Amy Racine-Poon
Pharmaceutical Division, CIBA-GEIGY AG, Basel, Switzerland CH-4002
Population modelling is widely used in Biometrical growth curve analysis, (see for example Berkey 1982), in pharmacokinetics and pharmacodynamic studies (see Sheiner and Beal 1980). The goals of population kinetics is to investigate the absorption, elimination and metabolism of a drug in the target patient population, and for the purpose of individual dose recommendation and to identify possible high risk subgroups. The population dynamic and kinetics studies is to related the observed efficacy or unwanted side effects of the individual patient to his observed plasma profiles for the purpose of optimal dosing for future patients. From a Bayesian perspective, both population kinetics and kinetics and dynamic models are variations from hierarchical modelling (Lindley and Smith 1972). Let Y denote the collection of the measured responses of the I patients (eg plasma concentrations and /or corresponding efficacy measurements), and let θ denotes the corresponding parameters of the I patients which defining the corresponding response pattern (eg plasma time profile, and / or the link function between kinetics and dynamic observations). Let φ be the hyper-parameter which defines the possible relationship between the individual parameters θ.
Population modelling therefore corresponds to the hierarchy of
(1) Individual measurements model
[yij I f(θi>tij)] j=l,..ni, i=l,..I. where
f a known parametric function, eg one compartment open model, θi the unknown parameter of the i-th individual.
(2) Between individual model
[θi I φ,zi],
where zi is the covariate of the i-th patient.
(3) Prior
[φ].
Here, we adopt the notation of Gelfand and Smith (1990), [ ] denote the density functions. The interest of population modelling are therefore inference of the population parameters φ or prediction of future patient’s parameter θ or profile y. Both aspects involve numerical integrations or some sort of approximations.
Numerical integrations are not always tractable. Approximations have not always been satisfactory, especially for sparse data sets. Diagnostics of the model assumptions have always been difficult. We illustrated how both the inference and the diagnostic aspect can be achieved by using Gibbs sampler (Wakefield et Al 1992). Two illustrative examples were used to demonstrate the outlier diagnostic aspect and model diagnostics. The first example was a dental measurement in children, in which the between children variation was modelled by a t density. Using the special property of t in addition to graphic tools, one can easily identify the outlying children. The second example of dynamic and kinetics studies in patients were used for the demonstration, we first illustrated the modelling aspect that is to link the observed concentration levels to the corresponding measure of efficacy using a effective compartment model, and then the computational and the validation aspects were also illustrated.
References:
[1]. Beal S. L. and Sheiner L. B. (1980), The NONMEM system, The American Statistician, 34,118-119.
[2]. Berkey, C.S. (1982), Bayesian approach for a nonlinear growth model, Biometrics, 38, 953-961.
[3]. Gelfand A E and Smith A F M (1990), Sampling-Based approach to calculating Marginal density, Journal of American Statistical Association, 85 , 398-409.
[4]. Lindley D and Smith. A F M (1972), Bayes estimates for the linear model (with discussion) J. R. Statistics Society Ser B. 34 1-41.
[5]. Wakefield J C , Smith A F M, Racine-Poon A and Gelfand A E (1992) Bayesian Analysis of Linear and Nonlinear Population Models using the Gibbs Sampler (submitted)
Reference: PAGE 1 (1992) Abstr 898 [www.page-meeting.org/?abstract=898]
Poster: oral presentation