Optimising Inductive Linearisation for nonlinear ODEs with an application to the Michaelis-Menten model: a stochastic simulation-estimation study

Background: Solving systems of nonlinear differential equations is of significant interest in applying pharmacokinetic-pharmacodynamic and systems pharmacology. Inductive Linearisation (IndLin) [1] is a numerical solver that has been developed for generating approximated solutions to nonlinear systems based on iterative linearisation to yield a linear time-varying (LTV) ODE. The method is then paired with the integrator eigenvalue decomposition (EVD) to solve the LTV system. Previously, the algorithm of the IndLin method has been optimised for a single subject simulation estimation study at three levels [2], (1) Stopping rule (for IndLin), (2) Fixed step size (EVD), and (3) smart update (for IndLin). Here, we optimise the EVD step size using an adaptive method and then evaluate these optimisation steps using stochastic simulation-estimation (SSE).

Objectives: This study aims to explore optimisation properties of the Inductive Linearisation method coupled with eigenvalue decomposition integration (EVD) as a solution to the Michaelis-Menten example using stochastic simulation and estimation.

Methods: An adaptive step size, where the step size changes over time based on the inverse of the rate of change in the response variable dydt-1 , was developed and compared with the two fixed step sizes (0.1  and 0.01  hours). The initial approximation of y0 , instead of being 0 for all t at each iteration of the estimation search, was replaced by the previous vector of y0(t)  from the previous iteration of parameter estimates (as per the smart update) and the tolerance of the inductive linearisation process was set to 1e-6. The optimised IndLin method with step sizes of 0.1, 0.01 and an adaptive step size were compared with the reference time-stepping ODE solver (ode45) for 1000 single-subject stochastic simulation estimation (SSE). The PK model was a single dose, first-order absorption, one compartment, Michaelis Menten output model. The dose was 3 mg and the between subject variability ( CV=31.6% ) was included on the parameters for simulation but not included on estimation as the simulated data were single subject. The run time (seconds), and the relative difference of the estimated parameters from their true values were compared for each of the methods

Results: The estimated parameter values were centred on the true values for the reference ode solver (ode45) and for IndLin-EVD with a step size of 0.01 and the adaptive step size. The relative differences (%) for the parameter estimates shown similar performance among the reference solution ode45, IndLin with fixed ss=0.01, and IndLin with adaptive step size. However, the larger step size of ss=0.1 produced obvious bias in its estimate of ka . The speed per estimation (mean ± standard deviation in seconds) of IndLin when using adaptive step size (0.857±1.00) was six times faster than ode45 (5.89±1.50). The speed of IndLin with the fixed ss=0.1 (3.21±1.95) was faster than that with the fixed ss=0.01 (31.1±24.3). The minimum least squares objective function value was found to be a function of the pre-set tolerance of IndLin and for ode45. For IndLin the OFV was similar for all step-size values (0.581 to 0.768) and slightly higher than that for ode45 (0.241). These differences were not associated with apparent differences in parameter estimates.

Conclusion: These results indicate that, for this particular example, coupling IndLin with EVD, as a pragmatic approach to solving the resulting LTV system, provided an acceptable solution to solving nonlinear ODEs. Including an adaptive step size approach was faster than and as accurate as the reference solution provided by ode45. The main difference and potential advantage of the IndLin method vs. numerical time-stepping integrators (e.g., LSODA, Runge-Kutta) is that IndLin is not a local search solution. Using this method, all perturbations that occur at any point in the time span are available and known to the solver and are solved simultaneously.


  1. 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
  2. Sharif, S., Hasegawa, C., Duffull, S. B. (2021), Exploring and optimising the computational efficiency of Inductive Linearisation for estimation. Population Approach Group of Australia and New Zealand (PAGANZ)


  • University ofOtago