Analysis of the Circulatory System Using Second-Order Differential Equations

Abstract

Mathematics offers more than abstract formalism; it gives a quantitative language for describing how the body actually works. In this paper, we apply second-order linear differential equations to the dynamic behaviour of the human circulatory system, treating the interaction between arterial wall elasticity, blood viscosity and the pulsatile output of the heart as a mechanical oscillator. Framed in this way, the natural oscillations of the pulse wave become a source of quantitative measures of vascular compliance and resistance, in line with the lumped-parameter and one-dimensional approaches long used in cardiovascular fluid mechanics. We analysed physiological data drawn from thirty individuals (n = 30) spanning a range of ages and clinical backgrounds — healthy subjects, hypertensive patients and individuals with diabetes — and calibrated the model to obtain, for each participant, a damping coefficient (b) and a stiffness parameter (k) reflecting the state of the vasculature. The sign of the discriminant of the characteristic equation separates the resulting dynamics into three regimes: underdamped (elastic, responsive vessels), critically damped (optimal vascular adaptation) and overdamped (stiff or pathologically remodelled arteries), a classification broadly consistent with the clinical picture of arterial stiffening described in the literature. The findings suggest that a second-order differential-equation model, calibrated from routinely available vital signs, can serve as a simple, non-invasive complement to established markers of arterial stiffness such as pulse wave velocity.