Ordinary differential equation approximation of gamma distributed delay model.

Area of Science:

• Pharmacology
• Mathematical Biology
• Computational Science

Background:

• Pharmacodynamic (PD) models often incorporate delays, frequently modeled using gamma distribution convolutions.
• Approximating these convolution integrals is crucial for computational efficiency and model accuracy.

Purpose of the Study:

• To develop and validate a novel approximation method for convolution integrals in PD models with delays.
• To assess the accuracy and error bounds of the proposed approximation.
• To apply the method to a real-world biological system: chemotherapy-induced myelosuppression.

Main Methods:

• Utilized a system of ordinary differential equations based on binomial series properties to approximate the convolution integral.
• Derived an estimate for the approximation error, dependent on the number of differential equations (n) and the gamma distribution shape parameter (k).
• Validated the approximation using input functions with known explicit convolutions and applied it to WBC count data.

Main Results:

• The approximation demonstrates uniform convergence on compact time intervals.
• Accurate approximations were achieved when the shape parameter k was small (k ≤ 2).
• For larger k values, the approximation error decreased slowly, potentially requiring a higher number of differential equations (n) for acceptable accuracy.

Conclusions:

• The developed ordinary differential equation-based approximation offers an effective method for modeling delays in pharmacodynamic systems.
• The accuracy is dependent on the interplay between the number of differential equations and the gamma distribution shape parameter.
• The method successfully estimated parameters for a distributed delay model of chemotherapy-induced myelosuppression, demonstrating its practical utility.