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O Secondary Models
Thomas Ross and Paw Dalgaard
CONTENTS
3.1 Introduction
3.1.1 Philosophy, Terminology, and Methodology
3.2 Secondary Models for Growth Rate and Lag Time
3.2.1 Square-Root-Type Models
3.2.1.1 Temperature
3.2.1.2 Water Activity
3.2.1.3 pH
3.2.1.4 Other Factors
3.2.2 The Gamma Concept
3.2.2.1 Expanding Existing Models
3.2.3 Cardinal Parameter Models
3.2.3.1 Secondary Lag Time Models and the Concept
of Relative Lag Time
3.2.4 Secondary Models Based on the Arrhenius Equation
3.2.4.1 The Arrhenius Equation
3.2.4.2 Mechanistic Modifications of the Arrhenius Model
3.2.4.3 Empirical Modifications of the Arrhenius Model
3.2.4.4 Application of the Simple Arrhenius Model
3.2.5 Polynomial and Constrained Linear Polynomial Models
3.2.6 Artificial Neural Networks
3.3 Secondary Models for Inactivation
3.4 Probability Models
3.4.1 Introduction
3.4.2 Probability Models
3.4.2.1 Logistic Regression
3.4.2.2 Confounding Factors
3.4.3 Growth/No Growth Interface Models
3.4.3.1 Deterministic Approaches
3.4.3.2 Logistic Regression
3.4.3.3 Relationship to the Minimum Convex Polyhedron
Approach
3.4.3.4 Artificial Neural Networks
3.4.3.5 Evaluation of Goodness of Fit and Comparison
of Models
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3.4.4 Experimental Methods and Design Considerations
3.4.4.1 Measuring Both Growth and Inactivation
3.4.4.2 Inoculum Size
3.4.4.3 Are There Absolute Limits to Microbial Growth?
3.4.4.4 Experimental Design
3.4.4.5 Conclusion
Appendix A3. 1 — Characterization of Environmental Parameters Affecting
Microbial Kinetics in Foods
A 3 . 1 . 1 Temperature
A3 . 1 . 2 Storage Atmosphere
A3. 1 .3 Salt, Water-Phase Salt, and Water Activity
A3. 1.4 pH
A3. 1 .5 Added Preservatives Including Organic Acids, Nitrate,
and Spices
A3. 1.6 Smoke Components
A3. 1 .7 Other Environmental Parameters
References
3.1 INTRODUCTION
Changes in populations of microorganisms in foods over time (i.e., "microbial
kinetics") are governed by storage conditions ("extrinsic" factors) and product
characteristics ("intrinsic" factors). Collectively these have been termed "environ-
mental parameters." They may represent simple situations, e.g., where the storage
temperature is the only important factor influencing microbial kinetics, but in many
foods the environmental parameters that influence microbial kinetics are complex
and dynamic and include the combined effects of extrinsic factors such as temper-
ature and storage atmosphere; intrinsic factors such as water activity, pH, naturally
occurring organic acids, and added preservatives; and interactions between groups
of microorganisms.
Consistent with the widely accepted terminology introduced by Whiting and
Buchanan (1993), we term those models that describe the response of microorgan-
isms to a single set of conditions over time as "primary" models (see Chapter 2).
Models that describe the effect of environmental conditions, e.g., physical, chemical,
and biotic features, on the values of the parameters of a primary model are termed
"secondary" models.
Knowledge of the environmental parameters that most in uence growth of
microorganisms in foods is essential for the development, as well as for the practical
use, of predictive microbiology models. Secondary models that do not include all
the environmental parameters important in a food are said to be "incomplete" (Ross,
Baranyi, McMeekin, 2000) and require expansion (or simple "calibration" if those
factors are constant) to accommodate their effect on microbial kinetics. The envi-
ronmental parameters that are important for particular foods, however, are not always
known. In those situations the systematic approach of predictive microbiology can
help to elucidate the microbial ecology of the product.
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In this chapter we consider a range of types of secondary models including
those that model the probability that a predicted kinetic response will occur. The
chapter includes descriptions and comparison of models, as considerations for
development of robust, secondary models. Appendix A3. 1 details methods to mea-
sure environmental factors of importance — an essential element of the application
of predictive microbiology.
3.1.1 Philosophy, Terminology, and Methodology
The history of predictive microbiology, including the philosophical motivations of
Roberts and Jarvis (1983), who first proposed the concept, w as traced by Ross and
McMeekin (1994). From a purely pragmatic perspective, predictive microbiology
aims to collect and make accessible computerized data on the behavior of microbial
populations in response to defined environmental conditions, but mathematical mod-
eling also provides a useful and rigorous framework for the hypothetico-deductive
scientific process. To develop a consistent framework that enables us to understand
and predict the microbial ecology of foods it is desirable to integrate the patterns
of microbial behavior revealed in predictive modeling studies with knowledge of
the physiology of microorganisms and physical and chemical processes and phe-
nomena that occur in foods and food processes (Ross, Baranyi, McMeekin, 2000).
Various types and categorizations of models are recognized. Empirical models
are, essentially, pragmatic and simply describe a set of data in a convenient mathe-
matical relationship with no consideration of underlying phenomena. Mechanistic
models are built up from theoretical bases and, if they are correctly formulated, can
allow the response to be interpreted in terms of known physical, chemical, and
biological phenomena. An advantage of mechanistic approaches is that they tend to
provide a better foundation for subsequent development and expansion of models;
i.e., taken to their logical extreme, models for specific situations would simply be
special, or reduced, cases of a much larger and holistic model that describes, quan-
titatively, the microbial ecology of foods. The process of developing models that are
able to be integrated readily with other models so as to describe more complex
phenomena has been termed "nesting" or "embedding." A fuller explanation of the
bene ts of that approach w as provided by Baranyi and Roberts (1995).
In one sense, a model is the mathematical expression of a hypothesis. If this
approach is adopted, it follows that the parameters in such models might be readily
interpretable properties of the system under study, and that the mathematical form of
the model would enable interpretation of the interactions between those factors.
Interpretability of model parameters is a feature highly valued by many authors in the
predictive microbiology literature (e.g., Augustin and Carlier, 2000a,b; Rosso et al.,
1993; Wijtzes et al., 1995). Although the development of predictive microbiology has
seen the embedding of more and more mechanistic elements, or at least models whose
structure and parameterization reflects known or hypothesized underlying phenomena,
in practice many models currently available in predictive microbiology are not purely
empirical, and none are purely mechanistic (Ross, Baranyi, McMeekin, 2000).
Another, often cited, advantage of mechanistic models is that if they are built on
sound theory they are more likely to facilitate prediction by extrapolation. Conversely,
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as none of the models in use in predictive microbiology can be considered to be
mechanistic, they can only be used to make predictions by interpolation. (Determi-
nation of the interpolation region encompassed by a model is discussed in Sections
3.2.5 and 3.4.3.4.) It is perhaps ironic, then, that 20 years of experience in predictive
microbiology has not demonstrated the practical usefulness of mechanistic models
that have been proposed to date (see Section 3.2.4). In general, even with good quality
data the mechanistic models do not provide better fit and are us ually harder to work
with than quasi-mechanistic or empirical models currently used.
Predictive microbiology is a specific application of the field of mathematical
modeling and, as such, the same rules of modeling as are applied in those other
disciplines are relevant to the development of predictive food microbiology models.
These have been discussed by various authors (Draper and Smith, 1981; McMeekin
et al., 1993; Ratkowsky, 1993), and an overview is presented in Table 3.1.
Experimental methods and design considerations relevant to kinetic models were
discussed in detail in McMeekin et al. (1993; Chapter 2), Davies (1993), and Legan
et al. (2002) and are also discussed in Chapter 1 . Two points that we feel are necessary
to reiterate are the limitations of the central composite design in predictive micro-
biology studies, and consideration of spoilage domains when growth of spoilage
microorganisms is studied. Legan et al. (2002) accentuated the importance of exper-
imental design in growth modeling studies stating:
in other disciplines, such as engineering, central composite designs are commonly used
for developing response surface models. For microbiological modeling, however, these
designs have serious limitations and should be avoided. Central composite designs
concentrate treatments in the centre of the design space and have fewer treatments in
the extreme regions where biological systems tend to exhibit much greater variability.
Microbial food spoilage is dynamic and in some cases relatively small changes in
environmental parameters cause a complete shift in the micro or a responsible for
product spoilage. Thus, to avoid modeling growth of spoilage microorganisms under
conditions where they have no in uence on quality , a product-oriented approach that
includes determination of the spoilage domain of specific micoor ganisms is often
required (Dalgaard, 2002).
We will not comment further on methodology appropriate to development of kinetic
models, other than to say that to develop reliable secondary models an understanding
of microbial physiology and its interaction with food environments and storage and
processing conditions must be borne in mind in the design of experiments including
the preparation of cultures and interpretation of measurements of population changes.
This issue is particularly explored and exempli ed in Section 3 .4.4.4 concerning exper-
imental considerations relevant to the development of growth limits models.
3.2 SECONDARY MODELS FOR GROWTH RATE AND
LAG TIME
Implicit in the appropriate development and use of secondary models in predictive
food microbiology is the ability to characterize foods, and the environment they
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TABLE 3.1
Some Considerations in the Selection of Models
Subject
Parameter estimation
properties
Stochastic assumption
Parameter
interpretability
Parsimony
Interpolation region
Correct qualitative
features
"Extendibility"
(embedding, nesting)
Reasons
Relates to the procedure and reliability of estimating the model parameters.
In general, models should have parameters that are independent, identically
distributed, normal or "iidn" (see, e.g., Ratkowsky, 1993)
The form of the model, and choice of response variables, should be such
that the difference between prediction and observations (or some
mathematical transformation of them) is normally distributed, and that the
magnitude of the error is independent of the magnitude of the response
modeled. If not, the fitting can be dominated by some data, at the expense
of other data
As noted in the text, it is useful if the parameters have biological/
physical/chemical interpretations that can be readily related to the
independent and dependent variables. This can simplify the process of
model creation and also aid in understanding of the model (This may be
less important than the behavior and performance of the model.)
Models should have no more parameters than are required to describe the
underlying behavior studied. Too many parameters can lead to a model that
ts the error in the data, i.e., generates a model that is specific to a particular
set of observations. Nonparsimonious models have poor predictive ability
No models in predictive microbiology can be considered to be mechanistic
and predictions can be made by interpolation only. Thus, the interpolation
region de nes the useful range of applicability of the model. The
interpolation region is affected by not only the range of individual variables,
but also the experimental design (see Section 3.2.5)
In mathematical terms, these are the analytical properties of the model
function. They include convexity, monotonity, locations of extreme, and
zero values. If biological considerations prescribe any of these, the model
should reflect those properties accurately
When a model is developed further (such as to include more or dynamically
changing environmental factors) the new, more complex model should
embody the old, simpler model as a special case
Source: Modi ed from Ross, T, Baranyi, J., and McMeekin, T A. In Encyclopaedia of Food Microbiology,
Robinson, R., Bart, C.A., and Patel, P. (Eds.), Academic Press, London, 2000, pp. 1699-1710.
present to contaminating microorganisms, in terms of those biotic and abiotic ele-
ments that affect the dynamics of the microbial population of interest. Methods to
characterize the physicochemical environment, including temperature, gaseous
atmosphere, salt and/or water activity, pH and organic acids, spices, smoke, and
other components, are discussed in detail in Appendix A3.1. These topics are also
considered in Chapter 5, including discussion of the in uence o f other organisms
and heterogeneity in the environment. Another important element is the ability to
characterize temporal changes in the environment. Techniques for modeling micro-
bial population dynamics under time -varying conditions are considered in Chapter 7.
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Within predictive microbiology the development and application of secondary
models for growth rates and lag times have been extensively reviewed (Buchanan,
1993b; Davey, 1999; Farber, 1986 ; ICMSF, 1996a,b; McDonald and Sun, 1999;
McMeekin et al, 1993; Ross, 1999a,b; Ross and McMeekin, 1994; Skinner et al.,
1994; Whiting, 1995). This section describes types of secondary growth rate and
lag time models that are currently available, but with particular focus on more
recent developments, and also includes a detailed tabulation of models available
for specific microorganisms.
3.2.1 Square-Root-Type Models
3.2.1.1 Temperature
As discussed later (Section 3.2.4), in many cases the classical Arrhenius equation
is inappropriate to describe the effect of suboptimal temperature on growth rates of
microorganisms because the (apparent) activation energy (E a ) itself is temperature
dependent. To overcome this problem Ratkowsky et al. (1982) suggested a simple
empirical model (Equation 3.1). When this model was fitted to experimental growth
rates the data were square -root transformed to stabilize their variance and this simple
model and its numerous expansions are named square-root-type, Ratkowsky -type,
or Beleradek-type models (McMeekin et al., 1993). These models, and the closely
related cardinal parameter models (see Section 3.2.3), are probably the most impor-
tant group of the secondary models within predictive microbiology.
Jii =b-(T-T . ) (3.1)
V " max v mm y v y
where b is a constant and T is the temperature. The parameter T min , a theoretical
minimum temperature for growth, is the intercept between the model and the tem-
perature axis (Figure 3.1). T min is a model parameter and its value can be 5 to 10°C
lower than the lowest temperature at which growth is actually observed. This inter-
pretation differs from that embodied in the cardinal parameter models, as discussed
in Section 3.2.3 and Chapter 4).
From growth rates measured at several different constant temperatures the
values of b and T min in Equation 3.1 can be determined by classical model tting
techniques (see Chapter 4). Recently it was suggested that b and T min could be
estimated from a single, optimally designed, experiment where growth resulting
from a temperature profile is recorded (Bernaerts et al., 2000). These authors
concluded that such an optimal, dynamic, one-step experiment would reduce the
experimental work required to develop a model signi cantly and would have sub-
stantial potential within predictive microbiology. So far this technique has not found
wider use within predictive microbiology and its ability to estimate model param-
eters accurately remains to be con rmed for different microorganisms and environ-
mental parameters.
Ratkowsky et al. (1983) expanded Equation 3.1 to include the entire biokinetic
range of growth temperatures (Equation 3.2, Figure 3.1). From this model the optimal
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Temperature (°C)
FIGURE 3.1 Simulation of Equation 3.1 (solid line) and Equation 3.2 (dashed line), b =
0.025 h°- 5 /°C, T ]nui = -8°C, c = 0.30°C- 1 ,and T max = 40°C.
growth temperature can be determined by solving the following equation: c x (T opt
- r min ) = exp[c x (r opt - r min )] - 1 (McMeekin et al., 1993).
Ai
M max =b-(T- TV ) • (1 - exp(c(r - T )))
(3.2)
where b and c are constants, T is the temperature, r min the theoretical minimum
temperature below which no growth is possible, and T max is the theoretical maximum
temperature beyond which growth is not possible.
While Ratkowsky et al. (1982, 1983) settled for an exponent of 2, the original
Beleradek models had a variable exponent value. Dantigny (1998) and Dantigny and
Molin (2000) used the concepts of dimensionless growth rate variables (effectively
the same as the gamma factor concept; see Section 3.2.3) to explore the most
appropriate value of the exponent for bacterial growth rate data using Beleradek-
type models. They reported a correlation between the estimate of T min and the
exponent value used and found that when T min and the exponent were simultaneously
fitted by nonlinear re gression, thermophiles had lower fitted e xponent values than
did mesophiles or psychrotrophic organisms. They reported that the use of the square-
root model leads to an underestimation of the minimum temperature for growth
when the exponent value is significantly less than 2.
3.2.1.2 Water Activity
McMeekin et al. (1987) found that growth responses of Staphylococcus xylosus
followed Equation 3.1 at different values of water activity. T min was constant and
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thus independent of water activity and Equation 3.3 was suggested to describe the
combined effect of temperature and water activity (McMeekin et al., 1987).
M =b-(T-T . )-a -a . (3.3)
" max v mm ' -\l w w mm v '
where b and T mm are as previously de ned, a w is the water activity, and a w min is the
theoretical minimum water activity below which growth is not possible.
Later, Miles et al. (1 997) suggested that Equation 3.4 be used to take into account
the effect of the entire biokinetic ranges of both temperature and water activity.
M =b-(T-T . )-Q-exp(c(T-T )))■ (a -a . )(1 - exp(d(a -a )))
"max v rmn- / v .rv \ max /7/ \f v w wmin /v -^ v v w wmax 777
(3.4)
where b, c, T, T min , T max , a w and a w min are as previously defined, d is a fitted constant,
and a w max is a theoretical maximum water activity beyond which growth is not
possible.
Most food-related microorganisms grow at water activities very close to 1.000
and in those cases the expanded water activity term (i.e., containing a w max ) in
Equation 3.4 is not needed to predict growth in foods. However, some microorgan-
isms, e.g., several marine bacteria, have a substantial requirement for minerals. To
model growth responses of these microorganisms, the inhibitory effect of high water
activities, i.e., low salt concentrations, must be taken into account. For the human
pathogen Vibrio parahae mo lyticus, a wmax has been determined to be 0.998. Some
seafood spoilage bacteria are more inhibited by high water activity; e.g., growth of
Halobacterium salinarium was only observed at a w values below 0.9 (Chandler and
McMeekin, 1989; Doe and Heruwati, 1988; Miles et al., 1997).
3.2.1.3 pH
Vox Yersinia enterocolitica, Adams et al. (1991) found that growth responses followed
Equation 3.1 at different values of pH. Again, T mm was constant and Equation 3.5
was suggested.
m =b-(T-T )JpH-pH . (3.5)
V " max v min y \j -* ■* min v y
where pH mm is the theoretical minimum pH below which growth is not possible and
other parameters are as previously defined.
On the basis of the observation of Cole et al. (1990) that growth rate was
proportional to hydrogen ion concentration, Presser et al. (1997) introduced the
following quasi-mechanistic term to describe the effect of pH on bacterial growth:
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P m ax = Pop,x(l-10^'"" p// ) (3.6a)
By analogy, another term was introduced for superoptimal (i.e., alkaline) pH con-
ditions, leading to the following model for the entire biokinetic pH range:
M max = IV x (1 " 1 pHmia ~ pH ) x (1 " 1 pH ~ pH ™ ) (3 .6b)
The validity of that term was evaluated against an extensive data set for Escherichia
coli growth, including variables of temperature, water activity, and lactic acid con-
centration for a range of acid and alkaline environmental pH levels (see Equation 3.10).
Wijtzes et al. (1995, 2001) continued the development of square -root-type mod-
els and suggested Equation 3.7 for growth responses of Lactobacillus curvatus at
different temperatures, a w values, and pH
V=b-(a w -a wm J-(pH-pH im J-(pH-pH m J-(T-T mn f (3.7)
3.2.1.4 Other Factors
Equation 3.8 was suggested to model the effect of carbon dioxide-enriched (%C0 2 )
atmospheres on growth of the specific spoilage organism Photobacterium phospho-
reum on sh (Dalgaard, 1995; Dalgaard et al., 1997). Later, similar but square -root-
transformed terms were used to model the effect of C0 2 and sodium lactate (NaL)
on growth of Lactobacillus sake and Listeria monocytogenes at a constant pH
(Equation 3.9; Devlieghere et al., 1998, 2000a,b, 2001).
(%ca -%ca)
M =b(T-T . ) x -^ — (3.8)
"max v min / (\/r^r\ v '
2 max
"V •"*■ max V min
>max = b
•(T-T . )
V mm '
mm
a w~ a ww* (3.9)
• [CO, -CO,
-v 2 max 2
• J NaL -NaL
v max
As noted above, a more comprehensive square-root-type model that includes the
effects of temperature, pH, water activity, and lactic acid has been suggested and
developed in a series of publications (Presser et al., 1997; Ross, 1993a,b; Salter et al.,
1998; Tienungoon, 1988) and has been applied to Listeria monocytogenes and Escher-
ichia coli growth rates. It was presented in its most complete form in Ross et al. (2003):
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U = C
r* max
x(T-T mm )x(l-exp(dx(T-T)))
max
x . fa ^a ~ x (1 - expfe x (a -a )))
^7 w wmin v r\&\w wmax ///
Xa/i-IO^""^
x A /i_io p// "- p// °'«
(3.10)
x 1-
x 1-
L4C
1/ ■ x(l + 10 p// " M ")
rain v J
LAC
D . x(l + 10
mm v
/* fl -pff
mm
)
where c, d, and g are fitted parameters, LAC is the lactic acid concentration (mM),
^min the minimum concentration (iriM) of undissociated lactic acid that prevents
growth when all other factors are optimal, D inin the minimum concentration (mM) of
dissociated lactic acid that prevents growth when all other factors are optimal, pK a is
the pH for which concentrations of undissociated and dissociated lactic acid are equal,
reported to be 3.86 (Budavari, 1989), and all other terms are as previously defined.
One of the advantages of the square -root-type models, and the cardinal param-
eters models, is that their form enables them to be readily simplified into models
for special cases; e.g., in Equation 3.10, if one factor is held constant then the terms
involving that factor simply reduce to constants.
