DDEXPAND interface for coding delay differential equations based models in NONMEM
Wojciech Krzyzanski (1), Robert Bauer (2)
(1) University at Buffalo, Buffalo, USA, (2) ICON, Gaithersburg, USA.
Objectives: Models using delay differential equations (DDEs) can be coded as ordinary differential equations (ODEs) with the method of steps [1]. A drawback of method of steps is the large number of ODEs and delays making the DDE model implementation strenuous to code. DDEXAPND is a program that uses the method of steps to expand the base ODE’s to include their time-delayed ODE’s, and generate a NONMEM control stream for the DDE model.
Methods: DDEXPAND is a utility program that expands an NM-TRAN-template control stream to form a functional NMTRAN control stream to be run by NONMEM. DDEXAPAND requires for input a template text file (*.dde) with base model equations and a regular NONMEM data file (*.csv). The program outputs a text file (*.ctl) with a NMTRAN control stream and a modified data file (*.csv). DDEs are implemented in the *.dde file using the NMTRAN syntax for ODEs with additional structures accounting for delays (TAUy) on states x (AD_x_y) and their past conditions (state values for negative times) (AP_x_y). Published DDE models of rheumatoid arthritis (RA) development in collagen-induced arthritic mice [2] and influenza A virus (IAV) infection in humans [3] were used to test DDEXPAND functionality. Time courses for model states were simulated using NONMEM 7.3 [4] and compared with time courses generated by DDE solver dde23 in MATLAB (MathWorks, Natick, MA).
Results: The *.dde RA model has 8 base ODEs with one delay TAU1, two delay states AD_1_1 and AD_6_1, and two past condition equations for the delay states, AP_1_1 and AP_6_1. The *.ctl RA model generated by DDEXPAND had 4 additional ODEs that used the original 8 user-defined ODE’s as their template. DDEXPAND also modified the *.csv file to include dose records for the additional ODEs. The *.dde IAV model has 5 base ODEs with one delay TAU1, one delay state AD_3_1, and one constant past equation for the delay state, AP_3_1. The *.ctl IAV model generated by DDEXPAND had 35 additional ODEs that used the original 5 user-defined ODE’s as their template to cover a simulation time of 8*TAU1. For both models, simulated time courses of the model states were no different from analogous ones obtained by the MATLAB dde23 solver.
Conclusion: DDE based models can be implemented in NONMEM using the method of steps. DDEXPAND provides a convenient tool for propagating and coding DDEs in NONMEM. DDEXPAND solutions of DDE models are equally accurate as solutions obtained by standard DDE solvers.
References:
[1] Perez-Ruixo JJ, Kimko HC, Chow AC, Piotrovsky V, Krzyzanski W, Jusko WJ, Population cell lifespan models for effects of drugs following indirect mechanism of action. J. Pharmacokin. Pharmacodyn. 32: 767-793 (2005).
[2] Koch G, Wagner T, Plater-Zyberk C, Lahu G, Schropp J, Multi-response model for rheumatoid arthritis based on delay differential equations in collagen-induced arthritic mice treated with an anti-GM-CSF antibody. J Pharmacokinet Pharmacodyn 39(1):55–65 (2012).
[3] Koch G, Krzyzanski W, Perez-Ruixo JJ, Schropp J, Modeling of delays in PKPD: classical approaches and a tutorial for delay differential equations. J. Pharmacokin. Pharmacodyn. 41:291–318 (2014).
[4] Beal SL, Sheiner LB, Boeckmann AJ & Bauer RJ (Eds.) NONMEM 7.3 Users Guides. 1989-2013. Icon Development Solutions, Gaithersburg, Hanover, USA.
