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Q Modeling the History
Effect on Microbial
Growth and Survival:
Deterministic and
Stochastic Approaches

Jozsef Baranyi and Carmen Pin

CONTENTS

9.1 Introduction

9.2 Modeling the History Effect at Population Level
(Deterministic Modeling)

9.2.1 Traditional Parameters

9.2.2 Replacing the Lag Parameter with a Physiological
State Parameter

9.2.3 A Rescaling of the Physiological Parameter Provides
a New Interpretation

9.3 Modeling the History Effect at Single-Cell Level
(Stochastic Modeling)

9.3.1 Nature of Stochastic Modeling

9.3.2 Physiological State and Lag Parameters for Single Cells

9.3.3 A Simulation Program

9.3.4 Lag/Survival Distribution of Single Cells and Lag/Shoulder
of Population Curves

9.3.5 History Effect on the Shoulder Periods of Survival Curves

9.4 Concluding Remarks
Acknowledgment
References

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9.1 INTRODUCTION

The objective of food safety microbiology is to eliminate or at least significantly
reduce the concentration of pathogenic microbes in food and to prevent their resus-
citation and multiplication. Predictive microbiology focuses on the quantitative
methods used to achieve this goal.

In a mathematical sense, a predictive model is a mapping between the environ-
ment of the microbial cells and their response. While primary models describe how
the microbial cell concentration changes with time in a constant environment (see
Chapter 2), a secondary model (such as D values vs. temperature; see Chapter 3)
expresses a typical parameter of the response as a function of the environmental
factors. Both the environment and the response are characterized by one or more,
possibly dynamic, variables (i.e., they may change with time during observation).
Temperature changing with time is an example of a dynamic environment (see
Chapter 7); growth/survival curves are examples of dynamic responses, either in
constant or dynamic environments.

In accordance with the standard approach to modeling, microbial responses are
studied from a distinguished time point — the zero time of observation. Naturally,
this zero time is when the environment of the cells undergoes a sudden change.
Researchers should know, or at least implicitly assume the conditions at this zero
time to provide initial values for modeling. Such initial values are, for example, the
initial composition of natural flora (in the case of food environment) or the inoculum
level and initial physiological state of the studied cells. Respectively, we can speak
about actual (current) and previous (historical or preinoculum) environments — the
initial values are the result of the previous environment.

A major obstacle in the development of predictive microbiology is that it is
difficult to acquire data at low levels of cell concentration. Currently, commonly
available counting methods can only measure cell concentrations of higher than
approximately 10 cells per gram of sample. Therefore, it is frequently problematic
to validate predictive microbiology models since most of them focus on situations
when the cell concentration is relatively low and the measurements are inaccurate.

Growth modeling concentrates on the early stages of microbial growth in the
actual environment. As mentioned, the previous environment affects growth in the
actual environment, which effect gradually diminishes after inoculation. This is called
the adjustment process; the time during which it happens is called the lag period.

Death modeling deals with decay in microbial concentration having two main
causes: thermal inactivation, when death is relatively fast; and nonthermal death,
which is generally a much slower process. Although the physiological mechanisms
of the two can be completely different, it is possible to construct mathematical
models for them in an analogous way. What is more, to some extent, growth and
death modeling can also be discussed analogously. Accordingly, we show some basic
techniques to model the effect of history on growth and death in a parallel way.

We also present a comparative review of the applied deterministic and stochastic
models. Deterministic models can always be treated as sort of "averaged out"
versions of stochastic models. However, "smoothing down" the biological variability
means a certain loss of information — the question is just how much. Biology itself

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always has stochastic elements, but sometimes they are small enough (or we hope
they are small enough) to be negligible. In this case, the applied model (and its one
or more variables and processes) is reduced to be deterministic, which is simpler
and easier to handle. However, such a reduction can be misleading when the random
elements are significant. For example, when modeling the lag period of a population
of relatively few cells (the "dormant" phase of the "log cone. vs. time curve" before
exponential growth), the random effects originating from the variability among the
cells cannot be ignored. Similarly, when modeling the number of survivors in an
adverse environment, the stochastic variability of the resistance of individual cells
can also become significant.