An example is a model developed for Listeria monocytogenes (Ross et al, in
press; WHO/FAO, in press), in which the superoptimal water activity term is not
relevant, and in which a term for the effect of nitrite on L. monocytogenes growth
rate was also included. That novel term was based on analysis of the predictions of
the Pathogen Modeling Program (Buchanan, 1993a; www.arserrc.gov/mfs/patho-
gen.htm). The fitted model is shown in Equation 3.11.
M
max
= 0.1626
x(T- 0.60) x (1 - exp(0. 1 29 x (T - 5 1 .0)))
xJ(a -0.925)
w
xJl-lO^-p*
x 1-
LAC
4.55x(l + 10 pH " 386 )
(
493 - NIT x
1 +
v
(6.5 -pH)
2
\
/
\
/
(3.11)
\
'493
/
where NIT is the concentration of nitrite and all other terms are as previously defined.
As shown in Figure 3.2, Equation 3.10 and Equation 3.11 represent a new
generation of square -root-type models where the level of lactic acid in uences the
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range of pH values for which growth is theoretically observed, reflecting the known
interaction between pH and undissociated lactic acid, and also the individual growth
rate suppressing effects of hydrogen ion concentration and undissociated lactic acid
concentration. This was not the case for the environmental parameters included in
Equation 3.1 to Equation 3.9. In those models, each term expressed how an envi-
ronmental factor reduced the growth rate of a microorganism. However, for those
models the expected multidimensional growth space was not influenced by levels
of the different environmental parameters. This limitation of predictive models for
growth rate has been recognized and has led, in part, to the development of growth/no
growth models (discussed in Section 3.2.3 and Section 3.4). To make accurate
predictions, a model can include terms to force the predicted growth rate to zero
(Augustin and Carlier, 2000b; Le Marc et al., 2002). Alternatively, the probability
of growth under the test conditions can first be assessed using a growth boundary
model. If growth is possible, a growth rate model in combination with a lag time
model can be used to estimate the extent of growth (Ross et al., in press; WHO/FAO,
in press).
It is also notable that the pH and lactic acid terms in Equation 3. 10 are effectively
gamma-model type terms (see Section 3.2.2), in which the effect of the level of
growth rate inhibitor is scaled between and 1, where 1 represents no inhibition,
i.e., the optimal level of that environmental factor. In the case of lactic acid, the
optimal level would be 0, while for pH the optimum is ~7. This illustrates the close
relationship between square-root-type models, and those that embody the gamma
concept, such as the cardinal parameter models.
3.2.2 The Gamma Concept
The concept of dimensionless growth factors, now known as the gamma (y) concept,
was introduced in predictive microbiology by Zwietering et al. (1992). Later, minor
changes and new developments were added (Wijtzes et al., 1998, 2001; Zwietering,
1999; Zwietering et al., 1996).
The gamma (y) concept relies on:
1. The observation (e.g., Adams et al., 1991; McMeekin et al., 1987) that
many factors that affect microbial growth rate act independently, and that
the effect of each measurable factor on growth rate can be represented by
a discrete term that is multiplied by terms for the effect of all other growth
rate affecting factors, i.e.:
(i, = /temperature) x/a w ) x/(pH) x /(organic acid)
xXother^ x/(other 2 ) x ... ../(otherj
2. That the effect on growth rate of any factor can be expressed as a fraction
of the maximum growth rate (i.e., the rate when that environmental factor
is at the optimum level)
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a)
in
6
><
to
E
zi.
4.0 4.5 5.0 5.5
6.0 6.5 7.0 7.5 8.0 8.5 9.0
PH
b)
in
6
X
TO
5
1.0-1
0.9-
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
• *
//
i — M-
l i
4.0 4.5 5.0 5.5
i i i i
6.0 6.5 7.0 7.5
i i i
8.0 8.5 9.0
PH
FIGURE 3.2 Simulation of Equation 3.7 (a) and Equation 3.10 (b) at a fixed temperature
and water activity. pH inin is 4.0 and pH^^ is 9.0. For Equation 3.10, U ir]hl = 10 mM and D max
= 1000 mM The concentrations of lactic acid (LAC) depicted are mM (dashed line), 50
mM (dotted line), and 100 mM (dash-dotted line).
Under completely optimal conditions each microorganism has a reproducible
maximum growth rate, notwithstanding the potential effect of strain variability. As
any environmental factor becomes suboptimal the growth rate declines in a predict-
able manner, and the extent of that inhibition can be related to the optimum growth
rate by calculating the relative rate at the test condition compared to that at the
optimum. Thus, under the gamma concept approach, the cumulative effect of many
factors poised at suboptimal levels can be estimated from the product of the relative
inhibition of growth rate due to each factor, as indicated by Equation 3.12. The
relative inhibitory effect of a specific environmental variable is described by a growth
factor "gamma" (y), a dimensionless measure that has a value between and 1 (e.g.,
Equation 3.13 to Equation 3.15).
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The relative inhibitory effect can be determined from the "distance" between
the optimal level of the factor and the minimum (or maximum) level that completely
inhibits growth by recourse to a predictive model. In the gamma model approach,
the reference growth rate is |i max , so that reference levels of temperature, water
activity, etc. are those that are the optimum for growth rate, usually represented as
T opt , # w opt , pH opt , etc. The combined effect of several environmental factors is then
determined by multiplication of their respective y factors (Equation 3.16).
Y =
Growth rate at actual environmental conditions
Growth rate at optimal environmental conditions
P m ax(7>„pH,etc.)
M
max opt
(3.12)
y(T) =
T-T
mm
\ 2
T -T -
y opt mm j
(3.13)
YK) =
a — a
w w mm
1 — a
w mm
y(pH) =
{pH-pH^ydpH-pH)
(P H n „, ~ P H mm ) • (P H n™ - P H o,J
opt
opt
(3.14)
(3.15)
P,»x = P
max opt
V(T)-y(aJ-y(pH)
(3.16)
The effect of environmental parameters like carbon dioxide, sodium lactate, and
nitrite has also been included in square-root-type models (see, e.g., Equation 3.8 to
Equation 3.11). The absence of these inhibitory substances is optimal for growth
and therefore the calculation of y factors requires information only about the lowest
concentration of each substance that prevents growth (or, similarly, the maximum
level that can be tolerated before growth ceases) analogous to minimum inhibitory
concentrations (MICs).
y(co 2 ) =
' /<>co 2max -%co 2 v -
V
%co, _ - %co, t
2 max 2 opt
r %co 2wK -%co 2 v
/
V
%co
2 max
J
(3.17)
3.2.2.1 Expanding Existing Models
Given that there is a finite number of models (see Table 3.5 and Table 3.6), and that
few models include factors of relevance to all foods, some workers have attempted
to integrate terms for specific variables from one model into another to suit a specific
2004 by Robin C. McKellar and Xuewen Lu
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food and the conditions of interest. Because of the assumption of independent action
of growth rate inhibitors, the dimensionless y factors can, in principle, be readily
exchanged between existing models and, at the time of writing, this is increasingly
being done. Values of parameters like \i max , [\ ovP T nmv T opt , a wmm , pH mm , pH opt , pH max ,
and %C0 2max from which gamma factors can be derived are known for a considerable
number of food-related pathogenic microorganisms. The approach was possibly
taken to its logical conclusion by Augustin and Carlier (2000a,b) who collated, and
integrated into a single model, literature data and observations for more than 15
factors in foods that affect the growth rate of L. monocytogenes.
For spoilage bacteria from chilled foods, growth kinetics at low temperatures
are often well characterized but values of (J max , |i opt , r opt , pH opt , and pH max are fre-
quently unknown or have not been determined accurately. This is the case, for
example, for the specific spoilage organisms Photobacterium phosphoreum,
Shewanella piitrefaciens, and Brochothrix thermosphacta. In this situation the clas-
sical gamma concept cannot be used to develop a secondary model. However, when
a simple square-root-type model including the effect of temperature and, e.g., C0 2 ,
has been developed for chilled product stored at a known pH (pH ref ) and water
activity (a w ref ) then these models can be expanded at suboptimal growth conditions
by addition of y-like factors, as shown in Equation 3.18 (Dalgaard et al., 2003).
Umax = b
• (T-T . )
v mm /
nun
^C0 lmax -%C0 2 )l%CQ lmm (3.18)
Ma - a . ) I (a , — a . )
A/ v w w mm / v w ref w mm y
(pH-pH m J/(pH ref - P H m J
Clearly, this approach should be used with some caution because the assumption
of independent action has not been tested for all environmental factor combinations.
Thus, the range of applicability of the expanded model should be evaluated, e.g.,
by comparison with data from challenge tests or naturally contaminated products
(Gimenez and Dalgaard, in press). (Section 3.2.5 discusses the expansion of existing
polynomial models.)
3.2.3 Cardinal Parameter Models
Cardinal parameter models (CPMs) were introduced to predictive microbiology in
1993 and have become an important group of empirical secondary models (Augustin
and Carlier, 2000a,b; Le Marc et al., 2002; Messens et al., 2002; Pouillot et al.,
2003; Rosso, 1995, 1999; Rosso et al., 1993, 1995; Rosso and Robinson, 2001).
The basic idea behind CPMs is to use model parameters that have a biological or
graphical interpretation. When models are fitted to experimental data by nonlinear
regression (see Chapter 4), this has the obvious advantage that appropriate starting
values are easy to determine. General CPMs rely on the assumption that the inhib-
itory effect of environmental factors is multiplicative, an assumption that was
2004 by Robin C. McKellar and Xuewen Lu
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formalized in the gamma (y) concept discussed above (Section 3.2.2). Thus, general
CPMs consist of a discrete term for each environmental factor, with each term
expressed as the growth rate relative to that when that factor is optimal; i.e., each
term has a numerical value between and 1 . At optimal growth conditions all terms
have a value of 1 and thus (i max is equal to |i. t (Equation 3.19).
Equation 3. 19 to Equation 3.21 show a CPM that includes the effect of temper-
ature (T), water activity (a w ), pH, inhibitory substances (c z ) and qualitative factors
(k) on (i max (Augustin and Carlier, 2000a). This extensive CPM was developed from
available literature data from many studies for growth of Listeria monocytogenes.
The inhibitory substances included (1) undissociated acetic acid, lactic acid, and
citric acid, (2) Na-benzoate, K -sorb ate, and the undissociated form of sodium nitrite,
and (3) glycerol monolaurin, butylated hydroxy anisole, butylated hydroxy toluene,
fer^-butylhydroquinone, C0 2 , caffeine, and phenol. In addition, the effect of com-
petitive growth of microorganisms and the inhibitory effect due to specific types of
foods were included in the model as qualitative factors.
n p
M IM x = K P , ■CM 1 {T)-CM i {aJ-CM l (pH)-Y[y(c l )-\\ k j ( 319 )
i=\
7=1
CM,
a
0.
(X-X )-(X-X . )
v may / V mm J
n
max
mm
.71-1
(x -x . ) -[(X t -x . yoc-x t )-(x t -x)
V nnt mm-' L\ nnt mm-' \ nnt / V nnt max-'
opt
mm
opt
mm
opt
opt
«n-\)-X t +X -«■!)]
vv y opt mm /J
0.
X < x„.
mm
(3.20)
X . <x<x
mm max
x>x
max
Yfe) =
(i-c./Mic.y
o.
c <MIC.
i i
c >MIC.
(3.21)
where X is temperature, water activity, or pH. X min and X max are, respectively, the
values of X t below and above which no growth occurs, X opt is the value at which
Umax i s equal to its optimal value (i opt . MIC t is the minimal inhibitory concentration
of specific compounds above which no growth occurs.
Within predictive microbiology various CPMs were developed during the 1 990s
and in the same period different cardinal parameter temperature models were inde-
pendently developed in other fields, e.g., to predict the effect of temperature on
growth rates (r) of crops (Equation 3 .22; Yan and Hunt, 1 999; Yin and Wallace, 1 995).
(
r = 7i
max
T-T
mm
\
T -T
V opt mm i
(
T -T
max
\
T -T
v max opt J
T —T
max opt
T -T ■
opt mm
(3.22)
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~V
TABLE 3.2
Parameter Values in Square-Root Type (Sqrt) and Cardinal Parameter
Models (CPM)
min
opt
^o
Pt
Organism
Sqrt CPM Sqrt CPM Sqrt CPM Sqrt CPM
Reference
Escherichia coli 2.9 4.9 41.0 41.3 49.2 47.5 2.3 2.3 Rosso et al. (1993)
Salmonella Typhimurium 3.8 5.7 39.8 40.0 51.1 49.3 1.7 1.7 Oscar (2002)
pH min
PH
c i
pH ma
R
opt
Sqrt CPM Sqrt CPM Sqrt CPM Sqrt CPM
Listeria monocytogenes 4.2 4.6 7.0 7.1 9.8 9.4 1.0 0.95 Rosso et al. (1995)
In several ways CPMs resemble square-root models and responses of the two
types of models can be practically identical, e.g., for the effect of temperature, water
activity, and pH (Oscar, 2002, Rosso et al., 1993, 1995). Parameters in the two types
of models are typically named T imn , T max , a w min , a w max , pH mm , and pH max . However,
these model parameters are not de ned in entirely the same way for CPMs and
square-root-type models. In fact, when identical data are fitted to the two types of
models square -root-type models estimate lower T irdn , # w Illin , and pH min values and
higher r max , # wmax , and pH max values (Table 3.2; see also Chapter 4).
r mm values estimated by CPMs and square-root-type models often differ by
~2°C as shown in Table 3.2. Table 3.3 shows that a 2°C difference of a r min value
has a pronounced effect on (J max values predicted by both a square -root-type model
and a CPM. Thus, parameter values estimated by using one of these types of models
TABLE 3.3
Effect of 7" min Values (-1°C and +1°C) on \i max Values Predicted by a
Square-Root and a Cardinal Parameter Model at 4, 8, and 12°C
Square-Root Model 3
Cardinal Parameter Model'
Temperature
(°C)
rhnax
(h" 1 )
/o
H"max
(h" 1 )
o/
/o
T = -1 °C
'mm ' *-
7min = +1°C
Difference
T = -1 °C
'mm ' *~
^mi„=+1°C
Difference
4
0.0216
0.0078
64
0.0216
0.0087
60
8
0.0700
0.0424
40
0.0718
0.0485
33
12
0.1461
0.1046
28
0.1549
0.1237
20
a The model of Ratkowsky et al. (1983) used with values of the model parameters b and c selected to
obtain a T opt value ~37°C and a U opt value of- 1.0 h 1 . T max was 45.0°C.
b The model of Rosso et al. (1993) used with T opt of 37°C, u opt 1.0 h 1 , and T max 45.0°C.
2004 by Robin C. McKellar and Xuewen Lu
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1 I
-10 -5
i
t 1 1 1 1 1 1 1 1 r
5 10 15 20 25 30 35 40 45 50
Temperature (°C)
FIGURE 3.3 Simulation of the model \i max = u opt x CM 2 (7), with CM 2 (T) given by Equation
3.23 and with 7^ of-6°C, T x of 1 °C, T c of 12°C, 7 0Pt of 37°C, and T max of 45°C. u opt was 1 .0 lr 1 .
cannot be used with the other type of model. This situation is similar to the
estimation of |i max values by some primary growth models. Modi ed Gompertz
models (Gibson et al., 1987; Zwietering et al., 1990), e.g., overestimate |i. max by
-15% (Dalgaard et al., 1994; Membre et al., 1999; Whiting and Cygnarowicz-
Provost, 1992) and their growth rate values should not be used together with the
exponential, the logistic, or other Richards family of growth models that rely on
accurate |i max values.
Classical CPMs (Equation 3.19 and Equation 3.20) as well as square-root-type
models describe a straight line relation between suboptimal temperatures and
|i max (Figure 3.1 and Figure 4.4 [Chapter 4]). It has been reported by Bajard et al.
(1996) that a different, biphasic, relationship can be observed for some strains of
Listeria monocytogenes. More recently, Le Marc et al. (2002) observed a biphasic
relationship for a strain of Listeria innocua. Le Marc et al. (2002) suggested an
expanded CPM (Equation 3.23) to simulate this type of growth response (Figure
3.3). In Equation 3.23, r c is the change temperature and T x corresponds to the T 1Xihl
value in a classical cardinal temperature model (Rosso et al., 1993). McMeekin et
al. (1993), however, cautioned against the interpretation of apparently continuously
curved relationships as the combination of two linear responses, and provided a
simple illustration of the effect. It should also be noted that other workers (e.g.,
Nichols et al., 2002a,b; see also Chapter 4) have not observed the "curvature" in
the low temperature region of L. monocytogenes growth.
2004 by Robin C. McKellar and Xuewen Lu
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CM (T) =
(T-T.) 2 (T-T )
v l 7 v max 7
(T -T)-[(T -T)-(T-T f )-(T -T )(T . + T -2T)]
v opt l 7 LV opt l 7 v opt-' v opt max 7 v opt 1 7J
(T -T) 2 (T-T )
v c l 7 v max 7
,T<T<T
in ax
- / J* _J- \ 2
min
<r„ - r,) ■ [(r w - r,) -(T e - r ,) - ( r , - r„) • (r , + r, - 2 ■ r c )]
opt
opt
opt
opt
T-T
\ c min J
mm
, T . <T<T
(3.23)
As stated above, general CPMs rely on the assumption that different environ-
mental parameters have independent and thereby multiplicative effects on (i max
(Equation 3.19). The successful use of many general CPMs and square-root-type
models has shown this assumption to be reasonable for wide ranges of environmental
conditions. However, numerous studies have shown that the growth range of a
microorganism to one environmental condition is affected by other environmental
factors (see Section 3.4). This suggests that the predictive accuracy of general CPMs
can be improved by taking into account interactions between environmental param-
eters, particularly where one factor is sufficiently stringent that it reduces the growth
range of the organism in other environmental "dimensions."
Various approaches have been suggested to describe growth limits under the
influence of multiple variables (see Section 3.4.4). Two such approaches have been
suggested for direct incorporation in CPMs and are discussed briefly here. Augustin
and Carlier (2000b) developed a global secondary model for L. monocytogenes,
including terms for interactions that prevented growth. Absolute minimal cardinal
values X^ [n were estimated by assuming that all inhibitory substances were absent.
Similarly, absolute minimal inhibitory concentrations MIC® were estimated for
optimal concentrations of other environmental parameters (X = X opt ). Then, interac-
tion between environmental parameters was taken into account by modifying each
of the X° in values (Equation 3.24) and the MIC® values, depending on levels of
other environmental parameters. After calculation of appropriate r min , a w min , pH min ,
and MIC f values, growth rates were then predicted by using Equation 3.19 to
Equation 3.21.
r
a
n
X ■ =X -(X -X°. )
mm opt x opt min 7
1-
C.
\
(
V
?=1
MIC.
' /
Y -Y
opt
-0
\
y - r u
. opt min j
r v^ 1/3
Z -Z
opt
z -z°.
V opt mill j
(3.24)
with X, 7, and Z being temperature, pH, or water activity.
A different approach was used by Le Marc et al. (2002) to model the interactive
effects of temperature, pH, and concentration of undissociated organic acids (HA)
on growth of Listeria innocua. Cardinal parameter values were kept constant and
the space of environmental factors was divided into (1) the independent effect space
(^ = 1), (2) the interaction space (0 < ^ < 1), and (3) the no growth space (^ = 0)
(Equation 3.25 to Equation 3.27).
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M max = K B , ■ CM i CO ■ CM, {pH) ■ x{\HA\) ■ %(T, P H, [HA])
(3.25)
Z,(<tfT,pH,ULA)) =
1 y<0
2(1 - v) 9 < y < 1
V|/>1
(3.26)
¥ =
I
<P,
2n (i -v
(3.27)
with (p r =(W CM 2CO)-> <? pH = {\-CM l (pH))\im& ^ UndissocjatedLacticAcid(JJLA)
= 1 - (ULA/MIC ULA )) and where e i are the environmental factors. For calculation of
CM 2 (T) and CM^pH), see Equation 3.20. Le Marc et al. (2002) selected a value of
0.5, which was used for 9.
The performance of the two approaches to model interaction between environ-
mental parameters is considered in greater detail in Section 3.4.4. As shown above,
CPMs that take into account the effect of interaction between environmental param-
eters are relatively complicated models. Thus, these models are not fully in agree-
ment with the originally cardinal parameter modeling approach, i.e., that CPM uses
only simple biological meaningful parameters that microbiologists are familiar with
and that are easy to use by biologists (Rosso et al., 1993), and raises questions about
whether those models are the most parsimonious forms available.
The model suggested by Augustin and Carlier (2000b) predicts the effect of
interaction between temperature, pH, and lactic acid concentration on growth of
Listeria monocytogenes to be more pronounced than the effect predicted for Listeria
innocua by the model of Le Marc et al. (2002). For example, the Augustin and
Carlier (2000b) model predicts no growth of Listeria monocytogenes at 8°C, pH
6.0, and with 200 mM of lactic acid, whereas at this condition the model of Le Marc
et al. (2002) predicts growth and also that there is no interactive effect of the
environmental factors (t > = 1). Recently, Gimenez and Dalgaard (in press) found the
model of Augustin and Carlier (2000b) to substantially underestimate growth of
Listeria monocytogenes in cold-smoked salmon. This could indicate that the model
is in fact overestimating the importance of the interaction between at least some sets
of environmental factors.
