hb or not hb - when and why accounting for background mortality in toxicological survival models matters?

Decisions in Environmental Risk Assessment (ERA) about impacts of chemical compounds on different species are based on critical effect indicators such as the 50% lethal concentration (LC50). Regulatory documents recommend concentration-response (or concentration-effect) model fitting on standard toxicity test data to get LC50 values. However, toxicokinetic-toxicodynamic (TKTD) models proved their efficiency to better exploit toxicity test data, at Tier-2 but also at Tier-1, delivering time-independent indicators. In particular, LC50 values can be obtained from the reduced General Unified Threshold model of Survival (GUTS-RED) with both variants, Stochastic Death and Individual Tolerance, that include parameter hb, the background mortality. Estimating hb during the fitting process or not depends on studies and fitting habits, while it may strongly influence the other GUTS-RED parameters, and consequently the LC50 estimate. We hypothesized that estimating hb from all data in all replicates over time should provide more precise LC50 estimates. We then explored how estimating hb impacted: (i) GUTS-RED model parameters; (ii) goodness-of-fit criteria (fitting plot, posterior predictive check, parameter correlation); (iii) LC50 accuracy and precision. We finally show that estimating hb does not impact the LC50 precision while providing more accurate and precise GUTS parameter estimates. Hence, estimating hb would lead to a more protective ERA. Graphical abstract Specifications table

dominant rate constant; and the background mortality rate hb, corresponding to the natural mortality that may occur even if the organisms are not exposed to a chemical stressor. Because of a presumed independence from parameters directly related to the potential toxicity of the chemical of interest (i.e., kd and the two others), the hb parameter is often fixed from the control data, while the other three GUTS parameters are estimated by fitting the model to all of the collected data. This way of doing has pros and cons which this paper aims at investigating with regards to a simultaneous fitting of GUTS providing estimates of the four parameters.
Independently setting the hb parameter before the fitting process means that only three of the four GUTS parameters will be estimated. Fixing the hb parameter may be a biological choice (e.g., no individual died until the end of the experiment under control conditions), or a computing-based choice (e.g., either to facilitate the convergence of the fitting process, or to increase the estimation precision, when data are sparse or when the experiment was not ideally designed). However, setting the h b parameter implies several implicit assumptions: (i) the h b parameter is not correlated to the three other GUTS parameters, and (ii) there is no additional information on the background mortality within the toxicity data. Consequently, setting the hb parameter amounts to neglect the fact that the experimental design may contain very low tested concentrations for which no effect can be observed, as under control conditions. Also, neglecting the potential correlation of hb with the three other parameters may impact their respective estimates, and subsequently the LC(x,t) estimate.
So, the aim of this paper is to evaluate the impact of estimating or not the hb parameter, on both the estimates of the three other GUTS parameters, and the subsequent estimation of the LC50 at final time. We performed our study with both variants of the reduced GUTS, namely GUTS-RED-SD and GUTS-RED-IT, on a collection of 38 data sets comprising different species-compound combinations. We first discussed the results from a modelling perspective, to statistically assess the impact of estimating or not estimating hb; then discussed in the perspective of regulatory ERA, to propose the best compromise between the rapid and daily completion of risk assessments and having sufficient scientific and statistical rigor so that the results on which decisions are based are reliable.

