of the 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 value of
*c* significantly greater than zero at the one-sided 0.10 level
and only 2 at a significance better than 0.05. The pooled root mean
square error (RMSE) for fitting equation
2 is 0.548 compared to
0.500 for equation 1. Thus,
it appears that, for individual survival curves, equation
2 does not generally
provide a significantly better fit than does equation
1. Hence, for this
analysis, equation 1 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
the log_{10} value would equal 0, using equation
1, the average predicted
log_{10} 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 log_{10} 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 log_{10}, 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 log_{10}. 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 log_{10} 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
log_{10} 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 log_{10} of the relative reduction range up to 1
log_{10}, 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 log_{10} 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 log_{10} 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 log_{10} 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
log_{10} 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 log_{10} 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.

## DISCUSSION

*D*
values are used often in reporting kinetic results of inactivation
studies, but many researchers
(3,
6,
7) have reported nonlinear
curves for *Salmonella* spp. Murphy et al.
(13) fit nonlinear curves
that have initial lag times, but reported only *D* values. Even
in papers that report *D* values, it is often stated that the
linear portion of the survival curve was used, implying that the actual
survival data indicated some type of nonlinear shape. Thus, comparisons
of predicted times needed to obtain specified lethalities using
nonlinear models with predicted times based on reported *D*
values may not be appropriate; it might be that the shoulders and the
tails of the survival curves are affected by the concentrations of the
factor being studied, for example, as seems to be the case regarding
the effect of fat concentrations on the heat resistance of
*Salmonella* in meat and poultry
(7,
9). In this work, the
survival curves were nonlinear and displayed tailing, and thus it was
not possible to report *D* values and use them for direct
comparisons with other published results. Hence, the comparisons that
are presented below need to be understood as only rough
approximations.

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-log_{10} 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-log_{10} 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-log_{10} 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 log_{10} 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 log_{10} 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 log_{10}, 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-log_{10} lethality could exceed 0.5
min.

The derived estimated times needed to obtain a
6.5-log_{10} 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.

## APPENDIX

Shown in Table A1are estimated parameters from individual regressions (equation 1).

## ACKNOWLEDGMENTS

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.

## FOOTNOTES

- Received 11 December 2002.
- Accepted 19 June 2003.

- Copyright © 2003 American Society for Microbiology