In a similar vein Ratkowsky and Ross (1995), recognizing the relationship
between absolute limits for each environmental factor and their relationship to the
parameters of square-root-type models and CPMs, experimented with the use of a
kinetic model as the basis of a growth boundary model using linear logistic regres-
sion. This approach is discussed later (see Section 3.4.3.2).
The classical CPMs, in particular those including the effect of temperature, water
activity, or pH, are now popular and used for many purposes within predictive
microbiology (see Table 3.5 and Table 3.6). As one example a cardinal temperature
and pH model has been combined with classical models of microbial kinetics, i.e.,
2004 by Robin C. McKellar and Xuewen Lu
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models that rely on yield factors and maintenance constants. In this way, production
of curvacin A by Lactobacillus curvatus LTH 1174 growing in MRS broth was
successfully modeled between 20 and 38°C and at pH values from 4.8 to 7.0
(Messens et al., 2002). Other examples include the use for CPMs to predict the
radial growth rate of molds on solidi ed laboratory media (Panagou et al., 2003;
Rosso and Robinson, 2001; Sautour et al., 2001). The ability of these models to
predict growth in foods deserves further study.
For practical use of secondary predictive models it is important to know the
precision of the predicted responses. With CPMs it has been suggested to determine
cardinal parameters values for a number of different strains within each of the
microbial species of interest (Membre et al., 2002). In this way a measure of intra -
species variability can be obtained. As an example, variability in the pH min value for
10 strains of E. coli was ±0.20 corresponding to approximately four times the
experimental error (Membre et al., 2002). More recently Pouillot et al. (2003)
suggested the use of a CPM together with a Bayesian procedure for parameter
estimation. This approach includes the use of hyperparameters and allows uncer-
tainty (due to imperfect knowledge or data) and true variability (e.g., due to differ-
ence between strains) to be determined separately (see also Chapter 4). The approach
seems most interesting and de nitely deserves to be studied further for different
secondary predictive models.
3.2.3.1 Secondary Lag Time Models and the Concept
of Relative Lag Time
When exponentially growing microorganisms are transferred from one environment
into another, similar environment, growth usually continues without delay, i.e., a lag
time is rarely observed. However, when the two environments differ, a lag time is
often observed. Similarly, when microorganisms in the lag or stationary phases are
transferred into identical or new environmental conditions a lag time may continue
or result, respectively. Depending on the physiological state of the microorganisms,
the magnitude of the shift in the environmental conditions, and the new environmental
conditions themselves, the duration of the lag time may range from to in nity.
Development of secondary lag time models is complicated by the fact that lag
time is in uenced not only by the actual environmental conditions but also by
previous environmental conditions and the physiological status of the cell, i.e., the
growth phase of microorganisms at the time of transfer between environments and
their "enzymatic readiness" to exploit the specific carbon and energy resources
within the new environment. Within predictive microbiology, two main approaches
have been used for development of secondary lag time models: (1) models where
lag time and growth rate are modeled independently and (2) models where lag time
is assumed proportional to the generation time. The latter group of models typically
rely on the assumption that microorganisms need to perform a given amount of work
to adapt to a new environment and that the rate at which this work can be done
depends on the growth rate potential of the organism in the new environment
(Robinson et al., 1998).
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In the former approach, lag times or lag rates (i.e., reciprocal of lag time) are
typically log-transformed to stabilize the variance of these data. Frequently, poly-
nomial models (see Section 3.2.5) or artificial neural networks (see Section 3.2.6)
have been used to develop independent secondary lag time models (Table 3.5). To
model the effect of temperature downshifts, temperature upshifts, and physiological
status of cells (e.g., exponential phase, stationary phase, starved, frozen, dried),
separate polynomial models have been used for the different physiological conditions
(Whiting and Bagi, 2002). When square-root-type and Arrhenius-type models are
used for lag time modeling, lag rates are modeled or reciprocal forms of the growth
rate models are used (see Section 3.2.1 and Section 3.2.4; Table 3.5 and Table 3.6).
Zwietering et al. (1994), e.g., used a square-root model (Equation 3.2) with
identical values of the parameters T min , c, and T msx to model lag time and growth
rate — only the value of b differed between the two models. Specific secondary lag
time models for particular environmental parameters have also been suggested, e.g.,
a hyperbola model for the effect of temperature (Equation 3.28; Oscar, 2002; Zwi-
etering et al., 1994):
X =
P
T-q
(3.28)
where X is the lag time, T the temperature, p the rate of change of lag time as a
function of temperature, q the temperature at which lag time is in nite, and m is an
exponent to be estimated.
Baranyi and Roberts (1994), Smith (1985), and McMeekin et al. (1993) have
observed that lag times for identical inocula introduced to (at least some) envi-
ronmental conditions are inversely proportional to growth rates and thus propor-
tional to generation times (T gen ). This generalization has limits, however, as dis-
cussed further below and probably is most relevant to changes in environmental
temperature . For example, Zwietering et al. (1994) showed that for the effect of
temperature on Lactobacillus plantarum the product of |i max and lag time (k) was
constant and had an average value close to 2. In these situations secondary lag
time models can be derived directly from a growth rate model by using the simple
concept of relative lag time (RLT; Equation 3.29) in common use but first defined
by Mellefont and Ross (2003). Clearly, RLT reflects the physiological status of
microorganisms introduced into a new environment as well as the difference
between their actual and their previous environments, and can be interpreted as
the amount of work the cell has to do to change its physiology (e.g., enzymes,
membrane composition, number of ribo somes) to be able to grow at |i max in that
new environment.
Baranyi and Roberts (1994) suggested a primary model to estimate lag times
from microbial growth curves and this model allowed determination of the para-
meters h , q , and a all of which reflect the physiological state of microorganisms
and, thereby, their readiness to grow in a given environment (Equation 3.29; Chapter
2). It can be seen that the parameter RLT is directly proportional to h .
2004 by Robin C. McKellar and Xuewen Lu
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^ = rlt x = **>t.W2)
gen r 1- max
(3.29)
( 1 "\
X-ii =RLT-\n(2) = h = In 1 + — =-ln(oc )
"max v y o v o J
where all parameters have meanings as indicated earlier.
Experimental methods to determine the physiological status of low levels of
microorganisms in foods remain to be developed. Thus, for the time being these
parameters have mainly theoretical importance.
The RLT concept is practically very useful for development of secondary lag
time models, but it should be used with caution. Delignette-Muller (1998) eval-
uated data from nine studies where the effect of temperature, pH, NaCl, and
NaN0 2 on lag time and generation time on different food-borne microorganisms
had been modeled independently. In four of the nine studies, RLT was constant
and an independent lag time model was not needed. However, primarily pH and
NaCl in uenced RLT in the remaining studies. On the basis of large amounts of
experimental data, Ross (1999a) showed the distribution of RLT of B. stearother-
mophilus, Clostridium perfringens, E. coli, L. monocytogenes, Salmonella, and
S. aureus included peaks in the range 3 to 6 under a very wide range of experi-
mental conditions. These distributions were similar to those presented by Augus-
tin and Carlier (2000a), who observed a median RLT of 3.09 for L. monocytogenes
(n = 1176). Using extreme environmental shifts, and severely growth-limiting
outgrowth conditions, the hypothesis that RLT values have an upper limit was
tested (Mellefont et al., 2003, in press). It was found that most RLTs were in the
range 4 to 6, and that RLTs greater than 8 could not be induced within the
experimental system employed. These observations suggest that while lag time
is apparently highly variable, RLT is more uniform and reproducible. Distribu-
tions of RLT can be used in stochastic modeling studies, for example, microbial
food safety risk assessments, where they could be used as plausible default
assumptions if specific lag time information was not available. This approach can
also simplify the growth modeling process because use of the RLT as a variable
enables the effects of growth rate and lag to be predicted by a single growth rate
model, as explained above.
The RLT concept implies that X is at a minimum value (A, min ) when the growth
rate is optimal (|i opt ). This relation has been used together with CPMs to obtain
simple secondary lag time models (Equation 3.30 and Equation 3.31; Augustin
and Carlier, 2000a; Le Marc et al., 2002; Pouillot et al., 2003; Rosso, 1995,
1999a,b).
X . -LI ,
X= — opt (3.30)
" max
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X = ™ (3.31)
CM 2 (J) ■ CM 2 (0" CM (pH)
For RLT models to be used in practice it must be known if, and to what extent,
abrupt or smooth shifts in environmental parameters like temperature, pH, and water
activity in uence RLT.
Data presented by Rosso (1999a,b) suggested that the effect of shifts in temper-
ature and pH on growth of E. coli during fermentation of yoghurt was appropriately
predicted by a CPM that relied on assumption of a constant RLT. Augustin et al.
(2000) suggested a model to take into account the effect of growth phase and
temperature history of L. monocytogenes on its RLT. For temperature downshifts
the RLT increased from ~0 for a temperature shift of 0-5 °C to ~2 for a downshift
of 30-3 5°C. To model the effect of temperature downshifts and upshifts on RLT of
L. monocytogenes, Delignette-Muller et al. (2003) recently used the data of Whiting
and Bagi (2002) and suggested simple biphasic linear models. Separate models were
used for inoculum with different physiological states. For E. coli, Mellefont and
Ross (2003) found a similar effect of temperature downshifts whereas temperature
upshifts had no systematic effect on RLT. For abrupt downshifts and upshifts in
water activity the data of Mellefont et al. (2003) suggest that simple biphasic linear
models, with different slopes for down- and upshifts, may be appropriate to predict
RLT of both Gram -negative and Gram-positive food-borne bacteria. The universality
of these responses remains unclear. For example, RLTs of S. aureus and L. mono-
cytogenes were largely unaffected by abrupt osmotic shifts over a wide range of salt
concentrations, whereas RLT of Gram-negative cells was strongly affected. More
research is required before models that are as reliable as existing growth rate models
can be developed for lag time, or RLT.
3.2.4 Secondary Models Based on the Arrhenius Equation
3.2.4.1 The Arrhenius Equation
The empirical Arrhenius-van't Hoff relationship:
rate = A exp(AE a I RT) (3.32)
or its mechanistic interpretation and modi cation due to Eyring (1935) based in
absolute reaction-rate theory:
rate = KTexpiAH* I RT) (3.33)
where the parameters may be interpreted as follows: A is a constant related to the
number of collisions between reactants per unit time, E a the activation energy, R the
gas constant (8.314 J/K/mol), T the temperature in Kelvin, K is similar to A but
includes steric and entropic effects, and A// J is the enthalpy difference between the
2004 by Robin C. McKellar and Xuewen Lu
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transition state complex and the reactants, are well established in chemistry to
describe the effects of temperature on the rate of chemical reactions. Taking the
logarithm of both sides of Equation 3.32:
In (rate) = ln(A) x AE/RT
and reparameterizing the equation becomes:
In (rate) = A' +
(
V
AE
R
\
J
x
r i a
\Tj
Thus, if ln(rate) is plotted against
the resulting plot is a straight line over
temperature ranges relevant to microbial growth and allows estimation of the "acti-
vation energy" of the reaction, as shown in Figure 3.4. The activation energy can
be used to characterize the reaction.
Temperature (°C)
76.84 66.84 56.84 46.84 36.84 26.84 16.84 6.84 -3.16 -13.16 -23.16
6 A 1 1 1 1 ! 1 1 ; 1 1 1-
0.00286
0.00306 0.00326 0.00346 0.00366
1/(Temperature [K])
0.00386
FIGURE 3.4 Diagram showing the effect of temperature on reaction rate predicted using the
Arrhenius model (Equation 3.33; solid line) and the effect of temperature on microbial growth
rate (dashed line) for a representative mesophilic organism. The "activation energy" is esti-
mated from the slope of the solid line, multiplied by the universal gas constant. Over a narrow
range of temperatures, the microbial growth rate follows the Arrhenius model prediction
(Equation 3.29). This range has been termed the "normal physiological range" (NPR). At
temperatures above or below the NPR, microbial growth rate deviates markedly from that
predicted by the Arrhenius model.
2004 by Robin C. McKellar and Xuewen Lu
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It has been argued that because all life processes are the result of chemical
reactions, the growth rate of organisms that cannot achieve thermal homeostasis
should also be described by Arrhenius kinetics. Within a narrow range of temperature
this is true. In practice, however, when microbial growth rate data for the full
biokinetic temperature range are presented as an Arrhenius plot, the data are far
from linear as shown in Figure 3.4, and con rmed by numerous studies (Heitzer et
al., 1991; McMeekin et al., 1993; Schoolfield et al., 1981).
A range of secondary models, based on adherence to the reaction kinetics
described by the Arrhenius model, but including terms to account for the observed
deviations, have been proposed. These models fall into two groups:
1. Those based on putative mechanistic modifications of the Arrhenius
models
2. Those based on empirical modifications
3.2.4.2 Mechanistic Modifications of the Arrhenius Model
Models in this category include those of Johnson and Lewin (1946) to describe the
high-temperature growth of bacteria, Hultin (1955) to describe rates of enzymatic
catalysis in the low temperature region, Sharpe and DeMichele (1977) who syn-
thesized these two equations to produce a model for the temperature dependence
of bacterial growth rate in the entire biokinetic region, the model of Schoolfield et
al. (1981), which is a reparameterization of the Sharpe and DeMichele model to
overcome difficulties in tting by nonlinear regression, and the models of
McMeekin et al. (1993) and Ross (1993a, 1999b). The latter models incorporate
contemporary knowledge of the thermodynamics of protein folding to overcome
failures in the Schoolfield et al. model related to unrealistic parameter estimate
(Ratkowsky et al, 1991).
The above models were originally developed to provide an interpretation of
microbial growth rates or enzyme-catalyzed reaction rates, in response to tempera-
ture but their mechanistic basis makes them attractive for use as secondary models.
This class of secondary models have previously been reviewed (McMeekin et
al., 1993; Ratkowsky et al., 1991; Ross, 1999b; Ross and McMeekin, 1994). In
summary, all of the models are based on the assumption that there is a single,
enzyme-catalyzed, rate-limiting reaction in any microorganism. This reaction is
characterized by an activation energy, which governs the rate of reaction in response
to temperature, according to Arrhenius kinetics. Enzymes are proteins, however, and
are themselves subject to the effects of temperature. The functional activity of
enzymes is dependent upon their shape, or conformation, but they are flexible —
the flexibility being required to achieve their catalytic function. Because temperature
affects the bonds in the molecule, if the temperature changes too much, the confor-
mation becomes so disrupted that denaturation takes place, both at high and low
temperatures. These denaturation events are reversible, but at high temperatures if
the temperature increases suf ciently, irreversible denaturation takes place (Ross,
1999b). Thus, these models include terms to model the probability, as a function of
temperature, that the enzyme is in its metabolically active conformation and use this
2004 by Robin C. McKellar and Xuewen Lu
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estimate to modify the predictions of the Arrhenius model. Equation 3.34 to Equation
3.36 are examples of this form of model.
Model of Hmshelwood (1946):
rate = A, cx V (-EJRT)-A 2 cx V (-E ahigh /RT
(3.34)
where R,T,A, and E Q have the same meaning as above. E aUgh is the activation energy
of the high-temperature denaturation of the rate-limiting enzyme.
Model of Schoolfield et al. (1981):
T
(25)
298
exp-
HJ I
R
1
\
,298 T j
K
1 + exp
T
\
J
+ exp
T
\
J
(3.35)
where T is the absolute temperature, R is the universal gas constant, and, for modeling
bacterial growth, the other parameters have been interpreted as follows: K is the
response (e.g., generation) time, p a scaling factor equal to the response rate (l/K)
at 25°C, H A the activation energy of the rate-controlling reaction, H L the activation
energy of denaturation of the growth-rate-controlling enzyme at low temperatures,
H H the activation energy of denaturation of the growth-rate-controlling enzyme at
high temperatures, T V2 the lower temperature at which half of the growth-rate-
Li
controlling enzyme is denatured, and T l/2 is the higher temperature at which half
of the growth-rate -controlling enzyme is denatured.
Model of Ross (1999b):
rate =
CT exp(A/f ! / RT)
1 + exp(-n(AH * -TAS* +AC [(T -T+ H )- T\n(T I T * g )]) / RT)
(3.36)
where C is a parameter whose value must be estimated, AH J the activation enthalpy
of the reaction catalyzed by the enzyme controlling the overall reaction rate, AC p
the difference in heat capacity (per mole amino acid residue) between the native
(catalytically active) and denatured state of the enzyme, T H * the temperature (K) at
which the AC p contribution to enthalpy is 0, T 5 * the temperature (K) at which the
AC p contribution to entropy is 0, AH * the value of enthalpy at T H * per mole amino
acid residue, AS* the value of entropy at T s * per mole amino acid residue, T the
temperature (K), R the gas constant (8.314 J/K/mol), and n is the number of amino
acid residues in the enzyme.
Equation 3.34 to Equation 3.36 include the simple Arrhenius model in the
numerator of the equation. The denominators in Equation 3.35 and Equation 3.36,
however, model the probability that the enzyme is in its active conformation. When
2004 by Robin C. McKellar and Xuewen Lu
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4-
0)
o
i-
U)
Q)
+■»
en
1 ■
260
!
i
i
i :
i \
i
i I
i •
i 1
i ;
' /
/
\ /© /
/ A
■■ 4
■■ 3
o
+■»
(0
c
£
o
c
a>
Q
2 ©
■■ 1
270
280 290 300
Temperature (K)
310
320
(0
>
FIGURE 3.5 Diagram showing the interaction of terms in mechanistic models for microbial
growth rate response to temperature. Curve 1 (dashed line) is the predicted growth rate in the
absence of master enzyme denaturation, i.e., Arrhenius kinetics as modeled by the numerator
of Equation 3.36. Curve 2 (dot-dash line) is the inverse of the probability of the "master
enzyme" being in the active conformation, i.e., the denominator of Equation 3.36. Curve 3
(solid line) is the overall predicted rate from the model, i.e., the quotient of values in Curve
1 divided by values in Curve 2.
that probability is high, the denominator takes values close to 1 , so that the overall
rate is close to that predicted by the Arrhenius equation in the numerator. When the
probability is lower, the value of the denominator increases, so that the observed
rate is lower than that predicted by the numerator alone. These relationships are
shown in Figure 3.5 for Equation 3.36, presented as rate vs. temperature for clarity
of interpretation.
In practice, few of these types of models have been routinely applied in predictive
microbiology, possibly because the models are highly nonlinear, and initial parameter
estimates are dif cult to determine. Furthermore, it is currently not possible to
independently measure the values of the parameters of the model because the putative
master reaction has not been identified, and the concept that a single reaction is rate
limiting under all environmental conditions seems improbable (Daughtry et al., 1 997;
Ross, 1999b). Finally, several workers (Heitzer et al., 1991; Ratkowsky, D.A.,
personal communication, 2003; Ross, 1993b) demonstrated that even with good
quality data, square-root-type models provide an equally good fit as those "mecha-
nistic" models, and are usually easier to work with. Examples of their use include
Broughall et al. (1983) and Broughall and Brown (1984) who used the Schoolfield
model, but also extended it to model the effect of water activity and pH, by replacing
some terms in the model with polynomial expressions in a w and pH. Adair et al.
(1989) used a reparameterized form of the Schoolfield et al. model, which was
2004 by Robin C. McKellar and Xuewen Lu
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essentially another form of the Johnson and Lewin (1946) model. Recent studies
(Ratkowsky et al., unpublished) have con rmed that Equation 3.36 does describe
bacterial temperature-growth rate curves well for a wide range of species, and, in
contrast with earlier models, that the estimated parameter values are realistic and
consistent with the theoretical bases of the model.
3.2.4.3 Empirical Modifications of the Arrhenius Model
A second class of Arrhenius-based models for growth rate and reciprocal of lag time
have been presented by Davey and coworkers. Davey (1989) introduced an Arrhe-
nius-type model for the effects of temperature and water activity, which is linear
and thus allows for explicit solution of the optimum parameter values. This model
has the form:
c a
In (rate) = C + -J- + -f + Cm + C A a* (3.37a)
T T
where T is temperature (K), a w has its usual meaning, and C , C 1? C 2 , C 3 , C 4 are
coef cients to be determined.
Davey (1989) reported that the model described well seven data sets from the
literature and subsequently demonstrated the ability of the model to also describe the
reciprocal of lag phase duration (Davey, 1991). Davey (1994) fitted a variation of the
model to the data of Adams et al. (1991) for Yersinia enterocolitica growth. The model
included terms for temperature and pH, and is analogous to Equation 3.37a:
C C
In (rate) = C + -± + -± + C 3 pH + C 4 pH 2 (3.37b)
where T is temperature (K), pH has its usual meaning, and C , C l5 C 2 , C 3 , C 4 are
coef cients to be determined.
