Introduction to predmicror
Vasco Cadavez
Ursula Gonzales-Barron
2022-06-28
Source:vignettes/predmicror.Rmd
predmicror.RmdIntroduction
Predictive Microbiology deals with the development of accurate and, at the same time, versatile mathematical models, able to describe the microbial evolution in food products as a function of environmental conditions, which are assumed to be measurable.
The predmicror (https://github.com/fsqanalytics/predmicror/) is a package for fitting the most widely used predictive microbiology models.
Primary growth models
The actual version includes primary growth models that describe microbial concentration as a function of time at constant environmental conditions. The model inputs are:
- \(t\): time, assuming time zero as the beginning of the experiment; and
- \(Y_{(t)}\): the natural logarithm of the microbial concentration \(X_{(t)}\) measured at time \(t\).
Users should make sure that the microbial concentration input is entered in natural logarithm, \(Y_{(t)} = ln(X_{(t)})\).
The number of model parameters is dependent upon the completeness of the microbial growth curve. The following parameters can be estimated using this web application:
- \(Y_0\): the natural logarithm of the initial microbial concentration at \(t=0\);
- \(\mu_{max}\): maximum specific growth rate given in time \(units^{-1}\);
- \(\lambda\): duration of the lag phase in time units; and
- \(Y_{max}\): the natural logarithm of the maximum concentration reached by the microorganism.
A full model should be adjusted to a complete
microbial curve, where the lag phase, exponential phase and stationary
phase can be identified. The predmicror can also fit
reduced models. A no-stationary phase model is to be
adjusted to an experimental microbial curve that presents lag phase and
exponential phase, whereas a no-lag phase model should
be adjusted to an experimental curve composed of exponential phase and
stationary phase. An experimental growth curve that presents only
exponential phase cannot be analysed using the
predmicror functions.
Full growth models
predmicror can adjust four nonlinear models to complete microbial growth curves: Huang model, Rosso model, Baranyi & Roberts model and the Zwietering reparameterised Gompertz model.
Huang model
The Huang growth model was developed by Huang (2008).
\[Y_{(t)} = Y_0 + Y_{max} -log \left( e^{Y_0} + (e^{Y_{max}} - e^{Y_0}) \times e^{-\mu_{max} \times B_{(t)}} \right)\]
\[B_{(t)} = t + \frac{1}{\alpha} \times log \left( \frac{1 + e^{-\alpha \times (t-\lambda)} }{1 + e^{\alpha\times \lambda}} \right)\]
After evaluating multiple growth data sets, Huang (2013) recommended fixing the parameter
\(\alpha\) to 4.0, thus
predmicror considers \(\alpha=4.0\).
Rosso model
The Rosso growth model is a simple two-phase model proposed by Rosso et al. (1996).
\[ Y_{(t)} = \begin{cases} Y_0 & \text{if t $\leq$ $\lambda$}\\ Y_{max}-log \left[1 +\left( \frac{e^{Y_{max}}}{e^{Y_0}}-1 \right) e^{-\mu_{max} (t-\lambda)} \right] & \text{if t > $\lambda$} \end{cases} \]
Baranyi & Roberts model
The original Baranyi & Roberts model attributes the lag phase to
the need to synthesise an unknown substrate q that is
critical for growth, whose initial value \(q_0\) is a measure of the initial
physiological state of the microbial cells (Baranyi & Roberts (1994)).
predmicror implements the Baranyi & Roberts model with basis on the transformation \(h_0 = \mu_{max} \times \lambda\), in order to estimate \(\lambda\). Thus, the model parameterisation used is:
\[Y_{(t)} = Y_0 + \mu_{max} \times A_{(t)} - \frac{1}{m} \times log \left[ 1 + \frac{exp(m \times \mu_{max} \times A_{(t)})-1}{exp(m \times (Y_{max}-Y_0)) } \right]\]
\[A_{(t)} = t + \frac{ log \left[ exp(-\mu_{max} \times t) + exp(-\mu_{max} \times \lambda) - exp(-\mu_{max} \times t-\mu_{max} \times \lambda) \right] }{\mu_{max}}\]
Most of the times, the parameter m, which characterises
the curvature before the stationary phase is assumed to be 1.0. The
predmicror simplifies this model by assuming \(m=1.0\).
No stationary phase growth models
predmicror can adjust three nonlinear models to microbial growth curves without stationary phase: reduced Huang model, reduced Baranyi & Roberts model and two-phase linear growth model.
Huang model
This model is a special case of the complete Huang model, suitable for experimental growth curves that do not reach stationary phases.
\[Y_{(t)} = Y_0 + \mu_{max} \times \left[ t + 0.25 \times log \left( \frac{1 + e^{-4\times(t-\lambda)}}{1 + e^{4\times\lambda}} \right) \right]\]
No lag phase growth models
predmicror can adjust two nonlinear models to microbial
growth curves that do not show lag phase: Richards
model and Fang model.
