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REGULARITY - 2011

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Section: New Results

General models for drug concentration in multi-dosing administration

Participants : Lisandro Fermin, Jacques LÃ©vy VÃ©hel.

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 ${(J_{n})}_{n \in 𝐍}$ an irreducible Markov chain taking values in the state space $K = {1, ..., k}$ with initial law $α_{i} = ℙ (J_{0} = i)$ for all $i \in K$ and transition probability matrix $Q = {(q_{i j})}_{i, j \in K}$ . We denote by ${(T_{n})}_{n \in 𝐍}$ the sequence of the random time doses and ${(S_{n})}_{n \in 𝐍}$ 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 ${(J_{n})}_{n \in 𝐍}$ 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 $λ_{J_{n}} \in {λ_{i}, i \in K}$ , where the jump rate $λ_{i}$ 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 ${(ν_{t})}_{t \in 𝐑}$ by $ν_{t} = \sum_{n \geq 0} J_{n} 1 l_{[T_{n}, T_{n + 1} [} (t)$ . We denote by ${(C_{t})}_{t \in 𝐑}$ the drug concentration stochastic process which take values on $𝐑_{+}^{*} =] 0, \infty [$ , we suppose that $ℙ (C_{0} = x) = 1$ . Between the jumps, the dynamical evolution of the continuous time process $(C_{t})$ is modeled by the flow $φ (t, x) = x exp {- k_{e} t}$ . Thus, the sample path of the stochastic process ${(C_{t})}_{t \in 𝐑_{+}}$ with values in $𝐑_{+}^{*}$ starting from a fixed point $x$ is given by

C_{t} = x e^{- k_{e} t} + \sum_{i \geq 1} d_{J_{i}} e^{- k_{e} (t - T_{i})} 1 l_{(t \geq T_{i})} .

(26)

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

𝒰 f (x, i) = - k_{e} x \frac{d}{d x} f (x, i) + λ_{i} \sum_{j \in K} q_{i j} (f (x + d_{j}, j) - f (x, i)),

(27)

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

The characteristic function of the concentration

The characteristic function $ϕ_{θ} (t, x, i)$ of $C_{t}$ , given the starting point $(x, i)$ , is the unique solution of the following system

\{\begin{matrix} \frac{\partial ϕ_{θ}}{\partial t} (t, x, i) = - k_{e} x \frac{\partial ϕ_{θ}}{\partial x} (t, x, i) + λ_{i} \sum_{j \in K} q_{i j} (e^{i θ d_{j} e^{- k_{e} t}} ϕ_{θ} (t, x, j) - ϕ_{θ} (t, x, i)), \\ ϕ_{θ} (0, x, i) = e^{i θ x} . \end{matrix}

(28)

Variability of the concentration

From (28 ) we have that the expectation $m (t, x, i) = 𝔼_{(x, i)} [C_{t}]$ of $C_{t}$ , given the starting point $(x, i)$ , is given by

m (t, x, i) = x e^{- k_{e} t} + \sum_{ν, j \in K} λ_{ν} q_{ν j} d_{j} \int_{0}^{t} e^{- k_{e} (t - s)} P_{i ν} (s) d s,

(29)

where $P_{i ν} (t) = ℙ (ν_{t} = ν | ν_{0} = i)$ . The variance $V a r (t, i)$ of $C_{t}$ , given the initial state $i$ , is given by

\begin{matrix} V a r (t, i) & = \sum_{ν, j \in K} λ_{ν} q_{ν j} d_{j}^{2} \int_{0}^{t} e^{- 2 k_{e} (t - s)} P_{i ν} (s) d s - {(\sum_{ν, j \in K} λ_{ν} q_{ν j} d_{j} \int_{0}^{t} e^{- k_{e} (t - s)} P_{i ν} (s) d s)}^{2} \\ + 2 \sum_{ν, j \in K} \sum_{ν^{'}, j^{'} \in K} λ_{ν} q_{ν j} d_{j} λ_{ν^{'}} q_{ν^{'} j^{'}} d_{j^{'}} \int_{0}^{t} \int_{0}^{t - s} e^{- k_{e} (t - s)} P_{i ν} (s) e^{- k_{e} (t - s - τ)} P_{j ν^{'}} (τ) d τ d s . \end{matrix}

(30)

The distribution of limit concentration

The characteristic function $ϕ (θ, i)$ of the limit concentration $C$ , given the starting state $i$ , satisfies

- k_{e} θ \frac{d}{d θ} ϕ (θ, i) + \sum_{j \in K} λ_{j} q_{j i} e^{i θ d_{i}} ϕ (θ, j) - λ_{i} ϕ (θ, i) = 0 .

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

ϕ (θ) = \sum_{j \in K} ϕ (θ, j) .

Variability of the limit concentration

We denote by $m_{i}$ the mean of the limit concentration $C$ in the state $ν = i$ and $m = \sum_{i \in K} m_{i}$ the mean of $C$ and $V a r$ its variance. Then,

\begin{matrix} m & = & \frac{1}{k_{e}} \sum_{i, j \in K} π_{i} λ_{i} q_{i j} d_{j} . \\ m_{i} & = & \frac{1}{k_{e}} \sum_{j \in K} π_{j} λ_{j} q_{j i} d_{i} + \frac{1}{k_{e}} (\sum_{j \in K} λ_{j} q_{j i} m_{j} - λ_{i} m_{i}) . \\ V a r & = & \frac{1}{2 k_{e}} \sum_{i, j \in K} π_{i} λ_{i} q_{i j} d_{j}^{2} + \frac{1}{k_{e}} \sum_{i, j \in K} λ_{i} q_{i j} d_{j} (m_{i} - π_{i} m) . \end{matrix}

Regularity of the limit concentration

The characteristic function $ϕ$ satisfies

| ϕ (θ) | \sim {K | θ |}^{- μ_{m a x}}, θ \to \infty,

(31)

where $K$ is a positive constant and $μ_{m a x} = {max}_{{i \in K}} \frac{λ_{i}}{k_{e}}$ .

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

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