On the basis of these observations, Davey (1994) extended his earlier proposed
general model structure for linear Arrhenius models (Davey, 1989) to account for the
effect of multiple environmental factors affecting growth rate to the following form:
j
In (rate) = C + ]T (C 2 ,._^ +C 2 2 ,r) (3.37c)
1=1
where j environmental factors, V, act in combination to affect the growth of the
modeled organism, and C , C 1? C 2 , ... , C. are coefficients to be determined.
This general form was applied by Davey and Daughtry (1995) to data of Gibson
et al. (1988) for Salmonella growth in response to temperature, NaCl, and pH. Thus,
their equation had the form:
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c c
ln(rate) = C + -^ + -|- + C 3 S+ C 4 S 2 + C 5 /?# + C 6 /?# 2 (3.37d)
where S is salt concentration (% w/v).
While the above model forms are empirical, they also recognize implicitly the
temperature dependence of microbial growth rates. Daughtry et al. (1997) invoked
chemical reaction rate theory to develop an alternative mechanistic model based on
the Arrhenius equation. Those workers cited Levenspiel (1972) as stating that cur-
vature in Arrhenius plots can arise if there are two, or more, reactions that "compete"
to limit the reaction rate and dominate under different conditions so that the overall
effect of temperature is the synthesis of the individual activation energies for the
rate-limiting reactions at different temperatures. Daughtry et al. (1997) considered
that bacterial growth was likely to be such a system.
By assuming that the "heat of reaction" (equivalent to the activation energy or
activation enthalpy in the above discussion) is a polynomial function of temperature,
the following modi ed Arrhenius model was developed:
C
ln(rate) = C + -^ + C 2 lnT (3.38)
This model fitted experimental data as well as the temperature-only form of Equation
3.37a.
The "linear Arrhenius" or "Davey" models have been used to model growth of
molds on solid microbiological media (Molina and Giannuzzi, 1999; Panagou et al.,
2003). Panagou et al. (2003) preferred cardinal parameter and gamma-concept-type
models (see Sections 3.2.2 and 3.2.3) over the Davey model because of their inter-
pretable parameter values. Davey models have also been applied to UV and thermal
inactivation and data describing the combined effects of pH and water activity on
thermal inactivation, including vitamin denaturation (see Section 3.3), but they have
not been widely adopted by other workers. McMeekin et al. ( 1 993) and Davey (200 1 )
identi ed a close correlation between estimates of coef cients C 1 and C 2 , and C 3
and C 4 , of Equation 3.37a, suggesting that the model was overparameterized.
3.2.4.4 Application of the Simple Arrhenius Model
For the entire biokinetic temperature range, growth rates of microorganisms are
described less appropriately by the Arrhenius-type equations (Equation 3.34 to
Equation 3.36) than by square-root-type and cardinal parameter models (see Section
3.2.4.2; Rosso et al., 1993; Zwietering et al., 1991). However, Arrhenius-type models
remain useful as secondary kinetic models for less extensive ranges of storage
temperatures (Table 3.5 and Table 3.6). Koutsoumanis and Nychas (2000) used
Equation 3.32 to model the effect of temperatures between and 15°C on n max and
reciprocal lag time of naturally occurring pseudomonads growing aerobically on a
type of Mediterranean sh. Koutsoumanis et al. (2000) also expanded the classical
2004 by Robin C. McKellar and Xuewen Lu
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Arrhenius model to take into account the combined effect of temperature and C0 2
on growth rates of spoilage bacteria in modified atmosphere packed fresh fish
(Equation 3.39).
Infix ) =
^r max /
E
a_
R
t
x
1
V re f
J.
T
\
+ ln(a ,-dL, x%CO.)
CO,
(3.39)
/
where T, E a , and R have their usual meaning, %C0 2 is the equilibrium concentration
of C0 2 in the headspace gas, d co is a constant expressing the effect of C0 2 on
Mmax an d T Kf and |i Tef are temperature and maximum specific growth rate, respectively,
at 273 K and %C0 2 . The term including d co in Equation 3.39, describing C0 2
inhibition of growth rate, was previously suggested by Kalina (1993).
The simple Arrhenius model has also been used to calculate relative rates of
spoilage (RRS) (Equation 3.37). RRS for a food product is de ned as the shelf life
(determined by sensory evaluation) at a reference temperature (r ref ) divided by the
shelf life observed at the actual storage temperature (Equation 3.40).
RRS =
Shelf life at T
ref
Shelf life at T
= exp
E
(
A
R
x
1 1
\
V
T T
ref
J
(3.40)
where T, E a , and R have their usual meaning and T ief is a reference temperature at
which the shelf life is known.
RRS models are interesting because they enable shelf life to be predicted at
different temperatures and for products where the specific spoilage organisms or the
type of reaction responsible for spoilage are not known. For an unusually tempera-
ture-sensitive modified atmosphere packed shrimp product (E a > 100 kJ/mol), Equa-
tion 3.40 described the effect of temperature (0 to 25 °C) on shelf life more appro-
priately than a similarly formulated RRS model relying on the square-root model
(Equation 3.1). However, a simple exponential RRS model was as useful as Equation
3.40. That the Arrhenius and exponential RRS models performed better than the
square-root model was due to the fact that different groups of microorganisms were
responsible for spoilage at low and high storage temperatures, respectively (Dalgaard
and Jorgensen, 2000). This situation is common and a reason why entirely empirical
RRS models can be more appropriate for shelf-life prediction than kinetic models
relying on growth of known spoilage microorganisms. In fact, kinetic models for
growth of spoilage bacteria are generally useful only for shelf-life prediction within
the spoilage domain of a specific microorganism (Dalgaard, 2002).
3.2.5 Polynomial and Constrained Linear
Polynomial Models
Of the types of secondary models applied within predictive microbiology polyno-
mial models are probably the most common. As shown in Table 3.5 and Table 3.6,
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the effect of many different environmental parameters (e.g., temperature, NaCl/a^
pH, nitrite, C0 2 , organic acids, and natural antimicrobials) has been described by
these linear models. Polynomial models were extensively used during the 1990s
and they remain widely applied although square -root-type and CPMs are becoming
increasingly popular (Table 3.5). Polynomial models are attractive, rst, because
they are relatively easy to fit to experimental data by multiple linear regression,
which is available in most statistical packages. Second, polynomial models allow
virtually any of the environmental parameters and their interactions to be taken
into account. Thus, application of polynomial models is a simple way to summarize
information from a data set. Once the coefficients in a polynomial model have been
estimated, the information is easy to use particularly if the model is included in
application software. In fact, the application software packages Pathogen Modeling
Program and Food MicroModel rely primarily on the use of polynomial models
(www.arserrc.gov/mfs/pathogen.htm; Buchanan, 1993a; McClure et al., 1994a).
To illustrate the use of polynomial models a quadratic equation used by McClure
et al. (1993) is shown below (Equation 3.41):
\ny =p x + p 2 x { + p 3 x 2 + p 4 x 3 + p A x l x 1 + p 6 x^x 3
(3.41)
+p 7 x 2 x 3 + p s x 7 ~ + p 9 x\ + p l0 x 7 - + e
"max'
where In y denotes the natural logarithm of the modeled growth responses (y = (J,
lag time or maximum population density [MPD], or the modified Gompertz model
parameters B or AT): p, (i= 1, ..., 10) are the coefficients to be estimated; x 1 is the
temperature (°C); x 2 is the pH; x 3 is NaCl (% w/v); e is a random error supposed to
have zero mean and constant variance.
As shown by Equation 3.41 the same polynomial equation can be used to model
different microbial growth responses. Actually, many studies have modeled the effect
of environmental conditions on specific parameters in primary growth models, par-
ticularly B, M, and C in the modified Gompertz model. Measures of lag time, growth
rate, or time for, e.g., a 1000-fold increase in the cell concentration are then calculated
at specific environmental conditions from the predicted value of B, M, and C (Bucha-
nan and Phillips, 2000; Eifert et al., 1997; Skinner et al., 1994; Zaika et al., 1998).
Growth responses to be modeled are typically In- or log 10 -transformed (Equation
3.41) and it is common practice to transform the growth response without trans-
forming the model.
However, polynomial models have properties that limit their usefulness as
secondary predictive models. Polynomials include many coefficients that have no
biological interpretation. As an example, Equation 3.41 uses 10 coefficients to model
the effect of three environmental parameters. With four environmental parameters,
polynomials with 15 coefficients are frequently used. The high number of coeffi-
cients and their lack of biological interpretation make it dif cult to compare poly-
nomial models with other secondary predictive models. The important information
included in, e.g., the T min parameter of a square -root -type model, is not provided
by a polynomial model.
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Higher order polynomial models, e.g., cubic or quadratic models have been
criticized for being too flexible and for attempting to model, rather than eliminate,
experimental error (Chapter 4; Baranyi et al., 1996; Sutherland et al., 1996). Because
of the very flexible nature of higher order polynomial models they should not be
used as secondary models within predictive microbiology unless very high quality
experimental data are available and support the application of these models. Fur-
thermore, because quadratic polynomial models are highly flexible they should only
be used to provide predictions by interpolation. Baranyi et al. (1996) pointed out
that the interpolation region of a polynomial model is the minimum convex poly-
hedron (MCP) defined by the ranges of the environmental parameters used to develop
the model, i.e., the experimental design. These authors also stressed that the inter-
polation region (Figure 3.10) can be substantially smaller than the rectangular
parallelepiped whose sides are given by the endpoints of the ranges of environmental
parameters, termed the "nominal variable space" (Baranyi et al., 1996).
Determination of the interpolation region of a polynomial model is not self-
evident and requires information about ranges of the environmental parameters used
to develop the model. Pin et al. (2000) suggested a method to determine if a specific
environmental condition is inside or outside the interpolation region of a particular
polynomial model. This method relies on the iterative algorithm used by the Solver
add-in of Microsoft Excel and thus is readily accessible to many users. However,
we believe for it to become widely used the calculation of interpolation regions
should be included in dedicated predictive modeling application software.
To overcome the problem that quadratic polynomial models can be too exible,
and therefore in some situations provide predictions that are not logical, the appli-
cation of constrained polynomial models was recently suggested (Geeraerd et al.,
in press). With this approach, the basic idea is to combine a priori information about
the effect of environmental parameters on growth responses with classical polyno-
mial models. For example, at suboptimal conditions it was assumed that the growth
rate should always increase for increasing temperature and a w values and decrease
for increasing C0 2 levels. Thus, the partial derivative of the model with respect to
temperature and a w should always be positive whereas the partial derivative of the
model with respect to C0 2 should always be negative. Coef cients of the polynomial
model were then fitted with the constraints obeyed at all edges of the experimental
design. The constrained polynomial model was fitted by the usual process of min-
imizing the sum of squared errors and the tting was carried out using the Optimi-
zation Toolbox within the MatLab software (Geeraerd et al., in press). As compared
to classical polynomial models, constrained polynomial models have the clear advan-
tage of being more robust but the clear disadvantage of being substantially more
dif cult to t. Simpli cation of the tting process seems necessary before con-
strained polynomial models nd wide application in predictive microbiology.
Masana and Baranyi (2000a) described methods for integration of new data into
existing polynomial models, pointing out that the interpolation region of the newly
developed model can be unexpectedly small and also presenting methods for quan-
tifying the increased risk of inadvertent extrapolation (Baranyi et al., 1996). Poly-
nomial models feature many cross-product terms, making the addition of new terms
much more complex than with models embodying the gamma concept (Section
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3.2.2). Nonetheless, when expanding a model by the addition of data for a new
variable, Masana and Baranyi (2000b) demonstrated that the original model can be
retained as a special case of the expanded model, by holding the terms of the original
model, i.e., those that do not contain the new variable, as constants during the model
tting process for the expanded model.
3.2.6 Artificial Neural Networks
Arti cial neural networks (ANNs) are algorithms that can be used to perform
complex statistical modeling between a set of predictor variables and response
variables. Their particular advantage is that they have the potential to approximate
underlying relationships of any complexity between those variables. They have been
used to generate secondary models for microbial growth rates and lag times (Garcia -
Gimeno et al., 2002, 2003; Geeraerd et al., 1998a; Jeyamkondan et al., 2001; Lou
and Nakai, 2001; Najjar et al., 1997), growth under uctuating environmental con-
ditions (Cheroutre-Vialette and Lebert, 2000; Geeraerd et al., 1998a), microbial
inactivation (Geeraerd et al., 1998b), and have been proposed as an alternative to
logistic regression modeling techniques (Tu, 1996). Their potential to replace logistic
regression for growth limits modeling (see Section 3.4) has also been described
(Hajmeer and Basheer, 2002, 2003 a,b) in which context they have been termed
"probabilistic neural networks" (PNNs).
Hajmeer et al. (1997) and Hajmeer and Basheer (2002, 2003a) describe the
principles of ANNs and related technologies in the context of predictive microbiol-
ogy, and numerous texts are dedicated to the subject but the following is largely
drawn from the succinct and lucid description of Tu (1996).
Arti cial neural networks were conceived decades ago by researchers attempting
to reproduce the function of the human brain, i.e., its ability to learn and remember,
but it was only in the 1980s that the "back-propagation" technique was rediscovered,
enabling such computational systems to "learn" mathematical relationships between
input and output variables.
Neural networks are effectively a series of mathematical relationships between
predictor variables ("input nodes"), a series of hidden "nodes," and an output variable
("output node") (Figure 3.6). Each input node is related to each hidden node, and
each hidden node is related to the output node, by some mathematical function.
Each input is given a weight during the "training" routines, the value of each hidden
node being the sum of a weighted linear combination of the input node values. In
addition, bias values can be added to the weighted values of the inputs. These are
analogous to the intercept in regression equations, while the weights are analogous
to coef cients of the independent variables. The output node receives a weighted
input from each of the hidden nodes, to which is often applied a logistic transfor-
mation or other function (the "activation function") to determine the overall output.
A set of input and corresponding output values is presented to the network, the
error is evaluated, and the weights are then adjusted to minimize the difference
between the predicted output and that which was observed. This process of adjust-
ment of weights is the back-propagation step and involves algorithms based on
complex equations. Input data are continuously presented to the neural network until
2004 by Robin C. McKellar and Xuewen Lu
INPUT LAYER
and NODES
Temperature S X
► ( T* )
pH
■► f pH* )
HIDDEN LAYER
and NODES
*w
W
(T, HN1)
rafT
W
(aw,HN1
OUTPUT LAYER
>®
ro
I
O
o
3'
cro
o
<
a
a
w
a-
pa
c
<
o
3
cr
o
o
o
o
FIGURE 3.6 Diagram of an imaginary arti cial neural network that might be used in predictive microbiology. The output is the response of the population
of microorganisms to variations in the temperature, pH, and water activity of their growth medium. (The diagram is fully explained in the text.)
2004 by Robin C. McKellar and Xuewen Lu
1237_C03.fm Page 97 Wednesday, November 12, 2003 12:40 PM
~V
the overall error has been minimized, a process analogous to the iterative routines
employed in nonlinear regression software. Optimal training algorithms can, at this
time, only be determined empirically. Additionally, when using ANNs other elements
of the modeling require experimentation, including the number of training cycles
(too many can reduce predictive performance), the number of nodes in the hidden
layer, and the ideal learning rate (the magnitude of change in the weights for each
training case).
In Figure 3.6, the input, hidden, and output layers are shown, as well as the
connections between them. Nodes are represented by circles. The W^ terms indicate
the weight applied to the inputs to hidden nodes. (Not all weights are represented
in the diagram.) The hidden nodes have a transformation applied to them, e.g., a
logistic function represented by the functions hi, hi, etc. Thus, in the example:
hi = l/(l+exp(Bias 1 + W (T> HN1) x T* + W w HN1) x pH* + W (aw> HN1) x a w *))
and, similarly:
Output =1/(1+ exp(B2 + W (hl) + W (h2) ))
Suf cient data are required so that a subset of data can be used to train the ANN,
while the remainder is used to test the predictive ability of the ANN. One complete
cycle of the training data set is called an "epoch" and the duration of the training
is often described as the number of epochs required to minimize the error in the
training set.
Tu (1996) compared the advantages and disadvantages of the ANN approach to
those of traditional statistical regression modeling, as summarized in Table 3.4.
Evaluation of the approach as applied to predictive models for microbial growth is
presented below, and in relation to growth limits models in Section 3.4.2. Further
comment is provided in Chapter 4, Section 4.4.3.
The use of ANN in predictive growth modeling remains relatively little devel-
oped, and direct comparison of the performance of different ANN techniques is still
lacking. To describe growth curves, Schepers et al. (2000) concluded ANN was less
appropriate than classical nonlinear sigmoidal growth models. Cheroutre-Vi alette
and Lebert (2000), however, found a recurrent (i.e., back-propagation) ANN suitable
to predict growth of Listeria monocytogenes under constant and uctuating pH and
NaCl conditions. As shown in Table 3.5 and Table 3.6, several secondary ANN
models have been developed including models for Aeromonas hydrophila, Brocho-
thrix thermosphaca, Escherichia coli, lactic acid bacteria, Listeria monocytogenes,
and Shigella exneri. These secondary ANN models have been compared with
polynomial, square -root-type, and cardinal parameter models. The comparisons
showed ANN typically fitted experimental data better and in most cases provided
slightly more accurate predictions. Thus, in general, ANN may provide slightly
improved predictions. Commercial neural network software is available and devel-
opment of ANNs has become relatively easy. However, ANN is a data-driven
approach and this could be a drawback because a secondary model that can be
written as an equation with coef cients and parameters is not produced. The
2004 by Robin C. McKellar and Xuewen Lu
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1237_C03.fm Page 98 Wednesday, November 12, 2003 12:40 PM
TABLE 3.4
Advantages and Disadvantages of Neural Network Approaches to Modeling
Disadvantages
Neural networks are a "black box" and have limited
ability to specifically identify possible causal
relationships between predictor and response
variables
Neural network models may be more dif cult to
use in the field
Neural network modeling requires greater
computational resources
Neural network models are prone to over tting
Advantages
Neural networks require less formal statistical
training to develop
Neural network models can implicitly detect
complex nonlinear relationships between
predictor and response variables
Neural network models have the ability to detect
all possible interactions between predictor
variables
Neural networks can be developed using multiple
different training algorithms
Neural network model development is empirical,
and many methodological issues remain to be
resolved
Source: After Tu, J.V. J. Clin. Epidemiol, 11, 1225-1231, 1996.
incorporation of classical secondary models in user-friendly application software
has been essential for the usefulness of predictive microbiology in industry, teaching,
and research. It remains to be demonstrated whether successful ANN models can,
in a similar way, be communicated to and conveniently applied by wide groups of
users within predictive microbiology.
3.3 SECONDARY MODELS FOR INACTIVATION
There are relatively few models that consider the effects of multiple environmental
factors on the rate of death of microorganisms, and these are discussed in Chapter
2 and Chapter 5. Some available inactivation models are also summarized in Table
3.5 and Table 3.6.
3.4 PROBABILITY MODELS
3.4.1 Introduction
Models to predict the likelihood, as a function of intrinsic and extrinsic factors, that
growth of a microorganism of concern could occur in a food were rst explored in
the 1970s. Those models were concerned with prediction of the probability of
formation of staphylococcal enterotoxin or botulinum toxin within a specified period
of time under de ned conditions of storage and product composition (Genigeorgis,
1981; Gibson et al., 1987). Phenomena that have been modeled using this approach
2004 by Robin C. McKellar and Xuewen Lu
1237_C03.fm Page 99 Wednesday, November 12, 2003 12:40 PM
TABLE 3.5
Examples of Secondary Models for Growth of Pathogenic and
Indicator Microorganisms
Type of
Microorganisms and
Secondary
Response
References
Model
Variables
Aeromonas hydrophila
Palumbo et al.
Polynomial
GT, b lag
(1992) a
McClure et al.
Polynomial
GT, lag
(1994b)
Palumbo et al.
Polynomial
GT, lag
(1996) a
Devlieghere et al.
Square-root,
Hmax> ^g
(2000a)
polynomial
Jeyamkondan et al.
ANN
GT, lag
(2001)
Aspergillus spp.
Pitt (1995)
Kinetic with
Growth and
yield factors
a atoxin
formation
Molina and
Arrhenius
Colony growth
Giannuzzi (1999)
Rosso and
CPM
Colony growth
Robinson (2001)
Sautour et al. (2001)
CPM
Colony growth
Bacillus cereus
Benedict et al.
Polynomial
GT, lag
(1993) a
Sutherland et al.
Polynomial
GT, lag
(1996) c
Zwietering et al.
Gamma
r*max
(1996)
Chorin et al. (1997)
Polynomial
Growth rate,
lag
Singaglia et al.
Polynomial
Spore
(2002)
germination
Clostridium botulinum
Baker and
Polynomial
Time to toxin
Genigeorgis
formation
(1990) a
Graham et al.
