Solution and implementation of distributed lifespan models
Gilbert Koch (1) and Johannes Schropp (2)
(1) The State University of New York at Buffalo, Pharmaceutical Sciences, Buffalo NY 14214, USA (2) Department of Mathematics and Statistics, University of Konstanz, Germany
Objectives: A realistic assumption for a (cell) population is that every individual has its own and unique lifespan. Krzyzanski, Woo and Jusko [1] developed PKPD models where these lifespans are described by a continuous distribution, e.g. Lognormal or Weibull. In these models, the rate of change of the population is formulated based on the mathematical convolution operator. Unfortunately, standard PKPD software is not able to handle this convolution operator. Even in MATLAB the implementation is complex. The objective of this work is to derive and implement a new and handy solution representation of such distributed lifespan models (DLSM).
Methods: We implemented the solution representation of DLSMs, given in an integral form, by the Riemann sum in NONMEM and ADAPT. In MATLAB we present several approaches to solve this integral form and perform a run-time comparison.
Results: We calculated the solution representation of DLSMs. In the rate of change formulation these models use the convolution operator. It turns out that this operator vanishes in the solution representation. Instead the cumulative distribution function (CDF) of the distribution appears. Note that in case of the realistic Weibull distribution for lifespans, the CDF is an explicitly known function.
Conclusions: The solution representation of a DLSM does not contain the convolution operator and therefore could be easily implemented in standard PKPD software, e.g. in NONMEM or ADAPT.
References:
[1] Krzyzanski W, Woo S, Jusko WJ. Pharmacodynamic models for agents that alter production of natural cells with various distributions of lifespans. J Pharmacokin Pharmacodyn (2006) 33(2): 125-66.
