(295b) Systems Approach to Nonlinear Microvascualture Blood Flow

The control of intracellular oxygen tension in neural tissue is a complex compilation of neurovascular oxygen transport into the tissue, oxygen diffusion through intracellular spaces, neuronal cellular metabolism, and the neural release of vasomodulating signals which spatially control distributed oxygen supply from nearby blood vessels. In order to test the various theories regarding this tightly controlled system, it is necessary to construct a model that can accurately model these dynamically integrated elements. Such a model must be able to quickly solve large sets of differential equations, capture the true nature of the intracellular oxygen problem utilizing a 3D approach, and provide physiologic results for blood pressure, blood flow, red blood cell velocity and oxygen tension.

The need for parametrically driven clinical models for predicting the perfusion of oxygen in brain tissue is apparent in the patients suffering from cerebral ischemia in the emergency room, the angiography suite and the intensive care unit. Though many topologically constrained models give a broad overview of the sensitivity of the oxygen supply to blood pressure, pH and respiratory rate, these models are insensitive to the patient-specific morphology that can widely skew the quantitative results of a patient, blurring the differentiation between healthy and necrotic tissue. Conversely, the abundance of morphologically accurate models of a single vessel structure within the Circle of Willis or a handful of capillaries in the microvasculature suffer from boundary condition estimations and give a highly accurate but narrow view of the patient’s health.

By generating a holistic model of the human cerebral vasculature which couples the large vessel morphology with the microvasculature structures in the pial vessels and capillary bed, a hemodynamic equivalent emerges which can be selectively focused and reassessed to determine a patients’ cerebral perfusion based on the solution of first-principle scientific computing effort.

The large vessel anatomy of the cerebral vasculature model including both arteries and veins was captured from MRA and MRV images using a centerline extraction approach and diameter extrusion to create both 3D parametric mesh of the vessel morphology. As the resolution magnetic resonance images does not extend to any vessel smaller than 100um in the large viewing scope necessary to capture the entire brain domain, a geometric vessel sprouting algorithm was employed to complete the pial and capillary vessels that complete the cerebral vasculature structure.

An extensive literature review was conducted to determine the physiologic and topological parameters for constructing a human cerebral microvasculature model. Though many of these models incorporated topologically accurate dense microvasculature structures, venous, arterial or capillary vessels, consisting structure of extravascular space, computational blood flow and pressure predictions, or oxygen transport - no model existed which portrayed all of these components. Many variations and improvements of the analytical Krogh cylinder model have supported, however these analytical models are frequently hampered by their inability to capture axial diffusion of oxygen in the tissue compartment and are limited to simple structures.

What has been established is that random and tortuous blood vessel networks enhance oxygen distribution and mixing compared to evenly spaced, ordered vessel configurations. Additionally, several studies of cerebral microvasculature using confocal microscopy and synchrotron-CT methods have been undertaken in human, rat and macaque models to rigorously establish topological features. By matching our model to the these studies it can be shown that a length scale, vessel lumen fractions, diameter distributions, and angle bifurcations can be rigorously matched using a modified 3D implementation of the Constrained Constructive Optimization proposed by Schreiner and Karch, 1999.

However, in order to parametrically predict the transport of oxygen based on pH, temperature, and hemodilution considerations, it is necessary to depict the biphasic nature of blood flow. Though some models attempt to predict the oxygen supply without this physical property of the blood, these predictions will remain insensitive to these clinical parameters. Many models exist which vigorously compute the 3D behavior of a biphasic system in a small set of blood vessels, usually surrounding an area of interest. What this work provides is a rheological equivalent of the solved biphasic problem projected across the holistic domain of the cerebral vasculature.

The relationship between the viscosity of blood, and its consequent tubular resistance, to the volume ratio of erythrocytes to plasma (referred to as hematocrit) requires addressing this multiphase behavior of the blood stream. A library of empirically based mathematical approaches have been developed to address this problem (Guibert, 2010), most notably the Kiani-Hudetz (1991) semi-empirical model based on hydrodynamic considerations and Pries’ (1990) phenomological description of the in vitro relative apparent viscosity.

While this relationships gives a fairly accurate view of the relationship between apparent viscosity and hematocrit (at Hd<0.7), the behavior of blood plasma and erythrocytes at a bifurcation requires a different investigation. The work of Dellimore (1983), Fenton (1985) and Pries (1989) have elaborated at length the mathematical relationships of fractional outflows and discharge hematocrit using a set of conditional relationships. While these models have been shown to obey conservation balances, this local scope of addressing a global problem requires nearly every vessel bifurcation to be treated as a special case. Additionally, anastomoses or cyclic graphs, such as those present in the capillary bed, are increasingly difficult to solve using these methods. To this end, a modified global conservation balance method, as proposed by Guibert, 2010, was implemented.

With the flow field established, two approaches can be constructed to determine the distribution of hematocrit.Decoupling the equations allows for solution by linear algebra approaches, for which iterative parallel solvers implemented in PETSc excel in decreasing the solution time of this otherwise nonlinear set of differential equations. With the flow field established, conservation of mass to each node was applied to determine the distribution of hematocrit throughout the network.

These morphometric considerations are necessary for accurately capturing the fate of oxygen delivered in the perfusion of cerebral tissue. By constructing morphologically accurate models of a topologically complex biological system such as the cerebral microvasculature, the predictions of first-principle computations can be used to (a) improve the interpretation of in vivo open cranial measurements acquired in both human and rodent models, (b) better understand the role of topological features of the cerebral microvasculature that promote inferior collateral to shed light on perfusion clinical imaging modalities, (c) exploring intercellular transport phenomena at a length scale and scope that is not available to imaging methods, and (d) gain insight into the transport of oxygen in the intracortical regions of human patients for whom invasive measurements are not an option.