Transforming parts of a differential equations system to difference equations as a method for run-time savings in NONMEM VI
Petersson, K., Friberg, L.E., Karlsson, M.O.
Div. of Pharmacokinetics and Drug Therapy, Dept of Pharmaceutical Biosciences, Faculty of Pharmacy, Uppsala University, Sweden
Objectives: As computing power increases, model runtimes decreases given that model complexity remain the same. Increased computing power also gives us the possibility of building more complex models that more adequately describes the sometimes complex mechanisms of diseases and drug effects. Even with the modern computers of today these models may require quite substantial amount of computing time, and for a model to be widely useful long runtimes are not practical. Models with substantial runtimes are often defined as differential equations ($DES in NONMEM). In this work we aim to explore if updating parts of the functions dependent on compartment amounts in the differential equations at pre-specified intervals could shorten model runtimes without loosing model fit.
Methods: In NONMEM VI there is the possibility to update the system at non-event times using a function called MTIME. Different parts of the differential equations in nine models based on differential equations [1-7] were moved from $DES to $PK and MTIME was used to update $PK at given intervals. The intervals were increased to give as short runtimes as possible but the intervals were kept short enough to retain roughly the same fit (OFV).
Results: For five [1,4,6-7] of the nine models we were able to shorten the runtimes to a pronounced degree (59-96% reduction) while for four models it was not possible to decrease runtimes and keep a similar fit. For the prolactin model  which had a runtime of over one month using FOCE the runtime dropped to 24 h. The fixed effects parameter estimates for four of the models which could be expedited were within 12% from the estimates of the original model. For the last of the faster models  one parameter differed by 23% and one by 13%. The mean absolute error for the fixed effects parameters in the faster models were between 1.5% and 8.0% and the mean absolute errors for the variance parameters were between 1.1% and 9.9%. The difference in OFV compared to the original models ranged between -14.4 and 1.7 units.
Conclusions: Moving parts of or whole equations from differential to difference equations using MTIME can in some cases shorten runtimes substantially while model fit and parameter estimates are retained. This approach may for example be useful in covariate modeling and in exploring the random effects model (e.g. IIV, IOV and semi-parametric distributions ). To understand the mechanism behind when this approach is applicable needs further studies.
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