Section: New Results
General models for drug concentration in multidosing administration
Participants : Lisandro Fermin, Jacques LÃ©vy VÃ©hel.
In collaboration with P.E Lévy Véhel (University of NiceSophiaAntipolis and Banque Postale).
In the past two years, we have developed models for investigating the probability distribution of drug concentration in the case of noncompliance. 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 multiintravenous 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 multioral model, and deal with the complications brought by the firstorder 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 nonhomogeneous Poisson process whose jump rate is controlled by mean of a Markov chain. In this sense our PDMP model is a generalization to the continuosmodels studied in [29] . In the following we detail our PDM model and the results obtained in the multiIV 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 ${\left({J}_{n}\right)}_{n\in \mathbf{N}}$ 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 ${\left({T}_{n}\right)}_{n\in \mathbf{N}}$ the sequence of the random time doses and ${\left({S}_{n}\right)}_{n\in \mathbf{N}}$ 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 ${\left({J}_{n}\right)}_{n\in \mathbf{N}}$ considering the following assumptions:

The patient takes a dose ${D}_{{J}_{n}}\in \{{D}_{i},\phantom{\rule{0.277778em}{0ex}}i\in K\}$ 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 ${\lambda}_{{J}_{n}}\in \{{\lambda}_{i},\phantom{\rule{0.277778em}{0ex}}i\in K\}$, 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 ${\left({\nu}_{t}\right)}_{t\in \mathbf{R}}$ by ${\nu}_{t}={\sum}_{n\ge 0}{J}_{n}1\phantom{\rule{0.166667em}{0ex}}{\mathrm{l}}_{[{T}_{n},{T}_{n+1}[}\left(t\right)$. We denote by ${\left({C}_{t}\right)}_{t\in \mathbf{R}}$ the drug concentration stochastic process which take values on ${\mathbf{R}}_{+}^{*}=]0,\infty [$, 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 ${\left({C}_{t}\right)}_{t\in {\mathbf{R}}_{+}}$ with values in ${\mathbf{R}}_{+}^{*}$ starting from a fixed point $x$ is given by
${C}_{t}=x{e}^{{k}_{e}t}+\sum _{i\ge 1}{d}_{{J}_{i}}{e}^{{k}_{e}(t{T}_{i})}1\phantom{\rule{0.166667em}{0ex}}{\mathrm{l}}_{(t\ge {T}_{i})}.$  (26) 
The process ${({C}_{t},{\nu}_{t})}_{t\in {\mathbf{R}}_{+}}$ is a PDMP. From [49] , we have that the infinitesimal generator $\mathcal{U}$ of ${({C}_{t},{\nu}_{t})}_{t\in {\mathbf{R}}_{+}}$ is given by
$\mathcal{U}f(x,i)={k}_{e}x\frac{d}{dx}f(x,i)+{\lambda}_{i}\sum _{j\in K}{q}_{ij}\left(f(x+{d}_{j},j)f(x,i)\right),$  (27) 
with $(x,i)\in E={\mathbf{R}}_{+}^{*}\times K$ and $f\in \mathbb{D}\left(\mathcal{U}\right)$ 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
$\left\{\begin{array}{c}{\displaystyle \phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\frac{\partial {\varphi}_{\theta}}{\partial t}(t,x,i)\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}{k}_{e}x\frac{\partial {\varphi}_{\theta}}{\partial x}(t,x,i)+{\lambda}_{i}\phantom{\rule{0.166667em}{0ex}}\sum _{j\in K}\phantom{\rule{0.166667em}{0ex}}{q}_{ij}\left({e}^{\text{i}\theta {d}_{j}{e}^{{k}_{e}t}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}{\varphi}_{\theta}(t,x,j)\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}{\varphi}_{\theta}(t,x,i)\right),}\hfill \\ {\varphi}_{\theta}(0,x,i)={e}^{\text{i}\theta x}.\hfill \end{array}\right.$  (28) 
Variability of the concentration
From (28 ) we have that the expectation $m(t,x,i)={\mathbb{E}}_{(x,i)}\left[{C}_{t}\right]$ of ${C}_{t}$, given the starting point $(x,i)$, is given by
$m(t,x,i)=x{e}^{{k}_{e}t}+\sum _{\nu ,j\in K}{\lambda}_{\nu}{q}_{\nu j}{d}_{j}{\int}_{0}^{t}{e}^{{k}_{e}(ts)}{P}_{i\nu}\left(s\right)ds,$  (29) 
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
$\begin{array}{cc}\hfill Var(t,i)& =\sum _{\nu ,j\in K}{\lambda}_{\nu}{q}_{\nu j}{d}_{j}^{2}{\int}_{0}^{t}{e}^{2{k}_{e}(ts)}{P}_{i\nu}\left(s\right)ds{\left(\sum _{\nu ,j\in K}{\lambda}_{\nu}{q}_{\nu j}{d}_{j}{\int}_{0}^{t}{e}^{{k}_{e}(ts)}{P}_{i\nu}\left(s\right)ds\right)}^{2}\hfill \\ & +2\sum _{\nu ,j\in K}\sum _{{\nu}^{\text{'}},{j}^{\text{'}}\in K}{\lambda}_{\nu}{q}_{\nu j}{d}_{j}{\lambda}_{{\nu}^{\text{'}}}{q}_{{\nu}^{\text{'}}{j}^{\text{'}}}{d}_{{j}^{\text{'}}}{\int}_{0}^{t}{\int}_{0}^{ts}{e}^{{k}_{e}(ts)}{P}_{i\nu}\left(s\right){e}^{{k}_{e}(ts\tau )}{P}_{j{\nu}^{\text{'}}}\left(\tau \right)d\tau ds.\hfill \end{array}$  (30) 
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
$\left\varphi \left(\theta \right)\right\sim {K\left\theta \right}^{{\mu}_{max}},\phantom{\rule{2.em}{0ex}}\theta \to \infty ,$  (31) 
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.