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Applied and Environmental Microbiology, July 2004, p . 3925-3932, Vol . 70, No . 7
Analysis and Validation of a Predictive Model for Growth and Death of Aeromonas hydrophila under Modified Atmospheres at Refrigeration Temperatures
Carmen Pin,1* Raquel Velasco de Diego,2 Susan George,1 Gonzalo D . García de Fernando,2 and József Baranyi1
Institute
of Food Research, Norwich NR4 7UA, United
Kingdom,1
Departamento de Higiene y
Tecnología de los Alimentos (Nutrición y
Bromatología III), Facultad de Veterinaria, Universidad
Complutense de Madrid, Madrid 28040,
Spain2
Received 29 October 2003/
Accepted 1 April 2004
Specific
growth and death rates of Aeromonas hydrophila were measured
in laboratory media under various combinations of temperature, pH, and
percent CO2 and O2 in the atmosphere . Predictive
models were developed from the data and validated by means of
observations obtained from (i) seafood experiments set up
for this purpose and (ii) the ComBase database
(http://www.combase.cc; http://wyndmoor.arserrc.gov/combase/).Two
main reasons were identified for the differences between the predicted
and observed growth in food: they were the variability of the growth
rates in food and the bias of the model predictions when applied to
food environments . A statistical method is presented to quantitatively
analyze these differences . The method was also used to extend the
interpolation region of the model . In this extension, the concept of
generalized Z values (C . Pin, G . García de Fernando,
J . A . Ordóñez, and J . Baranyi, Food Microbiol.
18:539-545, 2001) played an important role . The extension
depended partly on the density of the model-generating observations and
partly on the accuracy of extrapolated predictions close to the
boundary of the interpolation region . The boundary of the growth region
of the organism was also estimated by means of experimental results for
growth and death
rates .
Most predictive models in food microbiology focus on the specific growth
and/or death rate (or the doubling time [D value]) of a
microorganism as a function of the main environmental factors, such as
temperature, pH, and others . These models are commonly based on
observations made in a well-defined and controlled laboratory
environment, using microbiological media . It is also vital to validate
the predictions in food environments, which can be highly complex and
sometimes difficult to characterize .
The overall error of a model
is defined by means of the mean square error (MSE) between predictions
and observations made in food
(19) . If extrapolations
are omitted from the predictions, as they should be, then the overall
error refers only to the interpolation region . Sometimes, depending on
the experimental design and available data, it is difficult to
determine the interpolation region of a multivariate empirical model
based purely on observations . Baranyi et al.
(3) defined it as a
minimum convex polyhedron (MCP), or convex hull, in the space of
environmental factors . As Fig.
1 shows, the MCP encompasses those combinations of the environmental
conditions for which observations were made to generate the model . Its
vertices can be calculated as described previously
(3) . Model predictions
outside the MCP are extrapolations .
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FIG . 1 . The
interpolation region of the model is described by the MCP, which
encompasses all of the observations used to fit the model . Those
environmental conditions outside the MCP are
extrapolations.
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Often, conditions observed in
food fall outside of the interpolation region but are close enough to
it that they can be useful for model validation . These observations can
also help to extend the interpolation region of the model .
In
this paper, we report new experimental data about the growth and death
rates of Aeromonas hydrophila which vary with temperature, pH,
and percent CO2 and O2 in the atmosphere . Both
death and growth data were used to estimate the growth-no growth
boundary of the organism . The growth data were used to generate a
predictive growth model, which was then extensively validated by
comparing its predictions with various observations in food . Some
observations were outside of but close enough to the interpolation
region of the growth model to be useful for the validation procedure.
We developed an algorithm to extend the interpolation region of the
model in order to utilize those originally extrapolated values . The
line of thought leading to the method can be summarized as
follows .
Predictive models are usually based on observations of
the response parameter (in this case, the logarithm of the specific
growth rate) in broth . Standard fitting methods assume that the bias is
0 and that the variance is constant throughout the interpolation region
when predictions and observations are compared in broth . A partition of
the MSE between predictions and observations in food is illustrated in
Fig.