Many authors have studied the effect of history on the lag time before growth
(Augustin et al., 2000a; Augustin and Carlier, 2000; Beumer et al., 1996; Breand et
al., 1999; Buchanan and Klavitter, 1991; Dufrenne et al., 1997; Gay et al., 1996;
Hudson, 1993; Mackey and Kerridge, 1988; Membre et al., 1999; Stephens et al.,
1997; Walker et al., 1990; Wang and Shelef, 1992) and on the shoulder period before
exponential death (Cotterill and Glauert, 1969; Manas et al., 2001; Ng et al., 1962;
Sherman and Albus, 1923; Strange and Shon, 1964). In this chapter, we attempt to
provide a simple modeling approach to take into account the history effect via the
concept of initial work/damage done before the exponential growth/death phase.
First, we apply a deterministic model at population level (i.e., when the population
size is represented by a single continuous variable), after which we discuss similar
questions at single-cell level, using stochastic processes, when the variability among
individual cells is also taken into account.

9.2 MODELING THE HISTORY EFFECT AT
POPULATION LEVEL (DETERMINISTIC
MODELING)

9.2.1 Traditional Rvrameters

Throughout this chapter, x(t) denotes the cell concentration at the time t, and y(t)
denotes its natural logarithm. The slope of the "y(f) vs. time" curve is the (instan-
taneous) specific rate, denoted by \l(t). Its maximum value, |i max , is the maximum
specific rate. It is positive for growth and negative for death (Figure 9.1a and Figure
9.1b). The maximum specific rate is measured at the inflexion point of the curve,
if it has a sigmoid shape (Figure 9.1a). If the model is biphasic (Figure 9.1b), then
this maximum does not necessarily exist. In this case, (J max is meant in an asymptotic
sense: \l(t) — > (J max as t — > °°. In fact, we define the biphasic class of models by
the criterion that \i(t) converges monotonically from a small, |i.(0) initial rate, to a
finite value.

In the case of growth, the most frequently modeled parameter is the maximum
specific growth rate, or one of its rescaled versions: the r max maximum rate in terms
of log 10 cell concentration, or the doubling time, T d . Note the relation between them:

r„« = lWln 10; T d = In 2/^ max (9.1)

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a)

/ ,m

t

FIG U RE 9.1 (a) Main parameters of a sigmoid growth curve. The maximum specific growth
rate |i. max can be measured at the inflexion point of the curve, (b) Main parameters of a biphasic
survival curve. The instantaneous specific death rate converges to a limit value |i r

'max.

In the case of survival (or inactivation), instead of the analogous "halving time,"
it is more common to use the time to a decimal reduction, called D value:

D = -In 10/fJ

max

(9.2)

The reason for this slight inconsistency is historical: the first predictive models were
created to describe thermal inactivation processes and, at high temperatures, the
decimal reduction times were more practical to use than halving times.

Some authors use the maximum specific rate and some the rate of the log 10 x(f)
function, which can be obtained by an ln(10) ~ 2.3 conversion factor from [{(t). Most
frequently, cell concentration data are given in terms of log 10 of the cell number per
volume. However, the maximum specific rate has a more universal theoretical mean-
ing as shown by the formulae below:

dy d(\n x) dx I x

dt

dt

dt

m

(9.3)

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that is,

— « ll At (9.4)

max
X

The left-hand side of the latter equation denotes a relative increase or decrease of
the population in At time. Therefore, the following interpretation can be given for
the maximum specific rate: the probability that, at the time t, a single cell divides
(growth) or dies (death), during the small [t, t + At] interval, is about \x(t)At; i.e.,
proportional to the length of At, with the factor \x(t). This is a fundamental link
between the specific growth rate measured at population level and the probability
of division at single-cell level.

Another important, frequently modeled parameter is the lag time (for growth),
or shoulder period (for survival), which is denoted by X throughout this chapter. For
sigmoid curves (Figure 9.1a), it is traditionally defined as the time when the tangent
of the y(t) curve with the maximum slope crosses the initial level, y = In x Q (Pirt,
1975). (It is irrelevant whether the natural logarithm or log 10 is used to define the
lag or shoulder.) For biphasic curves (Figure 9.1b), to find a good "geometrical"
definition is not so straightforward as pointed out below.

