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
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Date
2019
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Massey University
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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.
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Keywords
Respiration, Regulation, Mathematical models