Introduction
Physiologically-based kinetic (PBK) models encompass both physiologically-based pharmaco-kinetic and physiologically-based (eco-)toxico-kinetic models depending on the context.1,2 Hence, PBK models can refer either to therapeutic drug development3 or environmental risk assessment.4–6 All PBK models are compartment models employing ordinary differential equations (ODE) to quantify chemical absorption, distribution, metabolism (i.e., biotransformation), and excretion processes within living organisms when exposed to chemical substances.7,8 One fundamental aspect of PBK model complexity is the degree of compartmentalization (i.e., differentiation of an organism into various tissues or organs).9,10 This complexity translates into a high number of parameters usually valued from literature information or expert knowledge. Consequently, most PBK models have purely predictive usages, for example, for human health assessments where novel experiments are very limited, even impossible.11 Nevertheless, recent advances in the development of Bayesian inference tools make it possible to deal with complex models12–15 even with sparse data sets, which opens up new possibilities for PBK models to obtain parameter estimates associated with their uncertainties and to propagate this information to model-informed predictions. Also, as part of new approach methodologies , PBK models as well as in vitro-in vivo extrapolation approaches can be combined with bioactivity data, all of this helping prioritize thousands of “data poor” chemicals in human health risk assessment.10,16
Relying a priori on the anatomical and physiological structure of the body, PBK model compartments usually correspond to target organs or tissues, possibly all interconnected and related to the external exposure medium. However, many PBK models only relate organs or tissues to blood or lymph, with only one or two organs or tissues linked to the external medium.17,18 Unfortunately, such simplifications of PBK models are often justified based on technical bases rather than physiological ones. All PBK models are written as coupled ODE whose parameters correspond to fluxes between compartments, for which information is partly available in the scientific literature. The general tendency to develop more complex models to include as many physiological processes as possible, and the current computational capacities, may give the impression that the most complex models will be the most efficient. Nevertheless, with increasing parameters to calibrate, such complex models may be avoided in favor of balancing complexity and simplicity. Current regulatory documents for assessing the bioaccumulation of chemicals in organisms usually only require simple one-compartment models,19–23 even if it is now recognized that it is necessary to also consider internal concentrations within target organs in order to fully capture the chemical bioaccumulation behavior, the specific role of organs, and the dynamics of toxic effects.24
Multi-compartment models sometimes reveal the necessary, for example, to finely decipher the internal contamination routes of specific chemical compounds causing damage to only specific organs.25–27 Additionally, PBK models can be crucial to predicting organ-level concentration-time profiles in a situation where animal testing is now prohibited, using PBK model information from one chemical substance to inform the development or evaluation of a PBK model for a similar chemical substance.3 In the perspective to enlarge and facilitate the use of PBK models, namely to include more compartments, to better estimate parameter values from data, and to better support a fine deciphering of underlying contamination processes after chemical exposure, there is today a clear need for user-friendly tools. From an automatized implementation, fully transparent and reproducible, such tools should simplify the use of any PBK models, preventing users from investing in technicalities, whatever the required number of compartments to consider physiologically, whatever the number of connections to account for between compartments in pairs or between compartments and the external medium, and whatever the species-compound combination of interest. Such tools seem the only way to gradually achieve greater acceptability of complex PBK models, even in a regulatory context.10,28
Capitalizing on recent publications on TK models,15,22 we present in this paper a very innovative solution of a fully generic PBK model written as a set of ODE. Benefiting from an exact solution is a tremendous advantage in numerical implementation. Indeed, it avoids discretizing ODE as there is no more numerical approximation. The solution is directly used for the inference process and the subsequent simulations. The gain in calculation time is enormous (more than 100-fold), associated with fair use of computer resources. Moreover, the new modeling framework we propose makes it possible to account for an infinite number of compartments, with all possible connections between pairs of compartments and between compartments and the exposure medium, independently of the investigated species or chemical substance. Indeed, we found a particularly condensed way to write a linear ODE system, which is typical of PBK models, thus allowing us to fully and exactly solve the ODE system to write an exact generic solution. This exact solution is fascinating when estimating many parameters related to many state variables, for which experimental data may be sparse, with few replicates and high variability.
