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.

中文翻译：

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谱图理论有效地表征肺气道网络中的通气异质性

本文介绍了一个线性算子，用于量化阻力树内传输的频谱特性，例如肺气道网络中的气流。我们称之为 Maury 矩阵的算子只作用于树的终端节点，相当于一个完整图的邻接矩阵，它总结了所有终端节点对之间的关​​系。我们展示了 Maury 算子的本征模式作为网络的松弛模式或电阻模式具有直接的物理解释。我们将这些发现应用于肺气道树中理想化和基于图像的通气模型，并表明莫里矩阵的光谱特性比拉普拉斯模式更简洁地表征了这些网络中的流动不对称性，并且Maury谱中的特征向量中心性与气道狭窄或阻塞引起的通气异质性现象密切相关。该方法在肺力学模拟中的降维以及从医学图像中提取的气道树模型的表征中具有应用。