Investigating Stochastic Differential Equations Modelling for Levodopa Infusion in Patients with Parkinson's Disease

Levodopa concentration in patients with Parkinson's disease is frequently modelled with ordinary differential equations (ODEs). Here, we investigate a pharmacokinetic model of plasma levodopa concentration in patients with Parkinson's disease by introducing stochasticity to separate the intra-individual variability into measurement and system noise, and to account for auto-correlated errors. We also investigate whether the induced stochasticity provides a better fit than the ODE approach.

In this study, a system noise variable is added to the pharmacokinetic model for duodenal levodopa/carbidopa gel (LCIG) infusion described by three ODEs through a standard Wiener process, leading to a stochastic differential equations (SDE) model. The R package population stochastic modelling (PSM) was used for model fitting with data from previous studies for modelling plasma levodopa concentration and parameter estimation. First, the diffusion scale parameter (σ_w), measurement noise variance, and bioavailability are estimated with the SDE model. Second, σ_w is fixed to certain values from 0 to 1 and bioavailability is estimated. Cross-validation was performed to compare the average root mean square errors (RMSE) of predicted plasma levodopa concentration.

Both the ODE and the SDE models estimated bioavailability to be approximately 75%. The SDE model converged at different values of σ_w that were significantly different from zero. The average RMSE for the ODE model was 0.313, and the lowest average RMSE for the SDE model was 0.297 when σ_w was fixed to 0.9, and these two values are significantly different.

The SDE model provided a better fit for LCIG plasma levodopa concentration by approximately 5.5% in terms of mean percentage change of RMSE.