Initial exploration of numerical methods for fitting models to PK data

Background: There are well established programs for PK/PD analysis, however, models can get extensive and complex and for parameter estimation are often very time consuming. Therefore, there is a constant need for development of faster methods for model based analysis in terms of definition of the models and the modelling process. The solution to all first-order linear time-varying differential equations can be found using integrating factors and a numerical approximated solution can be found using quadrature. Gauss-Legendre quadrature is using optimal time points and approximates the solution efficiently with few quadrature steps. An inductive linearization method can be used to approximate nonlinear ODEs and combined with the quadrature to give a numerical approximated solution. These methods promise to be faster and more accurate than standard ode solving. Twalk is a new (2010) iterative MCMC method within the Bayesian framework that has been shown to be stable and fast by its developers. Combined with twalk could provide the statistical power needed for model evaluation.

Aim: To explore numerical techniques to try to improve the computational efficiency of PKPD modelling.

Methods: (1) The twalk method was compared to NONMEM and WinBUGS for a simple linear PK problem. A dataset from orally administered warfarin, single subject PK data was used and modeled as a 1 compartment, 1st order absorption model. (2) An inductive linearisation solution was used for exploring the solution to a nonlinear ODE. It was specified as a 1 compartment model, with 1st order absorption and nonlinear elimination. Data was then simulated using ode45 and pre defined parameters for the given model.. (3) Twalk and the inductive method were combined to assess performance. The methods were investigated in terms of speed and accuracy using MATLAB. Both models considered were for a single subject and hence did not contain between subject random effects. Linear and nonlinear ODE models were considered.

Results: (1) Twalk performed parameter estimation with more than satisfactory speed, taking 3 seconds for the closed form solution of a linear PK problem, with 19000 iterations, compared to 9 seconds using WinBUGS using the same number of iterations and 5 seconds using NONMEM using subroutine ADVAN2 TRANS2. The Gauss-Legendre quadrature approximated the exact solution for the linear problem with 2 and 4 quadrature steps and could be used with twalk for parameter estimation. (2) A model specific approximate inductive linearization method for the nonlinear problem was used, which converged to the same solution as provided by using the Runge-Kutta numerical method, ode45 in MATLAB (at a relative tolerance of 10-8). (3) The inductive linearisation method did not work with the twalk method and the reason for this incompatibility was not resolved. The inductive linearization and quadrature or ode45 were used for data fitting with fminsearch (an optimization tool in MATLAB) and managed to successfully fit the given data, as did NONMEM. The data fitting runtimes for the nonlinear problem were 11 seconds for NONMEM, 30 seconds for fminsearch with the inductive linearization and 3 seconds for fminsearch with the ode45 approximation.

Conclusions: A new MCMC method (twalk) may provide useful speed advantages over existing methods (e.g. WinBUGS) and even approach the speed of NONMEM when using the FO approximatoin. An inductive linearisation method provided an accurate solution to a nonlinear ODE. The inductive method was not compatible with twalk and also requires speed optimization.