A geometric framework for immersogeometric analysis

The purpose of this dissertation is to develop a geometric framework for immersogeometric analysis that directly uses the boundary representations (B-reps) of a complex computer-aided design (CAD) model and immerses it into a locally refined, non-boundary-fitted discretization of the fluid domain. Using the non-boundary-fitted mesh which does not need to conform to the shape of the object can alleviate the challenge of mesh generation for complex geometries. This also reduces the labor-intensive and time-consuming work of geometry cleanup for the purpose of obtaining watertight CAD models in order to perform boundary-fitted mesh generation. The Dirichlet boundary conditions in the fluid domain are enforced weakly over the immersed object surface in the intersected elements. The surface quadrature points for the immersed object are generated on the parametric and analytic surfaces of the B-rep models. In the case of trimmed surfaces, adaptive quadrature rule is considered to improve the accuracy of the surface integral. For the non-boundary-fitted mesh, a sub-cell-based adaptive quadrature rule based on the recursive splitting of quadrature elements is used to faithfully capture the geometry in intersected elements. The point membership classification for identifying quadrature points in the fluid domain is based on a voxel-based approach implemented on GPUs. A variety of computational fluid dynamics (CFD) simulations are performed using the proposed method to assess its accuracy and efficiency. Finally, a fluid--structure interaction (FSI) simulation of a deforming left ventricle coupled with the heart valves shows the potential advantages of the developed geometric framework for the immersogeomtric analysis with complex moving domains

Inferring Geodesic Cerebrovascular Graphs: Image Processing, Topological Alignment and Biomarkers Extraction

A vectorial representation of the vascular network that embodies quantitative features - location, direction, scale, and bifurcations - has many potential neuro-vascular applications. Patient-specific models support computer-assisted surgical procedures in neurovascular interventions, while analyses on multiple subjects are essential for group-level studies on which clinical prediction and therapeutic inference ultimately depend. This first motivated the development of a variety of methods to segment the cerebrovascular system. Nonetheless, a number of limitations, ranging from data-driven inhomogeneities, the anatomical intra- and inter-subject variability, the lack of exhaustive ground-truth, the need for operator-dependent processing pipelines, and the highly non-linear vascular domain, still make the automatic inference of the cerebrovascular topology an open problem. In this thesis, brain vessels’ topology is inferred by focusing on their connectedness. With a novel framework, the brain vasculature is recovered from 3D angiographies by solving a connectivity-optimised anisotropic level-set over a voxel-wise tensor field representing the orientation of the underlying vasculature. Assuming vessels joining by minimal paths, a connectivity paradigm is formulated to automatically determine the vascular topology as an over-connected geodesic graph. Ultimately, deep-brain vascular structures are extracted with geodesic minimum spanning trees. The inferred topologies are then aligned with similar ones for labelling and propagating information over a non-linear vectorial domain, where the branching pattern of a set of vessels transcends a subject-specific quantized grid. Using a multi-source embedding of a vascular graph, the pairwise registration of topologies is performed with the state-of-the-art graph matching techniques employed in computer vision. Functional biomarkers are determined over the neurovascular graphs with two complementary approaches. Efficient approximations of blood flow and pressure drop account for autoregulation and compensation mechanisms in the whole network in presence of perturbations, using lumped-parameters analog-equivalents from clinical angiographies. Also, a localised NURBS-based parametrisation of bifurcations is introduced to model fluid-solid interactions by means of hemodynamic simulations using an isogeometric analysis framework, where both geometry and solution profile at the interface share the same homogeneous domain. Experimental results on synthetic and clinical angiographies validated the proposed formulations. Perspectives and future works are discussed for the group-wise alignment of cerebrovascular topologies over a population, towards defining cerebrovascular atlases, and for further topological optimisation strategies and risk prediction models for therapeutic inference. Most of the algorithms presented in this work are available as part of the open-source package VTrails

Heart Valve Mathematical Models

Nearly 100,000 heart valve replacements or repairs are performed in the US every year. Mathematical models of heart valves are used to improve artificial valve design and to guide surgeons performing valve-repairing surgeries. Models can be used to define the geometry of a valve, predict blood flow dynamics, or demonstrate operating mechanisms of the valve. In this thesis we reviewed features that are typically considered when developing a model of a heart valve. The main modeling methods include representing a heart valve using lumped parameters, finite elements, or isogeometric elements. Examples of a lumped-parameter model and isogeometric analysis are explored. First, we developed a simulation for the lumped-parameter model of Virag and Lulić, and we demonstrated its ability to capture the dynamical behavior of blood pressures, volumes, and flows in the aortic valve region. A Newton-Krylov method was used to estimate periodic solution trajectories, which provide a basis for examining the response to perturbations about initial conditions. Next, an isogeometric model of a heart valve was constructed based on NURBS geometry. The mechanical stiffness of the valve was computed. We discussed how the isogeometric representation could be used in a more complex fluid-structure interaction model to measure surface shear and estimate fatigue failure

