This dissertation seeks to develop novel data-driven algorithms to automatically construct simulation-suitable meshes from volumetric medical image data to enable high-throughput, large-cohort analyses of patient cardiac function from medical image data. Image-based cardiac modeling derives geometries from patient medical image data, and can simulate a variety of aspects of cardiac function, including electrophysiology, hemodynamics, and tissue mechanics to explore improvements in cardiovascular diagnoses and treatments, and the biomechanical underpinnings of diseases. However, creating accurate models of the heart from patient image data requires significant time and human efforts and is the primary challenge of image-based modeling.

Deforming-domain computational fluid dynamics (CFD) simulations of the intracardiac hemodynamics, in particular, require both the geometry and the deformation of the heart from a sequence of image snapshots throughout the cardiac cycle. In the first part of the thesis, we present a two-stage approach to automatically generate CFD-ready left ventricle (LV) models from patient image data. This approach first uses deep-learning-based automatic segmentation and then uses geometry processing algorithms to robustly create CFD-suitable LV models from image data.

In the remainder of this dissertation, we developed end-to-end deep-learning algorithms to directly construct the surface meshes of the whole heart from volumetric medical image data without the need for a multistage approach. These algorithms leverage shape templates, shape deformation methods, and regularization functions during optimization to generate meshes that are suitable for computational simulations. Namely, we trained a graph convolutional neural network to deform a pre-defined mesh template to fit the whole heart in a target 3D image volume. Various mesh deformation methods were combined with deep learning to create whole heart surfaces, including direct displacements on meshes, free-form deformation by control lattices as well as control-handle-based deformation using bi-harmonic coordinates. We demonstrated the application of our method in constructing a dynamic whole heart mesh from time-series CT image data to simulate the cardiac flow driven by the cardiac motion. We also demonstrated the capability to switch template meshes to accommodate different modeling requirements.

The algorithms developed in this thesis were able to construct a 4D dynamic simulation-suitable mesh of the heart within a minute on a standard desktop computer whereas prior methods require hours of human efforts and multiple programs. The code was implemented in Python, fully open-sourced, and can be conveniently executed from the command line.

Heart disease is the leading cause of death worldwide, killing 17.9 million annually. Blood flow patterns in the heart have been shown to be associated with disease, but the adoption of blood flow diagnostics in the clinic has been limited by the difficulty of imaging blood flow inside the body. Color-Doppler ultrasound and 4D-flow Magnetic Resonance Imaging (MRI) are currently the two main tools used to image blood flow. Color-Doppler ultrasound is widely-accessible and relatively inexpensive, but traditional ultrasound only collects a single component of velocity on a single plane. 4D-flow MRI collects all three components of velocity in a 3D domain but is much more expensive and difficult to use. Here, we use computational tools to improve image processing and analysis of blood flow in the left ventricle for both color-Doppler ultrasound and 4D-flow MRI data. Specifically, we seek to develop methods for de-noising and reconstructing interior flow fields and for calculating kinetic energy and viscous dissipation as potential diagnostic tools from limited data.

First, we investigate tools to augment color-Doppler ultrasound data. We test multiple methods to calculate a second, in-plane component of velocity from the available velocity and geometry information. Previously, methods to reconstruct the second component of velocity were developed using an assumption that through-plane divergence was negligible. We introduce alternative methods to reconstruct the second component of velocity without making the through-plane assumption. However, when compared to previous methods, these alternative methods did not show an improvement in the accuracy of predicting the second component of velocity.

Following these results, we propose modifications to estimates of diagnostic measures from color-Doppler ultrasound data. For many diagnostic measures calculated from color-Doppler ultrasound data, the velocity is measured, a second component of velocity is calculated, and a 2D evaluation of the diagnostic measure is performed. Instead, we propose a 1D evaluation of diagnostic measures directly from the single, measured component of velocity. We introduce this metric for kinetic energy and viscous dissipation rate in the left ventricle, two measures that have been shown to be correlated with heart disease. Using computational fluid dynamics simulation results to obtain a true measurement from all three components of velocity in the entire ventricle, virtual ultrasound measurements were taken and the different types of estimates were calculated from the virtual ultrasound. These reduced dimensional estimates were then compared to the true, 3D values. Both the 1D and 2D estimates were correlated with the 3D values and kinetic energy was more robust to noise and lower grid resolution. These results indicate that 1D estimates, and kinetic energy especially, should be continued to be explored for further use in the clinic.

Next, we explore a modal analysis method to de-noise and reconstruct 3D flow fields that can be applied to 4D-flow MRI or color-Doppler ultrasound data. While many methods have been introduced to de-noise 3D velocity fields, this modal analysis method provides advantages because it results in a divergence-free flow field, satisfies necessary boundary conditions, and can be applied to multiple types of data. To test the method, it was applied to flow inside a cube, through a stenosis, around a cylinder, and inside the left ventricle using data from computational fluid dynamics simulations. The modal analysis method was shown to reduce noise from noisy velocity fields and to be able to adequately reconstruct velocity fields with missing data points or missing components of velocity. These results are promising for use with 4D-flow MRI data where a 3D flow field on a 3D domain is available. These results are also promising for use with more recent advancements of color-Doppler ultrasound that allow for measurements on multiple parallel planes and for developing capabilities of color-Doppler ultrasound that allow for measurement of a second, in-plane component of velocity.

Through reduced dimensional estimates of kinetic energy and viscous dissipation rate in the left ventricle that can be measured using color-Doppler ultrasound data and the introduction of a modal analysis technique to de-noise and reconstruct 3D flow fields, this dissertation advances image processing and analysis tools for blood flow in the left ventricle.

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 (CVD) is the leading cause of death worldwide. Image-based computational fluid dynamic (CFD) simulations have been widely used to characterize patient-specific blood flow features in the cardiovascular system. The simulation results provide high spatiotemporal resolution of functional indicators of CVD not revealed through imaging. Moreover, computational modeling provides the ability to virtually test interventions or devices. Despite this upside, patient-specific simulations remain computationally expensive, prone to numerical instabilities, and can be sensitive to method parameters, all of which have limited their clinical use. These limitations have also been prohibitive in research applications requiring parametric analyses such as uncertainty quantification, data assimilation, optimization, parameter tuning, etc. These factors motivate the need for reduced-order modeling (ROM) of blood flow to be used in place of, or in conjunction with, fully-resolved CFD modeling.

As such, this dissertation focuses on developing ROMs to efficiently compute blood flow in cardiovascular domains that can enable timely decision support, and applications requiring ensembles of numerical simulations. Through fundamental understanding of physics and physiology, we developed a novel physics-based ROM strategy, that not only calculates blood flow and pressure at a fraction of the computational cost compared to the current standard CFD, but can do so much more accurately than other existing ROM methods. The proposed method demonstrated consistent agreement with CFD simulations in terms of flow rate and pressure distribution over a broad range of hemodynamic conditions and vascular geometries. To evaluate the utility of this ROM methodology, we applied this framework to several translational and research applications.

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.