2 . Such partitioning is commonly used in statistics to estimate bias and
variance . Here we use this technique to analyze the error between
broth-generated model predictions and food observations . The bias is
due to the fact that the data for the model were generated in
laboratory media, generally producing higher growth rates than in food.
The other component of the error expresses the variability of the
predicted parameter in food . It is due to factors that are not taken
into account in the model, such as food structure or microbial
interactions . Assuming that the bias and variance in food are constant,
they can be estimated inside the interpolation region and then
extrapolated to those regions for which only food data exist . In
another words, we determine how far the model can be extended so that
the constant bias and variance, estimated inside the interpolation
region, still hold .
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FIG . 2 . Analysis
of the error . The MSE between food observations and predictions can be
broken down as the sum of two components,
Bias2(g,ffood) +
Var(ffood), where
Bias(g,ffood) denotes the bias of the
g model when applied to food and
Var(ffood) denotes the variance of the observations
in
food.
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The extended region was constructed from data
close to the boundary of the original MCP . The extension depends partly
on the density of model-generating data and partly on the accuracy of
extrapolated predictions . To do this objectively, we defined a distance
concept in the space of the environmental variables in such a way that
one unit change had a similar effect on the modeled response, whichever
variable had changed . This is, in a sense, equivalent to the
normalization of the variables . In our algorithm, we rescaled the
environmental variables according to this technique (C . Pin and J.
Baranyi, Abstr . 4th Int . Conf . Predictive Modelling in Foods, p . 147,
2003) .
Some strains of Aeromonas spp., especially those
of A . hydrophila, are enteropathogens with virulence
properties, such as the ability to produce enterotoxins, cytotoxins,
and hemolysins and/or the ability to invade epithelial cells
(15,
18) . Infection may
produce localized illnesses, mainly in the gastrointestinal tract, and
exceptionally may affect systemic processes and require
hospitalization .
The growth of A . hydrophila has been
reported for a variety of vacuum-packaged products stored between
–2 and 10°C, such as smoked cod
(4), cooked crayfish
(12), beef
(9), roast beef
(11), and pork
(24), as well as for
modified atmosphere-packaged foods
(5,
13,
14) . Some results were
occasionally contradictory
(24) . Apart from
extending the interpolation region of a predictive model, another aim
of the present work was to study the behavior of this organism in
culture medium under modified atmospheres by means of mathematical
models and to test how these results can be applied to modified
atmosphere-packaged meat and
fish .
Bacterial strain.
A . hydrophila CECT 398
(Spanish Type Culture Collection) was maintained at
–20°C . Immediately before the experiments, it was
subcultured three times consecutively in tryptic soy broth (CM129;
Oxoid) incubated at 25°C for 24
h .
Broth experiments.
Broth experiments were performed as
previously described
(16) . Bottles containing
200 ml of tryptic soy broth, with the pH adjusted to the target value
and with different atmospheric compositions, were prepared and stored
at the required experimental temperature . Each bottle was inoculated
with 1 ml of the appropriate dilution of the bacterial culture to give
a final concentration of ca . 103 CFU/ml . At each sampling
time, bacterial counts were estimated by plating samples onto tryptic
soy agar (CM131; Oxoid) . In this way, bacterial kinetic curves were
generated for 110 combinations of environmental conditions (Table
1) . The conditions were intended to be uniformly distributed in the
environmental region between 1.5 and 11°C and pHs 5.2 and 7.2
and in atmospheres containing 0 to 80% CO2 combined
with 0 to 80% O2, with balanced
N2 .
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TABLE 1 . Death and
growth rates of A . hydrophila under modified atmospheres in
tryptic soy broth
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Seafood experiments.
Seafood was
purchased from a local fishmonger in Madrid . Striped tunny (Sarda
sarda), hake (Merluccius merluccius), and salmon
(Salmo salar) were filleted under aseptic conditions . Whole
Kuruma prawns (Penaeus japonicus) and sole (Solea
solea) and the fillets were inoculated with A . hydrophila
by immersion in a bacterial suspension of ca . 104 CFU/ml.
