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## Section: New Results

### General models for drug concentration in multi-dosing administration

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 $K=\left\{1,...,k\right\}$ with initial law ${\alpha }_{i}=ℙ\left({J}_{0}=i\right)$ 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 ${D}_{{J}_{n}}\in \left\{{D}_{i},\phantom{\rule{0.277778em}{0ex}}i\in K\right\}$ 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 ${\lambda }_{{J}_{n}}\in \left\{{\lambda }_{i},\phantom{\rule{0.277778em}{0ex}}i\in K\right\}$, 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 ${\nu }_{t}={\sum }_{n\ge 0}{J}_{n}1\phantom{\rule{-0.166667em}{0ex}}{\mathrm{l}}_{\left[{T}_{n},{T}_{n+1}\left[}\left(t\right)$. We denote by ${\left({C}_{t}\right)}_{t\in 𝐑}$ the drug concentration stochastic process which take values on ${𝐑}_{+}^{*}=\right]0,\infty \left[$, we suppose that $ℙ\left({C}_{0}=x\right)=1$. Between the jumps, the dynamical evolution of the continuous time process $\left({C}_{t}\right)$ is modeled by the flow $\phi \left(t,x\right)=xexp\left\{-{k}_{e}t\right\}$. 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

 ${C}_{t}=x{e}^{-{k}_{e}t}+\sum _{i\ge 1}{d}_{{J}_{i}}{e}^{-{k}_{e}\left(t-{T}_{i}\right)}1\phantom{\rule{-0.166667em}{0ex}}{\mathrm{l}}_{\left(t\ge {T}_{i}\right)}.$ (26)

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

 $𝒰f\left(x,i\right)=-{k}_{e}x\frac{d}{dx}f\left(x,i\right)+{\lambda }_{i}\sum _{j\in K}{q}_{ij}\left(f\left(x+{d}_{j},j\right)-f\left(x,i\right)\right),$ (27)

with $\left(x,i\right)\in E={𝐑}_{+}^{*}×K$ 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 ${\varphi }_{\theta }\left(t,x,i\right)$ of ${C}_{t}$, given the starting point $\left(x,i\right)$, is the unique solution of the following system

 $\left\{\begin{array}{c}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\frac{\partial {\varphi }_{\theta }}{\partial t}\left(t,x,i\right)\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}-{k}_{e}x\frac{\partial {\varphi }_{\theta }}{\partial x}\left(t,x,i\right)+{\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 }\left(t,x,j\right)\phantom{\rule{-0.166667em}{0ex}}-\phantom{\rule{-0.166667em}{0ex}}{\varphi }_{\theta }\left(t,x,i\right)\right),\hfill \\ {\varphi }_{\theta }\left(0,x,i\right)={e}^{\text{i}\theta x}.\hfill \end{array}\right\$ (28)

Variability of the concentration

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

 $m\left(t,x,i\right)=x{e}^{-{k}_{e}t}+\sum _{\nu ,j\in K}{\lambda }_{\nu }{q}_{\nu j}{d}_{j}{\int }_{0}^{t}{e}^{-{k}_{e}\left(t-s\right)}{P}_{i\nu }\left(s\right)ds,$ (29)

where ${P}_{i\nu }\left(t\right)=ℙ\left({\nu }_{t}=\nu |{\nu }_{0}=i\right)$. The variance $Var\left(t,i\right)$ of ${C}_{t}$, given the initial state $i$, is given by

 $\begin{array}{cc}\hfill Var\left(t,i\right)& =\sum _{\nu ,j\in K}{\lambda }_{\nu }{q}_{\nu j}{d}_{j}^{2}{\int }_{0}^{t}{e}^{-2{k}_{e}\left(t-s\right)}{P}_{i\nu }\left(s\right)ds-{\left(\sum _{\nu ,j\in K}{\lambda }_{\nu }{q}_{\nu j}{d}_{j}{\int }_{0}^{t}{e}^{-{k}_{e}\left(t-s\right)}{P}_{i\nu }\left(s\right)ds\right)}^{2}\hfill \\ & +2\sum _{\nu ,j\in K}\sum _{{\nu }^{\text{'}},{j}^{\text{'}}\in K}{\lambda }_{\nu }{q}_{\nu j}{d}_{j}{\lambda }_{{\nu }^{\text{'}}}{q}_{{\nu }^{\text{'}}{j}^{\text{'}}}{d}_{{j}^{\text{'}}}{\int }_{0}^{t}{\int }_{0}^{t-s}{e}^{-{k}_{e}\left(t-s\right)}{P}_{i\nu }\left(s\right){e}^{-{k}_{e}\left(t-s-\tau \right)}{P}_{j{\nu }^{\text{'}}}\left(\tau \right)d\tau ds.\hfill \end{array}$ (30)

The distribution of limit concentration

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

$-{k}_{e}\theta \frac{d}{d\theta }\varphi \left(\theta ,i\right)+\sum _{j\in K}{\lambda }_{j}{q}_{ji}{e}^{\text{i}\theta {d}_{i}}\varphi \left(\theta ,j\right)-{\lambda }_{i}\varphi \left(\theta ,i\right)=0.$

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

$\varphi \left(\theta \right)=\sum _{j\in K}\varphi \left(\theta ,j\right).$

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,

$\begin{array}{ccc}m\hfill & =\hfill & \frac{1}{{k}_{e}}\sum _{i,j\in K}{\pi }_{i}{\lambda }_{i}{q}_{ij}{d}_{j}.\hfill \\ & & \\ {m}_{i}\hfill & =\hfill & \frac{1}{{k}_{e}}\sum _{j\in K}{\pi }_{j}{\lambda }_{j}{q}_{ji}{d}_{i}+\frac{1}{{k}_{e}}\left(\sum _{j\in K}{\lambda }_{j}{q}_{ji}{m}_{j}-{\lambda }_{i}{m}_{i}\right).\hfill \\ & & \\ Var\hfill & =\hfill & \frac{1}{2{k}_{e}}\sum _{i,j\in K}{\pi }_{i}{\lambda }_{i}{q}_{ij}{d}_{j}^{2}+\frac{1}{{k}_{e}}\sum _{i,j\in K}{\lambda }_{i}{q}_{ij}{d}_{j}\left({m}_{i}-{\pi }_{i}m\right).\hfill \end{array}$

Regularity of the limit concentration

The characteristic function $\varphi$ satisfies

 $|\varphi \left(\theta \right)|\sim {K|\theta |}^{-{\mu }_{max}},\phantom{\rule{2.em}{0ex}}\theta \to \infty ,$ (31)

where $K$ is a positive constant and ${\mu }_{max}={max}_{\left\{i\in K\right\}}\frac{{\lambda }_{i}}{{k}_{e}}$.

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