(666e) Parametric Cerebral Blood Flow Predictions

Abstract

The control of intracellular oxygen tension in neural tissue is the result of a complex interaction of transport across the vasculature endothelial wall, diffusion through intracellular spaces, neuronal cellular metabolism, and the release of vasomodulating signals. 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.

 1.    Introduction

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. Few models give context to oxygen supply sensitive to blood pressure, pH and hemoglobin oxygenation, and fewer models still incorporate patient-specific morphology that can widely skew the result. These topological parameters are critical for assessing quantitative results, allowing for the differentiation between healthy and ischemic 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.

 2.    Methods

A holistic model of the human cerebral vasculature was generated 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.

2.1 Topological Construction

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 a network representation of the cerebral vasculature. As the resolution magnetic resonance images does not extend to any vessel smaller than 100µm 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. This 3D topologically constrained vessel growth algorithm was modified from the Constrained Construction Optimization (CCO) algorithm introduced Schreiner and Karch (1999). By expanding the image-reconstructed network with these geometric space-filling methods, a morphologically accurate representation of the arterial and venous trees of the cerebral vasculature was constructed, containing the Circle of Willis, the left and right middle, anterior and posterior cerebral arteries, the Superior and Inferior Sagittal Sinus. Secondary branches arising from these major vessels were sprouted along the surface of the cerebral cortex geometry, which was obtained from T₂-weighted MRI. A modified Voronoi mesh constructed from this geometry, and was connected to these acyclic bifurcating trees as a microvasculature surrogate, completing the connectivity of the model. The final model consisted of over 100,000 vessel bifurcations and over 80,000 vessels with three inlet boundary vessels (left and right internal carotid, basilar artery) and two outlet boundary vessels (left and right jugular veins).

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. 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 these studies it can be shown that a length scale, vessel lumen fractions, diameter distributions, and angle bifurcations can be rigorously matched.

2.2 Blood Flow Equations

Blood flow was described using established hemodynamic network models established by Pries (1990). Flow resistance of each cylindrical segment was approximated by the Poiseuille law, where the pressure drop across a single vessel is equivalent to the product of the bulk blood flow through that segment and the segment’s vascular resistance. The continuity equation was applied at each vessel connection. The pulsatile nature of the microcirculation was ignored, and a quasi-steady flow assumed as low Womersley numbers (Wo<0.1) dominate in the microvasculature. Inertia is neglected in our simplified fluid flow model due to low Reynolds numbers (Re<1). The effect of biphasic blood rheology was approximated by the Fahraeus‑Lindqvist equation, which predicts the apparent viscosity as a function of vessel segment diameter and the volumetric ratio of erythrocytes to plasma volume, called hematocrit.

To parametrically predict the transport of oxygen as a function of systemic blood thinning, or a reduction of hematocrit, it is necessary to depict the biphasic nature of blood flow. A global method was implemented in this work to capture the relationship between the hematocrit, the vascular resistance and the oxygen delivery to tissue.  

2.3 Oxygen Delivery to Tissue

After blood flow was solved, convective transport equations were constructed across each vasculature bifurcation. Using the hematocrit phase alone for the blood flow driven transportation of oxygen throughout the cerebral vasculature, an inlet Neumann boundary condition was assigned to the arterial inlets and a draining flux was assigned at the venous outlet. Oxygen molar flux was computed across each vessel segment, oxygen tension was assigned to the node of each.

A dual mesh technique was employed using the above described vasculature network of cylindrical segments and a volumetric tetrahedral brain mesh constructed from the T₂-weighted MRI of the same patient. A tetrahedral tissue volume may contain multiple vascular bifurcation nodes; however each vascular bifurcation node is uniquely assigned to one tetrahedral tissue volume. The species conservation for each network node was balanced, and the overall mass transfer coefficient is a function of wall thickness and the oxygen permeability of epithelial tissue. Oxygen mass exchange is driven by the concentration difference between oxygen tension in the hematocrit and oxygen tension in the brain tissue.

The extravascular space modeled by the volumetric mesh is a modeled is a single continuum with extracellular and intracellular space lumped together, which was consumed at a constant rate. Finally, diffusion equations were assigned in this extravascular tissue space and solved to predict the transport of oxygen throughout the tissue domain. Though many variations and improvements of the analytical Krogh cylinder model have been proposed, 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. For these reasons, the discretized approach described above was employed.

2.4 Parallel Solvers

With the flow field established, linearization of the nonlinear biphasic fluid flow eqiautions was proposed to determine the distribution of hematocrit. Decoupling the flow and hematocrit transport equations allows for solution by linear algebra approaches, for which iterative parallel solvers implemented in PETSc excel in decreasing the solution time of this nonlinear set of ordinary differential equations. With the flow field established, conservation of mass to each node was applied to determine the distribution of hematocrit throughout the network.

 3.    Results

The pial arterial inlet pressure was set to 90mmHg, the pial venous drain had 10mmHg, as obtained from literature values, giving a volumetric flow rate of 750mL/min, which is comparable to measured values.  The oxygen tension in this whole brain model gradually dropped from 90mmHg in the internal carotid arteries down to 30mmHg in jugular veins. The redirected oxygen extraction compares well to cortical measurements, and radial oxygen perfusion profiles in the tissue were also in good agreement to measured values.

Radial oxygen tension in the molecular layer ranged from 80mmHg at a distance of 10μm, to 45mmHg at 50μm distance. These oxygen gradients seem to be a result of the capillary‑depleted regions adjacent to larger arterioles, which were captured in the detailed microvasculature surrogate. Our simulations for regions without larger vessels show shallow oxygen profiles, completely flattening about 30μm away from the nearest capillary, equivalent to healthy results measured in rat models.

 4.    Conclusion

This morphologically accurate approach is necessary for accurately capturing the fate of oxygen delivered in the perfusion of cerebral tissue.These models of topologically complex biological system such as the cerebral microvasculature and 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.