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Applied and Environmental Microbiology, November 1998, p. 4416-4422, Vol. 64, No. 11
0099-2240/98/$04.00+0
Copyright © 1998, American Society for Microbiology. All rights reserved.
Analysis of the Influence of Environmental
Parameters on Clostridium botulinum Time-to-Toxicity by
Using Three Modeling Approaches
Donald W.
Schaffner,1,*
William H.
Ross,2 and
Thomas J.
Montville1
Department of Food Science, Rutgers
The
State University of New Jersey, New Brunswick, New Jersey
08901-8520,1 and
Health Canada, Food
Directorate, Banting Research Center, Ottawa, Ontario K1A 0L2,
Canada2
Received 2 April 1998/Accepted 12 August 1998
 |
ABSTRACT |
This study used the technique of waiting time modeling to analyze
the combined effects of temperature, pH, carbohydrate, protein, and
lipid on the time-to-toxicity of Clostridium botulinum 56A. Waiting time models can be used whenever the time to the occurrence of
some event is the variable of interest. In the case of the time-to-toxicity data, the variable is the time from the beginning of an experiment until a tube is identified as
positive. The statistical analysis used the SAS procedure
LIFEREG and included determination of the form of the response surface,
identification of the error distribution, and simplification of the
response surface. We found that increasing the macromolecule
concentration decreased the probability of toxin formation. The
probability of toxin formation also decreased at lower temperatures and
at pHs further from the optimum. The waiting time modeling approach to
developing models for botulinal toxin formation compared favorably with
other approaches but had one specific advantage. Waiting time
models have the inherent advantage that safety concerns regarding
predictions are automatically quantified in the analysis by formally
identifying a distribution of times-to-toxicity. The use of this
time-to-toxicity distribution permits a customizable margin of safety
(e.g., one in a million) not possible with other approaches.
| |
INTRODUCTION |
Predictive food microbiology has
gained favor in recent years as a means of conducting initial microbial
safety estimates for food products. Many different approaches for
modeling the behavior of the important human pathogen Clostridium
botulinum have been used, but all of the approaches can be grouped
into three basic types: probability, primary-secondary (7),
and other approaches.
Probability models are used to estimate the probability of toxin
formation directly, using a polynomial expression incorporating the
environmental variables. The first model used to predict the expected
fraction of toxic samples was developed by Roberts et al.
(27). Those authors used a logistic model in which the
probability of toxin production was inversely related to a polynomial
expression describing the effect of the experimental variables.
Lindroth and Genigeorgis (23) modeled the probability that a
single spore would initiate growth and produce toxin, using a similar
expression. Those authors have also expanded this general model type
for a variety of other systems and experimental variables
(2, 12, 13, 15, 19, 21, 24). Data presented by Tanaka
et al. (33) were used to construct a probability model based
on the quadratic equation (unpublished report) which predicts the
time-to-toxin production by C. botulinum in processed cheese
as a function of percent moisture, sodium chloride, and emulsifying
salts. This model was later published (3) and corrected
(34). Dodds (9) modeled the probability of toxin
production and time-to-toxicity as a function of quadratic polynomial
expressions with interactions using time, pH, and water activity.
Primary-secondary models use (i) either a kinetic model to describe lag
time and growth of the organism or a probability model to predict the
chance of toxin formation over time and then (ii) another model to
predict the effect of environmental factors on the parameters of the
first model. Gibson et al. (16) developed a kinetics model
for the growth of C. botulinum type A in pasteurized pork
slurry by using logistic and Gompertz functions. The relationship between the time to reach the maximum rate of growth and incubation temperature and sodium chloride concentration was described
graphically. Whiting and Call (36) used nonlinear regression
to estimate the parameters of a primary model for probability of growth
at a given time and then used polynomial expressions containing
experimental variables to predict the parameters of the primary model.
This approach was expanded to develop a model for nonproteolytic type B
C. botulinum, where inoculum size and time-to-toxicity
confidence intervals were also included in the model (37).
