# Modelling approach

    library(biogrowth)

## Population growth in predictive microbiology

In predictive microbiology mathematical models are usually divided in primary and secondary models (Perez-Rodriguez and Valero, 2012). Primary models describe the variation of the population size against time. These models are largely empirical, so they have model parameters that depend on the environmental conditions. A typical example is the relationship between the specific growth rate of microorganisms and the storage temperature. Predictive microbiology uses so-called, secondary models to describe the relationship between the parameters of the primary model and the environmental conditions.

## Primary growth models included in biogrowth

### Modified Gompertz model under static conditions

Zwietering et al. (1990) proposed a reparameterization of the Gompertz model with more meaningful parameters parameters. This model predicts the population size \(N(t)\) for storage time \(t\) as a sigmoid using the following algebraic equation

\[ \log_{10} N(t) = \log_{10} N_0 + C \left( \exp \left( -\exp \left( 2.71 \frac{\mu}{C}(\lambda-t)+1 \right) \right) \right) \]

where \(N_0\) is the initial population size, \(\mu\) is the maximum growth rate, \(\lambda\) is the duration of the lag phase and \(C\) is the difference between the initial population size and the maximum population size.

### Logistic growth model

The logistic growth model can be parameterized by the following equation (Zwietering et al. 1990)

\[ \log_{10} N(t) = \log_{10} N_0 + \frac{C}{1 + \exp{ \left(\frac{4 \mu}{C}(\lambda - t)+2 \right) } } \]

where \(N_0\) is the initial population size, \(\mu\) is the maximum growth rate, \(\lambda\) is the duration of the lag phase and \(C\) is the difference between the initial population size and the maximum population size.

### Richards growth model

The Richards growth model can be parameterized by the following equation (Zwietering et al. 1990)

\[ \log_{10} N(t) = \log_{10} N_0 + C \left[1+\nu \cdot \exp{ \left(1 + \nu + \frac{\mu}{A}(1+\nu)^{1+1/\nu} (\lambda - t) \right)} \right]^{-1/\nu} \]

where \(N_0\) is the initial population size, \(\mu\) is the maximum specific growth rate, \(\lambda\) is the duration of the lag phase and \(C\) is the difference between the initial population size and the maximum population size, and \(\nu\) defines the sharpness of the transition between growth phases.

### Baranyi model under dynamic conditions

Baranyi and Roberts (1994) proposed a system of two differential equations to describe microbial growth:

\[ \frac{dN}{dt} = \frac{Q}{1+Q}\mu'\left(1 - \frac{N}{N_{max}} \right)N \]

\[ \frac{dQ}{dt}=\mu' \space Q \]

Note that the maximum specific growth rate is written as \(\mu'\). The reason for this is that biogrowth makes calculations in log10 scale for the population size. Therefore, for consistency with the equations for static conditions, we use a different notation in these equations. Both parameters are related by the identity \(\mu' = \mu \cdot \log(10)\).

In the Baranyi model, the deviations with respect to exponential growth are justified based on two hypotheses. It introduces the variable \(Q(t)\), which represents a theoretical substance that must be produced before the microorganism can enter the exponential growth phase. Hence, its initial value (\(Q_0\)) defines the lag phase duration (under static conditions) as \(\lambda = \frac{\log (1+1/Q_{0})}{\mu}\). On the other hand, the stationary growth phase is defined by the logistic term \((1-N/N_{max})\), which reduces the growth rate as the microorganisms reach the maximum count.

Note that the original paper by Baranyi and Roberts included an exponent, \(m\), in the term defining the stationary growth phase. However, that term is usually set to 1 by convention in predictive microbiology and, consequently, has been omitted from biogrowth. Also, in its original paper the specific growth rate of \(Q(t)\) was defined by a different parameter (\(\nu\)). However, because this variable does not correspond to any known substance, it is a convention in the field to set \(\nu = \mu\).