In the methods, we first detail our generic modeling framework, together with notations of parameters and variables, providing the generic solution at the end of section 2. Then, the generic modeling framework is applied to the particular context of bioaccumulating chemical substances within organisms. We detail how to write the ODE system for both accumulation and depuration phases of standard bioaccumulation tests and then how to get the final generic solution to simulate internal concentrations over time from parameter estimates. Results presents four different situations where simulations were useful to predict what happens within organs and/or tissues according to the species under consideration and the chemical substance to which it is exposed. These case studies were chosen from the literature to be diverse and complementary in terms of questions that a PBK model can help to investigate. We thus present two case studies for the species Gammarus fossarum exposed to cadmium, for which internal concentrations have been measured within four organs: one case study with one-compartment PBK models for each organ considered separately; the other case study with a four-compartment PBK model. These case studies illustrate the added value of considering one single four-compartment model rather than four one-compartment models for each organ. The third case study concerns the sea cucumber exposed to six different antibiotics. Furthermore, the fourth case study concerns the species Danio rerio exposed to arsenic, whose bioaccumulation process is described with a six organ-based compartment PBK model.
Results
This section presents four case studies with different numbers of compartments. Two case studies concern the species Gammarus fossarum exposed to cadmium (Cd).27,44 The third one concerns Apostichopus japonicus sea cucumber exposed to six different antibiotics;45 the fourth one concerns species Danio rerio exposed to arsenic (As).46 Gestin and colleagues27,44 used several one-compartment PBK models compared with one four-compartment PBK model to gain knowledge on the accumulation and fate dynamic of Cd in and between gammarids’ organs. Subsections give the generic solutions with median parameter values for this particular case study (Table 2).
Table 2Medians of parameters estimated by Bayesian inference from TK one-compartment models separately fitted to each organ of Gammarus fossarum exposed to dissolved Cd at 11.1 μg·L−1 for seven days before being placed for 14 days under depuration conditions
Process | Organ | Parameter | Median value (in [t]−1) |
---|
Uptake | Intestines | ku,1 | 1917 |
| Caeca | ku,2 | 1571 |
| Cephalons | ku,3 | 91.1 |
| Remaining tissues | ku,4 | 135 |
Elimination | Intestines | ke,1 | 0.506 |
| Caeca | ke,2 | 0.053 |
| Cephalons | ke,3 | 0.060 |
| Remaining tissues | ke,4 | 0.026 |
Case study with one compartment
Applying Equation (2) with only one compartment leads to two single equations:
{dcA,i(t)dt=ku,1cx−ke,1cA,i(t) when 0≤t≤tc (accumulation phase) (a)dcD,i(t)dt=−ke,1cD,i(t) when t>tc (depuration phase) (b)
System of Equations (22a) and (22b) can easily be solved with the method of the separation of variables (also known as the Fourier method), leading to the following system of solutions for both accumulation and depuration phases:
{ cA,i(t)=ku,ike,icx(1−e−ke,it) when 0≤t≤tc (accumulation phase) (a)cD,i(t)=ku,ike,icx(eke,i(tc−t)−e−ke,it) when t>tc (depuration phase) (b)
Getting the set of solutions (23) directly from the generic expressions given by the combination of both Equations (17) and (21) leads precisely to the same result. Indeed, with one compartment, matrix E = −ke,i and vector U = ku,i.
Inspired from Gestin et al.,27,44 when considering only solid black arrows, Figure 3 highlights the target organs that can correspond to one compartment according to i: intestine (i = 1); cephalon (i = 2); caeca (i = 3); remaining tissues (i = 4). Each compartment has its own parameter pair for uptake (ku,i) and elimination (ke,i) rates (Table 2). Model parameters were estimated under a unified Bayesian framework.20 In particular, parameters of PBK one-compartment models were fitted separately for each organ of G. fossarum exposed to dissolved Cd at 11.1 μg·L−1 for seven days before being placed for 14 days under depuration conditions. Getting median parameter values as given in Table 2 allows simulating what happens within the intestines when it is connected to all other organs, for example (see Supplementary File 1 for more details).
