Mathematical biology

Abstract

This paper considers a model of the human cardiovascular-respiratory control system with one and two transport delays in the state equations describing the respiratory system. The effectiveness of the control of the ventilation rate V̇A is influenced by such transport delays because blood gases must be transported a physical distance from the lungs to the sensory sites where these gases are measured. The short term cardiovascular control system does not involve such transport delays although delays do arise in other contexts such as the baroreflex loop (see [46]) for example. This baroreflex delay is not considered here. The interaction between heart rate, blood pressure, cardiac output, and blood vessel resistance is quite complex and given the limited knowledge available of this interaction, we will model the cardiovascular control mechanism via an optimal control derived from control theory. This control will be stabilizing and is a reasonable approach based on mathematical considerations as well as being further motivated by the observation that many physiologists cite optimization as a potential influence in the evolution of biological systems (see, e.g., Kenner [29] or Swan [62]). In this paper we adapt a model, previously considered (Timischl [63] and Timischl et al. [64]), to include the effects of one and two transport delays. We will first implement an optimal control for the combined cardiovascular-respiratory model with one state space delay. We will then consider the effects of a second delay in the state space by modeling the respiratory control via an empirical formula with delay while the the complex relationships in the cardiovascular control will still be modeled by optimal control. This second transport delay associated with the sensory system of the respiratory control plays an important role in respiratory stability. As an application of this model we will consider congestive heart failure where this transport delay is larger than normal and the transition from the quiet awake state to stage 4 (NREM) sleep. The model can be used to study the interaction between cardiovascular and respiratory function in various situations as well as to consider the influence of optimal function in physiological control system performance. J.J. Batzel: SFB “Optimierung und Kontrolle”, Karl-Franzens-Universität, Heinrichstraße 22, 8010 Graz, Austria. e-mail: jerry.batzel@uni-graz.at F. Kappel: Institute for Mathematics and Scientific Computing and SFB “Optimierung und Kontrolle”, Karl-Franzens-Universität, Heinrichstraße 36, 8010 Graz, Austria. e-mail: franz.kappel@uni-graz.at S. Timischl-Teschl: Fachhochschule Technikum Wien, Hoechstaedplatz 5, 1200 Vienna, Austria. e-mail: susanne.teschl@technikum-wien.at Supported by FWF (Austria) under grant F310 as a subproject of the Special Research Center F003 “Optimization and Control” Mathematics Subject Classification (2000): 92C30, 49J15