A nonlinear coupled model of capillary blood flow under external pressure considering vascular elasticity and collapse

External pressure alters microcirculatory blood flow, yet quantitative pressure-flow models remain limited. We develop a nonlinear three-stage coupling model unifying vascular elasticity and progressive collapse. A smooth transition function bridges the elastic, transitional, and collapse-dominant regimes, eliminating discontinuities in conventional piecewise models. The model reveals stage-specific control: stiffness dominates low-pressure flow, collapse pressure scale governs the transition, and attenuation coefficient controls high-pressure flow. Sensitivity analysis identifies nonlinear coupling between vessel radius and stiffness, together with synergistic geometric and collapse effects. These findings provide a quantitative framework for pressure-induced microvascular flow regulation and pressure-based therapeutic optimization.