Spectral Approximation for Quasiperiodic Jacobi Operators

Quasiperiodic Jacobi operators arise as mathematical models of quasicrystals and in more general studies of structures exhibiting aperiodic order. The spectra of these self-adjoint operators can be quite exotic, such as Cantor sets, and their fine properties yield insight into associated dynamical systems. Quasiperiodic operators can be approximated by periodic ones, the spectra of which can be computed via two finite dimensional eigenvalue problems. Since long periods are necessary to get detailed approximations, both computational efficiency and numerical accuracy become a concern. We describe a simple method for numerically computing the spectrum of a period-$K$ Jacobi operator in $O(K^2)$ operations, and use it to investigate the spectra of Schr\"odinger operators with Fibonacci, period doubling, and Thue-Morse potentials

Comparison of reduced models for blood flow using Runge-Kutta discontinuous Galerkin methods

One-dimensional blood flow models take the general form of nonlinear hyperbolic systems but differ greatly in their formulation. One class of models considers the physically conserved quantities of mass and momentum, while another class describes mass and velocity. Further, the averaging process employed in the model derivation requires the specification of the axial velocity profile; this choice differentiates models within each class. Discrepancies among differing models have yet to be investigated. In this paper, we systematically compare several reduced models of blood flow for physiologically relevant vessel parameters, network topology, and boundary data. The models are discretized by a class of Runge-Kutta discontinuous Galerkin methods

Numerical methods and applications for reduced models of blood flow

The human cardiovascular system is a vastly complex collection of interacting components, including vessels, organ systems, valves, regulatory mechanisms, microcirculations, remodeling tissue, and electrophysiological signals. Experimental, mathematical, and computational research efforts have explored various hemodynamic questions; the scope of this literature is a testament to the intricate nature of cardiovascular physiology. In this work, we focus on computational modeling of blood flow in the major vessels of the human body. We consider theoretical questions related to the numerical approximation of reduced models for blood flow, posed as nonlinear hyperbolic systems in one space dimension. Further, we apply this modeling framework to abnormal physiologies resulting from surgical intervention in patients with congenital heart defects. This thesis contains three main parts: (i) a discussion of the implementation and analysis for numerical discretizations of reduced models for blood flow, (ii) an investigation of solutions to different classes of models in the realm of smooth and discontinuous solutions, and (iii) an application of these models within a multiscale framework for simulating flow in patients with hypoplastic left heart syndrome. The two numerical discretizations studied in this thesis are a characteristics-based method for approximating the Riemann-invariants of reduced blood flow models, and a discontinuous Galerkin scheme for approximating solutions to the reduced models directly. A priori error estimates are derived in particular cases for both methods. Further, two classes of hyperbolic systems for blood flow, namely the mass-momentum and the mass-velocity formulations, are systematically compared with each numerical method and physiologically relevant networks of vessels and boundary conditions. Lastly, closed loop vessel network models of various Fontan physiologies are constructed. Arterial and venous trees are built from networks of one-dimensional vessels while the heart, valves, vessel junctions, and organ beds are modeled by systems of algebraic and ordinary differential equations

Numerical method of characteristics for one-dimensional blood flow

Mathematical modeling at the level of the full cardiovascular system requires the numerical approximation of solutions to a one-dimensional nonlinear hyperbolic system describing flow in a single vessel. This model is often simulated by computationally intensive methods like finite elements and discontinuous Galerkin, while some recent applications require more efficient approaches (e.g. for real-time clinical decision support, phenomena occurring over multiple cardiac cycles, iterative solutions to optimization/inverse problems, and uncertainty quantification). Further, the high speed of pressure waves in blood vessels greatly restricts the time step needed for stability in explicit schemes. We address both cost and stability by presenting an efficient and unconditionally stable method for approximating solutions to diagonal nonlinear hyperbolic systems. Theoretical analysis of the algorithm is given along with a comparison of our method to a discontinuous Galerkin implementation. Lastly, we demonstrate the utility of the proposed method by implementing it on small and large arterial networks of vessels whose elastic and geometrical parameters are physiologically relevant

Simulating Cardiac Fluid Dynamics in the Human Heart

Cardiac fluid dynamics fundamentally involves interactions between complex blood flows and the structural deformations of the muscular heart walls and the thin, flexible valve leaflets. There has been longstanding scientific, engineering, and medical interest in creating mathematical models of the heart that capture, explain, and predict these fluid-structure interactions. However, existing computational models that account for interactions among the blood, the actively contracting myocardium, and the cardiac valves are limited in their abilities to predict valve performance, resolve fine-scale flow features, or use realistic descriptions of tissue biomechanics. Here we introduce and benchmark a comprehensive mathematical model of cardiac fluid dynamics in the human heart. A unique feature of our model is that it incorporates biomechanically detailed descriptions of all major cardiac structures that are calibrated using tensile tests of human tissue specimens to reflect the heart's microstructure. Further, it is the first fluid-structure interaction model of the heart that provides anatomically and physiologically detailed representations of all four cardiac valves. We demonstrate that this integrative model generates physiologic dynamics, including realistic pressure-volume loops that automatically capture isovolumetric contraction and relaxation, and predicts fine-scale flow features. None of these outputs are prescribed; instead, they emerge from interactions within our comprehensive description of cardiac physiology. Such models can serve as tools for predicting the impacts of medical devices or clinical interventions. They also can serve as platforms for mechanistic studies of cardiac pathophysiology and dysfunction, including congenital defects, cardiomyopathies, and heart failure, that are difficult or impossible to perform in patients

A computational framework for generating patient-specific vascular models and assessing uncertainty from medical images

Patient-specific computational modeling is a popular, non-invasive method to answer medical questions. Medical images are used to extract geometric domains necessary to create these models, providing a predictive tool for clinicians. However, in vivo imaging is subject to uncertainty, impacting vessel dimensions essential to the mathematical modeling process. While there are numerous programs available to provide information about vessel length, radii, and position, there is currently no exact way to determine and calibrate these features. This raises the question, if we are building patient-specific models based on uncertain measurements, how accurate are the geometries we extract and how can we best represent a patient's vasculature? In this study, we develop a novel framework to determine vessel dimensions using change points. We explore the impact of uncertainty in the network extraction process on hemodynamics by varying vessel dimensions and segmenting the same images multiple times. Our analyses reveal that image segmentation, network size, and minor changes in radius and length have significant impacts on pressure and flow dynamics in rapidly branching structures and tapering vessels. Accordingly, we conclude that it is critical to understand how uncertainty in network geometry propagates to fluid dynamics, especially in clinical applications.Comment: 21 pages, 9 figure