Case study with four compartments
Applying the general matrix ODE system given by the set of Equations (15) and (16) to the particular case of four compartments connected by pairs (Fig. 3) leads to the following writing:
{dCA(t)dt=Ucx+ECA(t) when 0≤t≤tc (accumulation phase) (a)dCD(t)dt=ECD(t) when t>tc (depuration phase) (b)
with vectors and matrices defined as follows:CA(t)=CD(t)=(c1(t)c2(t)c3(t)c4(t)c5(t)) U=(ku,1ku,2ku,3ku,4ku,5) E=(e1,1k1,2k1,3k1,4k1,5k2,1e2,2k2,3k2,4k2,5k3,1k3,2e3,3k3,4k3,5k4,1k4,2k4,3e4,4k4,5k5,1k5,2k5,3e5,4e5,5)
and diagonal elements of matrix E defined by:{e1,1=−ke,1−(k2,1+k3,1+k4,1) (a)e2,2=−ke,2−(k1,2+k3,2+k4,2) (b)e3,3=−ke,3−(k1,3+k2,3+k4,3) (c)e4,4=−ke,4−(k1,4+k2,4+k3,4) (d)
The exact solution of the matrix ODE system (24) can be directly deduced from Equations (17) and (21):
{CA(t)=(etE−I)E−1Ucx when 0≤t≤tc (accumulation phase) (a)CD(t)=(etE−e(t−tc)E)E−1Ucx when t>tc (depuration phase) (b)
Developing the matrix Equations (27a) and (27b) finally provides the following sets of four equations for both accumulation and depuration phases:
For the accumulation phase ( 0 ≤ t ≤ tc):
{dc1(t)dt=ku,1cx−ke,1c1(t)+k1,2c2(t)+k1,3c3(t) +k1,4c4(t)−(k2,1+k3,1+k4,1)c1(t) (a)dc2(t)dt=ku,2cx−ke,2c2(t)+k2,1c1(t)+k2,3c3(t) +k2,4c4(t)−(k1,2+k3,2+k4,2)c2(t) (b)dc3(t)dt=ku,3cx−ke,3c3(t)+k3,1c1(t)+k3,2c2(t) +k3,4c4(t)−(k1,3+k2,3+k4,3)c3(t) (c)dc4(t)dt=ku,4cx−ke,4c4(t)+k4,1c1(t)+k4,2c2(t) +k4,3c3(t)−(k1,4+k2,4+k3,4)c4(t) (d)
For the depuration phase (t > tc):
{dc1(t)dt=−ke,1c1(t)+k1,2c2(t)+k1,3c3(t) +k1,4c4(t)−(k2,1+k3,1+k4,1)c1(t) (a)dc2(t)dt=−ke,2c2(t)+k2,1c,(t)+k2,3c3(t) +k2,4c4(t)−(k1,2+k3,2+k4,2)c2(t) (b)dc3(t)dt=−ke,3c3(t)+k3,1c1(t)+k3,2c2(t) +k3,4c4(t)−(k1,3+k2,3+k4,3)c3(t) (c)dc4(t)dt=−ke,4c4(t)+k4,1c1(t)+k4,2c2(t) +k4,3c3(t)−(k1,4+k2,4+k3,4)c4(t) (d)
The four-compartment model in Equations (27a) and (27b) based on the matrix ODE system in Equations (24a) and (24b), with its exact solution in (28) and (29), assumes that all compartments are connected to each other by pairs and with the external medium (Fig. 3). This means that the model is considering all incoming and outgoing arrows from all compartments. This model comprises a total of 20 parameters, plus the cx value for the exposure concentration. As given in Table 3, median parameter values were used to simulate what happens within each organ in terms of internal concentration dynamic and compared to the previous results with the four independent one-compartment PBK models. See Supplementary File 1 for details.
Table 3Parameter estimates (expressed as medians and 95% uncertainty intervals) of the four-compartment model corresponding to solid black arrows in Figure 3 as provided by1 in Table S6
Organ-Connection | Parameter | Median | Q2.5% | Q97.5% |
---|
Intestines (uptake) | ku,1 | 3342 | 2720 | 3707 |
Intestines (elimination) | ke,1 | 0.54 | 0.415 | 1.402 |
Intestines-Caeca | k21 | 0.873 | 0.603 | 1.739 |
Caeca-Intestines | k12 | 0.218 | 0.132 | 0.376 |
Intestines-Cephalons | k31 | 0.059 | 0.034 | 0.124 |
Cephalons-Intestines | k13 | 0.262 | 0.124 | 0.871 |
Intestines-remaining tissues | k41 | 0.069 | 0.049 | 0.126 |
Remaining tissues-Intestines | k14 | 0.14 | 0.086 | 0.238 |
Intestines | σ1 | 8.974 | 6.469 | 15.28 |
Caeca | σ2 | 17.94 | 13.07 | 26.84 |
Cephalons | σ3 | 1.223 | 0.863 | 1.818 |
Remaining tissues | σ4 | 1.468 | 1.06 | 2.242 |
As illustrated above, our generic modeling framework allows simulating complex situations involving several compartments, their connections in pairs and/or with the exposure media. Let us now relate what we did for simulations of four one-compartment models for each organ of G. fossarum exposed to Cd – with the four-compartment model developed by Gestion et al.27 Indeed, they showed that G. fossarum takes up and eliminates Cd rather quickly, with the intestines and the caeca accumulating and depurating the most compared to the cephalon and the remaining tissues. Gestin et al. also proved that a four-compartment model better describes the Cd internal contamination route than the single one-compartment model for each organ. Furthermore, they finally highlighted that the most parsimonious multi-compartments model corresponds to the solid black arrows in Figure 3.
