Stability of the human respiratory control system II. Analysis of a three-dimensional delay state-space model

A number of mathematical models of the hum;ln respiratory control system ha ve been developed since 1910 to study a wide range of features of this comp lex system. Among them, periodic breathing (including Cheyne-Stokes respira tion and apneustic breathing) is a collection of regular but involuntary br eathing patterns that have important medical implications. The hypothesis t hat periodic breathing is the result of delay in the feedback signals to th e respiratory control system has been studied since the work of Grodins et al. in the early 1950's [1]. The purpose of this paper is to study the stab ility characteristics of a feedback control system of five differential equ ations with delays in both the state and control variables presented by Kho o et al. [4] in 1991 for modeling human respiration. The paper is divided i n two parts. Part I studies a simplified mathematical model of two nonlinea r state equations modeling arterial partial pressures of O-2 and CO2 and a peripheral controller. Analysis was done on this model to illuminate the ef fect of delay on the stability. It shows that delay dependent stability is affected by the controller gain, compartmental volumes and the manner in wh ich changes in the ventilation rate is produced (i.e,, by deeper breathing or faster breathing). In addition, numerical simulations were performed to validate analytical results. Part II extends the model in Part I to include both peripheral and central controllers. This, however, necessitates the i ntroduction of a third state equation modeling CO2 levels in the brain. In addition to analytical studies on delay dependent stability, it shows that the decreased cardiac output land hence increased delay) resulting from the congestive heart condition can induce instability at certain control gain levels. These analytical results were also confirmed by numerical simulatio ns.