Nonstationary Model of Oxygen Transport in Brain Tissue.

The paper addresses the mathematical study of a nonstationary continuum model describing oxygen propagation in cerebral substance. The model allows to estimate the rate of oxygen saturation and stabilization of oxygen concentration in relatively large parts of cerebral tissue. A theoretical and numerical analysis of the model is performed. The unique solvability of the underlying initial-boundary value problem for a system of coupled nonlinear parabolic equations is proved. In the numerical experiment, the tissue oxygen saturation after hypoxia is analyzed for the case when a sufficient amount of oxygen begins to flow into the capillary network. A fast stabilization of the tissue oxygen concentration is demonstrated. The reliability of the results of the numerical simulation is discussed.

Accounting for Tube Hematocrit in Modeling of Blood Flow in Cerebral Capillary Networks.

The aim of this paper consists in the derivation of an analytic formula for the hydraulic resistance of capillaries, taking into account the tube hematocrit level. The consistency of the derived formula is verified using Finite Element simulations. Such an effective formula allows for assigning resistances, depending on the hematocrit level, to the edges of networks modeling biological capillary systems, which extends our earlier models of blood flow through large capillary networks. Numerical simulations conducted for large capillary networks with random topologies demonstrate the importance of accounting for the hematocrit level for obtaining consistent results.

Extended model of impaired cerebral autoregulation in preterm infants: Heuristic feedback control.

Cerebral autoregulation is the ability to keep almost constant cerebral blood flow (CBF) for some range of changing the mean arterial pressure (MAP). In preterm infants, this range is usually very small, even absent, and a passive (linear) dependence of CBF on MAP is observed. Also, variations of the partial CO2 pressure and intracranial/venous pressure result in fluctuations of CBF. The absence of cerebral autoregulation may be a cause of intracranial hemorrhages due to instability of cerebral blood vessels, especially in the so-called germinal matrix which exists in a developing brain from 22 to 32 weeks of gestation. In the current paper, a mathematical model of impaired cerebral autoregulation is extended compared with previous works of the authors, and a heuristic feedback control that is able to keep deviations from a nominal CBF within a reasonable range is proposed. Viability theory is used to prove that this control can successfully work against a wide range of disturbances.

Direct modeling of blood flow through the vascular network of the germinal matrix.

A premature birth, before completion of the 32nd pregnancy week, increases the risk of cerebral hemorrhage. The cause of brain bleeding is very often the germinal matrix of the immature brain. The germinal matrix consists of richly vascularized neuroepithelial cells and is located over the lower part of the head of the caudate nucleus. By 32-36 gestation weeks, the germinal matrix essentially disappears so that its hemorrhage is a disease of premature infants. The aim of this paper consists in developing a model of the brain vascular network and computing the pressure distribution in the germinal matrix, particularly near arterioles and venules, where cerebral hemorrhage may occur. Capillary networks consisting of several millions of vessels are directly simulated in the present study.