Stephen Duffull (1), Gareth Hegarty (1,2)
(1) School of Pharmacy, University of Otago; (2) Department of Mathematics and Statistics, University of Otago, New Zealand
Objectives: PK and PKPD models are often formulated as systems of ordinary differential equations (ODE). The most common nonlinearity in PK is due to Michaelis-Menten processes but other nonlinear structures are common in PKPD. Due to the nonlinearity it is not possible to solve these systems, except in the simplest cases, in closed algebraic form and iterative time-stepping algorithms are employed. These algorithms, e.g. the Runge-Kutta methods, although very effective general solutions may be slow, are prone to imprecision, provide solutions at discrete time points and require knowledge of the stiffness of the system. In this work we propose a rapid iterative solution that is exact to any arbitrary level of accuracy and is continuously differentiable over time.
Methods: If we have an ODE of the general form dy/dt=f(t,y)+A(t,y)y with defined initial conditions, then this system can be linearised to a time-varying linear system by plugging in our previous value of y{n-1} as the predictor of our updated value of y{n}, such that:
dy{n}/dt = f(t,y{n-1}) + A(t,y{n-1})y{n}.
Here we can see that the system is no longer nonlinear as the functions f and A do not depend on the current iteration of y{n}. The time-varying linear solution can now be solved using standard procedures such as integrating factors and (if needed) Gauss-Legendre quadrature.
This general method is applied to a first-order input Michaelis-Menten output model.
Results: The starting values of y for the times of interest were set to that provided by a linear first-order input-output system (i.e. when n=1, y{n-1} was given by the linear system). The solution is provided for 5 Gaussian quadrature steps with 7 iterations of the iterative linearization. It was shown that the relative error for successive linear approximations decayed exponentially indicating the solution was convergent and at these settings the relative error was 1e-5.
Discussion: A method for solving nonlinear ODEs is presented and illustrated with a simple example. Because the solution depends continuously on time and analytical derivatives available the method is particularly amenable to estimation and optimisation problems.
Reference: PAGE 24 () Abstr 3348 [www.page-meeting.org/?abstract=3348]
Poster: Methodology - Other topics