For each storage condition, 12 samples of each type of seafood (whole
fish or fillet) were individually packaged under modified atmospheres
as previously described
(16) . Table
2 shows the pH values of the seafood samples and the conditions in which
they were stored . The storage temperatures and atmospheric compositions
were chosen according to the most frequently used commercial conditions
to store that type of seafood . At suitable time intervals, one sample
was removed, homogenized, diluted, and plated onto both tryptic soy
agar, for total counts, and Aeromonas medium (RYAN agar, Oxoid
CM 833, SR136E), for A . hydrophila
counts .
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TABLE 2 . Growth and
death rates of A . hydrophila in seafoodd
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Predictive modeling of specific growth and death rates.
The
bacterial growth and death curves, generated either with broth or
seafood, were fitted by the model of Baranyi and Roberts
(2) . This estimated the
main parameter, the maximum specific growth or death rate (measured as
the maximum slope of the curve of the natural logarithms of cell
concentration versus time), for combinations of environmental
factors .
The natural logarithms of the maximum specific growth
and death rates obtained [ln(µg)
(growth) and ln(µd) (death)] were then
modeled separately by two multivariate quadratic polynomials of
temperature, pH, and percent CO2 and O2 in the
atmosphere . A stepwise procedure
(22) was used to remove
those coefficients that did not contribute significantly to the model
from both polynomials (P > 0.10) .
The effects of
the environmental variables on the growth and death rates were
quantified by the averages of the generalized Z values,
calculated by a Monte Carlo simulation for both models
(17) . The most important
feature of the generalized Z value can be summarized as
follows: if xi and Zi denote
the ith environmental variable and its generalized Z
value, respectively, then the effect of one unit change in the
xi/Zi normalized variable on
the modeled parameter is about the same, regardless of which
environmental variable was
considered .
Analysis of error of predictions.
In what
follows, bold, lowercase letters will denote vectors that are commonly
used in mathematical texts .
Let x =
(x1...xk)
denote the vector of the studied environmental factors (in this paper,
these are temperature, pH, CO2, and O2; thus,
k = 4) . Let fx be the
natural logarithm of the maximum specific rate observed at x (so fx is a random
variable and Efx is its expected value).
When fitting the secondary model, Efx is
described by the quadratic polynomial model
g(x): g(x) =
Efx .
The core assumption of the least
squares method when fitting broth data is that the
g(x) – fx
error has a zero mean and a constant variance, independent of x (i.e., g is unbiased and minimally distanced
from the data used to create it)
(6) . Therefore, the
collected observations of the natural logarithms of the specific rate
can be considered randomly in the environmental space and we can speak
simply about the mean square error of observed f values with
respect to the g predictions, as follows: MSE(g,f)
= E(f – g)2 .
For
this model, f is a random variable but g is not . If
f is restricted to observations made in food, then
MSE(g,ffood) =
E(ffood – g)2 is an
indicator of the accuracy of the model applied to food . For
example, the accuracy factor of the model g as introduced by
Ross (21) is
approximately Ag =exp[ MSE(g,ffood)] (inasmuch as
we accept that the root MSE and the arithmetical average are close to
each other) . The percent discrepancy of Baranyi et al.
(1) is %D
=
{exp[MSE(g,ffood)]
– 1} x
100 .
Extending the interpolation region.
The model and its
interpolation region are based on broth data . Inside the region, the
predictions generally overestimate the observations made in food . We
wished to consider as many food observations from outside of the
interpolation region as possible to extend the interpolation
(applicability) region of the model .
Rearrange the above
expression as follows:
Notice
that the term (g – Effood) is a
constant (not a random variable); therefore,
so
the cross product in the binomial expression is equal to 0 and the
equation can be rearranged as follows:
Since
(g – Effood) =
E(g – Effood), the first component
of the MSE is the bias of the model for food observations and the
second is the variance of food measurements (Fig.
