The nonlinear (finite) deformation of flow is studied from the geometric point of view. First- and second-order covariant rates of deformation tensors are derived in the context of an arbitrary connection on frame bundles with torsion and Riemannian curvature. Also, the compatibility conditions of continua are extended to deformations on Einstein manifolds using a pullback Ricci curvature. We relate our compatibility conditions with existing formulations on Euclidean space.

A particular interest in studying nonlinear deformation here is to develop evolution equations for the principle rates and directions of finite deformation, which are quantities that are widely used to understand the flow topology. To this end, we present a spectral decomposition of two-point tensors on Riemannian manifolds. We derive evolution equations for the spectral decomposition, namely for the eigenvalues and eigenvectors of deformation tensors on Riemannian manifolds in terms of arbitrary covariant rates and using intrinsic Lie derivatives. We demonstrate the analogy between the spectral decomposition of deformation and attitude kinematics of moving frames. Geometric numerical integration of the evolution equations is explored through Lie group actions, namely the Lie algebra of quaternions. Our formulation extends attitude kinematics of deformation with Euler angles on 2-manifolds with the Riemannian metric. Numerical and analytic examples for both Euclidean space and Hamiltonian flows on Kählerian (symplectic) manifolds are given.

Accompanying our theoretical framework, a comprehensive high-performance computing platform for the Lagrangian analysis of flow is developed. The platform has been integrated into an online gateway with server-client architecture and available online as a community resource. Our platform is capable of processing up to billion-point grids and has been employed in real-time field experiments. We demonstrate practical applications of globe scale geophysical flows using comprehensive datasets.

Vascular homeostasis is the preferred state that blood vessels try to maintain against external mechanical and chemical stimuli. The vascular adaptive behavior around the homeostatic state is closely related to cardiovascular disease progressions such as arterial aneurysms. In this work, we develop a multi-physics computational framework that couples vascular growth & remodeling (G&R), wall mechanics and hemodynamics to describe the overall vascular adaptive behavior. The coupled simulation is implemented in patient-specific geometries to predict aneurysm progression. Lyapunov stability analysis of the governing equations for vascular adaptation is conducted to obtain a stabilizing criterion for aneurysm rupture. Also, to facilitate patient-specific computations, an algorithm is proposed to generate vascular homeostatic states by incorporating non-uniform residual stress and specifying optimal collagen fiber deposition angles. Since the accuracy and effectiveness of the computational models relies on properly estimating the unknown/hidden model parameters, we also demonstrate recent progress on improving data filtering techniques for inverse problems derived from model calibration. First, iterative ensemble Kalman filter (IEnKF) is applied to solve the inverse problems. The convergence of standard IEnKF is discussed and we show that the poor convergence of IEnKF is caused by the covariance shrinkage effect of the standard Kalman updates. An ensemble resampling based method is proposed to resolve this issue by perturbing the covariance shrinking via ensemble resampling and ensuring "correct" update directions by keeping the first and second moments of the resampling distribution unchanged. In the case of ill-posed inverse problems, we demonstrate that solution non-uniqueness can be overcome by incorporating additional constraints in a Bayesian inference framework. Constraints are imposed by defining a constraint likelihood function, and the method is demonstrated in both exact Bayesian and approximate Bayesian inference scenarios.

Abdominal aortic aneurysm (AAA) is a permanent local enlargement of the abdominal aorta. Complex anatomies, presence of side branches, and pulsatility of blood flow creates a complex chaotic flow field in AAAs. The progression of AAA can lead to rupture, which is one of the leading causes of death in the elderly. In this study, the flow topology in AAAs, role of hemodynamics in AAA progression, complex vectorial wall shear stress (WSS) patterns, and near-wall transport in AAAs were investigated.

Patient-specific computational fluid dynamics (CFD) was used to obtain blood flow information. Lagrangian coherent structures (LCS) were computed to study the flow physics. The utility of these structures in studying chaotic mixing and transport, flow separation, and vortex wall interaction was demonstrated in different patients. The effect of exercise on flow topology and quantitative mixing was evaluated. The evolution of a systolic vortex formed in the proximal region, strongly influenced the flow topology in the aneurysms. Intraluminal thrombus (ILT) deposition and lumen progression were quantified in several patients using magnetic resonance imaging over a 2--3 year followup. Point-wise spatial correlation of hemodynamic parameters to ILT deposition, revealed a negative correlation between oscillatory shear stress and ILT deposition. This was attributed to persistence recirculation, which can lead to unidirectional backward WSS.

