Wojciech Krzyzanski (1), Robert Bauer (2)
(1) University at Buffalo, Buffalo, USA (2) ICON, Gaithersburg, MD, USA
Objectives: Models involving delay differential equations (DDEs) are increasingly popular in analysis of pharmacokinetic and pharmacodynamics data. A distinct feature of DDEs is a history that needs to be evaluated in order to determine the change of the state at a current time. This requires additional computational time that makes numerical DDE solvers to run longer compared to analogous solvers for ordinary differential equations (ODEs). Another distinct feature that is poorly handled by available DDE solvers is presence of bolus dosing events that introduces discontinuous input into DDE models and increases computational time. A technique called method of steps (MOS) has been introduced that transforms any system of DDEs into a system of ODEs [1]. A feature that impacts numerical performance of MOS is high dimensionality of ODE systems. Another practical issue is complexity of implementation of MOS into PKPD software. A program DDEXPAND has been developed for coding DDE based models in NONMEM [2]. DDEXPAND applies MOS to translate DDE model equations into a NONMEM control stream that can run using any of available ODE solvers. The objective of this work was to assess numerical performance of MOS implemented in NONMEM by DDEXPAND compared to well established DDE solver dde23 implemented in MATLAB [3].
Methods: Three previously published DDE models were selected for testing the MOS: delayed logistic growth (LOGISTIC), rheumatoid arthritis (RA), and tumor growth inhibition [4]. All models were implemented in NONMEM 7.4 by means of DDEXPAND with ADVAN 13 ODE solver. For each model 100 predictions were simulated for one subject using published typical values without residual variability. The CPU time was recorded for each simulation. Analogously, each model was coded in MATLAB R2017b. The model DDEs were solved by dde23 for an individual subject at 100 observation times identical with NONMEM models. The CPU time was recorded for each simulation. Additionally, each model was implemented in MATLAB using MOS equations based on NONMEM control stream generated by DDEXPAND. The ode45 ODE solver was used to solve model equations. Similarly, 100 predictions were simulated and CPU times were recorded. The mean of 10 CPU times was used as a measure of performance. The same ATOL and RTOL values were used for all solvers. Maximum absolute and relative differences between NONMEM and MATLAB solutions were used as metrics of MOS accuracy. Both NONMEM and MATLAB were run on a PC computer with Intel Fortran compiler v 11.1.
Results: The LOGISTIC model consisted of one DDE with constant past. MOS resulted in 12 ODEs. The mean CPU time for NONMEM was 0.0591±0.0097and MATLAB 0.05830±0.0183 (dde23) and 0.09236±0.0254 (ode45). The absolute and relative errors were 2.27E-03 and 2.28E-04, respectively. MOS became unstable if the number of ODEs exceeded 50. The original RA model consisted of two-compartment PK and three PD DDEs with non-constant past, and two outputs (total arthritic score,TAS, and ankylosis score, AKS). There were 8 MOS ODEs. The simulations were done for placebo, and doses 1, 10, and 100 mg/kg. The mean CPU time for NONMEM was 0.08890±0.0196 and MATLAB 0.7516±0.0537 (dde23) and 0.1794±0.0370 (ode45). The absolute and relative errors for 100 mg/kg were 1.77E-06 and 4.78E-07 (TAS), 8.36E-06 and 6.04E-05 (AKS), respectively. The TGI model consisted of one-compartment with first-order absorption PK, one control PD, and two PD DDEs with constant past, and 5 multiple doses. There were 36 MOS ODEs. The mean CPU time for NONMEM was 0.09360±0.0222 and MATLAB 6.644±0.0511 (dde23) and 0.7396±0.0364 (ode45). The absolute and relative errors were 2.34E-04 and 3.28E-04 (control), 1.46E-04 and 3.1E-04 (treatment), respectively.
Conclusions: For LOGISTIC model NONMEM is equally fast as MATLAB, but it is one order less accurate than MATLAB (accuracy was improved with small increased ATOL and RTOL without significantly affecting computation time). If the number of steps becomes large, MOS might become unstable. For RA model with single bolus injection NONMEM is 8.5 times faster than MATLAB and equally accurate. For TGI model with multiple drug administrations NONMEM is 71 times faster than MATLAB and equally accurate.
References:
[1] Smith H (2011) An introduction to delay differential equations with applications to the life sciences. Springer, New York
[2] Bauer B, Krzyzanski W, DDEXPAND interface for coding delay differential equations based models in NONMEM. PAGE 26 (2017) Abstr 7312 [www.page-meeting.org/?abstract=7312]
[3] Shampine LF, Thompson S (2001) Solving DDEs in MATLAB. Appl Numer Math 37:441–458
[4] Koch G, Krzyzanski W, Perez-Ruixo JJ, Schropp J (2014) Modeling of delays in PKPD: classical approaches and a tutorial for delay differential equations. J Pharmacokin Pharmacodyn 41:291–318
Reference: PAGE 27 (2018) Abstr 8746 [www.page-meeting.org/?abstract=8746]
Poster: Methodology - Other topics