### Baranyi model under static conditions

Oksuz and Buzrul (2020) calculated the solution of the Baranyi model for static conditions given by the following equation

\[ \log_{10} N = \log_{10} N_{max} + \log_{10}{ \frac{1 + \exp{ \left( \ln (10) \mu (t-\lambda) \right)} - \exp{- \ln (10) \mu \lambda}} {\exp \left( \ln (10) \mu (t-\lambda) \right)- \exp{ \left( - \ln (10) \mu \lambda \right) + 10^{\log_{10} N_{max} - \log_{10} N_0}} } } \]

where \(N_0\) is the initial population size, \(\mu\) is the maximum specific growth rate and \(N_{max}\) is the maximum growth rate, and \(\lambda\) is the lag phase.

#### Relationship between Q0 and the lag phase duration

In the Baranyi model, the duration of the lag phase is determined by the initial value of the ideal substance \(Q(t)\), \(Q_0\). Disregarding the stationary phase, the Baranyi model becomes

\[ \frac{dN}{dt} = \frac{Q(t)}{1+Q(t)}\cdot \mu \cdot N(t) \\ \frac{dQ}{dt} = \mu \cdot Q(t) \]

where \(\mu\) is in natural logarithmic scale. Considering that \(\mu\) is constant (e.g. in constant environmental conditions), the second ODE is the usual exponential growth

\[ Q(t) = Q_0 e^{\mu \cdot t} \]

So, the first differential equation becomes

\[ \frac{dN}{dt} = \frac{Q_0 e^{\mu \cdot t}}{1+Q_0 e^{\mu \cdot t}}\cdot \mu \cdot N(t) \] For convenience, we can convert it to natural logarithm

\[ \frac{1}{N} \frac{dN}{dt} = \frac{d}{dt} \ln N = \frac{Q_0 e^{\mu \cdot t}}{1+Q_0 e^{\mu \cdot t}}\cdot \mu \]

This equation also has an analytical solution

\[ \ln N = \ln N_0 + \ln \left( Q_0 e^{\mu t} + 1 \right) - \ln \left( Q_0 + 1\right) \]

In the exponential phase (i.e. outside of the lag phase), \(t>>\). Then, \(Q_0 e^{\mu t} >> 1\) and the equation becomes

\[ \ln N = \ln N_0 + \ln \left( Q_0 e^{\mu t} \right) - \ln \left( Q_0 + 1\right) \]

This can be rearranged as

\[ \ln N/N_0 = \ln Q_0 + \mu \cdot t - \ln \left(Q_0 + 1 \right) \]

This is the equation of a line tangent to the growth curve in the exponential phase. The lag phase duration is defined as the point where this line cuts the horizontal \(\ln N = \ln N_0\); i.e. \(\ln N/N_0 = 0\). Then, the lag phase duration (\(\lambda\)) is the solution of:

\[ \ln Q_0 + \mu \cdot \lambda - \ln \left(Q_0 + 1 \right) = 0 \]

That is,

\[ \mu \cdot \lambda = \ln \left(Q_0 + 1 \right) - \ln Q_0 = \ln \frac{Q_0 + 1}{Q_0} \]

Ergo, the lag phase is given by

\[ \lambda = \frac{1}{\mu} \cdot \ln \left(1 + 1/Q_0 \right) \]

and \(Q_0\) is calculated from \(\lambda\) by

\[ Q_0 = \frac{1}{e^{\mu\lambda} - 1} \]

where \(\mu\) is defined in natural (i.e. exp(1)) scale.

The package includes the functions `Q0_to_lambda` and `lambda_to_Q0` to perform these operations. Please check the vignette Models based on secondary models to predict growth under constant environmental conditions for some critical points when using them.

### Baranyi model under static conditions without stationary phase

The algebraic solution of the Baranyi model can be simplified when there is no stationary phase to

\[ \log N = \log N_0 + \mu A \]

where

$$ A = t + ( e^{-t} + e^{-} - e^{-t - } )

$$

### Baranyi model under static conditions without lag phase

The algebraic solution of the Baranyi model can be simplified when there is no lag phase to

\[ \log_{10} N = \log_{10} N_{max} + \log_{10}{ \frac{1 + \exp{ \left( \ln (10) \mu t \right)} - 1} {\exp \left( \ln (10) \mu (t-\lambda) \right) + 10^{\log_{10} N_{max} - \log_{10} N_0}} } \]

### Trilinear model under static conditions

Buchanan et al. (1997) proposed a trilinear model as a more simple approach to describe the growth of microbial populations. This model is defined by the piece-wise equation.

