**DOI:**10.1128/AEM.70.7.3925-3932.2004

## ABSTRACT

Specific
growth and death rates of *Aeromonas hydrophila* were measured
in laboratory media under various combinations of temperature, pH, and
percent CO_{2} and O_{2} 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.

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 CO_{2} and O_{2} 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.

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.

## MATERIALS AND METHODS

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. 10^{3} 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% CO_{2} combined
with 0 to 80% O_{2}, with balanced
N_{2}.

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. 10^{4} 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.

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(μ* _{g}*)
(growth) and ln(μ

*) (death)] were then modeled separately by two multivariate quadratic polynomials of temperature, pH, and percent CO*

_{d}_{2}and O

_{2}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 *x _{i}* and

*Z*denote the

_{i}*i*th environmental variable and its generalized

*Z*value, respectively, then the effect of one unit change in the

*x*/

_{i}*Z*normalized variable on the modeled parameter is about the same, regardless of which environmental variable was considered.

_{i}Analysis of error of predictions.In what follows, bold, lowercase letters will denote vectors that are commonly used in mathematical texts.

Let *x* =
(*x** _{1}*…

*x*) denote the vector of the studied environmental factors (in this paper, these are temperature, pH, CO

_{k}_{2}, and O

_{2}; thus,

*k =*4). Let

*f*

**be the natural logarithm of the maximum specific rate observed at**

_{x}*x*(so

*f*

**is a random variable and E**

_{x}*f*

**is its expected value). When fitting the secondary model, E**

_{x}*f*

**is described by the quadratic polynomial model**

_{x}*g*(

*x*):

*g*(

*x*) = E

*f*

**.**

_{x}The core assumption of the least
squares method when fitting broth data is that the
*g*(*x*) − *f*** _{x}**
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*,*f*_{food}) =
E(*f*_{food} − *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 *A _{g}* =exp[√MSE(

*g,f*

_{food})] (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,f*

_{food})]− 1} × 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* − E*f*_{food}) 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* − E*f*_{food}) =
E(*g* − E*f*_{food}), 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):
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…|
(*x*** _{1}**…

*x*) notation in a similar way for cases when the indicator is calculated for the set of (

_{n}*x*

**…**

_{1}*x*) points.

_{n}Our basic assumption was that
Bias_{g}^{2}(*f*_{food}|*R _{g}*)
and Var(

*f*

_{food}|

*R*) are constant in an

_{g}*R*region which includes the original MCP of the

_{g}*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
*r*** _{1}**…

*r*denote those combinations of environmental factors for which measurements were made in broth and used for model creation:

_{n}*r*= (

_{j}*r*…

_{1j}*r*), where

_{kj}*j*= 1…

*n*.

For this model,
*k* is the number of environmental factors (temperature, pH,
etc. [in this paper, *k* = 4]).

Let
r̄ be the centroid of these points, defined as
follows:
In
other words, *r _{ij}* is the

*j*th observation of the

*i*th 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 r̄ 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*=
(*x*** _{1}**…

*x*) combination of environmental factors and the r̄ centroid, according to the formula where

_{k}*Z*is the average

_{i}*Z*value for the

*i*th environmental factor in the interpolation region, based on the model predictions.

Let
(*y*** _{1}**…

*y*) be the set of observations in food that are outside the MCP, sorted in increasing order according to their (normalized) distances from the centroid r̄ (Fig. 3). A subset of food observations,

_{m}*y*

**…**

_{1}*y*(1 ≤

_{j}*j*≤

*m*), will be used to extend the interpolation region if Var[

*f*

_{food}|(

*y*

**…**

_{1}*y*)]≤ Var[

_{j}*f*

_{food}|(

*r*

**…**

_{1}*r*)].

_{n}Because
of our basic assumption that the bias is constant, this is equivalent
to
MSE[*f*_{food}|(*y*** _{1}**…

*y*)]≤ MSE[

_{j}*f*

_{food}|(

*r*

**…**

_{1}*r*)].

_{n}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 r̄. 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 CO_{2} and O_{2} 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.

## RESULTS

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:
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 CO_{2} and
O_{2}, 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 CO_{2} in the atmosphere also had a greater effect on the
growth rate than on the death rate. An increase of 11% in the
CO_{2} concentration caused a 10% decrease in the growth
rate but only a 3% increase in the death rate. The effect of
O_{2} 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.

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. 10^{4} to 10^{5} 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).

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.

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.

## DISCUSSION

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 CO_{2}-enriched
atmospheres, low temperatures favor the dissolution of CO_{2}
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 CO_{2} was much larger on the growth rates
than on the death rates. The percent O_{2} 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 O_{2} 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.

## ACKNOWLEDGMENTS

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.

## FOOTNOTES

- Received 29 October 2003.
- Accepted 1 April 2004.
- ↵*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.

## REFERENCES

- American Society for Microbiology