Diseases like stroke, tumors and hydrocephalus affect brain dynamics in millions of people worldwide. This project aims at advancing mathematical methods for studying soft tissue fluid interaction in the brain. The mathematical models will address large brain deformations in dynamic interaction with blood and cerebrospinal fluid motion with three salient features:an Euler-Lagrangian moving grid formulation for soft porous brain tissues; integration of cerebral vasculature into a porous deformable cell matrix.; and advancement of direct numerical solution algorithms based on inexact Newton-type methods with Krylov subspace iterations.
For addressing the interdisciplinary nature of this project, the PI assembled a research team composed of biochemical engineers, two mathematicians, a neurosurgeon and an MRI physicist and biomedical engineer. Two major goals of the project are: Aim #1. Creation of a multi-scale fluid-structure interaction framework accounting for brain deformation mechanics with cerebrospinal fluid dynamics. Aim #2. We will incorporate realistic models of the cerebral vasculature.
Existing models for computing large brain deformation dynamics are still unsatisfactory. The proposed holistic model of intracranial dynamics addresses key deficiencies. Specific aims (i) bridge the gap between recent advancements in medical imaging and rigorous mathematical analysis, (ii) create novel mathematical formulations for fluid-solid interaction for deforming soft porous media, and (iii) account for dilations of the cerebral vasculature. Comprehensive multi-phase models of brain, blood and cerebrospinal fluid interaction derived from patient-specific images have never been attempted before to the best of our knowledge.
Currently quantitative analysis of medical images is slow, because there exists few mathematical methods for evaluating the interactions of blood, brain tissue and cerebrospinal fluid flow, distributed in the three-dimensional space and in time. Magnetic resonance imaging, angiography or computer tomography, will move forward biological and medical knowledge, when rigorous mathematical models to accurately predict transport phenomena become available. The proposed research will integrate existing medical imaging technology with mathematical modeling and scientific computing. The PI will share computational grids of human brains and symbolic codes for cerebral transport phenomena via the world-wide web. By this open internet approach, they will attract the interest of a growing scientific community. It is expected that this project will lay the mathematical foundations for a quantitative understanding of flow physics associated with the onset and progression of certain types of brain disorders. Thus, the project is prerequisite for generating better diagnostic and new treatment options in the future. The education plan provides specific steps for the dissemination of proposal outcomes and includes high school math and science education via the research experiences for teachers (RET), as well as outreach to minorities and underrepresented groups through undergraduate research (REU) mentorship.
NSF Report: Project Outcomes for the General Public Cerebral blood flow has profound effects on the overall system of brain mechanics and physiological functions. The initial scope of the proposed research to determine the soft tissue deformation dynamics under pulsating cerebral blood flow was overshadowed by the physiologic implications of patient-specific cerebral vascular morphology and the necessity of reconstructing patient-specific models over black-box geometric abstractions. Comprehensive predictive models are needed to systematically characterize the ability of the cerebral vasculature to distribute blood flow in response of such pathological damage to the blood vessel architecture. The fact that abnormal brain dynamics are clinically linked with circulatory disorders gives further support for including cerebral vasculature in mathematical models. The original project vision of reconstructing 3D, physiologically consistent, accurate, patient-specific brain geometries of functional regions needed for holistic model of intracranial dynamics relating pulsatile blood and cerebrospinal fluid dynamics was captured in the research endeavor. Research Objectives A morphologically accurate reconstruction of the vasculature structure that unites the large arterial vessel architecture in the Circle of Willis, the pial network along the cortical surface, penetrating arterioles, the capillary bed, and the draining penetrating venules, and the large venous confluences was constructed. Advanced image reconstruction algorithms utilizing the VMTK library were implemented and improved to achieve realistic 3D vasculature networks from medical images obtained from clinical MRA, MRV, CTA, 3DRA (DSA) images. The vessel centerlines extracted from these images capture the topology of the major blood vessels only, vessels below the resolution of 3T MRA (clinical MR) are unable to capture vessels with diameters < 100μm. These networks coupled with an improved vessel growth algorithm developed by the group to fill the gaps of microvessels not detectable in standard clinical medical images. This improved vessel growth algorithm is parametrically constrained, so microvessel geometry for any section of the brain, or for any species, could be constructed. We constructed a patient-specific human intracranial computer model which determined the volumetric blood flow rate in every landmark vessel of the cerebral vasculature down to the pial vessel network. A second, more detailed model capturing the architecture of a sub-cubic centimeter section of the human brain was constructed as well. This model was realized with continuous connectivity from left and right interior cerebral artery and basilar artery inlet to the left and right jugular venous outlet. Poiseuille blood flow equations were solved over the network using heterogeneous blood viscosity, and dynamical quasi-steady flow computations were solved for artificial angiography transit time throughout the cerebral vasculature. The sub-cubic centimeter human brain model was embedded in an unstructured extracellular brain tissue mesh and was enhanced with equations of oxygen transport between the two domains. A dual mesh approach was developed, and equations of convection were constructed for oxygen transport through the blood vessels coupled with transport diffusion equations in the extracellular tissue mesh where it was metabolized. This innovative full model of cerebral oxygen transport and metabolism was a significant contribution to the field of microvasculature modeling, resulting in a publication. Broader Impact These medical image reconstructions enhanced with vasculature growth are capable of constructing a holistic model of the entire cerebral vasculature including the cortical microvasculature. Computation of the hemodynamic state for even the most simplified version of this model requires over 500,000 equations, while the complete microcirculation with all capillaries ones require nearly 1 billion. Therefore progress has been made to parallelize the computations. The NSF resource XSEDE has been utilized and parallel computations of more detailed models exceeding 500,000 equations (both the whole brain model and sub-cubic centimeter model) have been computed using the PETSc library. Due to the large information content generated by such models, advances have been made in the visualization of this data. Using tools made available through the NIH funded vascular modeling toolkit (VMTK), immersive 3D environments have been implemented in partnership with the computer science department at our institution. By successfully integrating the expertise needed to execute multi-scale scientific computing, massive parallel mathematical libraries, and immersive 3D environment we have constructed a fully exploitable, patient?specific, morphologically accurate model of the human cerebral vasculature which predicts the volumetric blood flow rate, velocity, and blood pressure as well as oxygen transport to the cortical tissue.
