Model and problem definition

The mathematical definition of the problem is briefly described here.

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Let us consider an ODE model for a population of $n$ subjects. For subject $i$, with $i=1 \dots n$, this can be written:

\[ \frac{d X^i(t)}{dt}=f(X^i(t),\xi^i(t)) \]


\[ X^i(0)=h(\xi^i(0)) \]

where $\xi^i(t) = (\xi_1^i(t), ..., \xi_p^i(t))$ is a vector of $p$ individual parameters which appear naturally in the ODE system and have a biological interpretation; $X^i(t) = (X_{1}^i(t), \dots , X_{K}^i(t))$ is the vector of the K state variables (or compartments). We let $X(t, \xi^i(t)) = X^i(t)$ to underline that $\xi_i(t)$ completely determines the trajectories $X^i(t)$. We assume that $f$ and $h$ are possibly non-linear functions, twice differentiable with respect to $\xi^i(t)$.

Reparametrization of the system allows us to take constraints into account: we introduce one-to-one functions $\psi_l(.),$ $l=1 \dots p$ and defined transformed parameters $\tilde{\xi}_l^i(t)=\psi_l(\xi_l^i(t))$. For instance, biological parameters such as rates can be parametrized using a logarithmic transformation to ensure positivity, or parameters between 0 and 1 (such as probabilities or bioavailability) can be parametrized using a logistic transformation. A mixed effect model for the $\tilde{\xi}_l^i(t)$ allows introducing covariates and taking into account the between-subject variations. In this approach, random effects give each subject a different value for a subset $q \leq p$ of the biological parameters:

\[ \tilde{\xi_{l}}^i(t) = \phi_{l}+ \beta_{l} z_{l}^{i}(t) + u^i , \qquad u^i \sim \mathcal{N}\left(0,\sigma^2_l\right). \]

where $\phi_{l}$ is the intercept and $z^i _{l}$ is a vector of $n_e$ (possibly time-dependent) explanatory variables associated with the fixed effects of the $l^{th}$ biological parameter; $\beta_{l}$ is a vector of regression coefficients; $u^i$ is the individual vector of random effects with $\sigma^2_l=0$ for $l>q$.

In practice, we do not observe $X^i(t)$ directly, but we have discrete-time observations $Y^i(t_{ij})$ of some functions of $X^i(t)$. We assume that there are known link functions $g_m(.),$ $m=1,\dots , M$, leading to an additive measurement error model. For $i=1, \dots ,n$, $m=1,\dots , M$ and $j=1, \dots T_i$ (the number of observation time for subject $i$), we observe:

\[ Y^i_{m}(t_{ij})=g_m(X^i(t_{ij}))+\epsilon_{ijm}, \qquad \epsilon_{ijm}\sim \mathcal{N}\left(0,\sigma^2_m\right). \]

This observation scheme can be complicated by left-censoring due to detection limits We denote by $\theta$ the parameter vector to estimate of length $N=p+n_e+q+M$:

\[ \theta=\left( \left(\phi_{l}\right)_{l=1 \dots p}, \left(\beta_{l}\right)_{l=1 \dots n_e}, \left(\sigma_l\right)_{l=1 \dots q}, \left(\sigma_m\right)_{m=1 \dots M} \right). \nonumber \]