Modeling of delayed phenomena in PKPD by delay differential equations of lifespan type
Gilbert Koch (1), Johannes Schropp (1)
(1) Department of Mathematics and Statistics, University at Konstanz, Germany
Objectives: Delayed phenomena are common in PKPD modeling. Traditionally, transit compartments (TC) based on ordinary differential equations (ODE) are applied to handle delays and further also used to describe populations, e.g. cells, see  or . We investigate the relationship between TCs and delay differential equations (DDE) of lifespan type, see . Our aims are to rewrite TCs by a lifespan approach in order to reduce the amount of physiological non-interpretable stages, to apply DDEs for populations and, in general, to handle delays with DDEs. We present two applications and theoretical results.
We applied DDEs of the form
for PKPD modeling. In contrast to ODEs, such models consists of an explicit delay parameter T>0 and uses information from the past in the term x(t-T). In DDEs of lifespan type the parameter T describes the mean lifespan of individuals in a population.
Results: Our main theoretical result is that the totality of all objects of TCs with arbitrary initial values converges to a DDE of lifespan type, see . As a consequence one can substitute TCs by DDEs and also vice versa. In a first application we rewrote the TC structure of a standard tumor growth model (see e.g.  or ) by DDEs. The resulting model has exactly two states, one for proliferating cells and one for dying tumor cells, see . The second application dealt with arthritis development where increased cytokine concentration drives strongly delayed bone destruction, see . We directly applied DDEs of lifespan type to characterize this large delay and the PKPD model results in just three pharmacological meaningful states. Finally, we will characterize four typical structures of PKPD models by DDEs from a theoretical point of view.
Conclusions: DDEs of lifespan type are a serious alternative to traditional TCs of length n because the number of states to describe delays or populations reduces to exactly one. Further DDEs open the route to introduce information from the past of pharmacological processes into a PKPD model.
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