The definition of lag/shoulder does not have the strong basis that would connect
the parameter to a single-cell equivalent, as in the case of the specific rate. Baranyi
(1998) discussed the mathematical relation between the lag of the population (as
defined above) and the lag times of the participating individual cells (see later in
this chapter). He showed that the population lag depends on the subsequent specific
growth rate and the initial cell number, even if the individual lag times do not.
Another anomaly can be shown by means of biphasic models. It is natural to expect
that, if a series of tangents drawn to the y(t) curves converges to a limit value (see
Figure 9.1b), then the intersections with the initial v level also converge, in which
case the lag could be defined by the limit value. Baranyi and Pin (2001) showed a
biphasic death model where this was not true; the intersection, denoting the end of
the shoulder period, would not converge, as t — > °°, although the specific death rate
did converge.

9.2.2 Replacing the Lag Rrameter with a Rjysiological

Sate Parameter

For growth, the "waiting time" of the individual cells is due to an adjustment process
before the exponential growth. For survival curves, the shoulder period is due to an
initial resistance before exponential death (such as in the case of the multitarget
theory, Hermann and Horst, 1970). As shown by Baranyi (2002), it is not easy to
relate the distribution of the "waiting times" of individual cells to the apparent
lag/shoulder parameter of the population, measured in the traditional way.

Baranyi and Roberts (1994) suggested that the lag should be considered as a
derived parameter, a consequence of both the actual environment (via the specific
growth rate), and the history of the cells (via a newly introduced parameter, the
initial physiological state of the cells). Their method can be outlined as follows.

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c

3

o
o

time

FIGU RE 9.2 Interpretation of the oc physiological state. h = -ln(a ) = A,|i max is the "work
to be done," which is the same for different temperatures. h is an initial value, as is \nx .

Just as y is an initial value, we can introduce another value in order to quantify
the "suitability" of the population to the current environment (i.e., the history effect).
Let this quantity be denoted by oc (which comes from the term adjustment function,
in the original paper). Let it be a dimensionless number between and 1. For growth,
consider it as the fraction of the initial cells, which, without a lag, would be able to
produce the same final growth as the original x initial cells with a lag. (Although
the real situation is not like this, it is easier to think about a this way. In reality,
a is the mean value of the respective parameters of the individual cells. The point
is, as we will see later, that it does not matter if we think of it as "oc , the fraction
of the cells that are able to grow," or "each cell is oc -suitable")

By simple geometry (see Figure 9.2), it can be shown that

X =

-ln(cc n )

M

(9.5)

max

This formula expresses our expectation that the lag time depends on both the actual
environment, represented by |i max , and the history of the cells, characterized by oc .
The history effect appears as an initial value, and this will be called the initial
physiological state. Note that it does not represent simply the "health" of the cells,
rather their suitability to the actual environment. Its extreme values are a = 0, in
which case the lag is infinitely long, and oc = 1, when all the cells enter the
exponential phase immediately and the lag time is 0. The effect of the initial
physiological state on the lag is shown in Figure 9.3.

In terms of biological interpretability, the main advantages of using oc , instead
of lag, are:

1. It is an initial value, affecting the lag, and this is in accord with our
mechanistic thinking.

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time

FIG U RE 9.3 Depending on the cc physiological state, the lag can change while the maximum
specific growth rate remains the same.

2. It has a very strong connection with similar parameters of the individual
cells: the physiological state of the population is equal to the average of
the physiological states of the initial cells, irrespective of their number
(Baranyi, 1998).

3. It can be interpreted, via the individual cells, even if no traditional lag
can be measured on the bacterial curve (for example, when it does not
reach its inflexion point).

