Modelling uncertainty using stochastic differential equations – applications on drug action and spatial data
Background: Treatment in the later stage of Parkinson’s disease must be individualized in order to reduce both symptoms which occur as a result of insufficient levels of dopamine (‘off’) and involuntary movements (‘dyskinesia’) which occur due to excess levels. Since the disorder is progressive, the individualization must be repeated regularly. Several measures based on patient movements have been developed to assess the need for medication. But translating these measures directly to which medication level is required is not straight forward and requires modelling of levodopa in the body. Aim: To develop a refined mathematical model for levodopa in patients that would allow us to get better parameter estimates and a method for fitting the model. Proposed method: By using mixed effects modelling we can fit both measurement error and random effects on an individual level. By modelling the drug uptake, distribution and elimination processes as stochastic differential equations (SDE) instead of ordinary differential equations (ODE) we can also model the uncertainty in the dynamic model for drug by allowing a Wiener noise component. Using simulated data, we will investigate whether these sources of variability may be separated with the new model. The SDE based model will be then be used to find new parameter estimates that had been calculated using ODE based models in previously published studies. The parameter estimates from both studies will be compared along with their uncertainties to check whether the SDE based model is more refined and suitable.