Abstract
This paper introduces a linear operator for the purposes of quantifying the spectral properties of transport within resistive trees, such as airflow in lung airway networks. The operator, which we call the Maury matrix, acts only on the terminal nodes of the tree and is equivalent to the adjacency matrix of a complete graph summarizing the relationships between all pairs of terminal nodes. We show that the eigenmodes of the Maury operator have a direct physical interpretation as the relaxation, or resistive, modes of the network. We apply these findings to both idealized and image-based models of ventilation in lung airway trees and show that the spectral properties of the Maury matrix characterize the flow asymmetry in these networks more concisely than the Laplacian modes, and that eigenvector centrality in the Maury spectrum is closely related to the phenomenon of ventilation heterogeneity caused by airway narrowing or obstruction. This method has applications in dimensionality reduction in simulations of lung mechanics, as well as for characterization of models of the airway tree derived from medical images.
Highlights
In healthy human lungs, the airways form a bifurcating tree where, on average, around the first 16 generations of airways are purely conductive and serve to transport gas from the mouth to the alveolar region where the majority of gas exchange takes place
The Frobenius norm of the inverse Laplace operator is largely comprised of high-resistance modes, and so the corresponding eigenvectors are centralized on high-resistance motifs in the network
The mode reconstruction is more accurate in the image-based networks than the Horsfield networks for either operator, which is due to the increased variance of resistance in these networks; the effect is stronger for the Maury operator
Summary
The airways form a bifurcating tree where, on average, around the first 16 generations of airways are purely conductive and serve to transport gas from the mouth to the alveolar region where the majority of gas exchange takes place. The conducting airways terminate in approximately 30 000 respiratory units (or acini) [1], where diffusion becomes the dominant transport mechanism. Resistive flow in the conducting airways, as well as tissue compliance, determine the ventilation mechanics and require a high-dimensional mathematical model to be represented accurately [2,3,4]. The methods presented here provide a new approach to characterize the resistance structure and its effect on VH. This will be useful for visualization and characterization of complex airway models based on patient computed tomography (CT) images, lung casts or micro-CT scans of excised lungs
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