Polynomial
GT, time to
(1996) c
toxin
Independent Variables and Ranges
r(5-42°C); NaCl (0.5-4.5%); pH
(5.0-7.3); Na-nitrite (0-200 ppm);
anaerobic
T (3-20°C); NaCl (0.5-4.5%); pH
(4.6-7.0); aerobic
T (5-42°C); NaCl (0.5-4.5%); pH
(5.0-7.3); Na-nitrite (0-200 ppm);
aerobic
r(4-12°C); a w (0.974-0.992); C0 2
(0-2403 ppm); pH 6.12; nitrite (22
ppm)
Data from McClure et al (1994b)
T; # w ;pH; and colony size: limits not
specified in manuscript
T (25-36°C); propionic acid
(129-516 ppm)
T(25, 30, 37°C); a w (0.83-0.99); pH
(6.5); humectant: glucose/fructose
T(25°C); a w (0.88-0.99)
T (5-42°C); NaCl (0.5-5.0%); pH
(4.5-7.5); Na-nitrite (0-200 ppm);
aerobic
r(10-30°C); NaCl (0.5-10.5%); pH
(4.5-7.0); CO, (10-80%)
T (10-30°C); a w (0.95-1.00); pH
(4.9-6.6)
T (7-30°C); a w (0.95-0.991); pH
(5-7.5); humectant: glycerol
T (20-40°C); a w (0.94-0.99); pH
(4.5-6.5)
T (4-30°C); initial spore cone. (-2 to
+4 log cfu/g); initial aerobic plate
count (-2 to +3 log cfu/g)
T (4-30°C); NaCl (1.0-5.0%); pH
(5.0-7.3)
2004 by Robin C. McKellar and Xuewen Lu
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TABLE 3.5 (Continued)
Examples of Secondary Models for Growth of Pathogenic and
Indicator Microorganisms
Type of
Microorganisms and
Secondary
Response
References
Model
Variables
Whiting and
Polynomial
Time to
Oriente (1997) a
turbidity
Chea et al. (2000)
Polynomial
Spore
germination
Fernandez et al.
Polynomial
Time to
(2001)
turbidity
Clostridium perfringens
Juneja et al. (1996) a
Polynomial
GT, lag
Escherichia coli
Buchanan and Bagi
Polynomial
GT, lag
(1994) a
Sutherland et al.
Polynomial
GT, lag
(1995, 1997) c
Rasch (2002)
Polynomial
Growth rate
Ross et al. (2003)
Square-root
GT
Skandamins et al.
Vitalistic
Time to decline
(2002)
in cell
concentration
Garcia-Gimeno et
ANN
Growth rate,
al. (2003)
lag time
Whiting and Golden
Polynomial
Time to decline
(2003)
in cell
concentration
Listeria
monocytogenes 6
Buchanan et al.
Polynomial
Time to decline
(1997)
in cell
concentration
Razavilar and
Polynomial
Probability of
Genigeorgis
growth
(1998)
Cheroutre-Vialette
ANN
Absorbance at
and Lebert (2000)
600 nm
Independent Variables and Ranges
T (4-28°C); NaCl (0-4%); pH (5-7);
initial spore cone. (1-5 log cfu/g)
T (15-30°C); NaCl (0.5-4.0%); pH
(5.5-6.5)
T (5-12°C); NaCl (0.5-2.5%); pH
(5.5, 6.5); C0 2 (0-90%)
T (12-42°C); NaCl (0-3%); pH
(5.5-7.0.); Na-pyrophosphate
(0-3%)
T (5-42°C); NaCl (0.5-5.0%); pH
(4.5-8.5); Na-nitrite (0-200 ppm);
aerobic and anaerobic
T (10-30°C); NaCl (0.5-6.5%); pH
(4.0-7.0); Na-nitrite (0-200 ppm);
aerobic
T (10-30°C); NaCl (0.5-3.0%); pH
(4.5-6.5); reuterin (0-4 AU/ml)
T (7.6-47.4°C); a w (0.95 1-0.999);
pH (4.02-8.28); lactic acid (0-500
mM)
T (0-15°C); pH (4.0-5.0); oregano
essential oil (0.0-2.1%)
T (9-2 1°C); NaCl (0-8%); pH
(4.5-8.5); Na-nitrite (0-200 ppm)
T (4-37°C); NaCl (0-15%); pH
(3.5-7.0); Na-lactate (0-2%); Na-
nitrite (0-75 ppm)
T (4-42°C); NaCl (0.5-19%); pH
(3.2-7.3); lactic acid (0-2%); Na-
nitrite (0-200 ppm)
T (4-30°C); NaCl (0.5-12.5%);
methyl paraben (0-0.2%); pH (-5.9);
K-sorbate (0.3%); Na-propionate
(0.1%); Na-benzoate (0.1%)
T (20°C); pH (5.6-9.5); NaCl
(0-8%)
2004 by Robin C. McKellar and Xuewen Lu
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TABLE 3.5 (Continued)
Examples of Secondary Models for Growth of Pathogenic and
Indicator Microorganisms
Type of
Microorganisms and
Secondary
Response
References
Model
Variables
Bouttefroy et al.
Polynomial
Cell
(2000)
concentration
Rodriguez et al.
Arrhenius
Hm ax
(2000)
Augustin and
CPM
Umax, ^g
Carlier (2000a,b)
Ross et al. (in press) Square-root
Buchanan and
Phillips (2000)
Buchanan and
Phillips (2000)
Devlieghere et al.
(2001)
Le Marc et al.
(2002)
Polynomial
Polynomial
Square-root,
polynomial
CPM
Seman et al. (2002) Polynomial
K
GT, lag
GT, lag
Mmax» la §
Mmax> la §
Growth rate
Independent Variables and Ranges
T (22°C); pH (5.0-8.2); NaCl
(0-6%); curvaticin 13 (0-160
AU/ml)
T (4-20°C)
T (-2.7 to -45.5^);^
(0.910-0.997); pH (4.55-9.61);
acetic acid (0-20. 1 mM); lactic acid
(0-5.4 mM); citric acid (0-1.6
mM); Na-benzoate (0-0.7 mM); K-
sorbate (0-5.1 mM); Na-nitrite
(0-11.4 \\M); glycerol monolaurin
(0-118.5 ppm); butylated
hydroxyanisole (0-254 ppm);
butylated hydroxytoluene (0-48.7
ppm); terf-butylhydroquinone
(0-1400 ppm); C0 2 (0-1.64
proportion); caffeine (0-10.8 g/1);
phenol (0-12.5 ppm)
7(3-40 °C); a w (0.920-0.997); pH
(4.0-7.8); lactic acid (0-450 mM);
nitrite (0-150 ppm)
T (4-37°C); pH (4.5-7.5); NaCl
(0.5-10.5%); Na-nitrite (0-1000
ppm); aerobic
T (4-37°C); pH (4.5-8.0); NaCl
(0.5-5.0%); Na-nitrite (0-1000
ppm); anaerobic
T(4-12°C); tf w (0.9622-0.9883); Na-
lactate (0-3.0%); Na-nitrite (20
ppm); pH (6.2)
r(0.5-43°C); pH (4.5-9.4); acetic
acid (16-64 mM); lactic acid
(40-138 mM); propionic acid
(18-55 mM)
T (4°C); NaCl (0.8-3.6%); Na-
diacetate (0.0-0.2%); K-lactate
(0.15-5.6%); Na-erythrobate (317
ppm); Na-nitrite (97 ppm); Na-
tripolyphosphate (0.276%)
2004 by Robin C. McKellar and Xuewen Lu
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TABLE 3.5 (Continued)
Examples of Secondary Models for Growth of Pathogenic and
Indicator Microorganisms
Microorganisms and
References
Type of
Secondary
Model
Gimenez and Square-root
Dalgaard (in press)
Salmonella
Gibson et al. (1988) c Polynomial
Response
Variables
M n
GT, lag
Davey and
Arrhenius
Growth rate,
Daughtry (1995)
lag
Koutsoumanis et al.
Polynomial
Mm ax
(1998)
Oscar (1999)
Polynomial
Umax' la §
Oscar (2002)
Square-root,
CPM
Umax, ^g
Skandamins et al.
Vitalistic
Time to decline
(2002)
in cell
concentration
Shigella
Zaika et al. (1994,
Polynomial
GT, lag
1998) a
Jeyamkondan et al.
ANN
GT, lag
(2001)
Staphylococcus aureus
Ross and
Square-root
Growth rate
McMeekin (1991)
Buchanan et al.
Polynomial
GT, lag
(1993) a
Dengremont and
Square-root
r*max
Membre (1997)
Eifert et al. (1997)
Polynomial
Parameters in
primary
model
Vibrio spp.
Miles et al. (1997)
Square-root
GT
Independent Variables and Ranges
r(4-10°C); %WPS (2-6%); smoke
components/phenol (3-10 ppm);
pH (5.9-6.3); lactic acid (0-20,000
ppm); interaction with lactic acid
bacteria
T (10-30°C); NaCl (0.5-4.5%); pH
(5.6-6.8); aerobic
Data from Gibson et al. (1988)
T (22-42°C);pH (5.5-7.0);
oleuropein (0-0.8%); aerobic
T (15-40°C); pH (5.2-7.4); previous
pH (5.7-8.6); aerobic
T (8-48°C); aerobic
T (5-20°C); pH (4.3-5.3); oregano
essential oil (0.5-2.0%)
T (10-37°C); NaCl (0.5-5.0%); pH
(5.0-7.5); Na-nitrite (0-1000 ppm);
aerobic
Data from Zaika et al. (1994)
T (5-35°C); a w (0.848-0.997)
7 7 (12-45°C); NaCl (0.5-16.5%); pH
(4.5-9.0); Na-nitrite (1-200 ppm);
aerobic and anaerobic
T (10-37°C); NaCl (0-10%); pH
(5-8)
T (12-28°C); NaCl (0.5-8.5%); pH
(5.0-7.0); acidulants HC1, acetic
acid, or lactic acid; aerobic
T (8-45°C); a w (0.936-0.995);
aerobic
2004 by Robin C. McKellar and Xuewen Lu
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TABLE 3.5 (Continued)
Examples of Secondary Models for Growth of Pathogenic and
Indicator Microorganisms
Microorganisms and
References
Type of
Secondary
Model
Response
Variables
Yersinia spp.
Bhaduri et al.
(1995) a
Polynomial
GT, lag
Sutherland and
Bayliss (1994) c
Pin et al. (2000)
Polynomial
Polynomial
GT, lag
Umax' la §
Wei et al. (2001)
Square-root
Umax, ^g
Independent Variables and Ranges
T (5-42°C); NaCl (0.5-5.0%); pH
(4.5-8.5); Na-nitrite (0-200 ppm);
aerobic
T (5-30°C); NaCl (0.5-6.5%); pH
(4.0-7.0); aerobic
T (1-8°C); C0 2 (0-83%); 2
(0-60%)
r(4-34°C); air; vacuum; C0 2 100%
a Models included in the Pathogen Modeling Program, which is available free of charge at
www. arserrc . gov/mfs/PMP6_download.htm.
b Generation time = ln(2)/(i max .
c Model included in Food MicroModel. The values of model parameter are not included in the manu-
script.
d See Ross et al. (2000) for a list of Listeria monocytogenes growth models published prior to 2000.
include germination of spores, population growth, survival, and toxin formation.
These types of models became known as "probability" models.
In the latter part of the 1 990s it seemed that the only way to manage the risk to
consumers from certain pathogens was to ensure that the organism was never present
in foods, or to ensure that it was not able to grow in foods that could become
contaminated. The latter imperative led to the re-development of "growth/no-growth
boundary," or "interface" modeling.
This section is divided into three main parts. In the rst, Section 3.4.2, "tradi-
tional" probability modeling is brie y discussed. Section 3.4.3 presents and discusses
the newer growth/no-growth (G/NG) modeling approaches, while Section 3.4.4
considers methodological issues relevant to probability and G/NG modeling.
3.4.2 Probability Models
Several reviews of probability modeling were presented in the early 1990s (Baker,
1993; Baker and Gemgeorgis, 1993; Dodds, 1993; Lund, 1993; Ross and
McMeekin, 1994; Whiting, 1995) but, possibly because of the relative paucity of
new publications in this field since then, there has been no more recent dedicated
review. Whiting and Oriente (1997) and Zhao et al. (2001), however, provide
succinct updates.
2004 by Robin C. McKellar and Xuewen Lu
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TABLE 3.6
Examples of Secondary Models for Growth of Spoilage Microorganisms
Microorganisms and
Type of
Response
References
Model
Variables
Bacillus
stearothermophilus
Ng et al. (2002)
Polynomial
Growth rate
GOL a
Brochothrix
thermosphacta
McClure et al.
Polynomial
Mmax, la E
(1993)
Abdullah et al.
Polynomial
GT, lag,
(1994)
MPD
Geeraerd et al.
ANN
Mmax> ^g
(1998a)
Pin and Baranyi
Polynomial
Umax* la §
(1998)
Koutsoumanis et
Arrhenius
Hmax
al. (2000)
Jeyamkondan et al.
ANN
GT, lag
(2001)
Chryseomonas spp.
Membre and
Square-root
Mm ax
Kubaczka(1998)
Enterobacteriaceae
Pin and Baranyi
Polynomial
Mmax> ^g
(1998)
Lactic acid bacteria
Passosetal. (1993)
Kinetic
Mm ax
Ganzle et al.
Square-root
Hmax
(1998)
Ganzle et al.
CPM
Mm ax
(1998)
Devlieghere et al.
Square-root,
Mmax> ^g
(2000a,b)
polynomial
Lou and Nakai
ANN
Mmax> ^g
(2001)
Wijtzes et al.
Square-root
Mm ax
(2001)
Independent Variables and Ranges
T (45-60°C); NaCl (0-1.5%); pH (5.5-7.0)
r(l-30°C); NaCl (0.5-8.0%); pH (5.6-6.8);
aerobic
T (-2 to -10°C); C0 2 (2-40%); diameter of
meat particles (2-10 mm)
Data from McClure et al. (1993)
T (2-1 1°C); pH (5.2-6.4); aerobic
T (0-20°C); C0 2 (0-100%)
Data from McClure et al. (1993)
r(1.3-10°C); aerobic
T (2-1 1°C); pH (5.2-6.4); aerobic
pH (3.8-6.0); lactic acid (0-30 mM); acetic
acid (0-40 mM); NaCl (0-9%); cucumber
juice
T (3-41°C); aerobic
pH (4.2-6.7); ionic strength (0.0-1.97);
acetate (0-0.2 mM); aerobic
r(4-12°C); a w (0.962-0.9883); C0 2 (0-1986
ppm); Na-lactate (0.0-3.0 5); pH 6.2
r(4-12°C); a w (0.9 62-0.9883); CO, (0-241 1
ppm); pH 6.2
Subset of data from Devlieghere et al.
(2000a,b)
T (3-30°C); a w (0.932-0.990); pH (5.0-7.5)
2004 by Robin C. McKellar and Xuewen Lu
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TABLE 3.6 (Continued)
Examples of Secondary Models for Growth of Spoilage Microorganisms
Microorganisms and
Type of
Response
References
Model
Variables
Connil et al. (2002)
Polynomial
Umax, ^g
Messens et al.
CPM
Growth,
(2002)
bacteriocin
production
Leroy and De
CPM
Growth,
Vuyst (2003)
bacteriocin
production
Garcia-Gimeno et
ANN
Growth rate,
al. (2002)
lag
Messens et al.
CPM
Growth,
(2002)
bacteriocin
production
Garcia-Gimeno et
ANN
Growth rate,
al. (2002)
lag
Molds
Gibson et al.
Polynomial
Colony
(1994)
growth
Cuppers et al.
Square-root,
Colony
(1997)
CPM
growth
Valik et al. (1999)
Polynomial
Colony
growth
Batteyetal. (2001)
Polynomial
Probability
of growth
Panagou et al.
Polynomial,
Colony
(2003)
Arrhenius,
CPM
growth
Photobacterium
phosphoreum
Dalgaard et al.
Polynomial,
Mm ax
(1997) b
square-root
P seudomonas
Membre and
Polynomial
Mmax> ^g
Burlot (1994)
Neumeyer et al.
Square-root
GT
(1997) c
Pin and Baranyi
Polynomial
Mmax> ^g
(1998)
Independent Variables and Ranges
T (3-9°C); pH (2.5-6.5); glucose (0.2-0.6%)
7 1 (20-38°C); pH (4.8-7.0)
T 7 (20-37°C); pH (4.5-6.5)
T (20, 28°C); NaCl (0-6%); pH (4-7)
7 1 (20-38°C); pH (4.8-7.0)
T (20, 28°C); NaCl (0-6%); pH (4-7)
r(30°C); a w (0.810-0.995)
T (5-37°C); NaCl (0-7%)
T (25°C); a w (0.87-0.995); aerobic
T (25°C); pH (2.8-3.8); titratable acidity
(0.2-0.6%); sugar content (8-16°Brix); Na-
benzoate (100-350 ppm); K-sorbate
(100-350 ppm)
T (20-40°C); NaCl (2-10%); pH (3.5-5.0)
T (0-1 5°C); CQ 2 (0-100%)
T (4-30°C); pH (6-8); NaCl (0-5%)
T(0-30°C); a w (0.947-0.966)
T (2-1 1°C); pH (5.2-6.4); aerobic
2004 by Robin C. McKellar and Xuewen Lu
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TABLE 3.6 (Continued)
Examples of Secondary Models for Growth of Spoilage Microorganisms
Microorganisms and
Type of
Response
References
Model
Variables
Independent Variables and Ranges
Koutsoumanis et
Arrhenius
Mm ax
T (0-20°C); C0 2 (0-100%)
al. (2000)
Koutsoumanis
Square-root
Mmax> ^g
r(0-15°C)
(2001)
Rasmussen et al.
Process risk
GT
Data from Neumeyer et al. (1997)
(2002)
model
Shewanella spp.
Dalgaard (1993) b
Square-root
Mm ax
T (0-3 5 °C); aerobic, anaerobic
Koutsoumanis et
Arrhenius
rnnax
T (0-20°C); C0 2 (0-100%)
al. (2000)
Yeasts
Deak and Beuchat
Polynomial
Changes in
T (10-30°C); a w (0.93-0.99); pH (3.8-4.6
(1994)
conductance
K-sorbate (0-0.06%)
Passosetal. (1997)
Kinetic with
PfUW
T (30°C); pH (3.2-5.9); lactic acid (0-55
product
mM); acetic acid (0-35 mM); NaCl (0-6%
inhibition
cucumber juice; aerobic and anaerobic
Ganzle et al.
Square-root
Hm ax
T (8-36°C); aerobic
(1998)
Ganzle et al.
CPM
Mm ax
Ionic strength (0.0-3.2); acetic acid (0-90
(1998)
mM); aerobic
a GOL = germination, outgrowth, and lag time.
b Model included in the Seafood Spoilage Predictor (SSP) software available free of charge at
www.dfu.min.dk/micro/ssp/.
c Model included in the Food Spoilage Predictor (FSP) software.
3.4.2.1 Logistic Regression
Dodds (1993) explains that in relation to the hazard presented by Clostridium
botulinum in foods, the detection of the toxin is often more important than growth
and that while growth is continuous and fairly easily determined, the presence of
detectable toxin was seen as an "all-or-none" response. This led workers to seek
methods to predict the probability of production of detectable toxin levels in response
to the independent variables.
In probability models in predictive microbiology the data are usually that the
response (e.g., growth, detectable toxin production) is observed under the experi-
mental conditions, or that it is not. Responses such as detectable toxin production
can be coded as either (response not observed) or 1 (response observed) or, if
repeated observations have been made, as probability (between and 1). The prob-
ability is related to potential predictor variables by some mathematical function
using regression techniques.
2004 by Robin C. McKellar and Xuewen Lu
1237_C03.fm Page 107 Wednesday, November 12, 2003 12:40 PM
Logistic regression is a widely used statistical modeling technique — and is the
technique of choice — when the outcome of interest is dichotomous (i.e., has only
two possible outcomes). It is widely used in medical research (e.g., Hosmer and
Lemeshow, 1989). Because regression techniques do not exist for dichotomous data,
the regression equation is usually related to the log odds, or logit, of the outcome
of interest. This has the effect of transforming the response variable from a binary
response to one that extends from -oo to +°o re ecting the possible ranges of the
predictor variables, and has desirable mathematical features also (Hosmer and Leme-
show, 1989). The logit function is de ned as:
logit P = log(iV(l - P)) (3.42)
where P is the probability of the outcome of interest.
Logit P is commonly described as some function Y of the explanatory variables, i.e.:
logit P = 7 (3.43)
Equation 3.43 can be rearranged to:
1/(1 + e~ Y ) = P
or
e Y /(\ +e Y ) = P
where Y is the function describing the effects of the independent variables.
The latter parameterizations appear in some of the earlier probability modeling
literature.
Zhao et al. (2001) assessed the performance of linear and logistic regression to
model percentage data that are "bounded," and may be considered as rescaled
probability values. It was con rmed that logistic regression provided a much more
accurate description of percentage data than linear regression, which had the insur-
mountable problem of predicting values outside the range of the data (i.e., less than
0% or greater than 100%).