9.2.3 A Rescaling of the Rmio logical Parameter Riovides
a New Interpretatio n

An interpretation of the history effect is also possible by means of the inverse of
the physiological state. Introduce h Q = ln(l/a ) = -In a . It can be conceived as an
initial hurdle, or the "work to be done" during the lag (Robinson et al., 1998; see
Figure 9.2). As we see, it is the product of the lag and the maximum specific growth
rate. Observe that, using the doubling time, T d = In 2/|i. max relation, the following
formula can be obtained:

h Q =ln2

T,

(9.6)

The quantity X/T d is often called "relative lag" (McMeekin et al., 2002) and it is
essentially the same as h . Some authors have reported that it is unaffected by small
changes of temperature (Pin et al., 2002; Robinson et al., 1998) and pH (McKellar
et al., 2002a). This can be explained as follows: the more suitable the cells are to
the new environment, the less work they have to carry out to adapt. Therefore, the
lag is longer at lower temperatures because the "work to be done," after the inocu-
lation, is carried out more slowly, although the amount of work is the same (Figure
9.4). Authors studying the effect of history and actual growth conditions on the work
to be done during the lag include Delignette-Muller (1998), Robinson et al. (1998),
Pin et al. (2002), and Augustin et al. (2000a).

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lnx

ln(a

time

FIGURE 9.4 As the temperature decreases, the work to be done (the difference between
In x and ln(oc x )) is the same, while the maximum specific growth rate (i max becomes
smaller and the lag becomes longer. The work to be done during the lag is also carried out
at slower rates.

The parameter h = -In oc , the "work to be done" before the exponential phase,
plays the role of a bridge between the history and the actual environment. With an
analogy, imagine that cells are crossing this bridge. If its angle is horizontal (h Q =
0; no difficulty to overcome), then there is no lag. If it is vertical (h = <*>), then the
work to be done is infinitely big, the lag is infinitely long. On the other hand, if the
temperature is higher then every movement is quicker and so is the crossing (lag),
although the work to be done is the same (Figure 9.4). It can be expected that this
work is smaller when adjusting to a more favorable environment than the opposite
way (Delignette-Muller, 1998; Robinson et al., 1998). This theory explains why the
environmental effect on the lag has always been much less accurately predicted
(Dufrenne et al., 1997; McKellar et al., 2002b; McMeekin et al., 2002; Robinson et
al., 1998; Whiting and Bagi, 2002) than the specific growth rate. The lag also depends
on the history, the details of which often have not even been recorded, let alone
taken into account in the model.

9.3 MODELING THE HISTORY EFFECT AT SINGLE-CELL
LEVEL (STOCHASTIC MODELING)

9.3.1 Nature of Stochastic Modeling

Finer details of the effect of history on growth/death can be seen through a closer
look at the distribution of the division/survival times of individual cells. As men-
tioned in Section 9.1, if only deterministic models are used to describe responses
of microbial populations then the variability among single cells is ignored. This is
a problem because predicting microbial variability is increasingly important for
Quantitative Microbial Risk Assessment. This variability of the responses increases
under stress conditions, which makes predictions even less accurate.

A characteristic feature of stochastic modeling is that it sometimes provides
unexpected results. An example of this is the relation between the doubling time of
a growing population and the average generation time of the individual cells in the

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population. Depending on the distribution of the individual generation times, these
two quantities can be markedly different. If the generation times follow the classical
exponential distribution, then their average is l/|i max , while the doubling time is
known to be In 2/|i. max , where |i. max is the (constant) maximum specific growth rate
of the population. Baranyi and Pin (2001) described the exponential growth as a
Poisson birth process and showed that with this approach the doubling time depends
on the number of cells, from l/|i- max , when the number of cells is 1, converging to
In 2/|l max , when the number of cells increases.

9.3.2 Physiological Sate and Lag Parameters for
Sngle Cells

Let N(t) be the number of cells in a defined space, at the time t (remember, that x{f)
denoted the concentration of the cells). Let the lag for the ith cell of the initial
population be denoted by x, (/ = 1,2,..., N). We assume that they are identically
distributed independent random variables, and their mean value is x. Note that x, is
less than the time to the first division of the /th cell, which also includes the first
generation time.

Define the physiological states a, (i = 1,2,... JV) for the individual cells by a,=
exp (-|i raax X,), analogously to the deterministic theory. A basic theorem of Baranyi
(1998) is that the average of the individual physiological states is the same as the
physiological state of the population, irrespective of the distribution of the individual
lag times. Moreover, if ^(AO denotes the (traditionally defined) lag of the population
consisting of TV cells, then

N N

-Hi,.