Modelling
Both variants of the reduced GUTS, namely GUTS-RED-SD and GUTS-RED-IT, were systematically used, both under option 1 (hb simultaneously estimated with the other three parameters) and option 2 (hb estimated on control data only), as detailed below. The GUTS-RED-SD variant assumes that all individuals are identically sensitive to the chemical substance by sharing a common threshold internal dose, and that mortality is a stochastic process whose intensity is linearly increasing once this threshold is reached. In contrast, the GUTS-RED-IT variant is based on the critical body residue approach, which assumes that individuals differ in their thresholds following a given probability distribution and that they die as soon as the internal dose reaches the individual-specific threshold [5].
Option 1 (denoted O1) stands for the fitting of the two GUTS-RED variants in their complete form, both involving four parameters, among which kd and hb as described earlier. In addition, variant GUTS-RED-SD has parameters zw (the internal threshold from which effects may appear) and bw (the "killing rate", that is a parameter related to the intensity of the effect) to describe the instantaneous mortality rate as a function of the internal dose via a linear stepwise function [4]: where ( ) stands for the internal dose at time . Variant GUTS-RED-IT also has two additional parameters, related to the fact that the individual threshold is usually assumed to follow a log-logistic probability distribution, of median mw and spread β. This leads to the following cumulative distribution function: with 0< < ( ( )) = , the maximal internal dose reached during the exposure period.
Option 2 (denoted O2) stands for the fitting of the two GUTS-RED variants, the hb parameter being fixed from control data only, according to the following calculation: In equation (3), stands for the observed number of survivors either at initial time, ( ), or at final time, ( ), of the experiment. In the end, four situations were investigated crossing the two GUTS-RED variants (SD and IT) with the two options (O1 and O2). Hereafter, each situation is always associated with the same color code: dark green for SD-O1; light green for SD-O2; dark orange for IT-O1; light orange for IT-O2. Both GUTS-RED variants were fitted with the R-package "morse" [6] under R version 4.2.1 ("Funny-Looking Kid", 2022-06-23). In particular, function "survFit()" was used, with option "model_type" equal to "SD" (for variant GUTS-RED-SD) or "IT" (for variant GUTS-RED-IT) and option "hb_value" equal TRUE (for option 1) or FALSE (option 2). Under option 2, parameter hb was always calculated before the fitting process. Finally, function "doseResponse_survFitCstExp" of package "morse" was adapted to get the posterior probability distribution of the 50 estimate for each of the four situations SD-O1, SD-O2, IT-O1 and IT-O2.

Data sets
To ensure reliable results, the above-mentioned procedure was used with a total of 38 data sets (Table 1 ). Each data set was fitted under each of the four situations SD-O1, SD-O2, IT-O1 and IT-O2, leading to a total of 152 fitting outputs. For each data set, fitting outputs of SD and IT variants were stored in two separated PDF files, each including for both options 1 and 2: (i) fitting plots ; (ii) parameter estimates, expressed with the median, the 95% credible interval and the precision ; (iii) the fixed hb value (option 2 only) ; (iv) the posterior predictive check (PPC) ; (v) the comparison of prior and posterior marginal probability distributions for each estimated parameter ; (vi) the correlation matrix between estimated parameters ; (vii) the deviance information criteria (DIC) and the calculation of the difference between option 1 and option 2. Note that a negative value of the "Diff_DIC" indicates that the model with parameter hb simultaneously estimated with the three other parameters will be preferred. Finally, LC50 estimates are given for both options 1 and 2, expressed with the median and the 95% credible interval. All these output files were analyzed regarding the different goodness-of-fit criteria: (i) the visual fit, (ii) the PPC, (iii) the prior-posterior comparison, and (iv) the correlations between parameters.

