Background: Determining the optimal choice of a dose of medicine to meet the patient’s need is a complicated mathematical and statistical problem. Large between-subject variability in metabolic, excretion processes, and physiological systems involving drug effect, as well as complicated nonlinearities, are at the heart of the problem. This leads to nonlinear differential equations that need to be solved to design studies and estimate parameters. Inductive linearisation  is a new method to solve systems of nonlinear ODEs. It is an iterative method that converts a nonlinear ordinary differential equation (ODE) into a time-varying linear ODE, which can then be solved by standard integration techniques (e.g., eigenvalue decomposition). The Inductive Linearisation method has potential advantages over the commonly used numerical time-stepping methods. The method can provide an arbitrarily accurate solutions for both stiff and non-stiff systems. The solution of partial derivatives is analytically available. Several approaches, such as lumping and control can be easily applied to the linearised system. The motivating example in this work is a first-order input, Michaelis-Menten elimination pharmacokinetic example for which no closed form solution exists.
Objectives: We aim to optimise the Inductive Linearisation method for estimation of the parameters (k_a, V, V_max , and K_m) after fitting the model to one set of simulated observed data. ODE45 was the reference comparator method. The Levenberg-Marquardt algorithm was used for parameter estimation.
Methods: The Inductive Linearisation method was optimised at three levels: (1) No optimisation. We term this the naïve solution in which the number of iterations of the linearisation is fixed and the initial conditions for Inductive Linearisation remain unchanged during optimisation. (2) A stopping rule is added for the iterative linearisation based on a pre-defined error tolerance (Tol). (3) The stopping rule from (2) is combined with the use of informative initial conditions for the iterative linearisation. We term this a “smart update”. Using the smart update, the initial conditions for Inductive Linearisation are set to the previous concentration from the last vector of parameter values from the estimation algorithm. Two comparator metrics, the run time (seconds) and the accuracy of parameter estimation (% Relative Difference) were compared between each of the methods and the reference (ode45) method. Various combinations of Tol from 1e-3 to 1e-12 for Inductive Linearisation and step size for the paired integrator (in this case eigenvalue decomposition) were explored.
Results: The reference integration method (ode45) took 0.2 seconds for parameter estimation, returned an objective function of 0.303 and the relative error of the parameters ranged from 17 to 31 %. The naïve method (with the maximum number of iteration n=20 ) took 8.4 seconds for parameter estimation, returned an objective function of 0.467 and the relative error of the parameters ranged from -22 to 20 %. The stopping rule method (with the stopped number of iteration n=15 ) took 6.8 seconds for parameter estimation, returned an objective function of 0.467 and the relative error of the parameters ranged from -0.7 to 54 %. The smart update method (with the topped number of iteration n=15 ) took 3.5 seconds for parameter estimation, returned an objective function of 0.469 and the relative error of the parameters ranged from -1.8 to 36 %. These evaluations were performed at a Tol value of 1e-6 as the default Tol for the reference algorithm (ode45) and a step size of 1e-1 . The accuracy of Inductive Linearisation was equivalent to the reference when Tol = 1e-7 and step size = 1e-1 .
Conclusion: Our result has clearly shown that the smart update method was significantly faster and more accurate than the other Inductive Linearisation methods and was similar to ode45. It indicates that Tol and step size are two essential properties that can be optimised independently. We note that other integrators (other than eigenvalue decomposition) could be paired with Inductive Linearisation and yield equivalent or perhaps more efficient results. The next step will aim to expand the exploration to a stochastic simulation-estimation evaluation.
Duffull, S. B. and Hegarty, G. (2014), an inductive approximation to the solution of systems of nonlinear ordinary differential equations in pharmacokinetics-pharmacodynamics. J Theor Comput Sci, 1:4. http://dx.doi.org/10.4172/jtco.1000119