(M-027) Implementation of Distributed Delay Differential Equations in RADAU5 and RADAR5

Wojciech Krzyzanski – Pharmaceutical Sciences – University at Buffalo, Buffalo, New York, USA; Nicola Guglielmi – Division of Mathematics – Gran Sasso Science Institute, L’Aquila, Italy

Objectives: Distributed delay differential equations (DDDEs) are used to model delays in pharmacokinetic and pharmacodynamic (PKPD) models of drug effects on cell populations. DDDEs have been implemented in pharmacometrics software such as NONMEM (ICON llc) and Phoenix (Certara Inc). The major challenge for DDDE models is numerical integration of the delay distribution kernel, which results in prolonged running times. A technique approximating DDDEs with ordinary differential equations (ODEs) has been proven as one of the fastest numerical methods for solving DDDEs. RADAU5 is a Fortran program for solving stiff and differential-algebraic problems [1], while RADAR5 extends RADAU5 to discrete DDEs. ODEs and DDEs approximations of DDDEs have been recently implemented in RADAU5 and RADAR5 [2,3]. This study aims to assess the RADAU5/RADAR5 DDDE solvers' performance at estimation of PKPD model parameters from pre-clinical data.

Methods: Previously published data on cytotoxic effect of anticancer drug, 5FU, on white blood cells (WBCs) in rats were used in the study [4]. A published PKPD DDDE model of chemotherapy-induced myelosuppression was implemented in RADAU5 [2]. The PK model consisted of a one-compartment model with Michaelis-Menten elimination and IV bolus input of 5FU. The PD model consisted of proliferating cells in bone marrow (PROLs) and circulating WBCs. PROLs proliferated at a first-order rate k_prol that was equal to the elimination rate of WBCs. The maturation of PROLs to WBCs was described by the gamma distributed delay with shape ν and mean transit time MTT. The negative feedback of WBCs on k_prol was modeled by the power function with an exponent γ. The baseline WBC count 〖Circ〗_0 was used to calculate the past for delayed PROLs. The cytotoxic effect of 5FU was described by the second-order rate constant SLOPE. k_prol was calculated as ν/MTT. The WBC data was log-transformed and the additive residual error model was applied. The PK parameters were fixed at their published values. Naive-pooled data was used for parameter estimation. The likelihood of WBC observations (-2LL) was minimized by the Nelder-Mead algorithm AMOEBA [6]. The standard errors were calculated from the diagonal of inverse Fisher Information Matrix (FIM). RADAU5, RADAR5, -2LL, AMOEBA, and FIM were coded as subroutines for a program compiled and run by Intel Fortran Compiler 2024.0 using Microsoft Visual Studio 2022.

Results: The minimization of -2LL resulted in the following estimates (%RSE) of PD parameters: 〖Circ〗_0 = 14.7 (10.0) 109 cells/L, MTT = 53.8 (56.3) h, ν = 1.43 (165), γ = 0.455 (136), and SLOPE = 0.0143 (47.4). The observed vs predicted and weighted residuals vs. time diagnostic plots did not reveal any systematic misfits of data. The correlation coefficient between observed and predicted values was 0.81. The run time was 7.4 sec.

Conclusions: The DDDE model adequately described the time courses of WBC counts in rats. The parameter estimates were close to estimates reported in [5]. Rather high %RSEs of estimates were a consequence of limiting WBC data to placebo and single-dose groups. The short running time implies that the RADAU5/RADAR5 implementation of the DDDE solver shows potential for application of DDDE models for analysis of population clinical data.

Citations: [1] Hairer E and Wanner G (1996) Solving Differential Equations II. Stiff and Differential Algebraic Problems. Second Edition. Springer Verlag, Berlin.

[2] Guglielmi N and Hairer E (2025) Applying stiff integrators for ordinary differential equations and delay differential equations to problems with distributed delays. SIAM J Sci Comp 47: A102-A123

[3] Guglielmi N and Hairer E (2001) Implementing Radau IIA methods for stiff delay differential equations. Computing, 67:1-12.

[4] Friberg LE, Freijs A, Sandstrom M, Karlsson MO (2000) Semiphysiological model for the time course of leukocytes after varying schedules of 5-fluorouracil in rats. J Pharmacol Exp Ther 295:734–740

[5] Krzyzanski W, Hu S, Dunlavey M (2018) Evaluation of performance of distributed delay model for chemotherapy-induced myelosuppression. J Pharmacokin Pharmacodyn. 45:329 – 337

[6] Press WH, Teukolsky SA, Vetterling WT, Flannery BP (1992) Numerical Recipes in Fortran 77. The Art of Scientific Computing. Volume 1. Cambridge University Press, Cambridge.

Keywords: distributed delays, RADAU5, RADAR5