Population pharmacokinetics and stochastic differential equations: models and methods

Conclusion

Population pharmacokinetics and stochastic differential equations: models and methods

Maud Delattre (1,2), Marc Lavielle (1,2)

(1) University of Paris Sud, (2) Inria Saclay ˆIle-de-France

”Jeunes probabilistes et Statisticiens”, Marseille, 2012

1 Introduction

2 Model

3 Methodology

4 Conclusion

Conclusion

Introduction

Pharmacokinetics (PK)

study of the evolution of a drug in the body human body represented as a set of compartments transfers between compartments

statistical models based on ODEs Example:

Input ka ke

V

Our objectives:

to present new SDE (Stochastic Differential Equations)-based PK models in a population approach

to develop some specific estimation procedure for the parameters in these models

Conclusion

Fixed-effects model development

Example: bolus

Dose administered by rapid injection into a single compartment

Input

k V

Purpose:

model forC(t) (concentration of drug in the compartment at timet)

a) deterministic model Ordinary differential model

dC(t) =−k C(t)dt

Figure:Evolution of the concentration of a bolus represented by an ODE.

Conclusion

Fixed-effects model development: structural model

b) diffusion model

some perturbation in the evolution of the drug

diffusion model

dC(t) =−k C(t)dt+γ(C(t))dW(t) Example

dC(t) =−k C(t)dt+γC(t)dW(t)

b) diffusion model

dC(t) =−k C(t)dt +γ(C(t))dW(t)

This model is not realistic:

monotony, sign . . .

Conclusion

Fixed-effects model development: structural model

b) diffusion model

More realistic assumption:k is randomly perturbed and driven by a SDE dk(t) = b(k(t)) dt +γ(k(t))dW(t) dC(t) = −k(t)C(t) dt

The kinetics is observed at discrete time points: (y1, . . . ,yn)

dk(t) =b(k(t))dt + γ(k(t))dW(t) dC(t) =−k(t)C(t)dt

yj= logC(tj) +j, j ∼

i.i.d.N(0, σ²)

Conclusion

Mixed-effects model

Nkinetics

yi= (yi1, . . . ,yi,n_i): observations for individuali

Multiple sources of variability:

intra-individual variability kis a diffusion process residual errors between-subjects variability

the parameters of the model are random variables

Mixed diffusion model: example

dki(t) =α(ki(t), φi)dt +γ(ki(t), φi)dWi(t) dCi(t) =−k_i(t)Ci(t)dt

yij = log(Ci(tij)) +ij

ij ∼

i.i.d. N(0, σ²(φi))

φi ∼

i.i.d. N(φpop,Ω)

i = 1, . . . ,N, j = 1, . . . ,ni

Conclusion

General mixed-effects model

Mixed diffusion model: general form

dXi(t) =b(Xi(t), φi)dt +γ(Xi(t), φi)dWi(t), yij =g(Xi(tij), φi) +ij

ij ∼

i.i.d. N(0, σ²(φi))

φi ∼

i.i.d. N(φpop,Ω)

The objectives are

i)to estimateθ= (φpop,Ω) from the observationsy= (y1, . . . ,yN);

ii)to estimate the individual parametersφi,i= 1, . . . ,N;

iii)to recover the individual trajectoriesXi(t),i= 1, . . . ,N.

Estimation ofθ(MLE)

θˆ=argmax

θ

p(y;θ)

→SAEM combined with the extended Kalman filter Estimation ofφi (MAP)

φˆi=argmax

φ_i

p(φi|y_i; ˆθ) Estimation ofXi (MAP)

Xˆ(tij) =argmax

X(t_ij)

p(X(tij)|y_i; ˆφi)

→Kalman smoother

Conclusion

Likelihood

Mixed diffusion model

dXi(t) =b(Xi(t), φi)dt +γ(Xi(t), φi)dWi(t), yij =g(Xi(tij), φi) +ij

ij ∼

i.i.d. N(0, σ²(φi))

φi ∼

i.i.d. N(φpop,Ω)

Thelikelihoodof the observations has a very complex expression

p(Y;θ) =

N

Y

i=1

Z

p(y_i, φ_i;θ)dφ_i=

N

Y

i=1

Z

p(y_i|φ_i)p(φ_i;θ)dφ_i

and

p(yi|φ_i) =p(yi1|φ_i)

n_i

Y

j=1

p(yij|y_i,j−1, . . . ,yi1, φi)

Iterationk of SAEM algorithm:

Simulation of the non observed data

Fori= 1, . . . ,n, simulateφ^(k)_i underp(φi|y_i;θk−1)

→Monte Carlo Markov chains methods

Stochastic approximation of the log-likelihood

Q_k(θ) =Qk−1(θ) +γ_k

" _n X

i=1

logp

yi, φ^(k)_i ;θ

−Qk−1(θ)

#

Maximization

θk=argmax

θ

Qk(θ)

Conclusion

SAEM algorithm for mixed effects SDE models

The key of the SAEM algorithm is a quick approximate computation of each individual complete likelihood:

p(y_i, φ_i;θ) = p(y_i|φ_i;θ)p(φ_i;θ)

= p(yi|φi)p(φi;θ)

p(φ_i;θ) is explicitly known: φ_i∼ N(φpop,Ω)

Theextended Kalman filterallows to approximate the conditional distribution of the observationp(yi|φi)with Gaussian distributions

dX_i(t) =b(X_i(t), φ_i)dt +γ(X_i(t), φ_i)dW_i(t), yij =g(Xi(tij), φi) +ij

dki(t) = (k_i^∗−ki(t))dt+ γp

ki(t)dWi(t), dQi(t) =−ki(t)Qi(t)dt,

yij= log (Qi(t)/Vi) + ij, ij ∼

i.i.d.N(0,σ²), φi = (logk_i^∗,logVi) ∼

i.i.d.N(φpop,Ω), φpop= (logk,logV).

Conclusion

Simulation study: estimation of the population parameters

dlogki(t) = (logk_i^∗−logki(t))dt+ γdWi(t), dQi(t) =−ki(t)Qi(t)dt,

yij= log (Qi(t)/Vi) + ij, ij ∼

i.i.d.N(0,σ²), φi = (logk_i^∗,logVi) ∼

i.i.d.N(φpop,Ω), φpop= (logk,logV).

dki(t) = (k_i^∗−ki(t))dt+ γ q

ki(t)dWi(t), dQi(t) =−ki(t)Qi(t)dt,

yij= log (Qi(t)/Vi) + ij, ij ∼

i.i.d.N(0, σ²),

φi= (logk_i^∗,logVi) ∼

i.i.d.N(φpop,Ω), φpop= (logk,logV).

Conclusion

Conclusion

we have proposed a new category of dynamical models for PK application in a population approach;

extension to more general dynamical systems and other application is straightforward

we have proposed new estimation procedure for these models good practical properties

Next steps:

simulation studies involving more complex models application to real data