The lag phase is defined considering that, as long as \(t < \lambda\), there is no growth (i.e. \(N = N_0\)).

\[ \log_{10} N(t) = \log_{10} N_0; t \leq \lambda \] The exponential phase is described considering that during this phase, the specific growth rate is constant, with slope \(\mu_{max}\).

\[ \log_{10} N(t) = \log_{10} N_0 + \mu(t-\lambda); t\in(\lambda,t_{max}) \]

Finally, the stationary phase is modeled considering that once \(N\) reaches \(N_{max}\), it remains constant.

\[ \log_{10} N(t) = \log_{10} N_{max}; t \geq t_{max} \]

where \(t_{max}\), defined is the time required for the population size to reach \(N_{max}\) ($t_{max} = ( {10} N{max} - _{10} N_0 )/+ $).

### Bilinear model with lag phase

This model is a simplification of the trilinear model without a stationary phase. It is described by the following equations:

\[ \log_{10} N(t) = \log_{10} N_0; t \leq \lambda \\ \log_{10} N(t) = \log_{10} N_0 + \mu(t-\lambda); otherwise \]

### Bilinear model with stationary phase

This model is a simplification of the trilinear model without a lag phase. It is described by the following equations:

\[ \log_{10} N(t) = \log_{10} N_0 + \mu \cdot t; t < t_{max} \\ \log_{10} N(t) = \log N_{max}; otherwise \] where \(t_{max}\), defined is the time required for the population size to reach \(N_{max}\) ($t_{max} = ( {10} N{max} - _{10} N_0 )/$).

### Loglinear model

This model is a simplification of the trilinear model that only has an exponential phase. It is described by the following equation:

\[ \log_{10} N(t) = \log_{10} N_0 + \mu \cdot t \]

### Gathering primary models metadata directly from biogrowth

The biogrowth package includes the `primary_model_data()` function, which provides information about the primary models included in the package. It takes just one argument (`model_name`). By default, this argument is `NULL`, and the function returns the identifiers of the available models.

    primary_model_data()
    #>  [1] "modGompertz"          "Baranyi"              "Baranyi_noLag"
    #>  [4] "Baranyi_noStationary" "Trilinear"            "Logistic"
    #>  [7] "Richards"             "Loglinear"            "Bilinear_lag"
    #> [10] "Bilinear_stationary"

If a model identifier is passed, it returns a list with mode meta-information.

    meta_info <- primary_model_data("Trilinear")

It includes the full reference

    meta_info$ref
    #> [1] "Buchanan, R. L., Whiting, R. C., and Damert, W. C. (1997). When is simple good enough: A comparison of the Gompertz, Baranyi, and three-phase linear models for fitting bacterial growth curves. Food Microbiology, 14(4), 313-326. https://doi.org/10.1006/fmic.1997.0125"

or the identifiers of the model parameters

    meta_info$pars
    #> [1] "logN0"   "logNmax" "mu"      "lambda"

## Secondary growth models included in biogrowth

Although other approaches for secondary modeling exist in the literature, the biogrowth package uses the gamma approach proposed by Zwietering et al. (1992). This approach assumes that the effect of each environmental factor on the growth rate is multiplicative and independent. Hence, the specific growth rate (\(\mu\)) observed under a combination of environmental conditions (\(X_1\), \(X_2\),…,\(X_n\)) is the result of multiplying the value of \(\mu\) observed under optimal conditions for growth (\(\mu_{opt}\)) times one gamma factor for each environmental factor (\(\gamma(X_i)\)). Because gamma factors represent growth rates under sub-optimal conditions, they must take values between 0 and 1.