3.4.2.2 Confounding Factors
Probability modelers used logistic regression to de ne the probability that detectable
toxins would be produced within a specified period of time and under specified
product composition and storage conditions. Models were based on the idea that a
product was safe/acceptable or that it was not. Nonetheless, the responses measured
in "probability modeling" were related to a number of factors that were in turn
related to the growth of the organism under study and, in some cases, also included
elements of survival. This approach appears to have arisen from the ideas of Riemann
2004 by Robin C. McKellar and Xuewen Lu
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~V
(1967) that the success of a preservation method with regard to C. botulinum is
related to the probability that one spore will germinate and give rise to toxin in the
nished product. In general, to assess the effect of preservation conditions on
probability of toxin production, the probability of growth from a single cell is
estimated as the number of spores able to initiate growth under the test conditions
(usually determined by MPN [most probable number] methods) divided by the
number originally inoculated (Lund, 1993). Often a series of increasingly dilute
inocula are subjected to the test conditions to determine the minimum fraction able
to initiate detectable growth under the test conditions.
It might be expected that the probability of detection would increase with time.
Indeed, Lindroth and Genigeorgis (1986) recognized that the probability of growth
detection was also dependent upon the lag time of the inoculum, its initial density,
and the duration of the study. They introduced a modi cation to the logit model to
specifically model these effects. That model was subsequently used in a number of
other studies (Baker et al., 1990; Ikawa and Genigeorgis, 1987). Whiting and Call
(1993) criticized earlier models for probability of C. botulinum outgrowth and toxin
production because they did not specifically monitor the time at which growth/toxin
formation was rst detected, and specifically modeled the probability of formation
of toxin as a function of time and storage conditions using the logistic function, i.e.,
the probability of detectable growth, when plotted as a function of time, is a sigmoid
curve. That approach was further re ned (Whiting and Oriente, 1997 ; Whiting and
Strobaugh, 1 998) by inclusion of the inoculum density as an independent variable
in the model.
Clearly, the probability of the responses in many of these traditional probability
models is strongly related to the growth rate of the organism under the experimental
conditions, leading Ross and McMeekin (1994) to conclude that the distinction that
had traditionally been made between probability and kinetic models was an arti cial
one. Similarly, Baker et al. (1990) noted that "The rate of P increase ... expresses
the growth rate ... ."
However, under some experimental conditions P does not always reach an
asymptote of 1 . This is evident in the data of Whiting and colleagues (Whiting and
Call, 1993; Whiting and Oriente, 1997; Whiting and Strobaugh, 1998), of Chea et
al. (2000), and of Razavilar and Genigeorgis (1998). It had also been described
earlier by Lund et al. (1987) who introduced to predictive microbiology a model
that recognizes that under some conditions, no matter how long one waits, not all
samples will show growth/toxi genesis.
While the above studies considered spores, Razavilar and Genigeorgis (1998)
applied a logistic regression approach to the probability of growth initiation within
58 days of Listeria monocytogenes and other Listeria species in response to
combinations of pH, salt, temperature, and methyl paraben, sodium propionate,
sodium benzoate, and potassium sorbate (Table 3.5). Their results, also, suggested
that under near-growth-limiting environmental conditions the asymptotic probabil-
ity of growth (i.e., given in nite incubation time) was sometimes less than 1.
Stewart et al. (2001) also commented that while kinetic models predict the mean
growth rate, these estimates may be meaningless under stressed conditions owing
to natural variability in biological responses. Similarly, Lund (1993) employed the
2004 by Robin C. McKellar and Xuewen Lu
1237_C03.fm Page 109 Wednesday, November 12, 2003 12:40 PM
~V
Gompertz model (see Chapter 2) to model the time-dependent probability of growth
of L. monocytogenes Scott A as a function of environmental factors. Even at near-
growth-limiting pH (4.3), however, the asymptote of the log(P growth ) was still close
to 1.
The above studies suggest that as environmental conditions become more inhib-
itory to growth, not only does the probability that growth will be observed during
the course of the experiment decrease, but the probability that growth is possible also
decreases. This may be because the generation or lag time of all cells within the
inoculum becomes in nitely long. Under these conditions, one begins to identify the
absolute limits to microbial growth under combined stresses, i.e., the G/NG interface.
3.4.3 Growth/No Growth Interface Models
Microbial growth is restricted to nite ranges for any environmental factor, with
growth rate sometimes declining abruptly within a very small increment of change
of environment. Individual factor limits have been determined and collated (e.g.,
ICMSF, 1996a). That the growth range of microorganisms for one factor is reduced
when a second environmental factor is less than optimal is also well recognized, and
underlies the Hurdle concept (Leistner et al., 1985) also known as (multiple) barrier
technology, or combined processes (Gorris, 2000). While the physiological basis of
this synergy remains incompletely understood, the ability to de ne the limits to
growth under combined environmental factors has enormous practical application in
maintaining the microbial safely and quality of foods. Whether pathogens grow at
all and the position of the G/NG boundary are of more interest than their growth
rate because any growth implies a potential to cause harm to consumers. Similarly,
so-called shelf-stable foods are sold, stored, and consumed over long periods of time.
Therefore, the ability of spoilage organisms to grow at all implies that they have the
potential to multiply to suf cient numbers to cause spoilage (Jenkins et al., 2000).
In the early to mid-1990s, a vein of experimentation using logistic regression
techniques was begun with the aim of developing models that could de ne absolute
limits to microbial growth in multifactorial space, irrespective of time of incubation
or number of cells in the inoculum. One impetus for this research was the problem
of listeriosis (Parente et al., 1998; Tienungoon et al., 2000). Strategies proposed to
control the threat of listeriosis included "zero tolerance" (i.e., not detectable in a
25-g sample) of the presence of L. monocytogenes in foods that could support its
growth, or to limit levels of contamination at the point of consumption to less than
100 cfu/g. Thus, foods that did not support the growth of L. monocytogenes were
considered to pose signi cantly less risk and to require much less regulatory "atten-
tion" and testing. It was, therefore, of great commercial interest to be able to predict,
without the need for protracted and expensive challenge testing, the potential for
growth of specific bacteria within a particular food or, equivalently, product formu-
lation options that would preclude growth.
Models de ning combinations of environmental conditions that just prevent
growth have become known as "G/NG interface," "growth boundary," or more simply
"growth limits" models. The importance of growth boundary models for the design
of safe foods and setting of food safety regulations, for the design of shelf-stable
2004 by Robin C. McKellar and Xuewen Lu
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1237_C03.fm Page 110 Wednesday, November 12, 2003 12:40 PM
foods, and as a means of empowering the Hurdle concept by allowing it to be applied
quantitatively has been discussed by various authors (Masana and Baranyi, 2000b;
McMeekm et al., 2000; Ratkowsky and Ross, 1995; Schaffner and Labuza, 1997).
Various approaches have been suggested to de ne the G/NG boundary. For
convenience, these are discussed below under three broad groupings:
1. Empirical, deterministic, approaches
2. Logistic regression techniques
3. Arti cial neural networks
Table 3.7 provides an overview of G/NG models published since 1990.
3.4.3.1 Deterministic Approaches
The rst explicit de nition of a microbial G/NG interface appears to be Pitt (1992),
who derived regression equations from published data to describe the tempera-
ture/water activity interface for a atoxin production and Aspergillus spp. growth.
The equation used to describe the interaction between temperature and water activity
limits for growth was:
r (minmax) = 19 ^ ± /(g56.71 - 2289 X (1. 172 - tf ))
g v v ^ w
where J C 1 ™™ 8 *) are the upper and lower temperature limits for growth at the specified
water activity.
A similar equation was presented for a atoxin production. The predicted inter-
faces from both models are shown in Figure 3.7 .
To describe the pH/a w(NaC1) interface of the food spoilage organism Brocothrix
thermosphacta, Masana and Baranyi (2000b) derived the midpoints of growth and
no-growth observations by interpolation and fitted a polynomial function to those
data. They noted that under some conditions, the interface was completely dominated
by one factor or the other, so that their nal model consisted of a pH vs. a w parabolic
curve and a NaCl-constant line. They also considered the effects of inoculum level
on the interface, which was determined at 25°C for up to 24 days of incubation.
Examples of the interface are shown in Figure 3.8.
Membre et al. (2001) estimated levels of sorbate that prevented growth of
Penicillium brevicomp actum in bakery products containing various levels of benzoate
by extrapolation of kinetic data. Equations were derived to de ne growth-preventing
combinations of sorbate and benzoate and were used to limit the range of predictions
from the kinetic model they developed for P. brevicomp actum growth rate.
Other workers have noted that the form and parameters of CPMs imply absolute
limits to microbial growth, and suggested approaches to de ning the G/NG interface
based on estimates of cardinal parameters. In this vein Ratkowsky and Ross (1995),
recognizing the relationship between absolute limits for each environmental factor
and their relationship to the parameters of square-root-type models and CPMs,
experimented with the use of a kinetic model as the basis of a growth boundary
2004 by Robin C. McKellar and Xuewen Lu
TABLE 3.7
Summary of Published Growth Boundary Models
Experimental Design
Environmental
■ VII
.. & «-o
Reference
Organism
Strain
Medium
Factors
Lower
Upper
Lev
Presser et al.
Escherichia
M23 (non-
Nutrient
Temperature
10
37
6
(1998)
coli
pathogenic)
Broth
a w (NaCl)
pH
Lactic acid
(mM)
0.955
2.8
0.995
6.9
500
4
>10
6
Parente et al.
Listeria
Scott A, V7,
Tryptone
Soy
Nisin (IU/ml)
1
2100
(1998)
mono-
cytogenes
and LI 1
Broth + 0.6%
Yeast Extract
Leucocidin F10
(AU/ml)
1
2100
pH
4.7
6.5
NaCl (% w/v)
0.7
4.5
EDTA (mmol)
0.1
0.9
Inoculum density
1.6 x 10 3
7.9 x 10 7
Validation
Nisin (IU/ml)
8
200
Set 1
Leucocidin F10
(AU/ml)
pH
NaCl (% w/v)
EDTA (mmol)
Inoculum density
8
4.7
0.7
0.08
0.6 x 10 3
200
6.5
4.5
4.72
2.5 x 10 7
Validation Set
Nism (IU/ml)
50
250
2
Leucocidin F10
(AU/ml)
pH
NaCl (% w/v)
EDTA (mmol)
Inoculum density
1
5.2
1.8
0.2
1 x 10 5
250
6
0.6
1 to 4
Total Data
Points
627
Measured by? Time Limit
OD Increase
(con rmedas
needed by
culture)
50 days
Other
Linear logistic
regression, SAS
PROCNONLIN
7 days
(@30°C)
Logistic regression
with polynomial
using LOGIT
1.14 module of
Systat
7 days
(@30°C)
10
7 days
(@30°C)
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2004 by Robin C. McKellar and Xuewen Lu
TABLE 3.7 (Continued)
Summary of Published Growth Boundary Models
Reference Organism
Bolton and
Frank
(1999)
Listeria
mono-
cytogenes
Strain
Mixture
(equal
numbers) of
Scott A, Brie
1,71
Switzerland,
2379 LA
Medium
Soft fresh
cheese
(similar to
"Mexican
style"
cheese)
Experimental Design
Environmental
Factors
Moisture
(% w/w)
salt (% w/w)
pH
Ranges
Lower Upper Levels Replicates
42
2
5
60
8
6.5
Total Data
Points
288
Measured by? Time Limit
Viable count
21 and 42
days
4
6
Other
Binary or "ordinal"
logistic
regression using
SAS PROC
LOGISTIC with
link functions.
For the latter,
three responses : P
of growth, stasis,
or death
(according to
change in viable
count; ±0.5 log
CFU) were
modeled
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Salter et al.
Escherichia
MR21
Nutrient
Temperature
7.7
37
60
(2000)
coli
(STEC)
Broth
a w (NaCl)
0.943
0.987
28
1-8, most 4 604
OD increase 50 days
Nonlinear logistic
regression, SAS
PROCLOGISTIC
and
PROCNONLIN
Jenkins et al.
Zygosaccha.ro
4637, history
Acidi edyeast
Salt (NaCl,
2.6
4.2
3
(2000)
myces bailii
unknown
nitrogen
%w/v)
broth
Sugar (fructose,
%w/v)
7
32
3
Total acetic acid
1.8
2.8
3
(%v/v)
pH
3.5
4
3
243
OD increase 29 days
SAS LIFEREG
2004 by Robin C. McKellar and Xuewen Lu
Tienungoon
et al.
(2000)
Listeria
mono-
cytogenes
(Scott A, L5
separate
models)
TSB-YE
Temperature 3.1
a w (NaCl) 0.928
pH 3.7
Lactic acid
(mM)
36.2
0.995
7.8
500
30
60
10
14
1 to 4
Lopez-Malo
Saccharo-
Not stated
^
0.93
0.97
3
et al.
myces
Model based
pH
3
6
4
(2000)
cerevisiae
on data of
Cerruti et al.
(1990)
K-sorbate (ppm)
1000
6
Masana and
Brocothrix
MR 165
Tryptone
Soya
NaCl
0.5
10
11
Bar any l
thermo-
Broth
pH
4.4
5.7
7
(2000b)
sphacta
Inoculum (cfu/
350 |ll)
10
1 million
3
2839
OD increase
90 days
Nonlinear logistic
regression, SAS
PROCLOGISTIC
and
PROCNONLIN
72 (x 2
Viable count,
50 h or
Logistic regression
observation
including
350 h
with rst-order
times)
decrease in
polynomial using
viable count
SPSS
Viable count
Polynomial fitted
to midpoints of
data-pairs of
adjacent growth
and no growth
observed
combinations
Fine grid
experiments
NaCl
pH
Inoculum (cfu/
350 fll)
Lanciotti et
al. (2001)
Bacillus
cereus
Staphylococ-
cus aureus
FG1
S33
BHI Broth
BHI Broth
Temperature
&w (glycerol)
Salmonella
enteritidis
B5
BHI Broth
pH
45 combinations at 5 levels of
NaCl and 5 levels of pH close
to the G/NG boundary
10 1000 2
10
450
Viable count
10
0.89
Ethanol (% v/v)
45
0.99
8
3
5
30 variables
2 x 30 for
OD increase
combina-
each strain
(600nm)
8
tions over
two
5
independent
trials for
each
5
organism
2-7 days
Generalized linear
logistic
regression,
Statistica
(Statsoft)
software
to
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2004 by Robin C. McKellar and Xuewen Lu
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TABLE 3.7 (Continued)
Summary of Published Growth Boundary Models
Experimental Design
Reference Organism
Strain
Stewart etal. Staphylococ- 5 strain
(2001) cus aureus cocktail
Medium
BHI Broth
Environmental
Factors
Ranges
Lower Upper Levels Replicates
a w (glycerol) 0.95
Initial pH 4.5
K-sorbate (ppm)
or
Total Data
Points Measured by? Time Limit
0.84
7
4
4
Ca-propionate
1000
3
(ppm)
Temperature
37
(°C)
McKellar
Escherichia
5 strain
TS Broth
Temperature
10
30
5
and Lu
coli 015 :H7
cocktail
(°C)
(2001)
Acetic acid
(modeled as
undissociated
form)
4%
8
NaCl
0.50%
16.50%
8
Sucrose
8%
3
pH
3.5
6.0
6
640
OD increase 168 days
1820
Visible
increase in
turbidity
3 days
Other
Toxin assayed
Modeled "time to
growth" using
LIFEREG
Linear logistic
regression used
(polynomial
form)
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2004 by Robin C. McKellar and Xuewen Lu
Membre et
Penicillium
Wild type
MY50 agar
pH
2.5
7.5
±6
1
76
Mycelial
75 days
Not directly
al. (2001)
compactum
from bakery
products
Sorbic acid
Propionic acid
Sodium benzoate
pH
Sorbic acid
(mg H)
Na-benzoate
(mg H)
0.0
0.0
1000.0
300.0
1
1
1
1
6
4
5
122
growth
modeled, growth
limits due to
sodium benzoate
and sorbic acid at
pH 5 were
derived by
extrapolation of
growth rate data
Commercial cakes
6
4
Uljas et al.
Escherichia
3 strain
Apple cider
pH
3.1
4.3
7
1600 x 3
Turbidity
12 hours
SAS
(2001)
coli 015 :H7
cocktail
(juice)
Temperature
Sodium benzoate
5
35
4
(growth
PROCLOGISTIC
In this case
the response modeled was P > 5 log inactivation
0.1%
3
within 48 hat
35°C) after
(dependency
modeled as simple
after various treatment times
/
c
dilution of
rst order
Potassium
0.1%
3
treated
equations of
sorbate
sample
predictor
Freeze-thaw
Not applied
Applied
3
756
variables, no cross
Ciders type
3
2 or 3
products)
Staphylococ-
5 strain
BHI Broth
a w (NaCl, or
0.84
0.95
4
8
OD increase
168 days
Toxin assayed
cus aureus
cocktail
sucrose-
fructose)
pH
K-sorbate (ppm)
4.5
7
1000
2
4
2
Modeled "time to
growth" using
LIFEREG
[Combined with data set of Stewart et al. (2001), 768 data]
1792
Listeria
ATCC 33090
BHI Broth
Temperature
43
16
(pH constant)
Turbidimetry
14 days
Novel term based
innocua
(+0.2% w/w
glucose,
+0 3% w/w
pH
4.5
9.4
15
(temperature
constant)
Viable count
by culture
1 month
on relative
inhibition of
growth rate-
yeast extract
Propionic acid
16
64
24
(pH varied
affecting factors
(mM)
from 5 to
data generated for
7.5)
combined kinetic
Lactic acid (mM)
20
138
27
(pH varied
from 4.8 to
7.1)
model that
predicts no growth
(NLINFIT in
MATLAB 5.2)
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2004 by Robin C. McKellar and Xuewen Lu
TABLE 3.7 (Continued)
Summary of Published Growth Boundary Models
Reference Organism
Battey and
Schaffner
(2001)
Spoilage
bacteria:
Acineto-
bacter
calco-
aceticus and
Gluconobac-
ter oxydans
a
JV
v5
V
Battey et al.
Spoilage
(2002)
yeasts:
Saccharo-
myces
cerevisiae,
Zygosaccha
romyces
bailii,
Candida
lipolytica
Hajmeer and
Data of Salter
Basheer
etal.(2000),
(2002,
see above
2003a,
2003b)
Strain
2 strain
cocktail
Medium
Cold lied,
ready to
drink,
beverages
3 strain
cocktail
Cold lied,
ready to
drink
beverages
Experimental Design
Environmental
Factors
pH
Titratable acidity 3
(%)
Sodium benzoate 3
(ppm)
Sugar content 3
(°C Brix)
Potassium 3
sorbate (pp)
pH 3
Titratable acidity 3
(%)
Sodium benzoate 3
(ppm)
Sugar content 3
(°C Brix)
Potassium 3
sorbate (pp)
Ranges
Lower Upper Levels Replicates
Total Data
Points
Measured by? Time Limit
2.8
3.8
0.2
0.6
100
350
8
16
100
350
2.8
3.8
0.2
0.6
100
350
8
16
100
350
84
Viable plate
count
8 weeks at
25°C
84
Viable plate
count
8 weeks at
25°C
Other
Model is based on
growth and
inactivation rates.
Included 14
duplicated
validation trials
(8 correctly
predicted from
model)
Included 14
duplicated
validation trials
(all correctly
predicted from
model)
Probabilistic
Neural Network
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model using linear logistic regression. This approach is discussed further below in
Section 3.4.3.2.
The approaches of Augustin and Carlier (2000b) and Le Marc et al. (2002) were
presented in Section 3.2.3. Essentially, these approaches are empirical. They are
based on assumed interactions between factors and are not fitted to G/NG data. An
example of the response predicted by these approaches is shown in Figure 3.11.
0.95-
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0.75
i i i i i i i i i
10 15 20 25 30 35 40 45 50 55
Temperature (°C)
FIGURE 3.7 Predicted temperature-water activity interface for mold {Aspergillus spp.)
growth (dashed line) and a atoxin production (solid line) from the model of Pitt (1992).
5-80
5-40
? 500
4-60
4-20
005
0-10
0-15
0-20
0-25
0-30
'w
FIGURE 3.8 Data and modeled growth/no-growth boundary for Brochothrix thermosphacta
i
in response to pH and water activity at 25°C. Water activity data were rescaled to b w
V
1-a
The data are for an inoculum of 1.5 x 10 6 cells/well (□), or for an inoculum of 1.5 x 10 1 and
1.5 x 10 3 cells/well (A). (Reproduced from Masana, MO. and Baranyi, J. Food Microbiol.,
17, 485^193, 2000b. With permission.)
2004 by Robin C. McKellar and Xuewen Lu
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15 20 25
Temperature (°C)
30
35
40
45
FIGURE 3.9 Data obtained from separate experiments for the growth/no- growth (G/NG)
boundary of Escherichia coli. Data are from Salter et al. (2000) (circles) and from unpub-
lished results of the authors (diamonds). Near the G/NG boundary, the data obtained from
discrete experiments do not form a smooth (monotonic) boundary, suggesting that small
differences in experimental procedures can signi cantly affect the position of the boundary.