Computational methods in cardiovascular mechanics

The introduction of computational models in cardiovascular sciences has been progressively bringing new and unique tools for the investigation of the physiopathology. Together with the dramatic improvement of imaging and measuring devices on one side, and of computational architectures on the other one, mathematical and numerical models have provided a new, clearly noninvasive, approach for understanding not only basic mechanisms but also patient-specific conditions, and for supporting the design and the development of new therapeutic options. The terminology in silico is, nowadays, commonly accepted for indicating this new source of knowledge added to traditional in vitro and in vivo investigations. The advantages of in silico methodologies are basically the low cost in terms of infrastructures and facilities, the reduced invasiveness and, in general, the intrinsic predictive capabilities based on the use of mathematical models. The disadvantages are generally identified in the distance between the real cases and their virtual counterpart required by the conceptual modeling that can be detrimental for the reliability of numerical simulations.Comment: 54 pages, Book Chapte

Dynamic and fluid–structure interaction simulations of bioprosthetic heart valves using parametric design with T-splines and Fung-type material models

This paper builds on a recently developed immersogeometric fluid–structure interaction (FSI) methodology for bioprosthetic heart valve (BHV) modeling and simulation. It enhances the proposed framework in the areas of geometry design and constitutive modeling. With these enhancements, BHV FSI simulations may be performed with greater levels of automation, robustness and physical realism. In addition, the paper presents a comparison between FSI analysis and standalone structural dynamics simulation driven by prescribed transvalvular pressure, the latter being a more common modeling choice for this class of problems. The FSI computation achieved better physiological realism in predicting the valve leaflet deformation than its standalone structural dynamics counterpart

Higher-order block-structured hex meshing of tubular structures

Numerical simulations of the cardiovascular system are growing in popularity due to the increasing availability of computational power, and their proven contribution to the understanding of pathodynamics and validation of medical devices with in-silico trials as a potential future breakthrough. Such simulations are performed on volumetric meshes reconstructed from patient-specific imaging data. These meshes are most often unstructured, and result in a brutally large amount of elements, significantly increasing the computational complexity of the simulations, whilst potentially adversely affecting their accuracy. To reduce such complexity, we introduce a new approach for fully automatic generation of higher-order, structured hexahedral meshes of tubular structures, with a focus on healthy blood vessels. The structures are modeled as skeleton-based convolution surfaces. From the same skeleton, the topology is captured by a block-structure, and the geometry by a higher-order surface mesh. Grading may be induced to obtain tailored refinement, thus resolving, e.g., boundary layers. The volumetric meshing is then performed via transfinite mappings. The resulting meshes are of arbitrary order, their elements are of good quality, while the spatial resolution may be as coarse as needed, greatly reducing computing time. Their suitability for practical applications is showcased by a simulation of physiological blood flow modelled by a generalised Newtonian fluid in the human aorta

Geometry and Mechanics

IASS-IACM 2008 Session: Geometry and Mechanics -- Session Organizers: Kai-Uwe BLETZINGER (TU Munich), Fehmi CIRAK (University of Cambridge) -- Keynote Lecture: "Modeling and computation of patient-specific vascular fluid-structure interaction using Isogeometric Analysis" by Yuri BAZILEVS , Victor M. CALO, Thomas J. R. HUGHES (University of Texas at Austin), Yongie ZHANG (Carnegie Mellon University) -- Keynote Lecture: "Optimal shapes of mechanically motivated surfaces" by Kai-Uwe BLETZINGER , Matthias FIRL, Johannes LINHARD, Roland WUCHNER (TU Munich) -- "Subdivision shells for nonsmooth and branching geometries" by Quan LONG, Fehmi CIRAK (University of Cambridge) -- "Water landing analyses with explicit finite element method" by John T. WANG (NASA Langley Research Center) -- "On a geometrically exact contact description for shells: From linear approximations for shells to high-order FEM" by Alexander KONYUKHOV, Karl SCHWEIZERHOF (University of Karlsruhe