Section: New Results

General models for drug concentration in multi-dosing administration

Participants : Lisandro Fermin, Jacques Lévy Véhel.

In collaboration with P.E Lévy Véhel (University of Nice-Sophia-Antipolis and Banque Postale).

In the past two years, we have developed models for investigating the probability distribution of drug concentration in the case of non-compliance. We have focused on two aspects of practical relevance: the variability of the concentration and the regularity of its probability distribution. In a first article [29] , in a series of three, is considered the case of multi-intravenous dosing using the simplest possible law to model random drug intake, i.e. a homogeneous Poisson distribution. In a second article [13] , we consider the more realistic multi-oral model, and deal with the complications brought by the first-order kinetics, which are essentially technical. Finally, in [12] , we put ourselves in a powerful mathematical frame, known as Piecewise Deterministic Markov process (PDMP), that allows us to deal with general drug intake schedules, going beyond the homogeneous Poisson case. We use a PDMP to model the drug concentration in the case of multiple intravenous doses. In this particular model, we consider that the doses administration regimen is modeled by a non-homogeneous Poisson process whose jump rate is controlled by mean of a Markov chain. In this sense our PDMP model is a generalization to the continuos-models studied in [29] . In the following we detail our PDM model and the results obtained in the multi-IV case, see [12] .

The model setting

Inspired by the PDMP model given in [47] , [48] , we consider a drug dosing stochastic regimen defined as follows.

Let us consider ${\left(J_n\right)}_{n\in 𝐍}$ an irreducible Markov chain taking values in the state space $K=\{1,...,k\}$ with initial law ${\alpha }_i=\mathbb{P}(J_0=i)$ for all $i\in K$ and transition probability matrix $Q={\left(q_{ij}\right)}_{i,j\in K}$. We denote by ${\left(T_n\right)}_{n\in 𝐍}$ the sequence of the random time doses and ${\left(S_n\right)}_{n\in 𝐍}$ the time dose intervals; i.e. $S_n=T_{n+1}-T_n$. We consider that the doses administration regimen is modeled by mean of the Markov process ${\left(J_n\right)}_{n\in 𝐍}$ considering the following assumptions:

• The patient takes a dose $D_{J_n}\in \{D_i,i\in K\}$ at the time $T_n$, where the doses $D_i$ are all different and different of zero.

• The time dose $S_n$ is a random variable with exponential law of parameter ${\lambda }_{J_n}\in \{{\lambda }_i,i\in K\}$, where the jump rate ${\lambda }_i$ of state $i$ is a positive constant.

We consider that these doses translate into immediate increases of the concentration by the value $d_i=\frac{D_i}{V_d}$ if $J_n=i$, where $V_d$ is the apparent volume of distribution . After that, the effect of the dose taken at time $T_n$ decreases exponentially fast with an exponential rate of elimination $k_e$.

We define ${\left({\nu }_t\right)}_{t\in 𝐑}$ by ${\nu }_t={\sum }_{n\ge 0}{J}_{n}1{\mathrm{l}}_{[T_n,T_{n+1}[}\left(t\right)$. We denote by ${\left(C_t\right)}_{t\in 𝐑}$ the drug concentration stochastic process which take values on ${𝐑}_{+}^{*}=]0,\infty [$, we suppose that $\mathbb{P}(C_0=x)=1$. Between the jumps, the dynamical evolution of the continuous time process $\left(C_t\right)$ is modeled by the flow $\phi (t,x)=xexp\{-k_{e}t\}$. Thus, the sample path of the stochastic process ${\left(C_t\right)}_{t\in {𝐑}_{+}}$ with values in ${𝐑}_{+}^{*}$ starting from a fixed point $x$ is given by

The process ${(C_t,{\nu }_t)}_{t\in {𝐑}_{+}}$ is a PDMP. From [49] , we have that the infinitesimal generator $𝒰$ of ${(C_t,{\nu }_t)}_{t\in {𝐑}_{+}}$ is given by

with $(x,i)\in E={𝐑}_{+}^{*}\times K$ and $f\in 𝔻\left(𝒰\right)$ the set of measurable and differentiable on the first argument.

The characteristic function of the concentration

The characteristic function ${\varphi }_{\theta }(t,x,i)$ of $C_t$, given the starting point $(x,i)$, is the unique solution of the following system

Variability of the concentration

From (28 ) we have that the expectation $m(t,x,i)={𝔼}_{(x,i)}\left[C_t\right]$ of $C_t$, given the starting point $(x,i)$, is given by

where $P_{i\nu }\left(t\right)=\mathbb{P}({\nu }_t=\nu |{\nu }_0=i)$. The variance $Var(t,i)$ of $C_t$, given the initial state $i$, is given by

The distribution of limit concentration

The characteristic function $\varphi (\theta ,i)$ of the limit concentration $C$, given the starting state $i$, satisfies

Thus, the random variables $C\left(t\right)$ converge in distribution, when $t$ tends to infinity, to a well defined random variable $C$ whose characteristic function is

Variability of the limit concentration

We denote by $m_i$ the mean of the limit concentration $C$ in the state $\nu =i$ and $m={\sum }_{i\in K}{m}_i$ the mean of $C$ and $Var$ its variance. Then,

Regularity of the limit concentration

The characteristic function $\varphi$ satisfies

where $K$ is a positive constant and ${\mu }_{max}={max}_{\{i\in K\}}\frac{{\lambda }_i}{k_e}$.

This result will allow us to describe in detail aspects of the limit distribution that are important for assessing the efficacy of therapy.