Open symbols denote no-growth conditions, and solid symbols indicate that growth was
observed.
The above approaches can be considered to be deterministic; i.e., they predict
only one position (-P grow th = 0.5) for the boundary, although the position of boundaries
can be adjusted by "weighting" data in the case of Masana and Baranyi (2000b) or
by selecting an appropriate value for in the case of the Le Marc et al. (2002)
approach (see Section 3.2.3). While the data of Masana and Baranyi (2000b) included
tenfold replication, the midpoints of the most "extreme" conditions that did allow
growth, and the least "extreme" conditions that did not allow growth were estimated
by interpolation and considered to represent 50% probability of growth. Other
workers have suggested that some problems require higher levels of con dence that
growth will not occur, so that methods that enable de nition of the interface at
selected levels of statistical con dence may have greater utility.
Another approach that implicitly characterizes the G/NG interface is that of
combined growth and death models in which the rate of growth and rate of death
under specified conditions are estimated simultaneously. The G/NG interface can be
inferred from those combinations of conditions where growth rate and death rate
are equal (see, e.g., Jones and Walker, 1993; Jones et al., 1994). A similar approach
is evident in Battey et al. (2001) who modeled both the rates of growth and rates
of death of spoilage molds in ready to drink beverages. The G/NG interface was
given, implicitly, as that set of conditions where the rate of growth was equal to the
rate of death.
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Ratkowsky et al. (1991) noted that as environmental conditions become more
inhibitory to microbial growth the variability in growth rates increases widely, which
implies that the probability that growth occurs at all becomes uncertain, because the
left-hand tail of the growth rate distribution falls below zero. This is supported in
the results of Whiting and colleagues (Whiting and Call, 1993; Whiting and Oriente,
1997; Whiting and Strobaugh, 1998), where P max (the proportion of spores that
successfully germinated and initiated growth) was shown to decline at increasingly
stringent conditions. Conversely, Masana and Baranyi (2000b) observed, as have
other workers (McKellar and Lu, 2001; Presser et al, 1998; Salter et al., 2000;
Tienungoon et al., 2000), that the difference in conditions that allow growth and
those that do not is usually abrupt, and often at or beyond the limits of resolution
of instruments commonly used to measure such differences. Thus, Masana and
Baranyi (2000b) questioned the need for approaches that model the transition
between conditions leading to high probability of growth and those leading to low
probabilities of growth. While this abrupt transition appears consistent within rep-
licated experiments it is less certain, however, that the same consistency is true
between experiments. Figure 3.9, showing experimental data, suggests that responses
near the boundary may be inconsistent when data from several discrete experiments
are combined. This may suggest subtle, but important, differences in response related
to the physiology of the inoculum, or its concentration. Furthermore, it suggests that
the ability to characterize probabilities of growth under specified sets of conditions
may be an important element of growth boundary models and that the boundary
may not be "absolute" but depend on the physiological state of the cell and, by
inference, on the size of the inoculum. This will be discussed further in Section 3.4.4.
3.4.3.2 Logistic Regression
Ratkowsky and Ross (1995) and others (Bolton and Frank, 1999; Jenkins et al.,
2000; Lanciotti et al, 2001; Lopez-Malo et al., 2000; McKellar and Lu, 2001;
Parente et al., 1998; Stewart et al., 2001, 2002; Uljas et al., 2001) reintroduced the
use of logistic regression to model categorical data (i.e., growth or no growth) in
predictive microbiology, enabling probabilistic determination of the G/NG bound-
ary. Use of the logit function enabled the probability of growth under specific sets
of conditions to be estimated, so that the G/NG boundary could be specified at
selected levels of con dence.
Ratkowsky and Ross (1995) aimed to model absolute limits to growth in mul-
tifactorial space, but only had available data based on a 72-h observation period.
While most other workers have preferred polynomial functions to describe the effect
of independent variables on the logit function, in the former approach a square -root-
type kinetic model was In-transformed and used as the basis of the function relating
the logit of probability of growth to independent variables, e.g., temperature, water
activity, pH. This approach was adopted in an attempt to retain some level of
biological interpretability of the models, a desire echoed by others (Augustin and
Carher, 2000a,b; Le Marc et al., 2002).
The form of the G/NG interface model of Presser et al. (1998) was derived from
the kinetic model of Presser et al. (1997) for the growth rate of E. coli (see Equation
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3.10). Novel data were generated specifically to assess the limits of E. coli growth
under combinations of temperature, pH, a w and lactic acid. The corresponding G/NG
model had the form:
LogitP = 28.0 + 8.90 ln(a„ - 0.943)
+ 2.01n(r-4.00) + 4.59 ln(l - 10 39 -p h ) (3.44)
+ 6.961n[l -LACI{\0.1 x (1 + 10p h " 3 - 86 ))]
+ 3.061n[l -LAC/(S23 x (1 + 10 386 -p h ))]
where all terms are as de ned in Section 3.2.1.
Some parameters in that model had to be determined independently, i.e., were
not determined in the regression, and were derived from the fitted values of square-
root-type kinetic models. Essentially the same approach was adopted by Lanciotti
et al. (2001) to develop G/NG models for B. cereus, S. aureus, and Salmonella
enteritidis. Ratkowsky (2002) commented on the increased exibility in being able
to determine all of the parameters in the model during the regression, and subsequent
studies developed the approach, eventually leading to a novel nonlinear logistic
regression technique (Salter et al., 2000; Tienungoon et al., 2000). Ratkowsky (2002)
pointed out that nonlinear logistic regression was a new statistical technique and
discussed bene ts and problems with that approach specifically in relation to growth
limits modeling. A problem with models of the form of Equation 3.44 is that for
conditions more extreme than the parameters corresponding to T min , pH min , a w min ,
etc., and which are tested experimentally though not expected to permit growth, the
terms containing those parameters would become negative. As all of those terms are
associated with a logarithmic transformation, the expression cannot be calculated
during the regression and such values are ignored in the model tting process, or
have to be eliminated from the data set before the tting process begins. This, in
turn, affects the values of the parameters of the fitted model. Ratkowsky (2002)
comments that an objective method for selection and deletion of such data is nec-
essary', but does not yet exist.
Bolton and Frank (1999) extended the binary logistic regression approach by
recoding growth and no growth data to allow a third category: survival, or stasis.
They termed this approach ordinal logistic regression. Parente et al. (1998)
"reversed" the response variable, and applied logistic regression techniques to the
probability of survival/no survival of L. monocytogenes in response to bacteriocins,
pH, EDTA, and NaCl. Stewart et al. (2001) modeled the probability of growth of
S. aureus within 6 months of incubation at 37°C, and at reduced water activity
achieved by various humectants. They also compared the growth boundary with the
boundary for enterotoxin production, and observed a close correlation between the
two criteria.
Growth limits models have also been developed for spoilage organisms including
Saccharomyces cerevisiae (Lopez -Malo et al., 2000) and Zygosaccharomyces bailii
(Jenkins et al., 2000) and cocktails of Saccharomyces cerevisiae, Zygosaccharomyces
bailii, and Candida lipolytica (Battey et al., 2002). Interestingly, the study of Jenkins
et al. (2000), while encompassing broader ranges of factor combinations, con rmed
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the simpler and earlier model of Tuynenburg Muys (1971). That model, which
specifies combinations of molar salt plus sugar and percent undissociated acetic acid
for stability of acidic sauces, still forms the basis of the industry standard for those
products. This observation suggests that limits to growth under combined conditions
can be highly reproducible.
3.4.3.3 Relationship to the Minimum Convex
Polyhedron Approach
The concept of the MCP was introduced by Baranyi et al. (1996) (see Figure 3.10)
to describe the multif actor "space" that just encloses the interpolation region of a
predictive kinetic model. If the interpolation region exactly matched the growth
region of the organism then the MCP would also describe the growth limits of the
organism. In practice, however, it would be impossible to undertake suf cient
measurements to completely de ne the MCP; i.e., the MCP has "sharp" edges
because of the method of its calculation, whereas from available studies (see Figures
3.8 and 3.9) the G/NG interface forms a continuously curved surface. However, it
might also be possible to use no-growth data to create a no-growth MCP and to
combine the growth MCP and no -growth MCP to de ne a region within which the
G/NG boundary must lie. This approach has been assessed and compared to a model
of the form of Equation 3.43 by Le Marc and colleagues (Le Marc et al., 2003).
These workers concluded that the logistic regression modeling approach produced
a smoother response surface, more consistent with observations, but that the MCP
approach had the advantage of being directly linked to observations and therefore
was not a prediction from a model.
FIGURE 3.1 Interpolation region (MCP) for a model that includes four-factor combinations
(7, pH, NaCl, NaN0 2 ). The interpolation region shown is that for NaCl = 0.5%, but is based
on the complete data set. Solid circles indicate conditions under which observations have
been made, while the lines represent the edges of the MCP. (From Masana, M.O. and Baranyi,
J. Food Microbiol. , 17, 367-374, 2000a. With permission.)
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3.4.3.4 Artificial Neural Networks
Recently, Hajmeer and Basheer (2002, 2003a,b) demonstrated the use of a Proba-
bilistic Neural Network (PNN) approach to de nition of the G/NG interface. PNNs
are a form of ANN (see Section 3.2.6). In a series of papers, based on modeling the
data of Salter et al. (2000) for the effects of temperature and water activity (due to
NaCl) on the growth limits of E. coli, Hajmeer and Basheer concluded that their
PNN models provided a better description of the data of Salter et al. (2000) than
did the nonlinear logistic regression method referred to above. Their conclusion is
considered in more detail in Section 3.4.3.5 below.
It should be noted that neither the logistic regression models described above,
nor the PNN, produce an equation that describes the interface. Rather, the output of
those models is the probability that a given set of conditions will allow growth. To
de ne the interface, it is necessary to rearrange the model for some selected value
of P to generate an equation that describes the G/NG boundary.
3.4.3.5 Evaluation of Goodness of Fit and Comparison
of Models
Methods for evaluation of performance of logistic regression models include the
receiver operating curve (ROC; also referred to as the concordance rate), the Hos-
mer-Lemeshow goodness-of- t statistic, and the maximum rescaled R 2 statistic.
These are considered in greater detail in Tienungoon et al. (2000).
Brie y, the ROC is obtained from the proportion of events that were correctly
predicted compared to the proportion of nonevents that were correctly predicted.
The closer the value is to 1 , the better the level of discrimination. In epidemiological
studies, ROC values > 0.8 are considered excellent. ROC values for G/NG models
are typically much higher.
The Hosmer-Lemeshow index involves grouping objects into a contingency
table and calculating a Pearson chi-square statistic. Small values of the index indicate
a good t of the model.
The maximum rescaled R 2 value is proposed for use with binomial error as an
analogy to the R 2 value used with normally distributed error. The closer the value
is to 1 , the greater is the success of the model in predicting the observed outcome
from the independent variables. Zhao et al. (2001) cite the deviance test and graphical
tools such as the index plot and half normal plot as methods for determining goodness
of t of linear logistic regression models.
Other methods based on calculation of indices from the "confusion matrix"
(Hajmeer and Basheer, 2002, 2003b) or the equivalent "contingency matrix"
(Hajmeer and Basheer, 2003a) were used to compare performance between models
derived from different approaches and applied to the same data.
Another method of evaluation is to compare the fitted model to independent data
sets (Bolton and Frank, 1999; Masana and Baranyi, 2000b; Tienungoon et al., 2000)
although, generally, such data are not readily available (see, e.g., McKellar and Lu,
2001). The model of Tienungoon et al. (2000) for L. monocytogenes growth bound-
aries showed very good agreement with the data of McClure et al. (1989) and George
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et al. (1988) despite that different strains were involved. There is also a remarkable
level of similarity between the observations of Tienungoon et al. (2000) and the
observations of Le Marc et al. (2002) on growth limits of L. innocna. Several
publications, however, report growth of L. monocytogenes at temperatures lower
than the minimum growth limit predicted by the Tienungoon et al. (2000) model,
possibly indicating strain variation or that the experimental design failed to recognize
important elements that facilitate L. monocytogenes growth at temperatures < 3°C,
i.e., that an inappropriate growth substrate was used. Similarly, McKellar and Lu
(2001) reported that their model failed to predict growth of E. coli 0157:H7 under
conditions where it had been previously reported, although it should be noted that
their model was limited to observation of growth within 72 h. Bolton and Frank
(1999) compared the predictions of their growth limits models for L. monocytogenes
in cheese to the data of Genigeorgis et al. (1991) for L. monocytogenes growth in
market cheese. The models predicted correctly in 65% of trials (42-day model) and
81% of trials (21 -day model).
Given the diversity of approaches, it is pertinent to ask: does one method for
de ning the G/NG interface perform better than another? As with kinetic models,
the ability to describe a specific experimental data set does not necessarily reflect
the ability to predict accurately the probability of growth under novel sets of con-
ditions. While measures of performance of logistic regression models are available,
they can be readily affected by the data set used. Perfect agreement between observed
and modeled data responses may not be possible if there are anomalies in the data.
Figure 3.11 provides a clear example. Nonetheless, for many growth limits models
high rates of concordance (typically >90%) have been reported. As noted earlier, in
epidemiological logistic regression modeling, rates higher than 70% are considered
to represent good ts to the data, implying that the limits to microbial growth are
highly reproducible when well -controlled experiments are conducted.
To date, only one direct comparison of G/NG modeling approaches has been
presented (Hajmeer and Basheer, 2002, 2003a,b) but this was based on one data set
only, i.e., that of Salter et al. (2000) for the growth limits of E. coli in tempera-
ture/water activity space. Only by comparing the performance of different modeling
approaches applied to multiple data sets does an appreciation of overall model
performance emerge. Nonetheless, to illustrate differences between models and give
some appreciation of their overall performance we compare several models using
the data of Salter et al. (2000) for the growth limits of E. coli R3 1 in response to
temperature and water activity. The model types compared are:
1. The PNN of Hajmeer and Basheer (2003a), which those authors were
able to summarize as a relatively simple equation
2. A model of the type of Equation 3.44 fitted to a subset of the Salter et
al. (2000) data set by Hajmeer and Basheer (2003a) (It should be noted
that, contrary to what is stated in that publication, the model presented
by Hajmeer and Basheer was not generated by nonlinear logistic regres-
sion but by a two-step linear logistic regression as described in Presser et
al., 1998)
2004 by Robin C. McKellar and Xuewen Lu
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0.99
0.98
>.
■«-»
>
o
ro 0.97-1
o
0.96-1
0.95
0.94
*♦> ♦ ♦ ♦ ♦ ooo d
%% $ t s°s
10
20 30
Temperature (°C)
40
50
60
FIGURE 3.1 1 Comparison of predicted no growth boundaries for four modeling approaches
applied to the data of Salter et al. (2000) (circles) for the growth limits of Escherichia coli
R31 in response to temperature and water activity (NaCl) combinations. Approach of Hajmeer
and Basheer PNN (heavy solid line); Linear Logistic /Equation 3.44 (heavy dashed line); Le
Marc et al. (2002) (light solid line); Augustin and Carlier (2000a) (light dashed line). The
data set was subsequently augmented with new data (diamonds), which reveals that none of
the models extrapolate reliably. (Solid symbols: growth; open symbols: no growth.)
3. A model of the type of Le Marc et al. (2002; Equation 3.25 to Equation
3.27), where T max = 49.23°C (to be consistent with the logistic regression
model parameter), a wmm = 0.948, and T mhl = 8.8°C, based on the minimum
water activity and temperature, respectively, at which growth were
observed
4. A model of the type of Augustin and Carlier (2000b; Equation 3.24)
assuming that T min = 8.8°C and a wmin = 0.948, consistent with the param-
eter values used for the Le Marc et al. (2002) model
The predicted interfaces are shown in Figure 3.1 1, together with the data used
to generate the models. (Note that the subsets of 143 of the 179 data of Salter et al.
(2000) used by Hajmeer and Basheer (2003a) to t the PNN and the Equation 3.44
type model were not identi ed.)
When compared to the full data set, the level of misprediction ranged from ~15
to 20 of the 179 data points for each of the models, suggesting that the level of
performance was not greatly different despite very different modeling approaches.*
A complication in the comparison of G/NG model performance is that most of the
* It should be noted that this analysis disagrees with the results of Hajmeer and Basheer (2002) who
reported only two to four mispredictions for the total (i.e., training and validation) data set.
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data are readily predicted, e.g., those that fall outside the known limits for growth
for individual environmental factors when all other factors are optimal. Such data
can "overwhelm" the data in which one is most interested, i.e., in the relatively
narrow region where factors interact to reduce the biokinetic ranges and, yet more
specifically, where the probability of growth rapidly changes from "growth is very
likely" to "growth is very unlikely." These data de ne the interface and, consequently,
data closest to the interface are more important when comparing model performance.
This has implications for experimental design, as discussed in Section 3.4.4 below.
To assess whether one model might be preferred on theoretical grounds, as
adjudged by its ability to extrapolate reliably, the predictions of all models in the
temperature range above 35°C can be compared to data subsequently generated,
shown in Figure 3.11, and not used to generate the models. Clearly, none of the
models extrapolate well.
From the above comparison, it appears that despite very different modeling
approaches and degrees of complexity of modeling, there is currently little to dif-
ferentiate those approaches on the basis of their ability to describe the G/NG interface
or on their ability to predict outside the interpolation region.
3.4.4 Experimental Methods and Design Considerations
As suggested above, currently there is little mechanistic understanding of how
environmental factors interact to prevent bacterial growth and it must be recognized
as a possibility that there is no single, common mechanism underlying the observed
boundaries for different factor combinations. Consequently, it is not possible from
rst principles to design the optimal experiment that captures the essential informa-
tion that will characterize the response and lead to reliable models. Instead, at this
time, experimental methods must be focused toward gaining enough data in the
interface region to be able to describe empirically the limits to growth.
First of all, two approaches may be distinguished that could affect the experi-
mental methods chosen. In one, the interest is in whether growth/toxin production,
etc. is possible within some specified time limit, which may be related to the shelf
life of the product. In other approaches, the objective is to de ne absolute limits to
growth, i.e., the most extreme combinations of factors that just allow growth. McKel-
lar and Lu (2001) argue that there is always a time limit imposed on G/NG modeling
studies. Strategies exist, however, that provide greater con dence that the "absolute"
limits to growth are being measured. Some of these are discussed below.
3.4.4.1 Measuring Both Growth and Inactivation
Several groups have assessed both growth and inactivation in their experimental
treatments (McKellar and Lu, 2001; Parente et al., 1998; Presser et al., 1999;
Razavilar and Genigeorgis, 1998). In this way the position of the boundary is inferred
from two "directions." If growth is not observed, an observer cannot be sure whether
growth is not possible or has not occurred yet. If it is known that at some more
extreme condition inactivation occurs, it can be inferred that the G/NG boundary
lies between those two sets of conditions.
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A potential problem with this strategy is that cultures can initially display some
loss of viability, but with survivors eventually initiating growth; i.e., population
decline cannot unambiguously be interpreted as "growth is not possible." Numerous
studies (e.g., Mellefont et al., 2003) have demonstrated that rapid transfer of a culture
from one set of conditions to another that is more stressful can induce injury and
death, but that survivors will eventually adjust and be able to grow. This has been
termed the Phoenix phenomenon (Shoemaker and Pierson, 1976). Such regrowth
has been reported in the context of G/NG modeling (Bolton and Frank, 1 999; Masana
and Baranyi, 2000b; Parente et al., 1998; Tienungoon et al., 2000).
Clearly, an experimenter interested in determining the "absolute" G/NG bound-
ary will need to maximize the resistance of the inoculum to stress on exposure to
the new, more stressful, environment. The use of stationary phase cultures as inocula
would seem to be a minimum requirement. It may be necessary to habituate cultures
to the test conditions (e.g., growth at conditions just less harsh) prior to inoculation
into the test conditions to maximize the chance that growth, if possible, will be
observed. One way to maximize the likelihood of observing the most extreme growth
limits would be to use cultures growing at the apparent limits as inocula into slightly
more stringent conditions. This also has the advantage of minimizing growth lags
on inoculation into a harsher environment.
3.4.4.2 Inoculum Size
Masana and Baranyi (2000b) indicated that inoculum size affected the position of
the boundary. Robinson et al. (1998) reported similar effects of inoculum density
on bacterial lag times. While it is clear that time to detection would depend on
inoculum density, growth detection methods were not cell-density -dependent in
those studies. Parente et al. (1998) also reported that a decrease in inoculum size
decreased the probability of survival. If the shock of transfer is known to inactivate
a xed proportion of the cells in the inoculum, to develop a robust model it will be
necessary to use an inoculum that ensures that even after inactivation there is a high
probability that at least one cell will survive.
The above observations lend support to the hypothesis that it is the distribution
of physiological states of readiness to survive and multiply in a new environment
that determines the position of the G/NG boundary, i.e., all other things being equal,
the more cells in the inoculum the more likely it is that there is one cell that has
the capacity to survive and grow. This also reinforces the equivalence between G/NG
boundary modeling and the modeling of conditions that lead to in nite lag times.