,1" ,1

e

X(N) = --\n-^ = - — ln-^ (9.7)

\L N \i N

This formula is demonstrated in Figure 9.5. The more cells there are in the inoculum,
the shorter is the expected population lag, converging to a limit value, A, min . Notice
that the correct model for single-cell-generated curve is bi-phasic; the curvature for
higher inoculum is caused by the distribution of the individual lag times.

Various authors have reported that at a very low inoculum (approximate to 1
cell per total volume of sample), the lag time is longer than expected (Augustin et
al., 2000b; Gay et al., 1996; Stephens et al., 1997). However, as the above formula
shows, this finding can also be explained by statistical means without extra biological
assumptions. The lag time decreases with the initial cell number because the expo-
nentially growing subgenerations of the cells with the shortest lag times will quickly
dominate the whole population.

In particular, Baranyi and Pin (2001) showed the mathematical relation between
the respective parameters (mean and variance) of the individual lag times and those
of the population consisting of TV cells. From these formulae, the distribution of
individual lag times is per se different from the distribution of the population lag
times. This was confirmed by the observation of Stephens et al. (1997) on heat-

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« 3

o
o

y =0

MD=Tl

time (h)

FIG U RE 9.5 Simulating four single-cell-generated growth curves (y = 0) growing separately,
and two separate growth curves generated by four cells each (y = In 4). The traditional lag
times X of the subpopulations are estimated by curve fitting. Observe that the X(N) mean
population lag converges to a limit value X Um while its variance (the spread of the growth
curves as a function of their inoculum) converges to zero (see Figure 9.6). Symbols: simulated
points; continuous thin lines: fitted curves; continuous thick lines: expected curves.

injured Salmonella cells. The calculations provided another reason why the lag time
should not be considered as a primary growth parameter but as the result of the initial
physiological state (oc ), the inoculum size and the maximum specific growth rate.
Smelt et al. (2002) estimated the population lag as the minimum of the individual
lag times. Our formula shows that the population lag is greater than the minimum
of the individual lag times, but closer to it than to the arithmetical mean.

9.3.3 A Simulation Rogram

To demonstrate the effect of the inoculum on the distribution of the population lag,
we developed a simulation program. One hundred growth curves from different
inoculum sizes were generated. The distribution functions used by the simulation
were based on the theory of Baranyi and Pin (2001).

We assumed that the individual lag times followed the gamma distribution, with
the parameters p and v, so the mean individual lag is T = /?/V. The biological
interpretation of this scenario is that the cells have to carry out p consecutive tasks,
and the times required by the individual tasks are independent, exponentially dis-
tributed random variables, with the parameter v (i.e., the average time required for
each task was 1/v).

The parameter p represented the amount of work to be done during the lag time
(h = p). Partly for simplicity, partly for mechanistic reasons, v = |l max was assumed.
The generation times, g j9 of individual cells, after the lag phase, followed the
exponential distribution with the parameter |l raax .

With the usual terminology for stochastic processes, the initial state of the system
is (M0), meaning that there are N cells in the lag phase and in the exponential
phase at the starting time (t ). The time t x of the first division in the population is

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that of the cell with the smallest value for the sum of its lag time, x is and its first
generation time, g t .

t x =min(x 1 + g u ... X N + g N )

After the first division, N- 1 cells remain in lag phase while 2 cells are in exponential
growth phase. Thus the state of the system is (N - 1, 2). The time t 2 of the second
division will be the minimum of the first division times among the N - I cells
remaining in the lag phase and the division times of the two daughter cells:

t 2 = min(x, + g l9 ... X N _ { + g N _ l9 t x + g N+v f, + g N+2 )
This second division can happen to a cell in exponential or in lag phase.