A1
Simulated with GUTS-RED-SD, true hb value = 0.01 [4] A2 Simulated with GUTS-RED-IT, true hb value = 0.02 [4] Outcomes In general, our results show that option 1 (that is estimating the hb parameter) provides more precise and accurate estimates of the three other parameters for both SD and IT GUTS-RED variants. Surprisingly, our results show that simultaneously estimating the four parameters of GUTS-RED variants does not provide better LC50 estimates. Consequently, in terms of ERA, if only the LC50 is of interest, fitting GUTS-RED variants by fixing hb (option 2) could be sufficient.
Influence of estimating hb on parameter estimates As illustrated in Figure 1, for both data sets A1 and A2 (Table 1), the estimated hb values under option 1 (hb,1) are closer to the true hb value than the calculated hb value under option 2 (hb,2). This results in favor of option 1 is also observed with the other data sets we analyzed (see supplementary information). For all data sets except one, whatever the GUTS-RED variant, values of hb,1 and hb,2 were different; the exception was obtained with data set TOX01-SPE12 (Table 1, Figure S1), a particular data set with only four time points and a very wide range of tested concentrations that could explain some fitting difficulties whether the h b parameter is estimated or not. As also illustrated in Figure  S1, option 2 tends to underestimate the background mortality, with 56/72 hb,1 estimates greater than hb,2 values. In addition, for all datasets, hb,1 estimates are quite precise, both with SD and IT. Even though it may seem obvious, estimating parameter hb with the others allow the quantification of its uncertainty in the same way.
Regarding the other parameters, Figure 1 shows that, under option 1, zw and bw (data set A1), as well as mw (data set A2) are closer to the true value than under option 2. Firstly, option 2 overestimates zw with GUTS-RED-SD (data set A1) and mw in GUTS-RED-IT (data set A2). This result for zw is confirmed with the remaining data sets, among which 14/36 data sets show a difference in zw estimation between option 1 (zw,1) and 2 (zw,2, Figure S3). Twelve out of these 14 data sets lead to a zw,1 value greater than the zw,2 value. The two exceptions concern data sets TOX01-SPE17 and TOX01-SPE18 with a long exposure period of 96 days during which 13 time points were distributed in triplets of closely spaced time points. Also, in these two data sets, the range of the tested concentrations is very large with quite widely spaced concentrations which could prevent a precise estimate of zw whatever the option. Similar results were obtained for mw with GUTS-RED-IT. Indeed, seven data sets show different mw estimates between option 1 (mw,1) and 2 (mw,2, Figure S5). Six out of these seven data sets show a mw,1 greater than the mw,2 value, the only exception being obtained with data set TOX01-SPE17 for which difficulties have already been identified (see above) for the estimation of zw. Because parameters zw and mw correspond to a No-Effect Concentration (NEC) within the model, this means that option 2 tends to overestimate the internal dose above which the chemical compound will have an effect on individuals.
Secondly, whatever the data sets, our results reveal that option 2 underestimates bw in GUTS-RED-SD (see Figure 1, for an illustration with data sets A1 and A2). Ten data sets led to bw estimates different between options 1 (bw,1) and 2 (bw,2, Figure S4). Out of these 10 data sets, eight data sets led to bw,2 less than bw,1. The two other data sets are TOX01-SPE03_a, and TOX05-SPE19 for which the dose-response curve at final time is very steep, making difficult the estimation of b w whatever the option. So, reminding that parameter b w stands for the intensity of the chemical effect on individuals, it appears that option 2 tends to underestimate this effect.