Such a situation corresponds to a nested model within the four-compartment ODE system as given by Equations (24a) and (24b). This model thus comprises only 12 parameters whose values are listed in Table 3. Figure 4 shows the simulated kinetics within the four organs. Our four curves exactly superimpose to the four median curves provided in Figure 3 by Gestin et al. Benefiting from this exact match between our exact generic solution and what was numerically integrated by these authors before the implementation of their inference process ultimately strengthens both approaches: the numerical integration (a basic Euler integration scheme with a time-step equal to 1/10 day); and the curve plotting from the exact solution. Nevertheless, to infer parameter values from observed data, there is absolutely no doubt that the exact solution will provide much better computational performance for implementing the Monte Carlo Markov Chain simulations needed to use Bayesian inference. Readers who would like to convince themselves of this added value of our generic solving can refer to our dedicated R-package named ‘rPBK,’ officially available from the official R CRAN website (https://CRAN.R-project.org/package=rPBK ) together with a very recent related paper.47 Thanks to the R-package ’rPBK’, we already experienced that using this exact generic solution divided at least by 100 the computation time of the whole process. Our first inference implementation required numerically integrating the original ODE system and running the MCMC algorithm to fit the numerical solution to the accumulation-depuration data sets corresponding to the different organs and tissues. The numerical integration step was highly computationally demanding. Avoiding this numerical step much faster now delivers relevant and precise parameter estimates.
Case study with five compartments
Zhu and colleagues45 studied the effect of six different antibiotics on the sea cucumber Apostichopus japonicus: sulfadiazine, trimethoprim, enrofloxacin, ofloxacin, clarithromycin, and azithromycin. All compartments (blood or organs) are internally related to the coelomic fluid. Except for the digestive tract, all compartments are externally related to seawater (Fig. 5). Based on the original paper,45 biological parameter values were extracted to calculate the model’s coefficients used for the simulations. All these parameters are provided in Table 4.
Table 4Parameter values for the five-compartment physiologically-based kinetic model on sea cucumbers, after recalculation from the initial parameters as given by45 in their supplementary information
Parameter | Sulfadiazine | Trimethoprim | Enrofloxacin | Ofloxacin | Clarithromycin | Azithromycin |
---|
ku,1 | 68.73 | 68.73 | 68.73 | 68.73 | 68.73 | 68.73 |
ke,1 | 68.73 | 5.61 | 4.01 | 4.89 | 5.35 | 5.64 |
ku,2 | 0.53 | 34.97 | 21.10 | 36.77 | 24.10 | 24.10 |
ke,12 | 0.06 | 285.76 | 234.84 | 229.46 | 177.87 | 209.72 |
ku,3 | 2.86 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
ke,3 | 0.45 | 68.73 | 68.73 | 68.73 | 68.73 | 68.73 |
ku,4 | 19.51 | 0.11 | 0.07 | 0.13 | 0.07 | 0.04 |
ke,4 | 4.11 | 0.51 | 0.59 | 0.88 | 0.22 | 0.14 |
ku,5 | 0.00 | 3.88 | 11.91 | 8.44 | 1.98 | 1.55 |
ke,t | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
k1,2 | 0.06 | 0.08 | 0.09 | 0.11 | 0.07 | 0.04 |
k2,1 | 0.50 | 0.40 | 0.92 | 0.70 | 0.26 | 0.20 |
k1,3 | 0.43 | 3.27 | 10.75 | 9.29 | 2.20 | 1.93 |
k3,1 | 2.70 | 0.40 | 1.16 | 1.06 | 0.48 | 0.25 |
k1,4 | 4.11 | 4.00 | 4.94 | 4.17 | 5.52 | 5.50 |
k4,1 | 19.51 | 27.22 | 32.72 | 29.37 | 28.15 | 35.42 |
k1,5 | 0.48 | 241.35 | 211.96 | 252.56 | 198.06 | 261.31 |
k5,1 | 2.46 | 29.69 | 29.03 | 26.96 | 25.34 | 25.65 |
Below are the model equations written within our new generic mathematical formalism, with the following correspondence between index i and the different compartments: coleomic fluid (CF, i = 1); the body wall (BW, i = 2), the mouth (MH, i = 3), the respiratory tree (RT, i = 4) and the digestive tract (DT, i = 5).
