Stability of a microvessel subject to structural adaptation of diameter and wall thickness

Vascular adaptation--or structural changes of microvessels in response to physical and metabolic stresses--can influence physiological processes like angiogenesis and hypertension. To better understand the influence of these stresses on adaptation, Pries et al. (1998, 2001a,b, 2005) have developed a computational model for microvascular adaptation. Here, we reformulate this model in a way that is conducive to a dynamical systems analysis. Using th ese analytic methods, we determine the equilibrium geometries of a single vessel under different conditions and classify its type of stability. We demonstrate that our closed-form solution for vessel geometry exhibits the same regions of stability as the numerical predictions of Pries et al. (2005, Remodeling of blood vessels: responses of diameter and wall thickness to hemodynamic and metabolic stimuli. Hypertension, 46, 725-731). Our analytic approach allows us to predict the existence of limit-cycle oscillations and to extend the model to consider a fixed pressure across the vessel in addition to a fixed flow. Under these fixed pressure conditions, we show that the vessel stability is affected and that the multiple equilibria can exist.