2):
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Our
basic assumption was that the bias and the variance in food, as defined
above, are constant not only in the interpolation region but also in
its extension, for which only food data exist . Because the logarithmic
transformation of the specific rate was found to be suitable for
modeling, the relative error of the specific rate observations in food
was constant, as was the variance of the log(specific rate) values
observed for food . We used this criterion to extend the interpolation
region: the extension is possible as long as the assumption holds.
Algorithmically, this means that we add those points to the MCP, for
which observations made in food do not contradict this
assumption .
The...|R notation is used
for situations when a statistical indicator is calculated in the
R region . We will use
...|(x1...xn)
notation in a similar way for cases when the indicator is calculated
for the set of
(x1...xn)
points .
Our basic assumption was that
Biasg2(ffood|Rg)
and Var(ffood|Rg) are
constant in an Rg region which includes the
original MCP of the g model .
Testing this assumption is
more reliable when many broth data are used for model creation . To
identify those regions, we introduced a normalized distance
concept .
Let the vectors r1...rn
denote those combinations of environmental factors for which
measurements were made in broth and used for model creation:
rj =
(r1j...rkj), where
j = 1...n .
For this model,
k is the number of environmental factors (temperature, pH,
etc . [in this paper, k = 4]) .
Let
 be the centroid of these points, defined as
follows:
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In
other words, rij is the jth observation of
the ith environmental factor (1
 i
k; 1
j
n), where
k is the dimension of the environmental space and n
is the number of those observations that were used for model creation.
Note that
is not the center of the interpolation region but is
the center of gravity of the data set used to fit the model, and
therefore its location is determined mainly by those intervals of the
environmental factors for which there are many observations .
A
normalized square distance is introduced between any x
=
(x1...xk)
combination of environmental factors and the
centroid,
according to the formula
 |
where
Zi is the average Z value for the
ith environmental factor in the interpolation region, based on
the model predictions .
Let
(y1...ym)
be the set of observations in food that are outside the MCP, sorted in
increasing order according to their (normalized) distances from the
centroid
(Fig.
3) . A subset of food observations, y1...yj
(1
j
m), will be used to extend
the interpolation region if
Var[ffood|(y1...yj)]
Var[ffood|(r1...rn)] .
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FIG . 3 . Relationship
between MSE for observations outside the interpolation region and the
distance between those observations and the centroid of the data set
used to fit the model . Numbers indicate the cumulative numbers of
observations located at a shorter or equal distance . (a) Analysis of
MSE . (b) Analysis of
variance.
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Because
of our basic assumption that the bias is constant, this is equivalent
to
MSE[ffood|(y1...yj)]
MSE[ffood|(r1...rn)] .
Therefore,
test points for which observations are made outside the original MCP
are used to extend the original MCP in a sequence defined by their
distances to the centroid
. For one test point, the MSE between
the predictions and observations is calculated by using all of those
test points outside the MCP that are closer to the centroid than to the
considered test point (Fig.
3) . The procedure stops
when the calculated MSE is larger than that measured inside the
original interpolation region . Once certain test points are accepted,
the extended MCP is determined as described previously
(3) .
Estimation of the probability of growth at the growth-no growth interface.
A logistic
regression model, as introduced previously
(20), was used to
describe the probability of growth, p, as a function of the
temperature, pH, and percent CO2 and O2 in the
atmosphere . Only the observations in laboratory medium were used (Table
1) . The fitted model
is
where
a, b, c, d, and e are the
parameters estimated by the maximum likelihood method .
The
predictive ability of this model was assessed by estimating the
percentage of concordance between predicted probabilities and observed
responses (22) . To
estimate this, we defined growth with a value of 1, while the value for
no growth was 0 . A pair of observations with different responses is
said to be concordant or discordant if the observation with the
response value 1 has a higher or lower predicted probability,
respectively .