Most recently Graham et al. (17) developed a kinetic model
for the growth of nonproteolytic C. botulinum from
spores by using the Gompertz (16) and Baranyi-Roberts
(5) models. Parameters of these sigmoid functions were
in turn modeled as a function of temperature, pH, and sodium
chloride concentration by using a quadratic polynomial.
The link between probability models (9, 23, 27) and kinetic
models (16, 17) was recognized implicitly by Baker et al.
(4) and discussed by Ross and McMeekin (30), and
the distinction between them may in many cases be an artificial one. If
C. botulinum can grow, it will probably produce toxin,
so extent and rate of growth and time-to-toxicity are closely linked.
Other approaches used to model C. botulinum behavior
include that of ter Steeg and Cuppers (34), who used a
waiting time modeling approach to develop expressions for the effect of
environmental parameters on time for 100-fold multiplication of
C. botulinum in a model processed cheese system.
Waiting time models can be used whenever the time to the occurrence of
some event is the variable of interest. In the case of the
time-to-toxicity data, this is the time from the beginning of an
experiment until a tube is identified as positive. The analysis of
time-to-toxicity data requires identification of the appropriate error
distribution. The error distribution describes the variability of the
waiting times in the experiment. The importance of proper error
distributions in predictive food microbiology has been noted by several
researchers (1), and proper distributions are essential if
models are to be used in quantitative risk analyses. Waiting time
modeling techniques are easily implemented by using most popular
statistical analysis software, are flexible, and are simple to
interpret, and the resulting models provide the user with a powerful
tool for prediction and risk assessment.
Rogers and Montville (29) used linear regression to model
the factors that influence the ability of nisin to inhibit
C. botulinum in a model food system. The difference in
time-to-toxin production between the nisin-containing and control
(nisin-free) conditions was modeled. The data for the influence of the
environmental factors on the time-to-toxin production in the absence of
nisin from these experiments were never analyzed independently. This study used the technique of waiting time modeling to analyze the combined effects of temperature, pH, carbohydrate, protein, and lipid
on the time-to-toxicity of C. botulinum for the
Rogers-Montville (29) data set.
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MATERIALS AND METHODS |
Microbiology.
The methods and techniques used to obtain the
data used in our analysis are summarized below and were fully detailed
by Rogers and Montville (29) and Rogers (28).
C. botulinum 56A spores were cultured in a basal medium
consisting of 1.0% (wt/vol) glucose (Sigma Chemical), 0.5% (wt/vol)
peptone (Difco), 0.5% (wt/vol) yeast extract (Difco), and 0.01%
(wt/vol) resazurin (Sigma). Protein, phospholipid, and soluble starch
were added to the medium according to a fractional factorial design for
a model food system containing three levels of five variables
(temperature, pH, protein, phospholipid, and carbohydrate) that was
generated by using ECHIP (Hockessin, Del.) statistical software. In
addition to the primary experiments (done in triplicate), the program
designated several conditions to be repeated (independently in
triplicate) so that the reproducibility and variability of the results
could be calculated. Each combination of food components for each
experiment was prepared separately, pH adjusted separately, and
inoculated separately. The macromolecule variables were protein
(chicken egg albumin at 0.075, 0.75, and 7.5% [wt/vol]), lipid
(lecithin at 0.075, 0.75, and 7.5% [wt/vol]), and carbohydrate
(cornstarch at 5, 17.5, and 30% [wt/vol]). The other variables were
pH (adjusted with HCl to 5.5, 6.0, or 6.5) and incubation temperature
(15, 25, or 35°C).
C. botulinum 56A spores were heat shocked at 80°C for
10 min and inoculated into each sample to give a final concentration of
104 spores/ml. Cultures were removed from incubation and
refrigerated when gas was observed. Botulinal counts on modified
McClung agar were determined by manual spread plating. Botulinal toxin
was determined by the U.S. Department of Agriculture enzyme-linked immunosorbent assay method, based on that described by Dezfulian and
Bartlett (8). The experiment was halted after 60 days of observation.
Modeling.
