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 (J n ) n𝐍 an irreducible Markov chain taking values in the state space K={1,...,k} with initial law α i =(J 0 =i) for all iK and transition probability matrix Q=(q ij ) i,jK . We denote by (T n ) n𝐍 the sequence of the random time doses and (S n ) n𝐍 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 ) n𝐍 considering the following assumptions:

  • The patient takes a dose D J n {D i ,iK} 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 {λ i ,iK}, 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 =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 ) t𝐑 by ν t = n0 J n 1l [T n ,T n+1 [ (t). We denote by (C t ) t𝐑 the drug concentration stochastic process which take values on 𝐑 + * =]0,[, we suppose that (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)=xexp{-k e t}. Thus, the sample path of the stochastic process (C t ) t𝐑 + with values in 𝐑 + * starting from a fixed point x is given by

C t =xe -k e t + i1 d J i e -k e (t-T i ) 1l (tT i ) .(26)

The process (C t ,ν t ) t𝐑 + is a PDMP. From [49] , we have that the infinitesimal generator 𝒰 of (C t ,ν t ) t𝐑 + is given by

𝒰f(x,i)=-k e xd dxf(x,i)+λ i jK q ij f(x+d j ,j)-f(x,i),(27)

with (x,i)E=𝐑 + * ×K and f𝔻(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

ϕ θ t(t,x,i)=-k e xϕ θ x(t,x,i)+λ i jK q ij e iθd j e -k e t ϕ θ (t,x,j)-ϕ θ (t,x,i),ϕ θ (0,x,i)=e iθx .(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)=xe -k e t + ν,jK λ ν q νj d j 0 t e -k e (t-s) P iν (s)ds,(29)

where P iν (t)=(ν t =ν|ν 0 =i). The variance Var(t,i) of C t , given the initial state i, is given by

Var(t,i)= ν,jK λ ν q νj d j 2 0 t e -2k e (t-s) P iν (s)ds- ν,jK λ ν q νj d j 0 t e -k e (t-s) P iν (s)ds 2 +2 ν,jK ν ' ,j ' K λ ν q νj d j λ ν ' q ν ' j ' d j ' 0 t 0 t-s e -k e (t-s) P iν (s)e -k e (t-s-τ) P jν ' (τ)dτds.(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 θd dθϕ(θ,i)+ jK λ j q ji e iθd 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)= jK 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= iK m i the mean of C and Var its variance. Then,

m=1 k e i,jK π i λ i q ij d j .m i =1 k e jK π j λ j q ji d i +1 k e jK λ j q ji m j -λ i m i .Var=1 2k e i,jK π i λ i q ij d j 2 +1 k e i,jK λ i q ij d j (m i -π i m).

Regularity of the limit concentration

The characteristic function ϕ satisfies

|ϕ(θ)|K|θ| -μ max ,θ,(31)

where K is a positive constant and μ max =max {iK} 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.