1. If a cell in lag phase divides, then the state of the system will be (N - 2,
4) and the time of the third division will be t 3 = min(x y + g l3 ..., T N _ 2 +

£/v-2> h + 8n+u h + 8n+2-> h + 8n+3> h + 8n+a)-

2. If a cell in exponential phase divides, then the system jumps into the

state (N - 1,3). Apart from numbering the daughter cells, the time to the
third division will be t 3 = min^ + g l9 ..., x N _ { + g N _ u t x + g N+2 , t 2 + g N+3 ,

h + ^^+4)-

The iteration can be continued in a similar manner and this is what our simulation
program did, until the population had a given number of cells. One hundred growth
curves were generated for each selected inoculum size. The parameters h and |l max
were chosen close to practical values: h = 4 and |l max = 0.5 h -1 . Thus the gamma
parameters to simulate the individual lag times were p = 4 and v = 0.5 Ir 1 (mean
individual lag = 8 h) and the parameter of the exponential distribution to simulate
the individual generation times was 0.5 Ir 1 (mean generation time = 2 h). The
traditionally interpreted lag time of each population curve was estimated by fitting
a biphasic (no upper asymptote) version of the model of Baranyi and Roberts (1994).
Figure 9.5 shows typical examples of the generated population curves (originating
from one and four cells). As can be seen, in the case of a single initial cell, when
the inoculum is In x = 0, the population lag (X, measured by the tangent to the
exponential phase) is close to the theoretical individual lag (x, = 8 h).
If the inoculum size increases then, from the formula

r

UN)

N->c

urn

Ln

M

1 +

v

MX
P

\

(9.8)

/

the population lag converges to A aim = 8 -In 2 = 5.54. This convergence is shown in
Figure 9.5. Figure 9.6 shows the distribution of the population lag times as a function
of the inoculum size.

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Frequency

Lag time

Inoculum

FIGU RE 9.6 As the inoculum size increases, the population lag time decreases, converging
to a limit value. Its variance converges to zero.

9.3.4 Lag/Survival Distribution of Sngle Cells and
Lag /Shoulder of R) pulatio n Curves

A general result from Baranyi (2002) connects the expected population growth curve
with the distribution of single-cell kinetics:

x(t) = N

ft \

je" ir - s) f(s)ds+jf(s)ds

Vo

(9.9)

/

where /(0 denotes the (common) probability density function (pdf) of the individual
lag times. It is analogous to the formula of Kormendy et al. (1998) for death curves:

x(t) = N

oo

J>*(

s)ds

(9.10)

where g(s) is the probability density function of the individual cells' death rate (i.e.,
the reciprocal of their survival time). The big difference is that while the probability
distribution of the individual survival times determines the whole death curve (actu-
ally by a Laplace transformation, as the formula shows) that of the individual lag
times does not. In both models, the first compartments are "dormant" states, but the
second ones are totally different: for growth, it is the state of exponential growth,
characterized by the specific growth rate; for survival curves, it is a "death phase"
without any parameter.

The formulae above represent a one-to-one mapping between the f(f) distribution
and the population growth/death curve. The problem is that, in practice, this mapping
works only in one direction. The expected population curve can be easily derived
from the distribution of the individual lag times, but vice versa it would need

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unrealistically accurate data. This is an example where the performance of stochastic
techniques is much better than that of their deterministic counterparts.

9.3.5 History Bfect on the Sho ulder Rriods
of Survival Curves

For survival curves, with the notations as above, let x t be the time for the zth cell to
die, and assume that it is distributed exponentially, with the mean value x (pure
Poissonian death process). In this case, F{t) = 1 - e~ t/x , so the survival curve is y(f)
= \n N - t/l. Therefore, in this case, the survival curve is linear and there is no
shoulder parameter: X = 0.

Analogously with our approach to modeling the distribution of individual lag
times for growth, consider now the situation when T i (i = 1,..., TV) follow the gamma
distribution, with the parameters p > 1 and v > 0, where the expected value of the
survival time for a cell is x = pN . This is interpreted by Baranyi and Pin (2001) in
the following way: the cell needs p damaging hits (see the multihit theory in Casolari,
1988), and the times ; (j = 1,...,/?) between the hits are independent, exponentially
distributed variables, with a common mean value = 7/V. Then the survival time
of the z'th cell is

X e ; (9-1D

x. =

therefore x, (i = 1,..., A/) are gamma distributed. In this case, as shown by the
authors, the derivative of y(t) converges to a constant, namely v, the limit shoulder
still does not exist if p > 1. In practice, the shoulder would be measured by v and
the smallest detectable value of y(t), which means that the shoulder measurement
would depend on the precision of the measurement method! This inconsistency
makes the shoulder parameter unsuitable for modeling. Instead, the p parameter
quantifying the "damage to be done" should be modeled, which is analogous to the
h parameter for growth curves.