Note that this approach does not consider interactions between the different terms. The reason for this is that, in the opinion of the authors, their implementation is still a research topic, without any model being broadly accepted. Therefore, the inclusion of interactions between terms is not in line with the philosophy of biogrowth, which aims at easing the application of broadly accepted modeling approaches.

\[ \mu = \mu_{opt} \cdot \gamma(X_1) \cdot \gamma(X_2) \cdot ... \cdot \gamma(X_n) \]

### Gamma factors by Zwietering

Zwietering et al. (1992) defined several functions for different environmental factors (temperature, pH and water activity). These functions can be generalized as

\[ \gamma(X) = \left( \frac{X-X_{min}}{X_{opt}-X_{min}} \right)^n \]

where \(X_min\) is the minimum value of \(X\) enabling growth, \(X_{opt}\) is the optimum value of \(X\) for growth and \(n\) is the order of the model. The latter parameter usually takes the values 1, 2 or 3, depending on the environmental factor, and the type of population studied.

### Cardinal Parameter Model (CPM)

Rosso et al. (1995) defined the following equation for gamma factors

\[ \gamma(X) = \frac{(X-X_{max})(X-X_{min})^n} {(X_{opt}-X_{min})^{n-1}( (X_{opt}-X_{min})(X-X_{opt}) - (X_{opt}-X_{max}) \cdot ((n-1)X_{opt} + X_{min}-n \cdot X) )} \] with \(\gamma(X) = 0\) for \(X \notin [X_{min}, X_{max}]\). In this equation, \(X_{min}\), \(X_{opt}\) and \(X_{max}\) are the minimum, maximum and optimum values of \(X\) for microbial growth (usually called cardinal parameters). The coefficient \(n\) is the shape parameter of the cardinal model (usually being fixed to 1, 2 or 3, depending on the case studied).

### (Adapted) Full Ratkowsky model

The full Ratkowsky model (Ratkowsky et al., 1983) is an extension of the square-root model by Ratkowsky (Ratkowsky et al., 1982) that accounts for the decline of the growth rate for temperatures higher than the optimal one. It is described by the following equation

\[ \sqrt{\mu} = b \left( X-X_{min} \right) \left(1-e^{c(X-X_{max})} \right) \] The application of the gamma concept requires that every gamma function takes values between 0 and 1, whereas the full Ratkowsky model takes values between 0 and \(\sqrt{\mu_{opt}}\). This can be adapted by defining a gamma function such as

\[ \gamma_{Ratkowsky}= \left( \frac{\sqrt{\mu}}{ \sqrt{\mu_{opt}} } \right)^2 \]

For that, we need to know the value of \(\sqrt{\mu_{opt}}\), which is not included as a parameter in the full Ratkowsky model. By setting the first derivative with respect to \(X\) to 0 and solving, we get

\[ X_{opt} = \frac{ W_n \left( e^{-cX_{min} +cX_{max} + 1} \right) + cX_{min} - 1 } {c} \]

where \(W_n\) is the Lambert-W function. Then, the maximum growth rate under optimal conditions is given by

\[ \sqrt{\mu_{opt}} = b \left( X_{opt}-X_{min} \right) \left(1-e^{c(X_{opt}-X_{max})} \right) \]

and the gamma function for the Ratkowsky model can be calculated as

\[ \gamma_{Ratkowsky}= \left( \frac{\sqrt{\mu}}{ \sqrt{\mu_{opt}} } \right)^2 = \left( \frac{ b \left( X-X_{min} \right) \left(1-e^{c(X-X_{max})} \right)} { b \left( X_{opt}-X_{min} \right) \left(1-e^{c(X_{opt}-X_{max})} \right)} \right)^2 = \left( \frac{ \left( X-X_{min} \right) \left(1-e^{c(X-X_{max})} \right)} {\left( X_{opt}-X_{min} \right) \left(1-e^{c(X_{opt}-X_{max})} \right)} \right)^2 \]

Note that parameter \(b\) banishes after doing the division, so this model has 3 model parameters: \(c\), \(X_{min}\) and \(X_{max}\).

