## Primary growth models included in biogrowth

### Modified Gompertz model under static conditions

Zwietering et al. (1990) proposed a reparameterization of the Gompertz model with parameters that have a biological interpretation for microbial growth. This model predicts the microbial count \(N(t)\) for storage time \(t\) as a sigmoid using the following algebraic equation

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

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

### Baranyi model under dynamic conditions

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

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

\[ \frac{dQ}{dt}=\mu \space Q \] In this 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 \(\lambda = \frac{\ln (1+1/Q_{0})}{\mu}\). 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 in predictive microbiology and, consequently, has been omitted from *biogrowth*. Also, in its original paper, the specific growth rate of \(Q(t)\) was given by a different growth rate (\(\nu\)). However, because this variable does not correspond to any known substance, this growth rate lacks a biological meaning and has been set to \(\mu\).

### Baranyi model under static conditions

For static conditions, the differential equation proposed by Baranyi and Roberts (1994) has as particular solution the sigmoid given by

\[ \log N(t) = \log N_0 + \mu \cdot A(t) - \frac{\log(1 + (\exp(\mu \cdot A(t)) - 1)}{ \exp(\log N_{max} - \log N_0))} \] where \(A\) is an adjustment function:

\[ A(t) = t + \frac{1}{\mu} \log(\exp(-\mu * t) + \exp(-h_0) - \exp(-\mu \cdot t - h_0)) \]

In these equations, where \(N_0\) is the initial microbial count, \(\mu\) is the maximum specific growth rate and \(N_{max}\) is the maximum growth rate. The parameter \(h_0\) is usually called the *work to be done* and is defined as \(h_{0} = \mu\cdot\lambda\), where \(\lambda\) is the lag phase. For compatibility with the other models, the functions in **biogrowth** take as input \(\lambda\) instead of \(h_0\).

### Trilinear model under static conditions

Buchanan et al. (1997) proposed a trilinear model as a more simple approach to describe microbial growth. 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. \(N = N_0\)).

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

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

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

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

where \(t_{max}\), defined is the time required for the microbial count to reach \(N_{max}\).

### Gathering meta data about primary models 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.

If, instead, one passess to this function any valid model identifier, the function returns the meta-information of that model.

That 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 name of the parameters of this model