Waiting time analysis was conducted by using the
SAS procedure LIFEREG. The process of selecting an appropriate model
and estimating its parameters requires three steps: determination of
the form of the response surface, identification of the error distribution, and simplification of the response surface.
(i) Response surface.
The form of the response surface
reflects the potential effects that the experimental variables may have
on the time-to-toxicity. Both cubic (18) and quadratic
(26) polynomial equations have been used to fit predictive
food microbiology data sets. If too many parameters are used to fit the
data, the risk of random fluctuations producing accidental and
erroneous effects is increased. Gauch (14) showed that
overfitting tends to increase the variance of the predicted values.
Based on this reasoning, we consider a quadratic response surface
sufficient to account for the effect of these variables on
time-to-toxicity, with all cross products included accounting for
potential interactions. The full model is as follows:
|
(1)
|
where
is the value of the response surface,
P is
percent protein (chicken egg albumin),
C is the percent
carbohydrate (cornstarch),
L is the percent lipid
(lecithin), pH is the initial pH of the
medium, and
T is the
incubation temperature. The subscripted
values represent regression
constants.
The quadratic term for carbohydrate concentration
(
7C2) was deleted from the model,
since the design for this experiment makes
it a redundant term. This
term is redundant because it can be
written exactly as a fixed, linear
combination of the other components
of the model. This redundancy is
simply a result of the particular
subset of the total number of
possible experimental conditions
that were chosen by Rogers and
Montville (
29). The redundant
term provides no additional
information beyond that which is given
by the other
terms.
(ii) Error distributions.
To identify an appropriate error
distribution, the full quadratic response surface model was fit twice
to the data, specifying first the generalized gamma distribution and
then the log-logistic distribution. The generalized gamma model is a
three-parameter distribution containing many of the common choices of
waiting time distributions as special cases (including the lognormal, Weibull, and gamma). The log-logistic distribution is an alternative two-parameter distribution with properties similar to the lognormal. The choice of error distributions was based on maximum-likelihood estimation techniques.
There are several reasons why the log-logistic distribution may be a
suitable choice for time-to-toxicity data. Each tube
will be inoculated
with many viable spores. If the spores act
independently, the time it
takes for a tube to become toxic is
related to the shortest time that
it takes an individual spore
in that tube to initiate growth and toxin
production. A mathematical
model for the distribution of shortest times
is the Weibull distribution.
However, if the ecology of each tube was
somewhat different (different
numbers of spores and variation in
physiology, etc.), then a slightly
different Weibull model may be
appropriate for each tube. The
log-logistic model incorporates the
Weibull distribution for individual
tubes together with between-tube
variation.
For the log-logistic model, the probability,
P(
t), that the
time-to-toxicity of any tube is less than or equal to
t is
given
by the following formula:
|
(2)
|
where
y = ln(
t),
is the response
surface, and
is a scale parameter for the error. The scale
parameter is proportional
to but not equal to the standard deviation.
Note that the response
surface calculates the natural logarithm of the
time at which
the probability of toxicity is 1/2 (the median of the
distribution).
The log-logistic model can also be written as follows:
|
(3)
|
where
= ln(
a) and
=
l/
b.
(iii) Response surface simplification.
The response surface
can often be simplified by removing individual terms without changing
the statistical significance of the fitted model. A backward selection
method was used to eliminate terms from the response surface in this
analysis. The final submodel must be consistent with the underlying
microbiology (i.e., it should not predict infinite rate increases with
increasing temperatures or increasing growth rates above the maximum
growth temperatures) (39).
The full response surface was fit to the data. The log-likelihood ratio
statistic for the full model,
LF, was recorded,
and
the individual terms are then ranked according to their reported
chi-squared statistics, with larger values (hence, smaller
P
values)
ranked highest. The term ranked lowest was removed from the
response
surface equation, and the new model was fit to the data. The
log-likelihood
ratio for this submodel,
LS, was
recorded, and the remaining terms
were ranked as before. The
log-likelihood ratio statistic,
=
2(
LF 
LS), was calculated and compared with the percentiles
of
the chi-squared distribution with degrees of freedom equal to
the
number of terms removed from the response
surface.
