Volterra series in pharmacokinetics and pharmacodynamics

Nonparametric black-box modeling has a long successful history of applications in pharmacokinetics (PK) (notably in deconvolution), but is rarely used in pharmacodynamics (PD). The main reason is associated with the fact that PK systems are often linear in respect to drug inputs, while the reverse is true for many PK/PD systems. In the PK/PD field existing non-parametric methods can deal with linear systems, but they cannot describe non-linear systems. Our purpose is to describe a novel implementation of a general nonparametric model which can represent non-linear systems, and in particular non-linear PK/PD systems. The model is based on a Volterra series, which is an integral series expansion of the response of a system in terms of its kernels and the inputs to the system. In PK we are familiar with the first term of the Volterra series: the convolution of the first kernel of the system (the so-called PK disposition function) with drug input rates. The main advantages of higher order Volterra representations is that they are general representations and can be used to describe and predict the response of an arbitrary (PK/PD) system without any prior knowledge on the structure of the system. The main problem of the representation is that in a non-parametric representation of the kernels the number of parameters to be estimated grows geometrically with the order of the kernel. We developed a method to estimate the kernels in a Volterra-series which overcomes this problem. The method (i) is fully non-parametric (the kernels are represented using multivariate splines), (ii) is maximum-likelihood based, (iii) is adaptive (the order of the series and the dimensionality of each kernel is selected by the method), and (iv) allows for non-equispaced observations (thus allowing a reduction of the number of parameters in the representation, and the analysis of, e.g., PK/PD observations). The method is based on an adaptation of Friedmans's Multivariate Adaptive Regression Spline method. Examples demonstrate the possible application of the approach to the analysis of different PK/PD systems.