A mathematical model of Cheyne-Stokes or periodic breathing

Abstract

Cheyne-Stokes Breathing is a periodic cycle of apnea followed by hyperventilation. A theory of this phenomenon is developed based on a minimal set of physiological assumptions. The rate of loss of CO₂ from venous blood is proportional to the CO₂ concentration in the lungs times the respiration rate; the respiration rate is a linear function of arterial CO₂ concentration above a threshold, and zero below that threshold. A time delay between blood in lungs and respiratory response allows the system to go into oscillation. These assumptions lead to a single nonanalytic delay-differential equation containing only three parameters, which we call respiratory recovery coefficients, $(\alpha ,\beta ,\gamma )$. A detailed study of the solutions to this equation is presented here. For $\beta$ below a first threshold, breathing becomes steady, and any disturbance recovers exponentially to the steady state (∼overdamped oscillator). Above the first threshold, breathing recovers to the steady state by decaying oscillations (∼underdamped oscillator). Above a second threshold, oscillations grow to reach a limit cycle, and when that cycle is sufficiently large, it represents the Cheyne-Stokes cycle of hyperventilation and apnea. Fourier analysis shows that the transition to growing oscillations is a forward or soft Hopf bifurcation. In the Cheyne-Stokes region (sufficiently large $\beta$), the equation predicts the shapes of the curves representing the time-dependence of arterial CO₂ and the respiration rate. From these shapes, we infer the values of the respiratory recovery coefficients for several groups of patients. With additional approximations, we infer the values of other physiological parameters, including cardiac output, CO₂ chemosensitivity, and volume of blood between lungs and detectors.