Healthy vasculature exhibits a hierarchical branching structure in which, on average, vessel radius and length change systematically with branching order. In contrast, tumor vasculature exhibits less hierarchy and more variability in its branching patterns. Although differences in vasculature have been highlighted in the literature, there has been very little quantification of these differences. Fractal analysis is a natural tool for comparing tumor and healthy vasculature, especially because it has already been used extensively to model healthy tissue. In this paper, we provide a fractal analysis of existing vascular data, and we present a new mathematical framework for predicting tumor growth trajectories by coupling: (1) the fractal geometric properties of tumor vascular networks, (2) metabolic properties of tumor cells and host vascular systems, and (3) spatial gradients in resources and metabolic states within the tumor. First, we provide a new analysis for how the mean and variation of scaling exponents for ratios of vessel radii and lengths in tumors differ from healthy tissue. Next, we use these characteristic exponents to predict metabolic rates for tumors. Finally, by combining this analysis with general growth equations based on energetics, we derive universal growth curves that enable us to compare tumor and ontogenetic growth. We also extend these growth equations to include necrotic, quiescent, and proliferative cell states and to predict novel growth dynamics that arise when tumors are treated with drugs. Taken together, this mathematical framework will help to anticipate and understand growth trajectories across tumor types and drug treatments.

中文翻译：

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使用分形几何和通用生长曲线作为诊断，比较肿瘤血管系统和代谢率与健康组织和预测对药物治疗的反应。

健康的脉管系统表现出分层的分支结构，其中，平均而言，血管半径和长度随分支顺序而系统地变化。相比之下，肿瘤血管系统在其分支模式中表现出较少的层次结构和更多的可变性。尽管在文献中强调了脉管系统的差异，但对这些差异的量化很少。分形分析是比较肿瘤和健康脉管系统的自然工具，特别是因为它已被广泛用于模拟健康组织。在本文中，我们对现有血管数据进行分形分析，并提出了一个新的数学框架，通过耦合预测肿瘤生长轨迹：（1）肿瘤血管网络的分形几何特性，(2) 肿瘤细胞和宿主血管系统的代谢特性，以及 (3) 肿瘤内资源和代谢状态的空间梯度。首先，我们对肿瘤中血管半径和长度比率的缩放指数的平均值和变化与健康组织有何不同提供了新的分析。接下来，我们使用这些特征指数来预测肿瘤的代谢率。最后，通过将此分析与基于能量学的一般生长方程相结合，我们得出了通用生长曲线，使我们能够比较肿瘤和个体发育的生长。我们还将这些生长方程扩展到包括坏死、静止和增殖细胞状态，并预测用药物治疗肿瘤时出现的新生长动态。综合起来，