The Warburg Geometry: Tumor Angiogenesis as a Thermodynamic Phase Transition

Tumor vasculature is characterized by chaotic, tortuous, and ”leaky” vessels that defy standard fluid-dynamicoptimization laws (Murray’s Law, α = 3.0). Here, we propose a thermodynamic explanation for this anomaly based onthe Theory of Thermodynamic Branching. We demonstrate that under conditions of metabolic stress (hypoxia/WarburgEffect), the vascular network undergoes a phase transition from minimizing hydrodynamic resistance (fluid transport)to minimizing diffusion impedance (nutrient exploration). Our simulations reveal that this shift in the cost functiondrives the optimal branching exponent from α ≈ 3.0 (Healthy) to exactly α ≈ 2.42 (Tumor). This suggests that tumorvessels are not merely ”broken pipes”, but geometrically approximate ”vascular dendrites”, adopting the same fractaldimension as cortical neurons to maximize nutrient uptake at the expense of flow efficiency.