Richards model
\[Y_{(t)} = Y_0 + \mu_{max} \times t - \frac{1}{m} \times log( 1 + \frac{exp(m \times \mu_{max} \times t) - 1}{exp(m \times(Y_{max}-Y_0))}\]
Fang model
Fang et al. (2012) and Fang et al. (2013) integrated the logistic growth model, producing a continuous model that is particularly suitable for growth curves without lag phase.
\[Y_{(t)} = Y_0 + Y_{max} - log \left[ e^{Y_0} + \left( e^{Y_{max}} - e^{Y_0} \right) \times e^{-\mu_{max} \times t} \right]\]
The cardinal parameter model
Predictive Microbiology deals with the development of accurate and versatile mathematical models, able to describe the evolution of microorganisms in food products as a function of environmental conditions, which are assumed to be measurable. Although there are a few classification schemes of predictive microbiology models, they have been traditionally classified into primary and secondary models.
predmicror can be used to fit cardinal parameter
models, which are secondary models that describe the growth
rate of microorganisms as a function of extrinsic and/or intrinsic
factors. These are models that estimate the optimum growth rate, and the
minimum, optimum and maximum values of extrinsic and intrinsic factors
(e.g. temperature, pH, water activity) that characterise the growth of a
given microbial strain.
The general cardinal parameter model used to describe and predict the effect of different environmental factors on the growth rate of a microorganism is based on a modular approach called the gamma concept (Zwietering et al., 1991), described as,
\[ \mu _{max}=\mu _{opt}\times \gamma \left(T \right)\times \gamma \left(pH\right)\times \gamma \left(aw\right)\times \gamma (Inh) \] where:
- \(\mu _{max}\): maximum growth rate (\(h^{-1}\) or \(day^{-1}\)) of the studied bacterial strain
- \(\mu _{opt}\): optimum growth rate (\(h^{-1}\) or \(day^{-1}\)) of the studied bacterial strain
- \(\gamma \left(T\right),\gamma \left(pH\right),\gamma \left(aw\right),\gamma (Inh)\): dimensionless functions describing the relative effects of temperature (T), pH, water activity (aw) and different measurable inhibitors (Inh) like undissociated organic acids or \(CO_2\).
The functions \(\gamma \left(T\right),\gamma \left(pH\right),\gamma \left(aw\right),\gamma (inh)\) have a range between 0 and 1, \(\gamma =0\) when growth is fully inhibited and \(\gamma =1\) when growth is not inhibited at all by the factor.
Cardinal Parameters
Many \(\gamma\)-type functions have
been proposed for temperature, pH, aw and lactic acid. The
predmicror adjusts the cardinal parameter model using the
general equation for \(\gamma\)
proposed by Rosso et al. (1995).
\[ \gamma (X)_n=\left\{\begin{matrix} 0, \qquad X \leq X_{min} \\ \frac{(X-X_{max})\times (X-X_{min})^n}{(X_{opt}-X_{min})^{n-1}\times \left[\begin{matrix}\left(X_{opt}-X_{min}\right)\times \left(X-X_{opt}\right)-\left(X_{opt}-X_{max}\right)\times \\ \left(\left(n-1\right)\times X_{opt}+X_{min}-n \times X\right)\end{matrix}\right]}, \qquad X_{min}<X<X_{max} \\ 0, \qquad X{\geq}X_{max}\end{matrix}\right. \]
- \(X\): intrinsic or extrinsic factor under study; temperature, pH or aw
- \(X_{min}\): value of the factor below which no growth occurs
- \(X_{max}\): value of the factor above which no growth occurs
- \(X_{opt}\): value at which bacterial growth is optimum
- \(n\): shape parameter ( \(n=2\) for temperature and water activity; and \(n=1\) for pH).
- \(X_{min}\), \(X_{max}\) and \(X_{opt}\) are known as cardinal parameters, and are estimated by fitting the cardinal parameter model to growth data from experiments carried out in broth.
More precisely, \(T_{min}\), \(T_{max}\) and \(T_{opt}\) are determined from
\(\gamma \left(T\right)\); \(pH_{min}\), \(pH_{max}\) and \(pH_{opt}\) are determined from \(\gamma \left(pH\right)\); and \(aw_{min}\), \(aw_{max}\) and \(aw_{opt}\) are determined from \(\gamma \left(aw\right)\). Often, when
adjusting the cardinal parameter model for aw, \(aw_{max}\) is set to one, because it is in
effect the maximum value of the water activity measurement.