Another well-known interpretation of the shoulder is the so-called multitarget
model (Hermann and Horst, 1970). According to that, a cell has p targets that are
being hit synchronously (not consecutively as in the previous case). The times needed
to destroy the targets, 6^ (j = 1,..., /?), are independent, exponentially distributed
variables with the common mean value of 1/v and the cell is live until all targets
are inactivated. Therefore, the survival time of the zth cell is

X.=max6. (9.12)

' \<j<p J

and F(f) = (1 - e~ vt ) p , where p > 1. The population survival curve is now

y(t) = In AT + ln(l - (1 - e- v 9") (9.13)

2004 by Robin C. McKellar and Xuewen Lu

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1237_C09.fm Page 298 Wednesday, November 12, 2003 1:08 PM

which converges to the

y a (t) = In N - v • (t - In pN) (9. 14)

linear asymptote. Therefore, the limit shoulder parameter can be calculated as

X = \np/v (9.15)

This shows another analogy to the modeling of the history effect on growth: the "In
/?" quantity could be used to characterize the initial "damage to be done," and the
shoulder period depends on the magnitude of the initial damage and on the rate as
the damage is being done.

9.4 CONCLUDING REMARKS

In this chapter, we showed that the history effect can be quantified either by the
"initial physiological state" or the "work to be done" parameters; they are simply
rescaled versions of each other. The traditional lag parameter has become a derived
one, useful for static conditions only. By introducing a variable for the rate at which
the work is carried out before the exponential phase, the model can be used for
dynamic conditions, too. This concept was successfully applied in a series of papers
(Baranyi et al., 1995, 1996) describing the entire bacterial growth profile (not only
the exponential phase) under fluctuating conditions.

The priority of a is purely a modeling concept; it does not mean that when
fitting a bacterial growth curve, oc and not the lag should be estimated. The formula
X = -lna /|i. max to calculate the lag expresses the view that both the history and the
current environment influence the lag. This has practical consequences when devel-
oping more complex, dynamic models or, for example, when developing predictive
software packages. These commonly require the user to provide values for the
environmental factors, from which the maximum specific rate (or one of its rescaled
versions) and a default lag are predicted. If the user gives the inoculum, then the
whole growth curve can be constructed, too. What our theory suggests is that it is
not only the inoculum that cannot be predicted from the current environment, but
also the initial physiological state, oc . This is shown in Figure 9.2: there are two
independent initial points on the y axis:

1. An inoculum value (y )

2. A point through which the tangent drawn to the inflexion will pass (which
is h = -lna lower than the inoculum)

Sometimes the value of the initial physiological state should be simply a = 1,
expressing the fact that the preinoculation conditions are the same as the current
environment.

As has been mentioned, many features of the shoulder period before the expo-
nential death phase can be discussed analogously to those of the lag period before

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~V

the exponential growth phase. Baranyi and Pin (2001) showed that the multitarget
theory of inactivation (Hermann and Horst, 1970), modeling the shoulder before the
exponential death rate, provides a parallel between the initial "damage to be done"
for death and the h Q "work to be done" for growth. The "bridge" between the history
and the current environment is characterized by h either for growth (Figure 9.2) or
death (the mirror image of Figure 9.2).

The question can be raised of why the distribution of individual lag times is
important once different distributions can result in practically the same population
curve. The answer lies in the interest of Quantitative Microbial Risk Assessment in
the distribution of the quantity "time to a certain level" (such as time to infective
dose or time to legally allowed concentration). The distribution of that time depends
heavily on the distribution of individual lag times.

Modeling the effect of history is more efficient with stochastic techniques, but
also more data- and maths-demanding. Deterministic models are able to produce
"averaged out" solutions only, and studying the variability around those predictions
is vital in Quantitative Microbial Risk Assessment.

ACKNOWLEDGMENT

The authors are indebted to Susie George for preparing the manuscript. This work
was funded by the EC project QLRTD-2000-01145 and the IFR project CSG
434.1213A.

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