Solving Delay Differential Equations in S-ADAPT by Method of Steps
Wojciech Krzyzanski(1), Robert J Bauer(2)
(1) Department of Pharmaceutical Sciences, University at Buffalo, Buffalo, NY 14260, USA; (2) ICON Development Solutions, Ellicott City, MD 21043, USA
Objectives: S-ADAPT is a version of ADAPT II that contains additional simulation and optimization abilities such as parametric population analysis . S-ADAPT utilizes LSODA to solve ordinary differential equations (ODEs), an algorithm designed for large dimension non-stiff and stiff problems. However, S-ADAPT does not have a solver for delay differential equations (DDEs). Our objective was to implement in S-ADAPT a DDE solver using the methods of steps.
Methods: The method of steps allows one to solve virtually any DDE system by transforming it to an ODE system . Fortran subroutines were added to the source S-ADAPT files which enabled it to solve DDE systems with multiple delay times and constant conditions for the past. The S-ADAPT DDE solver utilizes LSODA to obtain the solution. The solver was validated for systems of linear DDEs with one and two delay times and bolus inputs for which explicit analytic solutions were derived. Solving of nonlinear DDE problems were validated by comparing the solutions with ones obtained by the MATLAB DDE solver dde23 . The target mediated drug disposition PK and lifespan based indirect response PD models developed previously for recombinant human erythropoietin (rHuEPO) were used for tests . The performance of S-ADAPT for stiff problems was tested by increasing the erythropoietin receptor binding constant kon to values where the stiffness of the PK/PD model was anticipated.
Results: The user provided subroutines defining DDE problems for S-ADAPT resemble those for ODE problems with an addition of variables for delayed state variables. All necessary Fortran subroutines and global variables are stored in two files that need to be added to S-ADAPT directory. No re-installation is necessary. The comparison of S-ADAPT generated solutions for DDE problems with the explicit solutions as well as MATLAB produced solutions agreed up to number of significant digits set for LSODA by the constants RTOL and ATOL. The DDE solver was capable of solving the stiff PK/PD model with kon = 0.01, 0.1, 1, and 10 1/nM/h, the values that are up to 1000-fold larger than its estimate.
Conclusions: S-ADAPT is the first program designed for population PK/PD analysis that is capable of solving arbitrary DDE models with typical PK input consisting of multiple bolus injections and infusions. The performance of the S-ADAPT DDE solver is identical with the performance of the LSODA for large dimension systems by the virtue of the method of steps.
 S-ADAPT/MCPEM User's Guide [computer program]. Version 1.57. Berkeley, CA.; 2011.
 Driver R.D. Ordinary and Delay Differential Equations. Springer-Verlag, New York, 1977.
 L. F. Shampine, S. Thompson, Solving DDEs in MATLAB, Appl. Num. Math. 37: 441-458 (2001).
 S. Woo, W. Krzyzanski, W. J. Jusko, Target-mediated pharmacokinetic and pharmacodynamic model of recombinant human erythropoietin (rHuEPO). J. Pharmacokinet. Pharmacodyn. 34:849-868 (2007).