As already stated in Chapter 2, DEBtox models are rather complex mainly because they offer several ways of accounting for toxicant effect simultaneously on both growth and reproduction processes. Unlike for the GUTS models, this section cannot be comprehensive, because DEBtox model applications are still developed on a more case-by-case basis. However, to exemplify some aspects of the formulation and the implementation of a DEBtox model, published results by Billoir et al. (2011) are used to illustrate a case study about lethal and sublethal effects on daphnids (D. magna) exposed to time-varying cadmium exposure concentrations within a laboratory aquatic microcosm. It should be highlighted that the example included in this chapter illustrates the model formulation, implementation, and the results of the calibration phase. A proper validation with an additional data set was not performed, and therefore could not be included here.
5.1.1.
Model formulation
The dynamics of the scaled damage (called ‘scaled internal concentration’ in Billoir et al., 2011) within daphnids is derived from the total cadmium concentrations in water through a toxicokinetic model. Effects of cadmium on survival, growth and reproduction are then assumed to depend on this scaled damage. In the following, letter t refers to a general expression of time (in days), while tirefers
5.1.1.1. TK models
To model the exposure concentration time profile throughout the experiment, an empirical exponential decay model with a rate b (1/d) is used
Cj(t)¼ Cð7Þj expðb(t+7)), (23)
where Cj(t) is the time-variable exposure cadmium concentration (lg/L) at time t with j = 0,. . .,4 for the control and the four treatments. Parameters Cjð7Þ (j = 0,. . .,4) are fixed at nominal concentrations, 0, 10, 20, 40 and 80 lg/L, that is at concentrations initially introduced 7 days before the introduction of the organisms. Consequently, C0(t) = 0, whatever t.
A normal distribution links the exposure model to the measured cadmium concentrations
MCi;j;k Nðmean = CjðtiÞ; tau ¼ sEÞ for j = 1; :::; 4; (24)
where MCi,j,k are the measured cadmium concentrations (lg/L) at time-point ti, treatment j and
replicate k. ParametersEis the precision (1/variance) of the measurements ((lg/L)2).
Similarly to GUTS (equation 1), for each treatment, the scaled damage at time t, Dw(t), is linked to
Cj(t) through a one-compartment model with a dominant rate constant kD(1/d)
dDw(t)
dt ¼ kDðCj(t) Dw(t)), (25)
with the initial condition Dw(0) = 0. 5.1.1.2. TD models
According to the DEBtox modelling approach (Jager, 2017), survival, growth and reproduction are linked to the scaled damage through ‘linear-with-threshold’ relationships depending on respective so-called no-effect-concentrations, i.e. internal concentration thresholds below which no measurable effect can be detected (see Chapter 2). For the sake of simplicity, in this example, the NEC is assumed to be the same for growth and reproduction.
Survival is modelled with a GUTS-RED-SD model according to (equation 5) with parameter set h = (hb, bw, zw). As shown on Figure 27, the observed number of survivors at time-point ti, treatment
j and replicate k is modelled by variable MSi,j,kthrough a conditional binomial distribution according to
(equation 10) and (equation 11).
D. magna body growth is modelled using the von Bertalanffy growth model (Von Bertalanffy, 1938). Both the growth rate, c (1/d), and the maximum body length, Lm (mm), are assumed to be affected
by the scaled damage
dLj(t)
dt ¼ c(1 rGR (t)) (Lm(1 rGR(t)Þ Lj(t)), (26) withrGR(t)¼ minð1,bGRðDw(t) ZGRÞÞ (27)
and the initial condition Lj(0) = 1,
Figure 26: Observation time-points (in days) of cadmium concentrations (exposure), survival, growth and reproduction of D. magna over the course of the experiment (adapted from Billoir et al., 2011)
where Lj(t) is the body length (mm) at treatment j and time t, whereas rGR(t) is the ‘linear-with-
threshold’ relationship applying for both growth and reproduction with ZGR the NEC (lg/L) and bGR the
effect rate (lg/L per day) for both growth and reproduction. This strong assumption was made by Billoir et al. (2011), however it has to be reconsidered for other applications.
A normal stochastic link is used to relate the observed body length data to the growth model MLi;j;k Nðmean = Ljðti; tau ¼ sGÞ; (28)
where MLi,j,k are the observed body lengths (mm) at time-point ti, treatment j and replicate k.
ParametersGis the precision (1/variance) of the observations (mm2).
D. magna reproduction is modelled in accordance with DEB assumptions (Kooijman, 2000; see also Chapter 2). Indeed, the reproduction process depends on the growth process because organisms reproduce upon reaching their puberty length. Hence, the reproduction process is delayed when the growth process is affected by the cadmium concentration.
Among thefive assumptions of the DEBtox modelling for the way the toxicant affects the daphnid energetic budget (see Chapter 2), the effect model assuming an increase in maintenance energetic costs is used to describe the reproduction process as a function of both time and cadmium concentration. The following equation, derived by Billoir et al. (2011), is given for ad libitum food conditions Rj(t)¼ Rm 1 l3 p ð1 þ rGR(t)ÞðL2j(t)ð ð1 þ rGR(t)Þ1þ LjðtÞ 2 Þ l 3 pÞ if Lj(t) Lm [ lp , (29) else Rj(t)= 0. Rcumj(t)¼ Rcumjðt 1Þ þ Rj(t); (30)
with the initial condition Rcumj(0) = 0,
where Rj(t) is the daily reproduction rate per mother (#) at time t and treatment j, Rcumj(t) the time- cumulated number of offspring per mother (#) at time t and treatment j, lp the normalised length at puberty (Lp/Lm, dimensionless, with Lp the length at puberty), Rm the maximum reproduction rate
(1/d) and rGR(t) the‘linear-with-threshold’ relationship defined by (equation 27).
Accounting for the count status of reproduction data, and because an increased variability of the reproduction is observed with increasing mean values (namely an over-dispersion of the reproduction data), a negative binomial stochastic link is used to relate the observed time-cumulated number of offspring per mother to the reproduction model. It is parameterised so that its mean equalled the reproduction model output
MRcumi;j;k Negbinðr = RcumjðtiÞ
pR 1 pR
; p ¼ pRÞ;
where MRcumi,j,kare the time and cadmium concentration mother (#) at time-point ti, treatment j and
replicate k. Parameter pR() accounts for the overdispersion of the reproduction data.
5.1.2.
Model implementation in the R software
All endpoints (exposure, survival, growth and reproduction) werefitted simultaneously to estimate all 14 model parameters, since all corresponding observed data were linked together as shown in Figure 28. Parameters were estimated within a Bayesian framework via an R code based on JAGS and the R-package rjags (see Appendix A, SectionA.2.2for additional information). Prior probability distributions are those reported in Table 5 according to Billoir et al. (2011). Because it is not possible to numerically integrate differential equations in JAGS, models were implemented in discrete time with one day as time step.
Three independent MCMC chains were run in parallel. After an initial burn-in period of 5,000 iterations, the Bayesian algorithm was run 50,000 iterations and the corresponding sample of the joint parameter posterior distribution was recorded. The convergence of the estimation process was checked with the Gelman and Rubin statistics (Gelman and Rubin, 1992) (see Section 4.1.3.3).
The goodness-of-fit was assessed first by comparing prior and posterior distributions models, then with a graphical comparison of observed data and predictions in a way that account for parameter uncertainties and stochasticity of the model. Indeed, predictions are simulated at each MCMC iteration, along which the parameters vary according to their uncertainty, and with the same models than the ones considered to fit the observed data.
5.1.3.
Input data
As detailed in Billoir et al. (2011), experiments were conducted in laboratory microcosms (2-L beakers) filled with artificial sediment and gently aerated synthetic water in which cadmium (Cd) was introduced at nominal concentrations 10, 20, 40 and 80 lg/L, hereafter referred to as treatments 1 to 4. Four replicates were set up for each treatment. Microcosms were conditioned 7 days in the dark before introduction of organisms. Experiments were conducted under constant temperature, pH and light conditions.
Figure 27: Directed acyclic graph of the DEBtox model used to describe time-variable cadmium effect on survival, growth and reproduction of D. magna. This graph is adapted from by Billoir et al. (2011). Variable q stands for the quantity dH/dt as in equation (4)
At day 0 (7 days after the introduction of Cd), daphnids were introduced in all microcosms, then followed during 21 days. Numbers of survivors, body length and numbers of offspring were recorded at different time-points (Figure 26). The cadmium water concentration was also regularly measured.