If the log-likelihood ratio statistic was not significant, then the
last two steps (term removal and
calculation) were repeated
until
statistical significance was achieved. The resulting submodel
was the
one with the fewest remaining terms that were not statistically
significantly different from the full model. A 5% level of
significance
was used in this
analysis.
Comparison with other approaches.
Dodds (9)
defined lag time (LT) as the time until first toxin production for each
experimental setting, while Baker and Genigeorgis (2)
defined LT as the sampling period prior to toxin detection. In either
analysis, multiple linear regression models were fit to log(LT). The
use of the time to first toxin production or the lower censoring value
to define LT adds an element of safety to the analysis. The degree of
safety afforded by these LT definitions depends on the intensity of the
inspection protocol and the underlying variability of the
times-to-toxicity. Both of these approaches were applied to these data
and compared with the waiting time model.
An analysis using nonlinear regression to fit a modified logistic
cumulative distribution function to the data (
36) was
attempted, but this analysis was numerically unstable and failed
to
provide satisfactory results for many treatment
combinations.
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RESULTS |
For this data set, the log-logistic distribution yielded a larger
log-likelihood ratio statistic than other distributions and was
therefore considered to be the appropriate distribution. A record
of the steps in the backward selection procedure is given in Table
1. Note that the final term removed from
the model (LpH) contributed significantly to the model
(P < 0.05); however, the overall change in the
submodel from the full quadratic model was not significant, and so the
term was removed. Parameter estimates for the final model
are listed in Table 2.
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TABLE 1.
P values for variable removal with the
backward selection procedure of model parameters for the waiting
time model
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Explanation of the parameters and interactions.
The parameters
from equations 1 and 2 and their statistical significance (Tables 1 and
2) can be used to gain some insight into possible mechanisms
influencing C. botulinum toxin formation and their
suitability and importance for controlling the growth of this food
pathogen. All of the environmental factors studied here (pH,
temperature, and concentrations of protein, carbohydrate, and lipid)
influenced C. botulinum toxin formation to a
statistically significant extent, because every factor is represented
by at least one parameter in the final model for equation 4 (see below) (Table 2).
The easiest way to consider the net results of a series of complex
interactions (e.g., the lipid concentration, present in
three model
terms [
L2,
PL, and
CL] [Table
2])
is to present the results graphically.
Figure
1 shows the effect of varying each term
in the model under
an otherwise constant set of conditions
(
C = 30%,
L = 7.5%,
P = 0.075%, pH = 5.5, and
T = 15°
C; 10 days). The
y axis in
each
plot is the probability of toxin formation under the conditions
specified. Families of plots for other conditions look slightly
different but indicate the same trends (data not shown). The effect
of
pH on probability of toxin formation is shown in Fig.
1A, with
minimum
probabilities at pHs of 5.5 and 6.5 and an optimum occurring
between pH
6.00 and 6.25. Figure
1B shows the effect of temperature
on toxin
formation. Under the conditions selected, the model predicts
the
lowest probability of toxin formation at a temperature of
15°C and a
maximum probability of toxin formation above 30°C.
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FIG. 1.
Influence of intrinsic and extrinsic factors on the
waiting time model predicted probability of toxin formation
within 10 days by C. botulinum in a model
system (7.5% lipid, 0.075% protein, and 30% carbohydrate) at pH 5.5 and 15°C.
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The relationships between the lipid, carbohydrate, and protein
macromolecule concentrations and probability of toxin formation
all
show a similar trend (Fig.
1C, D, and E), with the highest
probability
of toxin formation at the lowest macromolecule concentration
and the
lowest probability of toxin formation at the highest macromolecule
concentration.
Comparison of predictions to observations.
For the
log-logistic model, the response surface defines the
median (50th percentile) of the time-to-toxicity distribution corresponding to each combination of explanatory variables. Observed toxicity times are compared to response surface estimates in Fig. 2. The 99.8th percentile (adjusted for
censoring and estimation error) is also plotted, as a dashed line.
