# Models

This document describes the statistical models used in morse to analyze survival and reproduction data, and as such serves as a mathematical specification of the package. For a more practical introduction, please consult the Tutorial vignette ; for information on the structure and contents of the library, please consult the reference manual.

Model parameters are estimated using Bayesian inference, where posterior distributions are computed from the likelihood of observed data combined with prior distributions on the parameters. These priors are specified after each model description.

# Survival toxicity tests

In a survival toxicity test, subjects are exposed to a measured concentration of a contaminant over a given period of time and the number of surviving organisms is measured at certain time points during exposure. In most standard toxicity tests, the concentration is held constant throughout the whole experiment, which is assumed for Analysis of target time survival toxicity tests in `morseDR` package, but not for Toxicokinetic-Toxicodynamic modeling which can handled time variable exposure. In the case of constant exposure, an experiment is generally replicated several times and also repeated for various levels of the contaminant. For time-variable exposure, a profile of exposure is usually unique, and the experiment is repeated with several profiles of exposures.

## Toxicokinetic-Toxicodynamic modeling

For datasets featuring time series measurements, more complete models can be used to estimate the effect of a contaminant on survival. We assume the toxicity test consists in exposing an initial number \(n_i^0\) of organisms to a concentration \(c_i(t)\) of contaminant (constant or time-variable), and following the number \(n_i^k\) of survivors at time \(t_k\) (with \(t_0 < t_1 < \dots < t_m\) and \(t_0 = 0\)), thus providing a collection \(D = {(c_i, t_k, n_i^k)}_{i,k}\) of experiments. In ‘morse’, we propose two Toxicokinetic-Toxicodynamic (TKTD) models belonging to the General Unified Threshold model for Survival (GUTS) [@jager2011; @Jager2018GUTSbook]. One is known as the reduced stochastic death model [@nyman2012] or GUTS-SD and the other is the reduced organism tolerance model or GUTS-IT, which we describe now.

Table: Parameters and symbols used for GUTS-SD and GUTS-IT models. Alternative symbols are used within pubications (see for instance [@jager2011; @delignette2017; @Jager2018GUTSbook]. The unit \([D]\) refers to unit of actual damage, \(n.d\) for non dimensional. For GUTS-IT model, we assume a log-logistic distributions, but other distributions are occasionally used [@albert2016].
Parameters | Symbols | Alternative symbols | Units | Models
--- | --- | --- | --- | ---
Background hazard rate | \(h_b\) | \(m_0\) | \(\text{time}^{-1}\) | SD and IT
Dominant toxicokinetic rate constant | \(k_d\) | \(\mbox{NEC}\) | \(\text{time}^{-1}\) | SD and IT
Threshold for effects | \(z_w\) | \(m_0\) | \([D]\) | SD
Killing rate constant | \(b_w\) | \(k_k\) | \([D]^{-1}\) | SD
Median of the threshold effect distribution | \(m_w\) | \(\alpha\) | \([D]\) | IT
Shape of the threshold effect distribution | \(\beta\) | \(-\) | \(n.d.\) | IT

#### GUTS Modelling

The number of survivors at time \(t_k\) given the number of survivors at time \(t_{k-1}\) is assumed to follow a binomial distribution: \[ N_i^k \sim \mathcal{B}(n_i^{k-1}, f_i(t_{k-1}, t_k)) \] where \(f_i\) is the conditional probability of survival at time \(t_k\) given survival at time \(t_{k-1}\) under concentration \(c_i(t)\). Denoting \(S_i(t)\) the probability of survival at time \(t\), we have: \[ f_i(t_{k-1}, t_k) = \frac{S_i(t_k)}{S_i(t_{k-1})} \]

The formulation of the survival probability \(S_i(t)\) in GUTS [@jager2011] is given by integrating the instantaneous mortality rate \(h_i\): \[ S_i(t) = \exp \left( \int_0^t - h_i(u)\mbox{d}u \right) \tag{2} \]

In the model, function \(h_i\) is expressed using the internal concentration of contaminant (that is, the concentration inside an organism) \(C^{{\scriptsize INT}}_i(t)\). More precisely: \[ h_i(t) = b_w \max(C^{\mbox{${\tiny INT}$}}_i(t) - z_w, 0) + h_b \] where (see Table of parameters):

- \(b_w\) is the and expressed in concentration\(^{-1}\).time\(^{-1}\) ;
- \(z_w\) is the so-called and represents a concentration threshold under which the contaminant has no effect on organisms ;
- \(h_b\) is the (mortality in absence of contaminant), expressed in time\(^{-1}\). \end{itemize}