{dCA(t)dt=Ucx+ECA(t) when 0≤t≤tc (accumulation phase) (a)dCD(t)dt=ECD(t) when t>tc (depuration phase) (b)
with vectors and matrices defined as follows:CA(t)=CD(t)=(c1(t)c2(t)c3(t)c4(t)c5(t)) U=(ku,1ku,2ku,3ku,4ku,5)E=(e1,1k1,2k1,3k1,4k1,5k2,1e2,2k2,3k2,4k2,5k3,1k3,2e3,3k3,4k3,5k4,1k4,2k4,3e4,4k4,5k5,1k5,2k5,3e5,4e5,5)
and diagonal elements of matrix E defined by:{e1,1=−ke,1−(k2,1+k3,1+k4,1+k5,1) (a)e2,2=−ke,2−k1,2 (b)e3,3=−ke,3−k1,3 (c)e4,4=−ke,4−k1,4 (d)e5,5=−k1,5 (e)
The exact solution of the matrix ODE system (30) can be directly deduced from Equations (17) and (21):
{CA(t)=(etE−I)E−1Ucx when 0≤t≤tc (accumulation phase) (a)CD(t)=(etE−e(t−tc)E)E−1Ucx when t>tc (depuration phase) (b)
The relationships between the initial and the recalculated parameters are the following, with the index SW referring to seawater:
ku,i=Di,SWVi and ke,i=Di,SWViPi,SW ∀i∈CF,BW,MH,RT,DTkCF,i=DCF,iVCF and ki,CF=DCF,iVCFPi,CF ∀i∈BW,MH,RT,DT
This leads to the first ODE of the five-compartment system for the sea cucumber:
dCCF(t)dt=ku,CFcw−(ke,CF+∑i≠CFki,CF) CCF(t)+∑i≠CF(kCF,iCi(t)) ∀i∈BW,MH,RT,DT
with cw = 10 μg·L−1 the concentration in the seawater to which they are exposed.kCF,i=DCF,iViPi,CF and ki,CF=DCF,iVj ∀i∈BW,MH,RT,DT
Finally, we get the four complementary equations of the system for the sea cucumber:
dCi(t)dt=ku,icw−(ke,i+kCF,i)Ci(t)+ki,CFCCF(t)
Given parameter values in Table 4, we performed simulations over time for the six antibiotics and the four internal organs. As shown in Figure 6, we reproduced the same median curves as those of 45 in Figure S10 of their supplementary information.
Case study with six compartments
Here is a final illustration of the usefulness of our generic solution to simulate any PBK model. We identically reproduced all the simulations provided by46 concerning exposure of zebrafish (Danio rerio) to arsenic (As). To this end, Zhang et al.46 proposed a six-compartments model with five compartments corresponding to organs: gills (i = 2), intestine (i = 3), liver (i = 4), head (i = 5) and carcass (i = 6). The sixth compartment corresponds to blood (i = 1). The five organs were connected to blood, while the gills and intestines were also connected to the external medium (contaminated water). These assumptions are translated in Figure 7, together with the different parameters used for the corresponding PBK model (Table 5).