Three of the 227 observations exceed this percentile. This would be
expected only 1% of the time, suggesting that one or more of these
three observations might be outliers. Refitting the model after removal
of these three data points had no effect on fitting the response
surface and only slightly affected the estimate of the scale parameter.
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FIG. 2.
Observed time-to-toxicity of C. botulinum in a model system versus median time-to-toxicity
predicted by using a waiting time model. The 99.8th percentile
(adjusted for censoring and estimation error) is plotted as a dashed
line.
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The probability of toxicity can be viewed as the probability of a
single package in one batch being toxic. For example, a
probability of
toxicity equal to 0.000001 (one in a million) can
also be thought of as
one package in one million being toxic.
The time at which the
probability of toxicity is any value of
interest (such as one in a
million) is easily calculated by using
this model. Letting
P0 denote the probability level of interest,
then the corresponding time,
t0, is
calculated by using the following
formula:
|
(4)
|
Contour lines in Fig.
3 indicate the
time (in days) at which the probability of toxicity is 0.000001 (one in
a million) for
various experimental conditions.
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FIG. 3.
Predicted time-to-toxicity (in days) of C. botulinum in a model system as influenced by macronutrients,
incubation temperature, and pH. Contour lines represent the predicted
time at which the probability of toxicity is one in a million. (A)
Predictions where the model system contains 0.075% (wt/vol) chicken
egg albumen, 30% (wt/vol) cornstarch, and 7.5% (wt/vol) lecithin. (B)
Predictions where the model system contains 7.5% (wt/vol) chicken egg
albumen, 30% (wt/vol) cornstarch, and 0.075% (wt/vol) lecithin.
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Comparison with other approaches.
Figure
4 displays a scatter plot comparing
log(LT) as predicted by the approach of Dodds (9) with the
median value predicted by the waiting time model. There is close
agreement between the median from the waiting time distribution and
most of the Dodds log(LT) predictions. If there were perfect agreement,
all of the data would fall along the (solid) line of equivalence.
However, for toxicity times close to 1 day, the log(LT) predicted
by the Dodds method exceeds the median of the data. This means
that actual times-to-toxicity would be less than predicted more than
50% of the time. A similar point can be noted by examining
the short-dashed line (a linear regression of the two
predictions). This regression line lies above the line of equivalence
at times less that 5 days but below the line of equivalence at
times greater than 5 days. All log(LT) predictions are on or above
the 10th percentile of the toxicity time distribution
(long-dashed line). We expect that future LT values would be less
than the predicted time-to-toxicity at least 10% of the time.
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FIG. 4.
Comparison of predicted times-to-toxicity of
C. botulinum obtained by using the model of Dodds
(9) and the waiting time model median values. The solid line
is the line of equivalence (perfect agreement). The short-dashed line
is the linear regression of the two predictions. The long-dashed line
is the 10th percentile of the toxicity time distribution.
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|
When LT is defined as by Baker and Genigeorgis (
2), there is
little difference between the median of the waiting time-to-toxicity
distribution and the predicted Baker-Genigeorgis LT (Fig.
5).
As with the previous case, the
Baker-Genigeorgis predicted log(LT)
exceeds the waiting time
distribution median for predictions close
to 1 day, but in this case
the short-dashed (regression) line
more closely approximates the line
of equivalence. All predictions
are well above the 10th percentile of
the time-to-toxicity distribution.
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FIG. 5.
Comparison of predicted times-to-toxicity of
C. botulinum obtained by using the model of Baker and
Genigeorgis (2) and the waiting time model median values.
The solid line is the line of equivalence (perfect agreement). The
short-dashed line is the linear regression of the two predictions. The
long-dashed line is the 10th percentile of the toxicity time
distribution.
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DISCUSSION |
Explanation of the parameters and interactions.
As one of
us has noted elsewhere (25), extreme care should be
used in interpretation of the statistical and practical significance of interaction effects, with an emphasis on avoiding
overinterpretation. This is especially important when the simple linear
terms (in this case P, C, L, pH, or T) do not
appear in the final model. For example, since the lipid term is
the first to be dropped from the model, a simplistic conclusion
would be that lipid concentration does not have a significant effect on
C. botulinum toxin formation. This is untrue, because
the effects of lipid concentration on toxin formation are incorporated
into the model by the L2, PL, and CL terms.
The effect of pH on microbial growth is known to show optimum,
minimum, and maximum values (
20). Our results show the
same
trends as those observed by others modeling the effect of pH on
growth of or toxin production by
C. botulinum (
9,
17,
31,
36) or on growth of other sporeformers (
25).
Enzymes are known
to have pH optima similar to those shown by whole
cells. pH is
also known to affect transport of nutrients into the cell.
Our
model, and the predictions shown in Fig.
1A, suggests that either
of these two mechanisms may be operating
here.
It is known that most simple chemical and biochemical reactions show a
dependence on temperature governed by Arrhenius rate
kinetics
(
38), which do not describe multienzyme microbial growth
very well. A complex version of the simple Arrhenius equation,
which
appears to describe the effect of temperature on microbial
growth
fairly well, has been developed (
32) and applied to
food-borne
pathogens (
1). This model supposes one key
enzyme (reversibly
inactivated at low and high temperatures) which
determines growth
rate as a function of temperature for a given
microbe. Because
of the nonlinearity shown in Fig.
1B, a similar sort
of key-enzyme
dependence may be governing the relationship between the
probability
of
C. botulinum toxin formation and
temperature. As with the pH
results, our results generally correspond
to those of others modeling
the effect of temperature on growth of or
toxin production by
C. botulinum (
13,
16,
21,
24,
36).
The relationship between macromolecule concentrations and probability
of toxin formation (Fig.
1C, D, and E) might at first
seem
counterintuitive, because higher nutrient concentrations
should lead to
better conditions for spore germination, more rapid
cell growth, and a
greater (not lesser) chance of toxin formation.
The results for lipid
concentration (Fig.
1C) are most easily
explained. Lecithin
(phosphatidylcholine) was used to change the
lipid concentration in the
model system. The purity of the lecithin
preparation (Sigma type XV-E,
from fresh frozen egg yolk; phosphatidylcholine
content, ~60%) was
such that free fatty acids (FFAs) were likely
present in the
mixture. FFAs are well-known germination inhibitors
(
22), so as the concentration of lipid (lecithin)
increased,
so did the concentration of inhibitory
FFAs.
The results for protein (Fig.
1D) were initially surprising, because
all clostridia are nutritionally demanding (
11) and
the
proteolytic strain of
C. botulinum used here would
proteolyse
proteins required to supply amino acids. However, a
literature
search revealed that egg white proteins (as were used in
this
study) inhibit
C. botulinum proteases
(
35) and would thus decrease
the supply of amino acids
available to support
growth.
Limitations of previous modeling approaches.
One potential
difficulty in applying the linear regression approach is the handling
of left- and right-censored waiting times. A left-censored waiting time
is created in these experiments when a sample contains toxin the first
time it is observed. A right-censored waiting time is created when an
experiment is concluded but a sample has yet to show toxin production.
For left-censored values, LT = 0, for which log(LT) is not
defined. As a result, these values must be removed from the analysis or
replaced by some arbitrary value. Since these values represent an
extreme for the response in the linear regression analysis, they
may have a substantial effect on model selection and parameter
estimation. This effect is demonstrated in Fig. 4 and 5, where short
time-to-toxicity values are generally overestimated by other approaches.
Similarly, when experiments are terminated after a fixed period
of time, the resulting LT values are restricted. The result
is to
artificially reduce the effect of important factors in the
linear
regression. This will again affect model selection and
parameter
estimation.
A further difficulty with the application of the linear regression
approach is the identification of the error distribution.
