A Mathematical Model of the Metabolic and Perfusion Effects on Cortical Spreading Depression

Abstract

Cortical spreading depression (CSD) is a slow-moving ionic and metabolic disturbance that propagates in cortical brain tissue. In addition to massive cellular depolarizations, CSD also involves significant changes in perfusion and metabolism—aspects of CSD that had not been modeled and are important to traumatic brain injury, subarachnoid hemorrhage, stroke, and migraine. In this study, we develop a mathematical model for CSD where we focus on modeling the features essential to understanding the implications of neurovascular coupling during CSD. In our model, the sodium-potassium–ATPase, mainly responsible for ionic homeostasis and active during CSD, operates at a rate that is dependent on the supply of oxygen. The supply of oxygen is determined by modeling blood flow through a lumped vascular tree with an effective local vessel radius that is controlled by the extracellular potassium concentration. We show that during CSD, the metabolic demands of the cortex exceed the physiological limits placed on oxygen delivery, regardless of vascular constriction or dilation. However, vasoconstriction and vasodilation play important roles in the propagation of CSD and its recovery. Our model replicates the qualitative and quantitative behavior of CSD—vasoconstriction, oxygen depletion, extracellular potassium elevation, prolonged depolarization—found in experimental studies. We predict faster, longer duration CSD in vivo than in vitro due to the contribution of the vasculature. Our results also help explain some of the variability of CSD between species and even within the same animal. These results have clinical and translational implications, as they allow for more precise in vitro, in vivo, and in silico exploration of a phenomenon broadly relevant to neurological disease.

Mathematical approaches to modeling of cortical spreading depression

“…II E 2, the interactions of the vascular effects and CSD have been modeled for the first time. 8 None of the models above for CSD include the important limitations that the vascular system places on the supply of oxygen and other chemicals that are needed by brain tissue during CSD. Energy consuming pumps utilize such chemicals and are responsible for returning the ionic concentrations in the ECS and the ICS back to their homeostatic states.…”

“…Energy consuming pumps utilize such chemicals and are responsible for returning the ionic concentrations in the ECS and the ICS back to their homeostatic states. Chang et al 8 (including the authors) focus on modeling important features essential to understanding the implications of neurovascular coupling during CSD. In this model, the sodium-potassium-ATPase, mainly responsible for ionic homeostasis and active during CSD, operates at a rate that is dependent on the supply of oxygen.…”

Focal cerebral ischemia and neurovascular protection

“…Depolarization is observed in the extracellular space as a prominent negative change of the slow potential, which may result from longitudinal gradients of depolarization along neurons, probably owing to zonal dendritic opening of ion channels that allow sustained influx of small cations in neurons (i.e., Na þ and Ca 2þ ). CSD can be modeled in a single neuron and it typically spreads at a rate of 2-6 mm/min [21]. CSDs are characterized by a breakdown of ion homeostasis that can be recovered by ion pumps if energy supply is adequate.…”

Towards dynamical network biomarkers in neuromodulation of episodic migraine

“…The advantage of such experiments is that the phenomenon of SD can be studied in isolation and the complex vasomotor feedback involved in SD can be neglected [41], though current modeling approaches include metabolic and perfusion effects on SD [42]. With this local focus set, SD is essentially a massive but temporary perturbation of ion homeostasis.…”