The data in Table
1 show the maximum
specific growth and death rates of A . hydrophila in broth . For
all broth experiments, when the pH was
6, the viable counts
decreased with time . The fitted model for the probability of growth,
p, was as follows:
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The
percentage of concordance between the predicted probabilities and
observations in broth used to fit the model was 91% . For
validation of this model, 250 observations in broth, with temperatures
of <11°C, water activities of >0.99, pHs of
<7, and different concentrations of CO2 and
O2, were selected from ComBase . The percentage with
concordance with these data was 85% . According to this model,
the probability of growth of A . hydrophila at pH 6 reaches 0.5
only at a relatively high temperature of ca . 8 to 9°C; the
probability is 0.7 at 11°C .
The effect of the
environmental variables in the region studied was quantified by using
the average Z values (Table
3), which were estimated from the models for the growth and the death rates
described in Table
4 . The number of model coefficients, which was originally 14 with a
standard quadratic surface, was reduced to 4 for the death model and 6
for the growth model . According to the Z values, a 4°C
decrease in the temperature caused a twofold decrease in the growth
rate . The death rate seemed to be unaffected by temperature . The pH
affected both growth and death, but had a larger effect on growth . A
1-U decrease in pH caused a twofold (or 100%) increase in the
death rate and a fourfold decrease in the growth rate . The percentage
of CO2 in the atmosphere also had a greater effect on the
growth rate than on the death rate . An increase of 11% in the
CO2 concentration caused a 10% decrease in the growth
rate but only a 3% increase in the death rate . The effect of
O2 was similar on both growth and death: an increase of 11
to 12% caused a 10% increase in the death rate and a
10% decrease in the growth rate .
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TABLE 3 . Average
Z values: changes in the environmental variables that cause a
twofold increase in the growth or death rate
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TABLE 4 . Parameter
estimates for the models for the natural logarithm of the maximum
specific growth and death rates
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The data in Table
2 show the rates observed
in seafood; A . hydrophila grew in all of the samples tested
except for sole . The soles were eviscerated, but their heads, gills,
and skin were left intact . The numbers of natural contaminating
bacteria were higher than in the other samples at the time of
inoculation, at ca . 104 to 105 CFU/g . A.
hydrophila is not a strong competitor when growing with other
bacteria (15), and the
inoculum could not colonize and compete with the well-established
indigenous flora . Consequently, these data were not used to estimate
the overall error of the model .
Five of the seafood conditions
lay outside of the interpolation region of the model (Table
2) . Of the rates measured
in fresh meat at temperatures up to 11°C in air, vacuum, and
modified atmospheres obtained from ComBase, 33 of 56 were produced
outside of the strict interpolation region of the model (Table
5) .
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TABLE 5 . Growth rates
observed for meat, obtained from ComBasea
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Figure 3 shows how
the extension of the interpolation region was carried out with the
ComBase data for meat . The error between predictions and observations
increased when the distance from the observations to the centroid of
the broth data increased . The MSE and the variance in food inside the
interpolation region were 0.223 (Fig.
3a) and 0.222 (Fig.
3b), respectively . The
closest observations to the centroid were a group of three observations
at an equal distance from the centroid . The MSE and the variance
calculated for these three observations were 0.119 and 0.0932,
respectively . The next step was a group of seven observations . The MSE
for all 10 outside observations was 0.208 and the variance was 0.207.
The next group of observations increased the MSE up to 0.394 and the
variance up to 0.341, which were already higher than the error and
variance inside the interpolation region and indicated a noticeable
increase in the bias . The seafood data were analyzed in the same way
(not shown) . Hence, the extension of the interpolation region was done
with 10 observations in meat selected as indicated above and with data
for all five seafood groups tested . The data in Table
6 show that the error of the model inside the original interpolation
region was practically equal to the error measured in the extended
MCP .