The internal concentration is assumed to be driven by the external concentration, following:

\[ \frac{\mathop{\mathrm{d}\!}C^{\mbox{${\tiny INT}$}}_i}{\mathop{\mathrm{d}\!}t}(t) = k_d (c_i(t) - C^{\mbox{${\tiny INT}$}}_i(t)) \tag{1} \]

We call parameter \(k_d\) of Eq.(1) the dominant rate constant (expressed in time\(^{-1}\)). It represents the speed at which the internal concentration in contaminant converges to the external concentration. The model could be equivalently written using an internal damage instead of an internal concentration as a dose metric [@jager2011].

If we denote \(f_z(z_w)\) the probability distribution of the no effect concentration threshold, \(z_w\), then the survival function is given by:

\[ S(t) = \int_0^t S_i(t) f_z(z_w) \mbox{d} z_w= \int \exp \left( \int_0^t - h_i(u)\mbox{d} u \right) f_z(z_w) \mbox{d} z_w \]

Then, the calculation of \(S(t)\) depends on the model of survival, GUTS-SD or GUTS-IT [@jager2011].

#### GUTS-SD

In GUTS-SD, all organisms are assumed to have the same internal concentration threshold (denoted \(z_w\)), and, once exceeded, the instantaneous probability to die increases linearly with the internal concentration. In this situation, \(f_z(z_w)\) is a Dirac delta distribution, and the survival rate is given by Eq.(2).

#### GUTS-IT

In GUTS-IT, the threshold concentration is distributed among all the organisms, and once exceeded for one organism, this organism dies immediately. In other words, the killing rate is infinitely high (e.g. \(k_k = + \infty\)), and the survival rate is given by: \[ S_i(t) = e^{-h_b t} \int_{\max\limits_{0<\tau <t}(C^{\mbox{${\tiny INT}$}}_i(\tau))}^{+\infty} f_z(z_w) \mbox{d} z_w= e^{-h_b t}(1- F_z(\max\limits_{0<\tau<t} C^{\mbox{${\tiny INT}$}}_i(\tau))) \] where \(F_z\) denotes the cumulative distribution function of \(f_z\).

Here, the exposure concentration \(c_i(t)\) is not supposed constant. In the case of time variable exposure concentration, we use an midpoint ODE integrator (also known as modified Euler, or Runge-Kutta 2) to solve models GUTS-SD and GUTS-IT. When the exposure concentration is constant, then, explicit formulation of integrated equations are used. We present them in the next subsection.

### For constant concentration exposure

If \(c_i(t)\) is constant, and assuming \(C^{{\scriptsize INT}}_i(0) = 0\), then we can integrate the previous equation (1) to obtain:

\[ C^{\mbox{${\tiny INT}$}}_i(t) = c_i(1 - e^{-k_d t}) \tag{4} \]

#### GUTS-SD

In the case \(c_i < z_w\), the organisms are never affected by the contaminant:

\[ S_i(t) = \exp( - h_b t ) \tag{3} \]

When \(c_i > z_w\), it takes time \(t^z_i\) before the internal concentration reaches \(z_w\), where: \[ t^z_i = - \frac{1}{k_d} \log \left(1 - \frac{z_w}{c_i} \right). \] Before that happens, Eq.(3) applies, while for \(t > t^z_i\), integrating Eq.(2) results in: \[ S_i(t) = \exp \left(- h_b t - b_w(c_i - z_w) (t - t^z_i) - \frac{b_w c_i}{k_d} \left(e^{- k_d t} - e^{-k_d t^z_i} \right) \right) \]

In brief, given values for the four parameters \(h_b\), \(b_w\), \(k_d\) and \(z_w\), we can simulate trajectories by using \(S_i(t)\) to compute conditional survival probabilities. In ‘morse’, those parameters are estimated using Bayesian inference. The choice of priors is defined hereafter.

#### GUTS-IT

With constant concentration, Eq.(4) provides that \(C^{\mbox{\){INT}\(}}_i(t)\) is an increasing function, meaning that:

\[ \max\limits_{0 < \tau < t} (C^{\mbox{${\tiny INT}$}}_i(\tau)) = c_i(1 - e^{-k_d t}) \]

Therefore, assuming a log-logistic distribution for \(f_z\) yields:

\[ S_i(t) = \exp(- h_b t) \left( 1 - \frac{1}{1+ \left( \frac{c_i(1-\exp(-k_d t ))}{m_w} \right)^{- \beta}} \right) \]

where \(m_w>0\) is the scale parameter (and also the median) and \(\beta>0\) is the shape parameter of the log-logistic distribution.