Complex vectorial variations in WSS was studied. Namely, variations in WSS magnitude, direction, and vector in space and time were quantified and compared. Several new WSS measures were introduced to better quantify WSS vectorial variations. The concept of Lagrangian wall shear stress structures (WSS LCS) was introduced. WSS was scaled to obtain a first order representation of near-wall velocity. Tracers representing biochemicals in thin concentration boundary layers were tracked on the aneurysm surface based on the WSS vector field. Formation of coherent structures from WSS tracers were shown. The WSS LCS organize near-wall transport in high Schmidt number flows and could be used to predict regions of high near-wall stagnation and concentration. A wall shear stress exposure time (WSSET) measure was introduced to quantify near-wall stagnation and concentration. Excellent agreement between WSSET and surface concentration obtained from 3D continuum mass transport was obtained. Finally, the important roles that WSS fixed points play in cardiovascular flows was discussed.

Cardiovascular disease has remained the leading cause of death worldwide. In addition, the 2015 Annual Data Report from the US Renal Data System shows that sudden cardiac death and arrhythmias are the leading cause of mortality for patients on routine hemodialysis for end-stage renal disease (ESRD). In this study, we aim to augment cardiac magnetic resonance (CMR) images obtained for such patients through a registration framework developed for analyzing two-dimensional (2D) CMR image sequences. The framework consolidates existing tools, such as segmentation and the large deformation diffeomorphic metric mapping (LDDMM) algorithm, and provides a systematic process to define the circumferential and radial directions in general left ventricle geometries, to quantify its myocardial strain throughout the cardiac cycle. While CMR image sequences are used in this work, the framework is designed to be agnostic to the imaging modality. In conjunction with this framework, validation tests were developed using synthetic deformation data generated through solid mechanics simulations. This was done for three analytical shapes: a box, thick-walled cylinder, and a thick-walled ellipsoid, all in three dimensions. This synthetic data was then used to provide a 2D input to the registration framework, which allowed a direct comparison of the deformation computed from the solid mechanics simulation and the registration framework. All of this was developed to use data that is generally readily accessible in clinical settings, as well as to provide a template for designing validation tests for myocardial strain computations.

In medical conditions with disrupted cerebrovascular flow, including chronic alterations throughout aging or acute events such as ischemic stroke, cerebrovascular modeling can provide mechanistic and treatment insights. Reduced-order cerebrovascular modeling can be particularly useful for patient-specific or population-averaged cerebrovascular hemodynamics, where the governing flow dynamics include physically-driven non-linear constraints with active homeostatic regulation as well as highly variable and interconnected cerebral blood vessel networks. A coupled one-dimensional lumped parameter network (1D-LPN) cerebrovascular model is described with myogenic and CO2 reactivity autoregulation modeling, circle of willis anatomical variations, cortical collateral flow compensation, cerebrospinal fluid generation/reabsorption, and intracranial pressure dynamics modeling. For parameter tuning of the numerous cerebrovascular model variables, a Bayesian optimization procedure establishes crucial parameter values with the remaining model parameters assigned through vascular aging and cerebral autoregulation quantitative population-averaged reviews. A specific focus on cerebrovascular aging alterations, a major source of vascular variance besides patient-specific variations, explores the parameter tuning procedure in a young population. The coupled 1D-LPN cerebrovascular model design, age-related vascular critical review, cerebral autoregulation critical review, and model parameter tuning procedure, comprises a prototype vascular test bed with the capability to model cerebrovascular-specific medical conditions while accounting for age-related vascular alterations.

Cardiovascular disease has remained the leading cause of death worldwide for the past 15 years, and organizations such as the American Heart Association (AHA) and the National Institute for Health (NIH) spend hundreds of millions of U.S. dollars annually to investigate heart disease and stroke. Local characteristics of blood flow in the heart and the rest of the cardiovascular system provide important information in both understanding progression of and diagnosis of cardiovascular diseases. Unfortunately, current medical imaging techniques cannot provide data with high enough temporal and spatial resolution to extract meaning- ful and accurate research conclusions. Thus, many researchers investigate cardiovascular diseases using a patient-specific blood flow simulation framework. In this framework, a pa- tient’s geometry is constructed on a computer from medical image data, and a numerical simulation, such as finite element analysis (FEA), is used to provide very high detail infor- mation. Typically, the most time consuming step and also one of the most crucial steps in this pipeline is constructing the geometry of interest from the image data. In addition, many of the tools to create an image-based model are commercial, not readily available, or dispersed amongst a variety of software packages. This dissertation discusses two main avenues of research: (1) the development of unique, customized, and open-source tools for vascular model construction and meshing for FEA and (2) the investigation into and the creation of novel model construction methods for an alternative of FEA called isogeometric analysis (IGA). All of the tools developed were implemented using open-source tools such as the Visualization Toolkit (VTK) and have been implemented into the software framework SimVascular. In addition, many of the methods developed were tested for applicability and robustness on the open-source vascular model repository, which is a large database of over 100 vascular models provided by the Open Source Medical Software Corporation (OSMSC).