Abstract
The safe and effective use of medications requires a careful selection of dose in order to attain the desired effects and minimise adverse effects. The choice of dose and dosing regimen is often based on mathematical models of the pharmacokinetic and pharmacokinetic-pharmacodynamic properties of the medication. These models are usually expressed as ordinary differential equations. Evaluation of these mathematical models is generally straightforward if the differential equation is linear; however, for nonlinear differential equations this can be complicated with issues with stability and efficiency of the solution. Current methods to solve these equations use time-stepping solvers. A new method, termed inductive linearisation, has been proposed recently. This solver has two components: (1) an iterative method that linearises the differential equation and (2) an integration method based on eigenvalue decomposition. The combination of both of these components is termed the Inductive Linearisation solver.
Optimisation of component one is based on determining a convergence criteria for the iterative linearisation process and for component two involves the choice of step-size to integrate the linear system. The solver as a whole also has the ability to remember previous iterations and hence it is possible to take advantage of repeated function calls that occur during estimation processes.
The work conducted in this thesis involves exploration and optimisation of the Inductive Linearisation solver. The solver is applied to a Michaelis-Menten elimination PK model as a non-stiff example (Chapters 2, 3, 5, 6) and the Van der Pol equation (Chapter 4) as a stiff example.
The first part of the thesis, Chapters 2 and 3, optimises two characteristics of the inductive technique. Initially, Chapter 2 presents the solver and explores the properties of both of its components. Chapter 3 then introduces and employs three optimisation methodologies and concludes with an evaluation of the performance of the optimised Inductive Linearisation solver for a single-subject simulation-estimation scenario.
In the second section of the thesis, Chapters 4, 5, and 6, application of the optimised Inductive Linearisation solver for three distinct scenarios is investigated. In Chapter 4, the impact of optimisation strategies for one simulation example of the Van der Pol stiff system is applied and evaluated. The repeatability of the oscillator system is then introduced as a step toward optimising the Inductive Linearisation solver for the oscillator system. In Chapter 5, the optimised inductive technique is applied to a single-subject stochastic simulation-estimation problem and its characteristics explored. Finally, the method is extended to a population simulation-estimation study in Chapter 6.
This work supports the widespread applicability of the optimised solver.