论文标题
使用图理论计算毛细管网络中血流模型中产生的拉普拉斯操作员
Using graph theory to compute Laplace operators arising in a model for blood flow in capillary network
论文作者
论文摘要
维持脑血流对于足够的神经元功能至关重要。先前的脑毛细管网络计算模型预测，异质性脑毛细血管流动模式会导致较低的脑组织部分氧气压力。有人提出，这可能导致许多疾病，例如阿尔茨海默氏病，急性缺血性中风，创伤性脑损伤和缺血性心脏病。我们以前已经开发了一种计算模型，该模型用于详细描述流动异质性对组织氧水平的影响。该论文的主要结果是，对于一类毛细管网络，段直径或电导的扰动总是会导致氧气水平降低。使用数值模拟和数学分析验证了该结果。但是，分析取决于与段流速有关的功能的拉普拉斯操作员的新颖猜想，以及它们如何依赖电导。本文的目的是为一般类网络的猜想提供数学上严格的证明。证明取决于确定由毛细管网络引起的某些图中的树木和森林的数量。
Maintaining cerebral blood flow is critical for adequate neuronal function. Previous computational models of brain capillary networks have predicted that heterogeneous cerebral capillary flow patterns result in lower brain tissue partial oxygen pressures. It has been suggested that this may lead to number of diseases such as Alzheimer's disease, acute ischemic stroke, traumatic brain injury and ischemic heart disease. We have previously developed a computational model that was used to describe in detail the effect of flow heterogeneities on tissue oxygen levels. The main result in that paper was that, for a general class of capillary networks, perturbations of segment diameters or conductances always lead to decreased oxygen levels. This result was verified using both numerical simulations and mathematical analysis. However, the analysis depended on a novel conjecture concerning the Laplace operator of functions related to the segment flow rates and how they depend on the conductances. The goal of this paper is to give a mathematically rigorous proof of the conjecture for a general class of networks. The proof depends on determining the number of trees and forests in certain graphs arising from the capillary network.
