Is our breathing optimal? : this dissertation is submitted for the degree of Doctor of Philosophy, School of Natural and Computational Sciences, Massey University, New Zealand

Abstract

One of the open questions in relation to the control of amplitude and frequency of breathing is why a particular pattern of breathing is observed. This thesis explores the hypothesis that the particular combination of breathing frequency and amplitude realised, is optimal with respect to some objective function. Several objective functions have been suggested in the literature, such as the rate of work during inhalation, the average force exerted by the respiratory muscles, and the weighted sum of volumetric acceleration and work during inhalation; all of these objective functions were studied using 1D models and all provided physiologically acceptable minima under normal conditions. The thesis investigates optimal solutions of mathematical models that range from 2D to 6D and reflect more accurately the coupling between lung mechanics and gas exchange. It shows how published 6D and 5D models can be reduced to new 3D and 2D models. At its simplest, the 2D model consists of two piecewise linear differential equations. The use of higher dimension models require a new definition of the optimization problem as minimizing a given objective function subject to several constraints, such as satisfying the differential equations and maintaining one of the variables at a given average value. The optimal problem can be solved analytically in the case of the simplest 2D model, using concepts from optimal control theory. The analytical solution is used to verify a numerical algorithm that is then used to solve the more complex models. Solutions of the optimization problem for the different objective functions, previously suggested in the literature have been calculated. In all the optimal solutions found in this thesis, the duration of inhalation is equal to the duration of exhalation. However, under normal conditions, the time duration of inhalation is expected to be shorter than that of exhalation. This might be resolved by imposing additional constraints or by proposing a different hypothesis to explain why a particular pattern of breathing is observed.