Table 5Parameter estimates for arsenic distribution in zebrafish (from46)
Connection | ℳℒ | Value | Unit | Connection | ℳℒ | Value | Unit |
---|
Water to gills | ku,2 | 5.28 10−5 | L.d−1 | Gill to water | ke,2 | 0.152 | d−1 |
Water to intestine | ku,2 | 1.52 10−4 | L.d−1 | Intestine to water | ke,2 | 0.672 | d−1 |
Blood to gills | k2,1 | 76.0 | d−1 | Gill to blood | k1,2 | 58.2 | d−1 |
Blood to intestine | k3,1 | 27.8 | d−1 | Intestine to blood | k1,3 | 3.94 | d−1 |
Blood to liver | k4,1 | 38.5 | d−1 | Liver to blood | k1,4 | 23.6 | d−1 |
Blood to head | k5,1 | 97.5 | d−1 | head to blood | k1,5 | 8.48 | d−1 |
Blood to carcass | k6,1 | 7.20 | d−1 | Carcass to blood | k1,6 | 0.317 | d−1 |
From the generic matrix form of a PBK model we present in this paper, the model writes as follows:
Cwhole(t)=1∑i=16wi¯(∑i=16wi¯ci(t))
with vectors and matrices defined as follows:C(t)=(c1(t)c2(t)c3(t)c4(t)c5(t)c6(t))T
where ci(t), ∀i = 1,6, are the variables corresponding to internal concentrations to be simulated. Variables ci(t) are equal to wiCi(t), ∀i = 1,6, with wi the mean wet weight of blood volume (when i = 1) or fish organs (∀i = 2,6). Variables Ci(t) are the measured concentrations of As in fish organs (∀i = 2,6, expressed in μg·L−1) at time t (in days). Variable cx is the exposure concentration of As in water (expressed in μg·L−1), assumed to be constant over time.From concentrations within organs, the concentration for the whole organism can be deduced as follows:
Cwhole(t)=1∑i=16wi¯(∑i=16wi¯ci(t))
According to our mathematical formalism, uptake parameters from water are included in the following vector:
U=(0ku2ku3000)T
while elimination parameters towards the water, as well as parameters corresponding to organ-organ connections, are gathered together within the following matrix:E=(−∑j=26kj,1k1,2k1,3k1,4k1,5k1,6k2,1−ke,2−k1,20000k3,10−ke,3−k1,3000k4,100−k1,400k5,1000−k1,50k6,10000−k1,6)
The matrix ODE system (36) finally leads to the following exact solution:
C(t)=(etE−I)E−1Ucx
where cx is the exposure concentration in water.The above matrix solution (41) can then be developed in order to retrieve the six-compartment PBK model as constructed by Zhang et al.46 Denoting the exposure concentration in water (variable cx in our modeling) by Cwater, we ultimately get:
{w1¯dC1(t)dt=∑j=27k1,jCj(t)wi¯−(∑j=27kj,1)C1(t)w1¯ (a)w2¯dC2(t)dt=ku,2Cwater+k2,1C1(t)w1¯−(ke,2+k1,2)C2(t)w2¯ (b)w3¯dC3(t)dt=ku,3Cwater+k3,1C1(t)w1¯−(ke,3+k1,3)C3(t)w3¯ (c)w4¯dC4(t)dt=k4,1C1(t)w1¯−k1,4C4(t)w4¯ (d)w5¯dC5(t)dt=k5,1C1(t)w1¯−k1,5C5(t)w5¯ (e)w6¯dC6(t)dt=k6,1C1(t)w1¯−k1,6C6(t)w6¯ (f)
Zhang et al. measured internal concentrations over time in each organ and blood during both accumulation and depuration phases. This data allowed them to estimate all their parameters. Using a Bayesian inference framework also provided them with quantifying the uncertainty of these parameters. Then they deliver parameter estimates as maximum likelihood (ℳℒ) values associated with a 95% credible interval, as partly reported in Table 5.
Based on parameter ℳℒ values in Table 5, using the fresh weight of the different organs as provided in Table S2 from Zhang et al.,46 we performed simulations of the internal concentrations within each of the five organs as well as in blood (variables ci(t), ∀i = 1,6). As shown in Figure 8, our generic solved PBK model (see Equation (41)) again exactly reproduce the median curves provided by Zhang et al. in Figure 1 (solid blue lines). Such an exact match between our curves and the authors’ ones again provides a complete check of the mathematical writing of our generic solution, together with the fact that it produces identical median curves. These results again reinforce the possibility of using this exact generic solution for the further implementation of inference processes with the guarantee of obtaining the parameter estimates much faster, avoiding the time-consuming step of the numerical integration of the ODE system.