Response of an oscillatory delay differential equation to a periodic stimulus

TL;DR

This paper models the oscillatory behavior of blood cell counts in periodic hematological diseases and chemotherapy using a delay differential equation, analyzing how periodic drug administration influences these oscillations to optimize treatment timing.

Contribution

It introduces a simple delay differential equation model with a periodic solution to study drug effects on oscillatory blood cell counts, aiding treatment timing decisions.

Findings

Model captures oscillations in blood cell counts due to periodic stimuli

Response analysis helps determine optimal drug dosing and timing

Insights into avoiding side effects through controlled perturbations

Abstract

Periodic hematological diseases such as cyclical neutropenia or cyclical thrombocytopenia, with their characteristic oscillations of circulating neutrophils or platelets, may pose grave problems for patients. Likewise, periodically administered chemotherapy has the unintended side effect of establishing periodic fluctuations in circulating white cells, red cell precursors and/or platelets. These fluctuations, either spontaneous or induced, often have serious consequences for the patient (e.g. neutropenia, anemia, or thrombocytopenia respectively) which exogenously administered cytokines can partially correct. The question of when and how to administer these drugs is a difficult one for clinicians and not easily answered. In this paper we use a simple model consisting of a delay differential equation with a piecewise linear nonlinearity, that has a periodic solution, to model the effect of a periodic disease or periodic chemotherapy. We then examine the response of this toy model to both single and periodic perturbations, meant to mimic the drug administration, as a function of the drug dose and the duration and frequency of its administration to best determine how to avoid side effects.