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Inert gases are used as markers or contrast agents in experimental and imaging methods that measure lung function. The premise is that by analysing the expired concentrations of inert gas, or by quantifying their distribution from \textit{in vivo} imaging, information can be gained about the ventilatory function of the lung. The inert gases that are used in the various methods can have very different physical properties, which potentially affects their distribution and hence the interpretation of the tests. Mathematical models have previously been derived to study expired inert gas washout, however they have generally had simple geometric structure and lack any contribution from airway resistance and tissue compliance. These models have provided much of the current understanding of the general features of expired gas washout tests, but they cannot be used to bridge to regional ventilation data acquired using contrast-enhanced \textit{in vivo} imaging. The goal of this thesis is to develop a mathematical model for predicting inert gas transport and mixing in the lung, such that the model includes the important mechanisms that contribute to the dynamics of the multiple breath nitrogen washout test and the distribution of xenon during contrast enhanced computed tomography imaging. A series of computational models are developed to simulate the distribution of inhaled gas, and its mixing in the periphery of the lung. The models are generated from anatomical and physiological data, and function within them is governed by physical laws. Advective flow distribution in the airways is dependent upon the density and viscosity of the gas being transported, the airway geometry, and the regional compliance of the parenchyma. One of the important features of a model is that it should not be more complicated than is necessary (Occams razor). The models that are developed here include complex geometry, however it is the effect of this complex geometry that it is necessary to study in linking the regional heterogeneity of ventilation to the lung geometry. Regional tracer gas concentration within the model geometry is determined by solving a one-dimensional equation that is dependent on the airway dimensions, the advective gas flow in the model and the diffusivity of the gas. Various models of advective flow are investigated beginning with a simple flow distribution based solely on parenchymal volume, progressing through flow distributions based on non-linear relationships between parenchymal pressure and parenchymal volume, to coupled models of the airway resistance and parenchymal distortion (including dependence on viscosity and density). This allows study of whether the gas transport properties influence the distribution of tracer gases in the lungs in a 1-D model. The mathematical model is demonstrated to be consistent with the current understanding of human lung physiology. It is physiologically consistent in terms of its predictions of ventilation distribution and gas mixing indices. The sigmoidal pressure-volume relationship that defines one of the tissue compliance models is shown to be both sufficient and necessary to develop a realistic degree of inter-regional heterogeneity in the human model. It is further shown that narrowed airways can produce an increase in inter-regional heterogeneity, concurrently with producing ventilation defects that are consistent with those seen in imaging studies of asthma. Heterogeneity in tissue compliance is shown to be sufficient for maintaining inter-regional heterogeneity in ventilation in microgravity. The mathematical model for inert gas transport is implemented in an anatomical model of the ovine lung, to which it is shown to be generally applicable. Based on model analysis of experimental measurements of inert gas distribution in the ovine lung, gas viscosity and advective-diffusive interaction are demonstrated to play no significant role in generating differences between the rate of washin and washout of inert gas in sheep. Dynamic pulmonary hyperinflation is also excluded as a factor. Due to advective-diffusive interaction in the periphery of the lung, the distribution of specific volume change predicted in the ovine computational model is shown to not necessarily match the distribution of specific ventilation estimated by a xenon-enhanced computed tomography imaging method. |
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