On the rupture of an aneurysm.

The intracranial aneurysm, with an estimated occurrence of up to 4% in the general population, belongs among the most dangerous of cerebrovascular diseases. Although less than one-fifth of these cases results in a subarachnoid haemorrhage, the resulting disability and mortality rate is too high. More understanding is needed regarding aneurysm rupture and its deleterious effects. This study presents a physiologically-feasible explanation for the development of an enormous haemodynamic stress in the vicinity of the aneurysm in the cerebrovascular bed. Using a computer circulation model, this build-up of pressure was developed by sequentially imposing a series of interventions on the normal system.

Natural and surgically imposed anastomoses of the circle of Willis.

The efficacy of the circle of Willis as a flow equalizer is well known. Most cerebral macrovasculatures also contain other natural anastomoses which are activated in times of stenotic stress. For the past several decades, neurosurgeons have surgically augmented the cerebral network with additional vessels which further increase the flow of blood to a defrauded region of the brain. It is desirable to know quantitatively what role these anastomoses play in the delivery of blood. Apart from computer simulation, such knowledge remains out of reach to the medical community but with modern simulation techniques, a wealth of information can be made available. This paper presents both time-dependent and period-averaged results of a detailed study of cerebral anastomoses. Four different models of the macrovasculature in the circle of Willis vicinity have been developed, two of which contain an extracranial-intracranial (EC-IC) anastomosis. Five cases were developed to show how the amount of blood flow is related to the sizes of the anastomoses. Since the EC-IC bypass is only marginally beneficial in those patients whose cerebral circulations are well-equipped with naturally occurring anastomotic vessels, procedures should be developed to screen for their presence or absence. The fluid mechanics associated with the EC-IC bypass operation dictate a beneficial result. Since the surgical procedures fail to consistently show reduction in risk even when good grafts have been made, there is an enigma in the study group results.

A theoretical model for stress-generated fluid flow in the canaliculi-lacunae network in bone tissue.

A mathematical model was developed to study stress-induced fluid flow in the canaliculi-lacunae system in an osteon. The effect of canaliculi diameters on the magnitude and depth of penetration of squeeze flow through the canaliculi system was investigated. An optimal canaliculus diameter (which would maximize the fluid velocity through the canaliculi) was determined. For canaliculi diameters of 0.2 micron, squeeze flow can nourish four to five concentric layers of osteocytes in an osteon. It is possible that such stress-induced flow may be important in bone remodeling, and that lack of such flow may be one cause for producing osteoporosis due to immobilization.

A multicomponent, random walk model of transport and metabolism inside a neuron.

A model of multicomponent transport, consumption, and production of metabolites inside a neuron containing discrete mitochondria and glycolytic enzymes is developed using a random walk model of molecular transport. The ratio of anaerobic to aerobic metabolism which maximizes ATP production under normal, ischemic, and anoxic conditions is calculated. The ratio of the number of mitochondria to glycolytic enzymes which maximizes ATP under normal conditions is also calculated. Because the volume of the neuron is fixed, the sum of the number of mitochondria and glycolytic enzymes is fixed. This constraint is incorporated in the optimization process as an interior penalty function. Some of the advantages of employing the random walk technique are simple stoichiometry can be used to model consumption and production of metabolites, the geometry of the enzyme system and their active sites can be easily included in the model, and saturation of enzymes can be more easily modeled.

Mathematical analysis of transport and consumption of molecules in heterogeneous brain tissue (methodology).

A computer model of metabolite transport and consumption in heterogeneous brain tissue, using a combination of probabilistic and deterministic techniques is being developed. The metabolites are put into two separate classes: (I) those that have reached a membrane for the first time during a small time step, delta t, and (II) those that have not yet reached a cell membrane for the first time during that time step. The time dependent spatial distribution of class (I) molecules is determined using random walk theory, which takes into account the actual paths of the molecules. The variation of the spatial distribution of class (II) molecules with time is determined using the time dependent diffusion equation with a boundary condition of zero concentration on the enclosing membrane boundaries.

A circle of Willis simulation using distensible vessels and pulsatile flow.

The development of a one-dimensional numerical (finite-difference) model of the arterial network surrounding the circle of Willis is described based on the full Navier-Stokes and conservation of mass equations generalized for distensible vessels. The present model assumes an elastic wall defined by a logarithmic pressure-area relation obtained from the literature. The viscous term in the momentum equation is evaluated using the slope of a Karman-Pohlhausen velocity profile at the vessel boundary. The afferent vessels (two carotids and two vertebrals) are forced with a canine physiologic pressure signature corresponding to an aortic site. The network associated with each main efferent artery of the circle is represented by a single vessel containing an appropriate amount of resistance so that the mean flow through the system is distributed in accordance with the weight of brain irrigated by each vessel as determined from a steady flow model of the same network. This resistance is placed a quarter wave-length downstream from the heart to insure proper reflection from the terminations, where the quarter wavelength is determined using the frequency corresponding to the first minimum on an input impedance-frequency diagram obtained at the heart. Computer results are given as time histories of pressure and flow at any model nodal point starting from initial conditions of null flow and constant pressure throughout the model. Variations in these pressure and flow distributions caused by the introduction of pathologic situations into the model illustrate the efficacy of the simulation and of the circle in equalizing and redistributing flows in abnormal situations.