The importance of the distribution of lag times on the observed lag times of whole
populations has been discussed by Baranyi (1998).
There may be more involved reasons for inoculum density -dependent responses
also, such as chemical messaging between cells (see, e.g., Miller and Bassler, 2001 ;
Winans and Bassler, 2000).
In conclusion, if the aim is to determine absolute limits to growth, a higher
number of cells is preferable. Masana and Baranyi (2000b) stated that growth
boundaries "represented the chance of growth for each sample; therefore, to assure
a low probability of growth in many samples, it will be more relevant to consider
2004 by Robin C. McKellar and Xuewen Lu
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boundaries for high inoculum levels." Equally, as noted above, steps to maximize
the cell's chances of survival and growth in the environment are also recommended.
There is potentially a caveat, however, that needs to be applied. Maximum
population densities of batch cultures are reported to decline under increasingly
harsh growth conditions. Thus, the use of high inocula may mask the true position
of the G/NG boundary if the inoculum used is already denser than the MPD of the
organism in a very stressful test environment.
3.4.4.3 Are There Absolute Limits to Microbial Growth?
In the above discussion it has been implicitly assumed that there are absolute limits
to microbial growth under combined environmental stresses. It is pertinent to exam-
ine this assumption.
Numerous authors have noted that, within an experiment, the transition between
conditions that allow growth, and those that do not, is abrupt and that usually all
replicates at the last growth condition grow, while all the replicates at the rst-growth-
preventing condition do not (Masana and Baranyi, 2000; Presser et al. , 1 998; Tienungoon
et al., 2000). McKellar and Xu (2001), for example, reported that of 1820 conditions
tested, all ve replicates of each condition either grew or did not grow. This abruptness,
however, is not always evident in the modeled results (Tienungoon et al., 2000).
Conversely, between experiments by the same researcher, using the same meth-
ods and the same strain, results are not always reproducible. Figure 3.9 provides an
example and Masana and Baranyi (2000b) make the same observation of their data
for Brochothrix thermosphacta. At the same time, however, there is evidence of
excellent reproducibility of boundaries between independent workers, using different
strains, and different methods in different locations. The results of Tienungoon et
al. (2000) were highly consistent between two strains tested, and more notably, with
those of George et al. (1988) and Cole et al. (1990) presented a decade earlier,
including different strains in one case. There is also a remarkable similarity between
the pH/temperature G/NG interface of Listeria innocua reported by Le Marc et al.
(2002) and the same interface for two species of L. monocytogenes presented in
Tienungoon et al. (2000).
Jenkins et al. (2000) noted that the boundary they derived for the growth limits
of Zygosaccharomyces baillii in beverages was very consistent with a model devel-
oped 30 years earlier for the stability of acidic sauces.
Stewart et al. (2002) noted that with S. aureus, as conditions became increasingly
unfavorable for growth, the contour lines (P grow th) they generated drew closer and
closer together, suggesting that conditions were approaching absolute limits that do
not allow growth. Conversely, there are examples where one group's observations
do not agree well with another's for an analogous organisms/environmental pair
(e.g., Bolton and Frank, 1999; McKellar and Xu, 2001). Delignette-Muller and Rosso
(2000) reported strain variability in the minimum temperature for growth.
While the above discussion points to heterogeneity in the physiological readiness
of bacteria to grow in a new environment, Masana and Baranyi (2000b) also infer
that differences in microenvironments, particularly within foods, could also be a
source of heterogeneity in observed growth limits.
2004 by Robin C. McKellar and Xuewen Lu
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~V
In conclusion, there is a body of experimental evidence that suggests that growth
boundaries, if carefully determined, might be highly reproducible. Conversely, coun-
terexamples exist. It remains to be determined whether the incongruous results arise
from signi cant and measurable differences in methodology, e.g., the detection time
used in the respective studies, or are due to uncontrollable sources (Table 3.7).
3.4.4.4 Experimental Design
As noted above, it is not possible from rst principles to design the optimal exper-
iment that captures the information to characterize the G/NG boundary. Various
authors have suggested physiological interpretations (Battey et al, 2001; Battey and
Schaffner, 2001; Jenkins et al., 2000; Lopez-Malo et al., 2000; McMeekin et al.,
2000) but none have yet been experimentally tested.
Thus, an empirical approach that aims to collect as much information in the
region of most interest, i.e., the G/NG interface, is recommended by most workers.
Several groups of researchers have indicated that they use a two-stage modeling
process. The rst uses a coarse grid of conditions of variables to roughly establish
the position of the boundary. The second phase monitors responses at conditions
near the boundary and at close intervals of the environmental parameters. Variable
combinations far from the interface, at which growth is either highly likely or highly
unlikely, do not provide much information to the modeling process, which seeks to
de ne the interface with a high level of precision. Equally, it is ideal to use a design
that gives roughly equal numbers of conditions where growth is, and growth is not,
observed (Jenkins et al., 2000; Legan et al., 2002; Masana and Baranyi, 2000b, Uljas
et al., 2001). Pragmatically, Legan et al. (2002) recommend setting up "marginal"
and "no-growth" treatments rst because these treatments will run for the longest
time (possibly several months). Those conditions in which growth is expected to be
relatively quick can be set up last because they only need monitoring until growth
is detected.
The nature of these studies necessarily involves long incubation times. Legan
et al. (2002) noted that particular care must be taken to ensure that the initial
conditions do not change over time solely as a result of an uncontrolled interaction
with the laboratory environment. Prevention of dessication or uptake of water vapor
requires particular attention. Changes resulting from microbial activity may, how-
ever, be an important part of the mechanism leading to growth initiation and should
not be stabilized at the expense of growth that would naturally occur in a food.
Legan et al. (2002) comment that, for example, maintaining the initial pH over time
is typically neither possible nor practical, even in buffered media, and that allowing
a change in pH due to growth of the organism more closely mimics what would
happen in a food product than maintaining the initial pH over time.
3.4.4.5 Conclusion
From the above discussion, unambiguous de nition of the G/NG boundary of an
organism in multidimensional space presents several paradoxical challenges. While
an experimenter will do well to remember these considerations in the interpretation
2004 by Robin C. McKellar and Xuewen Lu
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of his/her results, it seems probable that methods that have been used to date will
have come close to identifying the "true" G/NG boundary, and that the position of
the boundary will move only slightly if an experimenter acts to control all of the
above variables and to maximize the potential for the observation of growth in the
chosen experimental system.
While the discussion has not focused specifically on appropriate methods for
probability of growth within a defined time, many of the same principles and
considerations will apply.
Moreover, the field of growth limits modeling, while having an equally long
history as kinetic modeling, now seems to be quite disjointed, with little rigorous
comparison of approaches, let alone agreement on the most appropriate model
structures or experimental methods. In particular, the earlier work in probability
modeling seems to have been ignored by some more contemporary workers, without
reasons being indicated.
The results of G/NG studies are clearly of great interest to food producers and
food safety managers. It is perhaps time, then, that the G/NG modeling community
seeks to find common ground and to begin to develop a rigorous framework for the
development, and interpretation, of growth limits studies.
APPENDIX A3.1 — CHARACTERIZATION OF
ENVIRONMENTAL PARAMETERS AFFECTING
MICROBIAL KINETICS IN FOODS
A3. 1.1 Temperature
In most situations, temperature is the major environmental parameter in uencing
kinetics of microorganisms in food and its effect is included in most predictive
microbiology models. During processing, storage, and distribution the temperature
of foods can vary substantially, frequently including periods of temperature abuse
for chilled foods (see, e.g., Audits International, 1999; James and Evans, 1990;
O'Brien, 1997; Sergelidis et al., 1997). Thus, it is an important property of secondary
models that they can predict the effect of changing temperatures on microbial
kinetics and application of these models relies on information about product tem-
perature and its possible variation over time. Numerous types of thermometers,
temperature probes, and data loggers are available (McMeekin et al, 1993, pp.
257-269; seagrant.oregonstate.edu/extension/ sheng/loggers.html) to measure the
temperature of foods or food processing equipment. Infrared non-contact thermom-
eters are often appropriate for foods but their use is limited for process equipment
with stainless surfaces.
A3. 1.2 Storage Atmosphere
Foods are typically stored aerobically, vacuum packed, or by using modi ed atmo-
sphere packing (MAP). "Controlled atmosphere packaging" can be considered a
special case of MAP. MAP foods are exposed to an atmosphere different from both
2004 by Robin C. McKellar and Xuewen Lu
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air and vacuum packed usually involving mixtures of the gasses carbon dioxide
(C0 2 ), nitrogen (N 2 ), and oxygen (0 2 ).
2 and C0 2 in uence growth of most microorganisms and secondary predictive
models must take their effect into account. The solubility of 2 in water, and thereby
into the water phase of foods, is low (~0.03 1/1) but it can be important for growth
and metabolism of microorganisms in both aerobic and MAP-stored products
(Dainty and Mackey, 1992). Numerous techniques and instruments are available to
determine 2 in the gas phase or dissolved in food. Microelectrodes to determine
gradients of dissolved 2 in foods are available (www.instechlabs.com/oxy-
gen.html; www.microelectrodes.com/) but models to predict the effect of such
gradients remain to be developed. To account for the effect of aerobic or vacuum
packed storage of foods a categorical approach has been used within predictive
microbiology. For aerobic conditions growth media with access to air have been
agitated. For vacuum packed foods microorganisms typically have been grown
under 100% N 2 .
C0 2 inhibits growth of some microorganisms substantially and, to predict micro-
bial growth in MAP foods, it is important to determine the equilibrium concentration
in the gas phase or the concentration of C0 2 dissolved into the foods water phase.
At equilibrium, the concentration of C0 2 dissolved into the water phase of foods
is proportional to the partial pressure of C0 2 in the atmosphere surrounding the
product. Henry's law (Equation A3.1) provides a good approximation for the solu-
bility of C0 2 .
CO A— =K H PC0 2 (A3.1)
In Equation A3. 1, K H is Henry's constant (mg/l/atm) and pC0 2 is the partial pressure
(atm) of C0 2 . Between and 160°C the temperature dependence of the Henry's
constant can be predicted by Equation A3. 2:
r-i _^_ -i
K H (mg - 1 - atm ) =
101325-2.4429 ( A3 - 2 )
exp(-6.8346 + 1 .2817 ■ 10 4 / K - 3.7668 • 10 6 / K 2 + 2.997 • 10 3 / K 3 )
where K is the absolute temperature (Carroll et al., 1991). Those authors expressed
K u as MPa/mole fraction. In Equation A3. 2 the constants 101,325 Pa/atm and 2.4429
was used to convert this unit into mg C0 2 /1 H 2 0/atm.
For MAP foods in exible packaging the partial pressure of C0 2 is conveniently
determined from the percentage of C0 2 inside the pack. A range of analytical
methods is available to determine C0 2 concentration in gas mixtures or concentra-
tions of dissolved C0 2 (Dixon and Kell, 1989; www.pbi-dansensor.com/Food.htm).
As shown from Equation A3.1 and Equation A3. 2, the concentration of C0 2
dissolved in the water phase of a MAP food with 50% C0 2 in the headspace gas at
equilibrium is 1.67 g/1 at 0°C and 1.26 g/1 at 8°C. Because of the high solubility of
2004 by Robin C. McKellar and Xuewen Lu
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C0 2 in water the gas composition in the headspace of MAP foods changes after
packaging. The equilibrium gas composition is in uenced by several factors, e.g.,
the percentage of C0 2 in the initial headspace gas (%C0 2 Imtial ), the initial gas/product
volume ratio (G/P), temperature, pH, lipids in the food, respiration of the food, and,
of course, permeability of the packing lm. Different mass-balance equations to
predict the rate of adsorption and solubility of C0 2 have been suggested (Devlieghere
et al, 1998; Dixon and Kell, 1989; Gill, 1988; Lowenadler and Ronner, 1994;
Simpson et al., 2001 a,b; Zhao et al., 1995). In chilled foods the rate of absorption
of C0 2 is rapid compared to growth of microorganisms. Therefore, to predict micro-
bial growth in these MAP foods it is suf cient to take into account the equilibrium
concentration of C0 2 .
Devlieghere et al. (1998) suggested Equation A3. 3 to predict the concentration
of C0 2 in the water phase as a function of %C0 2 Imtial and G/P. In Equation A3. 3,
d C02 is the density of C0 2 (1.976 g/1).
/
V
G
P
\
■ dC0 2 + K H
)
(G
\(~ir} "1 Equilibrium
L 2-1 aqueous
V
P
■ dC0 2 + K H
)
(
4
V
100
K H —- %CO* nitial ■ dC0 2
P
J
2
(A3.3)
Equation A3. 3 does not take into account the effect of the storage temperature and
Devlieghere et al. (1998) developed a polynomial model to predict the concentration
of dissolved C0 2 as a function of %C0 2 Imtial , G/P, and temperature. If, for example,
%C0 2 Imtial is 25, the polynomial model predicts that a G/P ratio of three results in
higher concentration of dissolved C0 2 than does a G/P ratio of 4, which is not
logical. In contrast we have found that the combined use of Equation A3. 2 and
Equation A3. 3 provides realistic predictions for concentrations of dissolved C0 2 . It
also seems relevant to include the effect of product pH on dissolved C0 2 , and thereby
the equilibrium concentration of C0 2 in the gas phase of MAP foods.
A3. 1.3 Salt, Water-Phase Salt, and Water Activity
While temperature is the single most important storage condition in uencing growth
of microorganisms in foods, NaCl is the most important product characteristic in
many foods. The concentration of NaCl in foods can be determined as chloride by
titration (Anon., 1995a). Instruments to determine NaCl indirectly from conductivity
measurements are available but extensive calibration for particular types of products
may be required. In fresh and intermediate moisture foods, NaCl is dissolved in the
water phase of the products.
To predict the effect of NaCl on growth of microorganisms in these products
the concentration of water-phase salt (WPS) or relative humidity, i.e., the water
activity (a w ) must be determined (Equation A3. 4 to Equation A3. 7).
Water-phase salt can be calculated from Equation A3. 4:
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% Water phase salt =
%NaCl (w/v) x 100/(100 - % dry matter +%NaCl (w/v)) (A3.4)
Water activity is a fundamental property of aqueous solutions and is de ned as:
a = -5- (A3.5)
w
Po
where p is the vapor pressure of the solution and p is the vapor pressure of the
pure water under the same conditions of temperature, etc.
For mixtures of NaCl and water there is a direct relation between the WPS
content and a w (Resnik and Chirife, 1988; Equation A3. 6 and Equation A3. 7). For
cured foods where NaCl is the only major humectant these relations are valid as
documented, e.g., for cold-smoked salmon (Jorgensen et al., 2000) and processed
"delicatessen" meats (Ross and Shadbolt, 2001). To determine water activity of
foods, instruments relying on the dew point method are now widely used because
of their speed (providing results within a few minutes), robustness, and reliability
but other methods and instruments are available (Mathlouthi, 2001).
a w = 1-0.0052411 -%WPS- 0.00012206 -%WPS 2 (A3.6)
% WPS =8-140.01 -(a -0.95) -405. 12 -(a -0.95) 2 (A3.7)
w / v w
A3.1.4 pH
For many microorganisms, small pH variations in the pH range ~6 to -7 have very
little or no effect on population kinetics. In more acidic foods, however, pH per se
can greatly in uence microbial kinetics but can also accentuate the effect of other
added preservative compounds. The pH of solid foods is often determined by homog-
enizing 10 g of a sample with 10 to 20 ml of distilled water and measuring the pH
of the suspension using a standard combined electrode.
A3. 1.5 Added Preservatives Including Organic Acids,
Nitrate, and Spices
High concentrations of organic acids occur naturally in some foods and various
organic acids including acetic acid, ascorbic acid, benzoic acid, citric acid, lactic
acid, and sorbic acid are frequently added to foods. Organic acids can inhibit growth
of microorganisms markedly and secondary models to predict their inhibitory effect
are frequently needed. As for NaCl the secondary models must take into account
the concentration of organic acids in the water phase of products. In addition,
secondary models may need to describe the combined effect of organic acids and
other environmental parameters particularly the pH.
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In solution, organic acids exist either as the dissociated (ionized) or undissociated
species. The Henderson-Hasselbalch equation (Equation A3. 8) relates the proportion
of undissociated and dissociated forms of organic acid to pH and pK a according to
the following expression:
[A-]/[HA] = 10 H -P K a (A3.8)
where [HA] is the concentration of undissociated form of the acid, [A~] the concen-
tration of dissociated (ionized) form of the acid, and pK a is the pH at which the
concentrations of the two forms are equal.
While both the dissociated and the undissociated forms of organic acids have
inhibitory effects on bacterial growth the undissociated form is more inhibitory,
usually by two to three orders of magnitude, than the dissociated form (Eklund, 1989).
Cross-multiplying and rearranging Equation A3. 8 to solve for [HA] gives:
[HA] = [LAC]/(1 + 10P H "P K a) (A3. 9)
where [LAC] is the total lactic acid concentration and all other terms are as previously
de ned.
As the concentration of an undissociated acid increases the growth rate of
microorganisms decreases, eventually ceasing completely at a level described as the
MIC. This behavior, and its dependence on the interaction of pH and total organic
acid concentration, is included explicitly in several secondary models (Augustin and
Carrier, 2000a; Presser et al., 1997).
Simple enzyme kits are available to determine several of the organic acids that
are important in foods. Simultaneous determination of a range of organic acids is
possible by HPLC analysis and is often an appropriate method to use (Dalgaard and
Jorgensen, 2000; Pecina et al., 1984).
Nitrite can be added to some types of meat products and its concentration in the
water phase of products must be taken into account when secondary predictive
models for these products are developed. Colorimetric methods are available to
measure the concentration of nitrite in foods (Anon., 1995b; Karl, 1992).
Spices and herbs can have substantial antimicrobial activity and appropriate
terms may need to be included in secondary models (Koutsoumanis et al., 1999;
Skandamis and Nychas, 2000). The concentration of active antimicrobial compo-
nents in spices, herbs, and essential oils can vary substantially as a function, e.g.,
of geographical region and season (Nychas and Tassou, 2000; Sofos et al., 1998).
Therefore, the development of accurate secondary predictive models most likely will
have to rely on the concentration of their active antimicrobial components. Recently,
Lambert et al. (2001) showed the antimicrobial effect of the oregano essential oil
quantitatively corresponded to the effect of its two active components, i.e., thymol
and carvacrol. To quantitatively determine active components in spices, herbs, and
essential oils appropriate extracts can be analyzed by GC/MS techniques (Cosentino
et al., 1999; Cowan, 1999).
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A3. 1.6 Smoke Components
It has long been known that high concentrations of smoke components have strong
antimicrobial activity (Shewan, 1949). Today many meat and seafood products are
smoked but typically less intensively than some decades ago. However, even mod-
erate concentrations of smoke components can in uence growth rates, growth
limits, and rates of death/inactivation of microorganisms in foods (Leroi et al.,
2000; Leroi and Joffraud, 2000; Ross et al., 2000b; Sunen, 1998; Thurette et al.,
1998). Thus, to obtain accurate prediction of microbial kinetics in smoked foods
it is important to include terms for the effect of smoke components in secondary
models. Phenols are important antimicrobials in wood smoke, or in liquid smokes,
and a few secondary models include the total phenol concentration as an environ-
mental parameter (Augustin and Carlier 2000a,b; Gimenez and Dalgaard, in press;
Membre et al., 1997).
Classical colorimetric methods can be used to determine the total concentration
of phenols in smoked foods. These methods rely on formation of colored complexes,
e.g., between phenols and Gibb's reagent (2,6-dichloroquinone-4-chloroimide) or 4-
aminoantipyrine (Leroi et al., 1998; Tucker, 1942). The total phenol concentration
is a crude measure of how intensely foods have been smoked. By using GC/MS
techniques more detailed information about specific smoke components can be
obtained (Guillen and Errecalde, 2002; McGill et al., 1985; Toth andPotthast, 1984).
In the future, secondary models may be developed to include the effect of specific
phenols, other specific smoke components, and possibly their interaction with NaCl.
During the smoking of foods, phenols and other smoke components are mainly
deposited in the outer 0.5 cm of the product (Chan et al., 1975). Modeling the effect
of the spatial distribution in foods is another challenge.
A3. 1.7 Other Environmental Parameters
The environmental parameters discussed above include those that are of major
importance in traditional methods of food preservation. Many modern methods of
food preservation also rely on combinations of these environmental parameters.
However, the effect of a few well-known and several emerging food processing
technologies relies on the antimicrobial effect of other environmental parameters,
e.g., bacteriocins, gamma irradiation, high electric field pulses, high pressure, and
UV light. Secondary models for the effect of some of these environmental parameters
have been developed but will not be discussed here in detail. Other environmental
parameters related to food structure and to the effect of microbial metabolism on
changes in environmental parameters are discussed in Chapter 5 whereas the effect
of time- varying environmental parameters is discussed in Chapter 7.
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