Figure 1: Marginal posterior probability distributions of both GUTS-RED variants, SD (in green), IT (in orange), for both options 1 (dark colors) and
While the β parameter does not inspire any specific comment, the kd parameter that corresponds to the toxicokinetic part of the model, also appears close to the true value but only with GUTS-RED-SD, indeed GUTS-RED-IT clearly underestimates kd (Figure 1, data set A1). Similar results were obtained with the other data sets. A total of 18/36 data sets fitted with both GUTS-RED-SD and GUTS-RED-IT gave different kd estimates whatever the option ( Figure S2). For these 18 data sets, the kd estimate with GUTS-RED-IT was systematically lower than the one obtained with GUTS-RED-SD. This means that for half of the data sets we analyzed, the speed of the kinetic is underestimated, either by GUTS-RED-SD or GUTS-RED-IT. This corroborates the current well-known inability to anticipate the use of one or the other GUTS-RED variants for a particular species-compounds combination [2].
Our results on parameter estimates lead us to the conclusion that estimating the background mortality (option 1) provides hb values closer to the reality. Because option 2 tends to overestimate zw and mw, and to underestimate b w , using option 1 will provide more protective predictions in terms of ERA.
Influence of estimating hb on goodness-of-fit criteria Looking at fitting plots of posterior predictive checks for all data sets shows no difference whether parameter h b is estimated or not. Looking at parameter correlations seems to be the most convenient way to highlight the impact on estimating hb or not on the estimation of the three other parameters. Indeed, fixing hb before the inference process leading the other three estimates (option 2) means completely ignoring any correlation of hb with the other three parameters. However, as illustrated in Figure 2, not estimating hb hardly changes correlations between the other three parameters. In addition, we can see that when estimated, hb is only weakly correlated with the other three parameters. This result holds true for all other datasets we analyzed. Finally, whatever the GUTS-RED variants we used, whatever the datasets we analyzed, whatever the goodness-of-fit criteria we looked at, there was virtually no difference between the options 1 and 2. Nevertheless, because the parameters are better estimated under option 1 (see abovementioned arguments and Figure 1), we discuss only based on option 1 hereafter. Figure 3 confirms that hb is poorly correlated to the other three parameters for all data sets. Correlation coefficients rarely exceed 0.25, and when exceeding 0.25, it happens more often with GUTS-RED-IT than with GUTS-RED-SD. With the exception of two or three data sets, parameters of the GUTS-RED-IT variant are always positively correlated, while correlation coefficients are distributed around 0 with GUTS-RED-SD. The hb parameter is most often highly correlated with the β parameter, and the correlations of hb with both kd and mw is very similar most of the time; this is due to a strong correlation between kd and mw that inherently exists in GUTS-RED-IT [18]; see Figure S8 to visualize this for all the data sets we analyzed in this study. With variant GUTS-RED-SD, the correlation coefficients are more often negative between the parameters hb and kd than with the two other parameters; also, when positive, this correlation is very small. Biologically speaking, this means that the background mortality rate is more correlated to the toxicokinetic than to the toxicodynamic of the compound. Such a result is particularly interesting because it highlights a possible trade-off between the survival of organisms and their bioaccumulation of chemical substances.
Influence of estimating hb on the LC50 estimate As illustrated on Figure 4, whatever the data set, with one or two exceptions, there is almost no difference between the LC50 with both GUTS-RED-SD and GUTS-RED-IT whether parameter hb is estimated or not. However, LC50 estimates may sometimes strongly differ whether GUTS-RED-SD or GUTS-RED-IT is used (see for example Figure 4, on the right). Two LC50 estimates were considered significantly different when their 95% credible intervals did not or very few overlap. Hence, in total, only 6 out of the 38 data sets exhibited significantly different LC50 values. This result is consistent with other works also illustrating the impossible choice between both GUTS-RED variants [2].
Looking at Figure S8, in addition to the fact that the LC50 estimates are not very different, their precision also is similar both between option 1 and option 2, and between the use of GUTS-RED-SD and GUTS-RED-IT. This result is quite counterintuitive with the fact that option 1 provides more precise parameter estimates. An explanation may be that the hb parameter is poorly correlated to the other three parameters. Consequently, in terms of ERA, if only LC50 estimates are needed to decide, estimating hb or not does not matter a lot.

Conclusion
Our study clearly illustrated that background mortality (hb) values were almost always different between options 1 and 2. When estimated (option 1), parameter hb was always closer to the expected background mortality, and its estimate was quite precise. On the other hand, when hb was calculated from control data (option 2), it was always underestimated. We also noticed that option 2 tended to provide underestimated no-effect concentrations: parameter zw in GUTS-RED-SD or parameter mw in GUTS-RED-IT. Parameter bw displayed a similar pattern for GUTS-RED-SD, meaning that option 2 underestimated the intensity of the chemical effect on individuals. Regarding kd estimates (i.e., the kinetic quickness), there was no difference between options 1 and 2. Nevertheless, kd estimates differed according to the GUTS-RED variant, sometimes underestimated, either with GUTS-RED-SD or GUTS-RED-IT. This confirms our inability to anticipate which of the GUTS-RED variants would be more appropriate for a particular species-compound combination. Furthermore, we showed that neither the GUTS-RED variant nor the choice between option 1 and 2 had an impact on any of the goodness-of-fit criteria. Estimating hb or not only had tiny effects either on LC50 medians or its precision, while the choice of the GUTS-RED variant could sometimes provide different LC50 estimates. Altogether, for ERA mainly needing precise (and accurate) LC50 estimates, estimating the background mortality or not is of little importance. In contrast, benefiting of the most precise parameter estimates for GUTS-RED models may be of strong interest if the objective is to safely predict a survival probability under an environmentally realistic scenarios along which the exposure concentration is varying over time.