In effect,
linear regression analysis assumes that waiting times
satisfy a
lognormal probability law. For the analysis presented
here, the model
fit was improved by considering alternative error
distributions. In
addition, it should be noted that the inspection
protocol restricts the
possible values of LT. For example, Dodds
(
9) used six
inspection times over a period of 60 days, thereby
restricting LT to be
one of only six possible values. Experimental
settings for which no
tubes turned positive within 60 days were
excluded from the
analysis.
Whiting and Call (
36) used nonlinear regression to apply a
modified logistic cumulative distribution function (CDF) as the
primary
model for waiting times. This is a two-step procedure
that first fits a
parametric CDF to the proportion of positive
tubes at each inspection
time separately for each treatment. For
each experimental setting, the
response variable of the nonlinear
regression analysis (the
cumulative proportion of positive tubes)
is correlated. As a result,
the reported standard errors of the
parameter estimates are
incorrect. This may also introduce additional
bias in parameter
estimation. The overall effect on model selection
is
unknown.
The waiting time model alternative.
Waiting time models
provide a flexible approach to analyzing the effects of various
treatments on time-to-toxicity while identifying an appropriate error
distribution and accounting for any censoring introduced by the
inspection protocol. The ease of conducting a statistical analysis by
using this approach compares favorably with that for linear regression
methods, because most popular statistical software packages contain
methods for analyzing for waiting time. The results of such an analysis
are directly applicable to prediction and to the assessment of the risk
of toxicity occurring in a particular batch. Safety concerns regarding
predictions are automatically introduced into the analysis by formally
identifying a distribution of times-to-toxicity. This allows the
analyst to choose the margin of safety (for example, one in a million).
Unlike linear regression based on LT, this approach directly quantifies the level of safety in the analysis.
The waiting time approach does have several important limitations that
should be noted. First, additional laboratory work
may be required to
collect the data used for modeling. In particular,
the nonlinear
regression step requires a sufficient number of
tubes at each treatment
level indicating toxicity at different
inspection times. Second, the
statistical phase of the analysis
may also be more labor-intensive, as
each error distribution determines
a different CDF, which requires a
separate nonlinear regression
for each treatment combination. Finally,
in fitting several different
models, numerical stability and failure of
the fitting algorithm
to converge are also an
issue.
It is also important to note that other factors not present in the
Rogers-Montville (
29) data set may influence the variability
of the results. For example, it has been shown previously that
the
germination times of individual spores of
C. botulinum
may
differ significantly (
6). This germination time
difference
means that larger inoculum sizes (e.g., as used by Rogers
and
Montville [
29]) can result in a smaller
variability of time-to-toxicity
(
31) than for food
samples, which typically contain smaller
numbers of spores
(
10). Thus, the models presented here may
predict less
variability in time-to-toxicity than expected in
naturally contaminated
food
samples.
Conclusions.
We have provided preliminary evidence that higher
concentrations of macromolecules (protein, lipid, and
carbohydrate) may reduce the probability of toxin formation by
C. botulinum. This research has also confirmed that
lower temperatures and pHs farther from the optimum reduce the
probability that C. botulinum will form toxin within
the time period studied. We have also demonstrated that the waiting
time modeling approach to developing models for toxin formation
by C. botulinum is possible and compares
favorably with approaches used by others.
| |
ACKNOWLEDGMENTS |
This work was supported by funds from the USDA National Research
Initiative (96-35201-3458), the U.S. Hatch Act, and the state of New Jersey.
| |
FOOTNOTES |
*
Corresponding author. Mailing address: Department of
Food Science, RutgersThe State University of New Jersey, 65 Dudley
Rd., New Brunswick, NJ 08901-8520. Phone: (732) 932-9611, ext. 214. Fax: (732) 932-6776. E-mail: schaffner{at}aesop.rutgers.edu.
Publication D-10122-1-97 of the New Jersey Agricultural Experiment Station.
| |
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Applied and Environmental Microbiology, November 1998, p. 4416-4422, Vol. 64, No. 11
0099-2240/98/$04.00+0
Copyright © 1998, American Society for Microbiology. All rights reserved.
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Abstract
Introduction
