Do Vascular Networks Branch Optimally or Randomly across Spatial Scales?

Modern models that derive allometric relationships between metabolic rate and body mass are based on the architectural design of the cardiovascular system and presume sibling vessels are symmetric in terms of radius, length, flow rate, and pressure. Here, we study the cardiovascular structure of the human head and torso and of a mouse lung based on three-dimensional images processed via our software Angicart. In contrast to modern allometric theories, we find systematic patterns of asymmetry in vascular branching, potentially explaining previously documented mismatches between predictions (power-law or concave curvature) and observed empirical data (convex curvature) for the allometric scaling of metabolic rate. To examine why these systematic asymmetries in vascular branching might arise, we construct a mathematical framework to derive predictions based on local, junction-level optimality principles that have been proposed to be favored in the course of natural selection and development. The two most commonly used principles are material-cost optimizations (construction materials or blood volume) and optimization of efficient flow via minimization of power loss. We show that material-cost optimization solutions match with distributions for asymmetric branching across the whole network but do not match well for individual junctions. Consequently, we also explore random branching that is constrained at scales that range from local (junction-level) to global (whole network). We find that material-cost optimizations are the strongest predictor of vascular branching in the human head and torso, whereas locally or intermediately constrained random branching is comparable to material-cost optimizations for the mouse lung. These differences could be attributable to developmentally-programmed local branching for larger vessels and constrained random branching for smaller vessels.

当前推导代谢率与体重间异速生长关系（allometric relationship）的模型，均基于心血管系统的架构设计，并假定同一母血管分出的姊妹血管（sibling vessels）在半径、长度、流速与压强上均保持对称。本研究基于我们开发的Angicart软件处理的三维图像，对人类头部、躯干的心血管结构以及小鼠肺部的血管网络展开分析。与现有异速生长理论相悖，我们发现血管分支存在系统性不对称模式，这或可解释此前被报道的、代谢率异速标度（allometric scaling）预测结果（幂律或凹曲率）与实测数据（凸曲率）之间的矛盾。为探究血管分支系统性不对称的成因，我们构建了一套数学框架，基于已被提出的、在自然选择与发育过程中受青睐的局部分支节点级最优性原则推导预测结果。当前最常用的两类最优性原则分别为材料成本最优（material-cost optimizations，包括构建材料成本或血液容积最优），以及通过最小化功率损耗实现高效血流的流动最优。研究表明，材料成本最优原则下的解与整个血管网络的不对称分支分布相符，但与单个分支节点的分支模式匹配度欠佳。据此，我们还探索了尺度范围覆盖局部（分支节点级）至全局（整个网络）的约束型随机分支模型。我们发现，材料成本最优原则可最佳预测人类头部与躯干的血管分支模式，而对于小鼠肺部，局部或中等尺度约束的随机分支模型与材料成本最优原则的预测效果相当。上述差异或可归因于：大血管采用发育程序化的局部分支模式，而小血管则采用约束型随机分支模式。