Recommendations to address uncertainties in environmental risk assessment using toxicokinetic-toxicodynamic models

Providing reliable environmental quality standards (EQSs) is a challenging issue in environmental risk assessment (ERA). These EQSs are derived from toxicity endpoints estimated from dose-response models to identify and characterize the environmental hazard of chemical compounds released by human activities. These toxicity endpoints include the classical x% effect/lethal concentrations at a specific time t (EC/LC(x, t)) and the new multiplication factors applied to environmental exposure profiles leading to x% effect reduction at a specific time t (MF(x, t), or denoted LP(x, t) by the EFSA). However, classical dose-response models used to estimate toxicity endpoints have some weaknesses, such as their dependency on observation time points, which are likely to differ between species (e.g., experiment duration). Furthermore, real-world exposure profiles are rarely constant over time, which makes the use of classical dose-response models difficult and may prevent the derivation of MF(x, t). When dealing with survival or immobility toxicity test data, these issues can be overcome with the use of the general unified threshold model of survival (GUTS), a toxicokinetic-toxicodynamic (TKTD) model that provides an explicit framework to analyse both time- and concentration-dependent data sets as well as obtain a mechanistic derivation of EC/LC(x, t) and MF(x, t) regardless of x and at any time t of interest. In ERA, the assessment of a risk is inherently built upon probability distributions, such that the next critical step is to characterize the uncertainties of toxicity endpoints and, consequently, those of EQSs. With this perspective, we investigated the use of a Bayesian framework to obtain the uncertainties from the calibration process and to propagate them to model predictions, including LC(x, t) and MF(x, t) derivations. We also explored the mathematical properties of LC(x, t) and MF(x, t) as well as the impact of different experimental designs to provide some recommendations for a robust derivation of toxicity endpoints leading to reliable EQSs: avoid computing LC(x, t) and MF(x, t) for extreme x values (0 or 100%), where uncertainty is maximal; compute MF(x, t) after a long period of time to take depuration time into account and test survival under pulses with different periods of time between them.


Mathematical notations in reduced GUTS models
In this Supplementary Material concerning the derivation of mathematical equations, we use the following notations:  The dynamic of the internal damage (i.e. the toxicokinetic) is given by: We also suppose that D w (0) = 0. Therefore, when the external concentration is constant (i.e. C w (t) = C w , we have: Otherwise, if the external concentration is varying with time, we have:

Mathematical analysis of LC(x, t)
The lethal concentration makes sense only in the case of constant exposure profile (i.e. C w (t) = C w , ∀ t).
Definition 1 (LC(x, t)). The lethal concentration for x% of organisms at time t is the function denoted LC(x, t) and defined (using notations provided in Table S1) as:

) for model GUTS-RED-SD
For GUTS-RED-SD model (with S SD (C, t) the survival rate for GUTS-RED-SD), the LC SD (x, t) is given by: Then, with t z , the time at which the internal concentration is equal to the threshold, D w (t z ) = z, using (S1), we have: And equation (S6) can be developped as: The expression of t z prevents to have an explicit formulation of LC SD (x, t).
We can use equation (S7) to see that: Combining this result with equation (S8), we obtain:

Convergence of LC SD (x, t) when t tends to infinity
We assume the threshold concentration, z, is reached in a finite time, which mean that lim t→+∞ t − t z = +∞.
To compute lim t→+∞ LC SD (x, t), we have for one part of the equation: And for the other part of the equation, we have: As a consequence, we obtain:

Sensitivity:
To study the influence of x on the LC SD (x, t), we compute the sensitivity, which is simply the derivative of

Elasticity:
The elasticity of LC SD to parameter x represents the proportional change in LC SD in response to a proportional change in the parameter x and is given by:

Convergence of LC IT (x, t) when t tends to infinity
Using equation (S16), we can direclty see that For the specific case of x = 50%, we have: lim t→+∞ LC IT (50, t) = m w (S18)

Sensitivity:
As for the LC SD (x, t), we compute the sensitivity of LC IT (x, t) for parameter x by mean of the derivative:

Elasticity:
Here, for this model, we can simplify the equation of sensitivity by computing the elasticity which is given by: The elasticity of LC IT (x, t) to parameter x represents the proportional change of LC IT (x, t) in response to a proportional change in x. Here, the elasticity of LC IT (x, t) to x is only function of β and x.

Mathematical analysis of M F (x, t)
Contrary to the lethal concentration LC(x, t), the multiplication factor, denoted M F (x, t) makes sense for any type of exposure profile (constant or time-variable). F (x, t)). The multiplication factor leading to a reduction of x% of the survival rate at time t is the function denoted M F (x, t) and defined (using notations provided in Table S1) as:

Definition 2 (M
where C w (τ ) is the exposure profile along the continuous period from 0 to t. The expression is the survival rate at time t when the internal concentration is the exposure profile C w (τ ) multiplied by the constant multiplication factor M F (x, t).

Internal concentration scaled by Multiplication Factor
The internal damage (or scaled internal concentration) at time t, D w (t), is given by equation (S1) using the profile of external concentration, C w (τ ), with τ going from 0 to t.
If the profile of external concentration is multiplied by a factor M F (x, t) (assuming it reduces the survival rate by x% at time t), then, the internal concentration, denoted D M F w (t), is also multiplied by this factor. From equation (S1) we get: what leads to: Therefore, the internal damage referenced to the external concentration D w (t) is linearly scaled by the multiplication factor M F (x, t) whatever the exposure profile C w (τ ).

M F SD (x, t) for model GUTS-RED-SD
Using the definition of the multiplication factor, we have the following formulation for the GUTS-RED-SD model: When the external concentration is constant (C w (τ ) = C w ), we can obtain M F SD (x, t) by using the explicit expression of D w (t) given by (S2). And so, we can define t z and t z,M F such as: As for the LC SD (x, t), the expression of t z,M F prevents to have an explicit formulation of M F SD (x, t).

M F IT (x, t) for model GUTS-RED-IT
Using the definition of the multiplication factor, we have the following formulation for the GUTS-RED-IT model: Therefore, from a GUTS-RED-IT model, solving the toxicokinetic part giving max 0<τ <t (D w (τ )) is enough to find any multiplication factor for any x at any t.
When the external concentration is constant (i.e., C w (τ ) = C w ) we get:

Definition of WAIC and LOO-CV
For more information about information criteria and cross-validation, see [?]. Here we give a brief description to compute the Widely Available/Applicable Information Criterion (WAIC) and the Leave-One-Out Cross-Validation (LOO-CV).

WAIC: Widely Available/Applicable Information Criterion
The first step is to compute the log pointwise predictive density: Then the variance of the log pointwise predictive density: To finally compute the WAIC as:

LOO-CV: Leave-One-Out Cross-Validation
Cross-validation is like WAIC, applied several time on a subsample (S-1) drawn from the posterior.
Then the first order bias correction To finally compute the LOO-CV as: