Abstract
This paper gives the full analytical solution of the generic set of ordinary differential equations that define one-compartment toxicokinetic models. These models describe uptake and elimination processes taking place within living organisms when exposed to chemical substances. The models solved in this paper consider living organisms as a unique compartment, into which a parent compound enters via several possible exposure routes and from which it is eliminated as well as its potential metabolites. Benefiting from generic solutions of one-compartment toxicokinetic models is particularly useful when fitting them to experimental data, facilitating the writing of the inference algorithms leading to parameter estimates. Additionally, these models are of crucial interest in environmental risk assessment for the calculation of bioaccumulation metrics as required by regulators in support of decision making when they evaluate dossiers for marketing authorisation of active substances.
1. Introduction
In this paper, we consider a very generic one-compartment toxicokinetic (TK) model describing uptake and elimination processes taking place within living organisms when exposed to chemical substances. On a general point of view, TK models are of particular interest in Environmental Risk Assessment [6], for example when the calculation of bioaccumulation metrics is required by regulators in support of decision making when they evaluate dossiers for marketing authorisation of active substances.
The generic TK model considered in this paper is a one-compartment model considering organisms as a whole in which compounds may enter and from which these compounds can be eliminated). Note that there exists more complex TK models refining the description of contamination pathways within organisms, distinguishing organs and tissues from physiological hypotheses on potential targets of exposure compounds; such physiologically-based TK (PBTK) models mainly apply for humans [9, 8] or big mammals [3, 5]. The generic one-compartment TK model considered in this paper has the advantage of accounting for several exposure sources (e.g., by water, sediment and/or food), several elimination processes (e.g., direct elimination, dilution by growth and/or biotransformation) and several potential metabolites of the parent chemical compound to which organisms are exposed [7]. This generic TK model is composed of two sets of ordinary differential equations (ODE), one set for the accumulation phase (that includes both bioaccumulation and elimination processes, as those related to the bio-transformation of the parent compound into metabolites) and during which organisms are exposed to a given compound; the second one for the depuration phase (with only elimination processes including biotransformation) and during which organisms are transferred into a clean medium. The transition from one set of ODE to the other takes place at time tc corresponding to the duration of the accumulation phase (see Table 1).
In practice, the exposure concentration to which organisms are exposed may vary over time as in realistic environments, but then, there is no analytical solution of the TK model; only a numerical solution can be calculated in this case with an appropriate algorithm. Our paper thus assumes that the exposure concentration remains constant over time whatever the exposure source. Such an experimental condition can be ensured for most of the chemical compound when performing laboratory experiments. In addition, this assumption allows to provide the exact solution of the full one-compartment TK model by considering as many routes of exposure and as many elimination processes as desired, as well as an infinite number of phase I metabolites, i.e., directly derived from the parent compound to which organisms are exposed. Note that [4] already provided a partially resolved one-compartment TK model but only for the accumulation phase and one exposure route.
2. The one-compartment TK set of ODE
2.1. Accumulation phase (0 ≤ t ≤ tc)
The set of ODE describing the accumulation phase (0 ≤ t ≤ tc) writes as follows:
All parameters and variables, with their meaning and units when applicable, are gathered together in Table 1.
Equation (1.a) for the parent compound is a linear first-order ODE with constant coefficients and a second member. Equation (1.a) admits as a particular solution. Equation (1.a) without its second member writes:
Given the initial condition C(t = 0) = C0 (C0 ≥ 0), we finally get the full analytical solution of Equation (1.a), providing the internal concentration of the parent compound over time during the accumulation phase (0 ≤ t ≤ tc) as follows:
See Table 1 for the definition of parameter R.
Equation (1.b) is also a linear first-order ODE with constant coefficients and a second member, with the following analytical solution when removing its second member:
The method of variation of a constant consists of writing the general solution of Equation (1.b) as: and to find function K(t) by deriving and re-injecting the result into
Equation (5). The derivative from Equation (1.b) writes: while the re-injection into Equation (1.b) leads to: which integrates into:
See Table 1 for the definition of parameter Dℓ.
The general solution of Equation (1.b) finally writes as follows:
Let’s consider the following initial condition , biologically meaning that when the accumulation phase starts, organisms are only exposed to the parent compound, so that there are no metabolites within. Then, we finally get the full analytical solution of Equation (1.b) providing the internal concentration of metabolite ℓ over time during the accumulation phase:
2.2. Depuration phase (t ≥ tc)
The set of ODE describing the depuration phase (t ≥ tc) writes as follows:
All parameters and variables, with their meaning and units when applicable, are gathered together in Table 1.
Equation (11.a) is a linear first-order ODE without a second member, so that it has a general solution of the following form:
For the depuration phase and the parent compound, the initial condition comes from the calculation of the internal parent compound concentration at the end of the accumulation phase (i.e., at time t = tc) thanks to solution (3):
From the general analytical solution given by Equation (12), we get leading to a constant K in Equation (12) equals to:
Then, the final analytical solution of Equation (11.a) providing the internal concentration of the parent compound over time during the depuration phase (t ≥ tc) writes:
For simplicity reasons, Equation (15) above can be written as Cp(t) = Qe−(E+M)×t with Q as defined in Table 1.
Equation (11.b) is a linear first-order ODE with constant coefficients and a second member. The analytical solution of Equation (11.b) without its second member writes:
As previously, the method of the variation of a constant provides the general solution of Equation (11.b) as , requiring to search for function K(t).
The derivative of from Equation (16) writes:
The re-injection of derivative (17) into Equation (11.b) leads to: which integrates into: finally leading to the general analytical solution of Equation (11.b):
Constant C can be determined from the initial condition, i.e., from the internal concentration of metabolite ℓ at t = tc both at the end of the accumulation phase and at the beginning of the depuration phase. From Equation (20), we get , and from Equation (10), we get . Finally, we get the following expression for constant C:
Replacing constant C in Equation (20) gives the final analytical solution of Equation (11.b) providing the internal concentration of metabolite ℓ for the depuration phase (t > tc) as follows:
Replacing constant Q by its own expression (Table 1) leads to:
3. The generic set of solutions
Reminding the following intermediate notations and we finally obtain the full set of analytical solutions corresponding the whole one-compartment TK set of ODE describing the time-course for both the accumulation and the depuration phases of the parent compound and its potential metabolites.
The analytical solution for the internal concentration of the parent compound during the accumulation phase, previously referred as Equation (3):
The analytical solution for the internal concentration of metabolite ℓ during the accumulation phase, previously referred as Equation (10):
The analytical solution for the internal concentration of the parent compound during the depuration phase, previously referred as Equation (15):
The analytical solution for the internal concentration of metabolite ℓ during the depuration phase, previously referred as Equation (23):
We could fully finish the writing of the very final generic analytical solution of the one-compartment TK model with all the parameters to estimate from observed data using an inference process by replacing constants U, E and M by and , respectively.
4. Model simulations
Model simulations of the generic set of solutions (Equations (3), (10), (15) and (23)) were performed under the R software with 500 time points in [0; tf] where tF stands for the final time of each simulation. These simulations illustrate the toxicokinetic in three case studies where different compounds are bioaccumulated by different species. In order to proceed, the time duration of the accumulation phase (parameter tc) is required, as well as the exposure concentrations in the media and the model parameter values (Table 2). Figures 1 to 5 show the simulations of internal concentrations over time for the three species-compound combinations considered in case studies described below. All necessary parameter values were directly picked-up from the MOSAICbioacc (https://mosaic.univ-lyon1.fr/bioacc) example files. For each case study, TK model parameters were varied one-at-a-time with 20, 50 and 80% of increase from the original value.
4.1. The very simplest one-compartment TK model
Equations (3) and (15) are simulated for a simple case study where fish are exposed to a highly hydrophobic chemical contaminated water. Only natural excretion is considered [2] (Figure 1 and Table 2). The corresponding inputs are the exposure concentration cw (referring to ci and I = 1), the uptake rate from water (referring to and I = 1) and the excretion rate (referring to and J = 1). When parameter increases, internal concentrations are higher (e.g., blue curve in Figure 1.a) than with the original value (orange curve in Figure1.a). Biologically speaking, the higher the uptake rate is for a given substance, the more it is bioaccumulated by organisms. Conversely, an increase in parameter leads to a lower internal concentration (e.g., blue curve in Figure 1.b), consistent with the known underlying biological mechanism: the faster a contaminant is eliminated, the quicker its concentration decreases in organisms.
4.2. A one-compartment TK model with several exposure routes, no metabolites
Equations (3) and (15) are simulated for a freshwater shrimp exposed to an organic chlorine compound by contaminated sediment and food. Only natural excretion is considered as elimination process [7] (Figure 2 and Table 2). The corresponding inputs are the exposure concentration via sediment cs and via food cf (referring to ci and I = 2), the two uptake rates from sediment and food, and respectively (referring to and I = 2) and the excretion rate (referring to and J = 1). When parameter increases, internal concentrations are higher (e.g., blue curve in Figure 2.a) than with the original value (orange curve in Figure 2.a). Biologically speaking, the higher the uptake rate from sediment is for a given substance, the stronger it is bioaccumulated by organisms. Regarding the exposure from food, as parameter is low, its variation does not significantly influence the internal concentration of the contaminant (Figure 2.b). Biologically speaking, this means that the exposure via sediment is the major route of contamination for these organisms. Besides, as previously shown in section 4.1, an increase in parameter leads to a faster decreasing concentration (e.g., blue curve in Figure 2.c).
4.3. A one-compartment TK model with one exposure route, several metabolites
Equations (3), (10), (15) and (23) are simulated for a freshwater shrimp exposed to an organic biocide by contaminated water. Three metabolites derived from the parent compound are considered [1] together with the natural excretion (Figures 3 to 5 and Table 2). The corresponding inputs are the exposure concentration via water cw (referring to ci and I = 1), the up-take rate (referring to and I = 1), the excretion rate (referring to and J = 1), the three metabolization rates and (referring to and L = 3) and the three elimination rates of the metabolites , and (referring to and L = 3). When parameter increases, in-ternal concentrations are higher for metabolite 1 (e.g., blue curve in Figure 3.b) than with the original value (orange curve in Figure 3.b). Conversely, internal concentrations are lower for the parent compound (e.g., blue curve in Figure 3.a). Biologically speaking, the more the biotransformation rate for a given metabolite is increasing, the higher is its concentration within or-ganisms due to the highly biotransformation of the parent compound. This leads to a lower internal concentration of the parent compound than with the original value of (e.g., blue curve, Figure 3.a). An increase in also induces a decrease in the internal concentrations of the other metabolites (Figure 3.c and d). Besides, when parameter increases, this only affects the internal concentration of metabolite 1 (Figure 4.b). In addition, as previously viewed (sections 4.1 and 4.2), an increase in parameter will induce high internal concentrations for both the parent compound and its metabolites (Figure 5). Indeed, the more the internal concentration of the parent compound is increasing, the more the biotransformation process will intensify, leading to high internal concentrations for each metabolite.
5. Acknowledgments
The authors would like to express their sincere thanks to Miléna Kaag, Yacout Lahlou and Nino Molin who designed the MOSAICbioacc prediction tool as part of their 4th year study project at the National Institute of Applied Sciences (INSA) in Lyon (France). This work was made under the umbrella of the French GDR “Aquatic Ecotoxicology” framework aiming at fostering stimulating scientific discussions and collaborations for more integrative approaches. This work was financially supported by the Graduate School H2O’Lyon (ANR-17-EURE-0018) and “Université de Lyon” (UdL), as part of the program “Investissements d’Avenir” run by “Agence Nationale de la Recherche” (ANR).