Cardinal parameter models available in predmicror
Cardinal parameter model for temperature
The predimicror adjusts the Cardinal Temperature Model
with Inflection (CTMI) (Rosso1993?) to determine
optimum growth rate ( \(\mu_{opt}\))
and the cardinal parameters \(T_{min}\), \(T_{max}\) and \(T_{opt}\).
\[ \sqrt{\mu _{max}}=\sqrt{\mu _{opt}\times \gamma \left(T\right)} + \varepsilon \]
\[ \gamma (T)=\left\{\begin{matrix} 0, \qquad T \leq T_{min} \\ \frac{(T-T_{max}) \times (T-T_{min})^2}{\left(T_{opt}-T_{min}\right)\times\left[\begin{matrix}\left(T_{opt}-T_{min}\right)\times \left(T-T_{opt}\right)-\left(T_{opt}-T_{max}\right) \times\\ \left(T_{opt}+T_{min}-2T\right)\end{matrix}\right]}, \qquad T_{min} < T < T_{max}\\ 0, \qquad T \geq T_{max}\end{matrix}\right. \]
To fit this model to the growth data, the response variable, maximum growth rate ( \(\mu _{max}\)), is square-root transformed to reduce heterocedasticity. The residuals are represented by \(\varepsilon\), and are assumed to follow a normal distribution with mean zero and variance \(\sigma^2\).
Cardinal parameter model for pH
To determine optimum growth rate (\(\mu_{opt}\)) and the cardinal parameters
\(pH_{min}\), \(pH_{max}\) and \(pH_{opt}\), predmicror adjusts
the cardinal model for pH proposed by (Lemarc2002?).
\[ \sqrt{\mu _{max}}=\sqrt{\mu _{opt}.\gamma \left(pH\right)}+\varepsilon \]
\[ \gamma (pH)=\left\{\begin{matrix}0,\qquad pH{\leq}pH_{min}\\\frac{(pH-pH_{max}){\times}\left(pH-pH_{min}\right)}{\left[\left(pH_{opt}-pH_{min}\right).\left(pH-pH_{opt}\right)-\left(pH_{opt}-pH_{max}\right).\left(pH_{min}-pH\right)\right]},\qquad pH_{min}<pH<pH_{max}\\0,\qquad pH{\geq}pH_{max}\end{matrix}\right. \]
Cardinal parameter model for Aw
predmicror adjusts the cardinal model for water activity
from (Rosso1993?) to
extract optimum growth rate (\(\mu_{opt}\)) and the cardinal parameters
\(aw_{min}\) and \(aw_{opt}\).
\[ \sqrt{\mu _{max}}=\sqrt{\mu _{opt} \times \gamma \left(aw\right)} + \varepsilon \]
\[ \gamma (aw)=\left\{\begin{matrix} 0, \qquad aw \leq aw_{min} \\ \frac{(aw-1.0)\times(aw-aw_{min})^2}{\left(aw_{opt}-aw_{min}\right)\times\left[\begin{matrix}\left(aw_{opt}-aw_{min}\right){\times}\left(aw-aw_{opt}\right)-\left(aw_{opt}-1.0\right)\times \\ \left(aw_{opt}+aw_{min}-2aw\right)\end{matrix}\right]}, \qquad aw_{min}<aw<1.0 \\ 0, \qquad aw \geq 1.0 \end{matrix} \right. \]
To fit this model to the growth data, the response variable, maximum growth rate ( \(\mu_{max}\)), is square-root transformed to reduce heterocedasticity. The residuals are represented by \(\varepsilon\), and are assumed to follow a normal distribution with mean zero and variance \(\sigma^2\).
Cardinal parameter model for inhibitory substance
For the inhibitors (Inh) including undissociated organic acids, \(CO\) and others, the cardinal parameter
model proposed by (Lemarc2002?))and (Coroller2005?) can be fitter
by the predmicror.
\[ \sqrt{\mu_{max}}=\sqrt{\mu_{opt} \times \gamma \left(inh\right)}+\varepsilon \]
\[ \gamma \left(Inh\right)=1-\left(\frac{Inh}{MIC}\right)^\alpha \]
-
Inh: concentration of the inhibiting substance or compound (e.g. undissociated organic acid (mM), CO (%) -
MIC: Minimum Inhibitory Concentration (mM or %, accordingly) - \(\alpha\): shape parameter of the curve (\(\alpha=1\) the shape is linear; \(\alpha>1\) the shape is downward concave; and \(\alpha<1\) the shape is upward concave).
To fit this model to the growth data, the response variable, maximum growth rate ( \(\mu_{max}\)), is square-root transformed to reduce heterocedasticity. The residuals are represented by \(\varepsilon\), and are assumed to follow a normal distribution with mean zero and variance \(\sigma^2\). The model parameters are MIC and \(\alpha\).