## Secondary model by Aryani

The secondary model by Aryani et al. (2016) was defined for the relationship between the growth rate and pH. It is defined as

\[ \gamma(X) = 1 - 2^{- \left( X - X_{min} \right)/ \left( X_{1/2} - X_{min} \right) } \]

where \(X_{min}\) is the minimum value of \(X\) enabling growth, and \(X_{1/2}\) is the value of \(X\) where \(\gamma(X) = 0.5\).

## Secondary model for water activity by Rosso

The secondary model for water activity by Rosso (et al. 2001) is defined as

\[ \gamma(X) = \frac{X-X_{min}}{1 - X_{min}} \]

where \(X_{min}\) is the minimum value of \(X\) enabling growth. Please note that this model is defined between 0 and 1, so it specially tailored for the effect of water activity.

## Secondary model for inhibitory compounds

The general model for inhibitory compounds is defined as

\[ \gamma(X) <- 1 - \left( \frac{X}{MIC} \right)^\alpha \]

where \(MIC\) is the minimum inhibitory concentration and \(\alpha\) is a shape factor.

### Gathering meta data about secondary models directly from biogrowth

The biogrowth package includes the `secondary_model_data()` function, which provides information about the secondary models it implements. It takes just one argument (`model_name`). By default, this argument is `NULL`, and the function returns the identifiers of the available models.

    secondary_model_data()
    #> [1] "CPM"           "Zwietering"    "fullRatkowsky" "Aryani"
    #> [5] "Rosso_aw"      "Inhibitory"

If, instead, one passes any valid model identifier, the function returns its meta-information.

    meta_info <- secondary_model_data("CPM")

This includes the full reference of the model

    meta_info$ref
    #> [1] "Rosso, L., Lobry, J. R., Bajard, S., and Flandrois, J. P. (1995). Convenient Model To Describe the Combined Effects of Temperature and pH on Microbial Growth. Applied and Environmental Microbiology, 61(2), 610-616."

or the identifiers of its model parameters

    meta_info$pars
    #> [1] "xmin" "xopt" "xmax" "n"

## Summary of the model parameters

Quantity | Identifier | Description
--- | --- | ---
\(\log_{10} N\) | `logN` | Decimal logarithm of the population size.
\(\log_{10} N_0\) | `logN0` | Decimal logarithm of the initial population size.
\(N\) | `N` | Population size.
\(N_0\) | `N0` | Initial population size.
\(t\) | `t` | Elapsed time.
\(C\) | `C` | \(\log_{10} N_{max} - \log_{10} N_0\).
\(\mu\) | `mu` | Specific growth rate.
\(\lambda\) | `lambda` | Duration of the lag phase.
\(\nu\) | `nu` | Sharpness of the transition between growth phases in the Richards model.
\(Q\) | `Q` | Variable of the Baranyi model describing the lag phase
\(Q_0\) | `Q0` | Initial value of \(Q\).
\(X_{min}\) | `xmin` | Minimum value of \(X\) enabling growth.
\(X_{opt}\) | `xopt` | Optimum value of \(X\) for growth.
\(X_{1/2}\) | `xhalf` | Value of \(X\) where \(\mu = \mu_{opt}/2\).
\(X_{max}\) | `xmax` | Maximum value of \(X\) enabling growth.
\(MIC\) | `MIC` | Minimum inhibitory concentration.
\(alpha\) | `alpha` | Shape factor.
\(n\) | `n` | Order of the secondary model.
\(\mu_{opt}\) | `mu_opt` | Value of \(\mu\) under optimal environmental conditions for growth.

## Numerical methods

The biogrowth package uses FME for model fitting and deSolve for estimating population growth under dynamic conditions. Namely, the differential equations required for dynamic calculations are solved using the `ode()` function from deSolve, which implements several numerical methods. Model fitting is done using the `modFit()` function from FME for linear regression and the `modMCMC()` function from the same package for MCMC fitting. The functions in biogrowth use the default settings of these functions, although this can changed using the `...` argument included in every relevant function.

## References

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