Solving Semi-Delay Differential Equations in NONMEM
Gilbert Koch (1), Wojciech Krzyzanski (1) and Johannes Schropp (2)
Institution: (1) Department of Pharmaceutical Sciences, State University of New York at Buffalo, USA, (2) Department of Mathematics and Statistics, University of Konstanz, Germany
Objectives: Delay differential equations (DDEs) are a growing tool to describe delays and lifespans in pharmacokinetic and pharmacodynamic (PKPD) modeling [1]. Such equations are e.g. used to describe maturation of blood cells, incubation times in epidemics or strongly delayed phenomena like bone destruction in arthritis. In contrast to its ordinary differential equation (ODE) counterpart, a DDE describes a delay or lifespan with an explicit delay parameter T, is able to produce complex oscillating behavior and also includes information from the past, more precisely, the time before the PKPD system gets started. Currently, DDEs in its general form with a single delay
x'(t) = f(t,x(t),x(t-T)) , x(t)=x0(t) for t≤0 (1)
could not be directly solved in NONMEM. However, we identified an important sub-class of DDEs (1), calling them Semi-DDEs, which are often used in PKPD modeling [1] and can be easily implemented in NONMEM. The general structure of Semi-DDEs with a single delay reads
u'(t) = g(t,u(t)) , u(t)=u0(t) for t≤0 (2)
v'(t) = h(t,u(t),u(t-T),v(t)) , v(0)=v0 (3)
where (2) is an ODE but equipped with a past u0(t) for t≤0 and (3) uses the delayed state u(t-T) given by (2). Roughly speaking, Semi-DDEs (2)-(3) do not permit the rate of change of a state to be described by a right hand side which depends on its own delayed state.
Results: We will demonstrate that Semi-DDEs (2)-(3) could be simply rewritten by two systems of ODEs, one system for the time before delay T and one for the time after the delay. Applying the ALAG command and a case-by-case analysis, Semi-DDEs could be solved with NONMEM. As example we consider a Semi-DDE based PKPD model for rheumatoid arthritis [2] where inflammation and strongly delayed ankylosis was modeled.
Conclusions: We identified an important sub-class of DDEs, the so-called Semi-DDEs, which could be solved with standard ODE solver and therefore be implemented in NONMEM.
References:
[1] Koch G, Krzyzanski W, Perez-Ruixo JJ, Schropp J (submitted) Modeling of delays in PKPD - Classical approaches and a tutorial for delay differential equations
[2] Koch G, Wagner T, Plater-Zyberk C, Lahu G, Schropp J (2012) Multi-response model for rheumatoid arthritis based on delay differential equations in collagen-induced arthritic mice treated with an anti-GM-CSF antibody. J Pharmacokinet Pharmacodyn 39(1):55-65
