The Geometry of Circulatory Shock: A Conceptual Multi-Scale Lagrangian Framework for Physiology-Informed Hemodynamic Phenotyping

Background: Hemodynamic failure remains a major determinant of mortality in critical illness, yet its detection is often delayed because conventional monitoring relies predominantly on Eulerian measurements that quantify pressure and flow magnitude without resolving the spatial and temporal organization of circulation. Consequently, clinically significant states of dysfunction may persist despite apparently stable hemodynamic indices. The Geometry of Shock is a conceptual and hypothesis-generating multi-scale framework intended to integrate established cardiovascular physiology with emerging computational approaches for the analysis of circulatory dysfunction. Framework: The proposed framework combines Guytonian venous return physiology and cardiopulmonary interactions with Lagrangian flow topology, geometric representations of circulatory equilibrium, topological data analysis, and physics-constrained inverse modeling. Rather than focusing exclusively on static thresholds of pressure and flow, the framework proposes a structural interpretation of circulation centered on the dynamic organization and coherence of blood transport across cardiovascular domains. Within this paradigm, under-recognized hemodynamic phenotypes—including stressed volume failure, oscillatory shock during spontaneous breathing, macro–microcirculatory decoupling, and pulmonary vascular pressure–flow dissociation—may emerge from disrupted coupling between vascular, cardiac, pulmonary, and microcirculatory systems. These states may represent reversible structural transitions in venous return geometry and cardiopulmonary interaction preceding overt circulatory collapse. Conclusions: By reframing shock as a disorder of circulatory structure and coherence rather than solely a deficit in flow, this framework proposes a mechanistic foundation that may support future approaches aimed at earlier recognition of instability, improved physiological characterization of hemodynamic phenotypes, and future development and prospective validation of physiology-informed computational decision-support strategies in critical care. These concepts remain exploratory and hypothesis-generating rather than clinically validated.