Mathematical Modeling of Neurovascular Coupling

Abstract

Local regulation of cerebral blood flow is essential for providing active regions of the brain with sufficient oxygen and nutrients. Blood flow to a given region depends on the diameters of vasoactive arterioles upstream of the capillaries that perfuse the region. Mechanisms to coordinate vascular responses are necessary to direct blood flow to areas of demand, a phenomenon known as neurovascular coupling. This coordination is achieved through conducted responses, electrical signals propagating along vessel walls. The goal of this work is to employ theoretical modeling techniques to gain an improved understanding of the mechanisms and behaviors of conducted responses and their effects on blood flow in the cerebral microvasculature. In chapter 2, the propagation of conducted responses along the vessel wall is explored. Previous models for conducted responses have utilized standard cable theory, which predicts exponential decay of the signal from the site of stimulus. Experimental studies of these signals demonstrate passive decay in some cases, but propagation of conducted responses without decay has also been observed. The activity of potassium inward rectifier (Kir) channels have been identified as a possible reason for this behavior. Neuronal activity leads to the release of potassium into the extracellular space, which increases the activity of Kir channels. This provides a basis for neuronal initiation of conducted responses. A cable-theoretic model of the vascular endothelium incorporating Kir channels is developed and analyzed in order to explore these phenomena. Our results indicate that non-decaying signals in the form of traveling waves can occur. Such behavior occurs only for a limited range of parameter values. Outside this range, responses decay with distance, but the nonlinear properties of the Kir channel can still enhance propagation distances. In chapter 3, the role of these conducted responses in the regulation of cerebral blood flow is examined. A model is developed that includes the metabolic conducted response, together with responses to circumferential wall tension (myogenic response) and wall shear stress. A compartmental model for flow regulation is used, with compartments connected in series representing arterial and venous vessels together with capillaries. The geometrical parameters of this model are derived from network structures observed in the cerebral cortex. A single compartment consisting of relatively short penetrating arterioles is assumed to be vasoactive. Capillaries are stimulated with varying concentrations of extracellular potassium. This model indicates that physiologically reasonable levels of stimuli can account for the amount of flow increase observed during neurovascular coupling despite the short lengths of vasoactive arterioles. The brain also tends to maintain blood flow constant despite changes in arterial blood pressure, a phenomenon known as autoregulation. The model including the myogenic response predicts a much weaker level of autoregulation than is observed experimentally. The metabolic conducted response is not triggered in this case. A vasoconstrictive response to increased shear stress has been observed experimentally in brain microvessels. Inclusion of this response enhances the autoregulation predicted by the model. In summary, this work presents a quantitative framework for understanding the mechanisms underlying cerebral blood flow regulation.