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TABLE 6 . MSE(g,ffood),
Bias2 (g,ffood), and
Var(ffood) of the growth model on the original and
extended MCP for seafood and meat
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The MSE, estimated for the predictions and observations that
were used to fit the data, was 0.22 . Since the model is unbiased for
these data and assuming that the dependence of the growth rates on the
environment is described well enough by the model, the MSE is due to
the variance of the bacterial response in broth . The analysis of the
error when applying the model to food is shown in Table
6 . Because the bias of the
model is very small, the main source of error can be identified as the
variance in food . Moreover, note that the variances of the bacterial
responses in food and in laboratory media were very
similar .
In experiments at pHs of
6 selected from the ComBase database
(http://www.combase.cc; http://wyndmoor.arserrc.gov/combase/),
A . hydrophila died in 31 but grew in 80 . pH values close to 6
may be inhibitory enough to prevent the growth of the organism at
refrigeration temperatures . In addition, in CO2-enriched
atmospheres, low temperatures favor the dissolution of CO2
as carbonic acid into the medium, and consequently the pH value drops
(8) . On the other hand, as
indicated by the Z values in Table
3, the pH had a
significant effect not only on the growth but also on the death of
A . hydrophila . According to the Z values, in the
range from 1.5 to 11°C, a decrease in temperature does not
noticeably increase the rate of death . Low refrigeration temperatures
can prevent bacterial growth, but they do not accelerate bacterial
death . The effect of CO2 was much larger on the growth rates
than on the death rates . The percent O2 had a noticeable
effect on both growth and death . It has also been reported that the
growth of A . hydrophila is slower in air than in atmospheres
saturated with nitrogen
(10) . In situations of
oxygen stress, as reported for Escherichia coli
(7), the growth rate can
be limited by the low intracellular level of superoxide dismutase,
which provides effective protection against superoxide ion toxicity.
Both the faster death rates and the slower growth rates observed with
increasing O2 concentrations in the atmosphere can be
attributed to the toxic consequences of oxygen metabolism
(23) .
The data in
Tables 2 and
5 show the probability of
growth of A . hydrophila and the predicted maximum specific
rate under different conditions . The dominant part of the original
interpolation region was in the environmental space where the
probability of growth was close to or higher than 0.5 . After the
extension, reliable predictions could also be obtained for conditions
with lower probabilities of growth . The extension of the model was
carried out in a region for which a relatively high number of
model-generating data existed . However, this does not imply that the
probability of growth in this region is the highest . In fact, we could
not generate reliable predictions of the specific growth rate in the
region of the highest probability of growth because of the lack of
growth data for that region .
As shown in Fig.
3a, the further the
predictions were from the interpolation region, the higher the MSE was.
Compared to the increase in the MSE, the increase in the variance was
relatively small (Fig.
3b), so we deduced that
when we extrapolated, the increase in the bias was the major component
responsible for the increase in the MSE .
If laboratory media
simulate the conditions in food perfectly, then the MSE between food
observations and model predictions is equal to the error obtained when
fitting the model to laboratory data . When the latter error is smaller,
it indicates more variability in the growth parameters in food and/or
the bias of the model when applied to food . This paper focused on
quantification of the components of the error of a predictive model for
A . hydrophila . The model presented here was practically
unbiased for both meat and seafood and the variances of the bacterial
response in food and in laboratory media were very similar . As a
consequence, the data generated in laboratory media can be utilized
efficiently to study bacterial responses to food
environments .
We acknowledge the ComBase
consortium for making data available .
The support of the European
Commission, Quality of Life and Management of Living Resources, Key
Action 1 (KA1) on Food, Nutrition and Health, project no.
QLK1-CT-2002-300513 is thankfully acknowledged . G.D.G.F . acknowledges
the support of the Comisión Interministerial de Ciencia y
Tecnología (CICYT, Spain) through project ALI99-0405/98 and
project
AGL2000-0692 .
* Corresponding
author . Mailing address: Institute of Food Research, Norwich Research
Park, Norwich NR4 7UA, United Kingdom . Phone: 44 1603 255021 . Fax: 44
1603 255288 . E-mail:
carmen.pin{at}bbsrc.ac.uk .
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