### Inference

Posterior distributions for all parameters \(h_b\), \(b_w\), \(k_d\), \(z_w\), \(m_w\) and \(\beta\) are computed with JAGS [@rjags2016]. We set prior distributions on those parameters based on the actual experimental design used in a toxicity test. For instance, we assume \(z_w\) has a high probability to lie within the range of tested concentrations. For each parameter \(\theta\), we derive in a similar manner a minimum (\(\theta^{\min}\)) and a maximum (\(\theta^{\max}\)) value and state that the prior on \(\theta\) is a log-normal distribution [@delignette2017]. More precisely: \[ \log_{10} \theta \sim \mathcal{N}\left(\frac{\log_{10} \theta^{\min} + \log_{10} \theta^{\max}}{2} \, , \, \frac{\log_{10} \theta^{\max} - \log_{10} \theta^{\min}}{4} \right) \] With this choice, \(\theta^{\min}\) and \(\theta^{\max}\) correspond to the 2.5 and 97.5 percentiles of the prior distribution on \(\theta\). For each parameter, this gives:

- \(z_w^{\min} = \min_{i, c_i \neq 0} c_i\) and \(z_w^{\max} = \max_i c_i\), which amounts to say that \(z_w\) is most probably contained in the range of experimentally tested concentrations ;
- similarly, \(m_w^{\min} = \min_{i, c_i \neq 0} c_i\) and \(m_w^{\max} = \max_i c_i\) ;
- for background mortality rate \(h_b\), we assume a maximum value corresponding to situations where half the indivuals are lost at the first observation time in the control (time \(t_1\)), that is: \[ e^{- h_b^{\max} t_1} = 0.5 \Leftrightarrow h_b^{\max} = - \frac{1}{t_1} \log 0.5 \] To derive a minimum value for \(h_b\), we set the maximal survival probability at the end of the toxicity test in control condition to 0.999, which corresponds to saying that the average lifetime of the considered species is at most a thousand times longer than the duration of the experiment. This gives: \[ e^{- h_b^{\min} t_m} = 0.999 \Leftrightarrow h_b^{\min} = - \frac{1}{t_m} \log 0.999 \]
- \(k_d\) is the parameter describing the speed at which the internal concentration of contaminant equilibrates with the external concentration. We suppose its value is such that the internal concentration can at most reach 99.9% of the external concentration before the first time point, implying the maximum value for \(k_d\) is: \[ 1 - e^{- k_d^{\max} t_1} = 0.999 \Leftrightarrow k_d^{\max} = - \frac{1}{t_1} \log 0.001 \] For the minimum value, we assume the internal concentration should at least have risen to 0.1% of the external concentration at the end of the experiment, which gives: \[ 1 - e^{- k_d^{\min} t_m} = 0.001 \Leftrightarrow k_d^{\min} = - \frac{1}{t_m} \log 0.999 \]
- \(b_w\) is the parameter relating the internal concentration of contaminant to the instantaneous mortality. To fix a maximum value, we state that between the closest two tested concentrations, the survival probability at the first time point should not be divided by more than one thousand, assuming (infinitely) fast equilibration of internal and external concentrations. This last assumption means we take the limit \(k_d \rightarrow + \infty\) and approximate \(S_i(t)\) with \(\exp(- (h_b + b_w(c_i - z_w))t)\). Denoting \(\Delta^{\min}\) the minimum difference between two tested concentrations, we obtain: \[ e^{- b_w^{\max} \Delta^{\min} t_1} = 0.001 \Leftrightarrow b_w^{\max} = - \frac{1}{\Delta^{\min} t_1} \log 0.001 \] Analogously we set a minimum value for \(b_w\) saying that the survival probability at the last time point for the maximum concentration should not be higher than 99.9% of what it is for the minimal tested concentration. For this we assume again \(k_d \rightarrow + \infty\). Denoting \(\Delta^{\max}\) the maximum difference between two tested concentrations, this leads to: \[ e^{- b_w^{\min} \Delta^{\max} t_m} = 0.001 \Leftrightarrow b_w^{\min} = - \frac{1}{\Delta^{\max} t_m} \log 0.999 \]
- for the shape parameter \(\beta\), we used a quasi non-informative log-uniform distribution: \[\log_{10} \beta \sim \mathcal{U}(-2,2)\]