An isogeometric analysis framework for ventricular cardiac mechanics

The finite element method (FEM) is commonly used in computational cardiac simulations. For this method, a mesh is constructed to represent the geometry and, subsequently, to approximate the solution. To accurately capture curved geometrical features many elements may be required, possibly leading to unnecessarily large computation costs. Without loss of accuracy, a reduction in computation cost can be achieved by integrating geometry representation and solution approximation into a single framework using the Isogeometric Analysis (IGA) paradigm. In this study, we propose an IGA framework suitable for echocardiogram data of cardiac mechanics, where we show the advantageous properties of smooth splines through the development of a multi-patch anatomical model. A nonlinear cardiac model is discretized following the IGA paradigm, meaning that the spline geometry parametrization is directly used for the discretization of the physical fields. The IGA model is benchmarked with a state-of-the-art biomechanics model based on traditional FEM. For this benchmark, the hemodynamic response predicted by the high-fidelity FEM model is accurately captured by an IGA model with only 320 elements and 4,700 degrees of freedom. The study is concluded by a brief anatomy-variation analysis, which illustrates the geometric flexibility of the framework. The IGA framework can be used as a first step toward an efficient workflow for an improved understanding of, and clinical decision support for, the treatment of cardiac diseases like heart rhythm disorders

Efficient isogeometric thin shell formulations for soft biological materials

This paper presents three different constitutive approaches to model thin rotation-free shells based on the Kirchhoff-Love hypothesis. One approach is based on numerical integration through the shell thickness while the other two approaches do not need any numerical integration and so they are computationally more efficient. The formulation is designed for large deformations and allows for geometrical and material nonlinearities, which makes it very suitable for the modeling of soft tissues. Furthermore, six different isotropic and anisotropic material models, which are commonly used to model soft biological materials, are examined for the three proposed constitutive approaches. Following an isogeometric approach, NURBS-based finite elements are used for the discretization of the shell surface. Several numerical examples are investigated to demonstrate the capabilities of the formulation. Those include the contact simulation during balloon angioplasty.Comment: Typos are removed. Remark 3.4 is added. Eq. (18) in the previous version is removed. Thus, the equations get renumbered. Example 5.5 is updated. Minor typos in Eqs. (17), (80), (145) and (146), are corrected. They do not affect the result

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

Fluid–structure interaction modeling and simulation of transcatheter heart valves

Bioprosthetic heart valves (BHVs) are prostheses fabricated from xenograft biomaterials for treating valvular disease. While these devices have mechanical and blood flow characteristics similar to the native valves, the durability remains limited to 10-15 years with device failure continues to result from leaflet structural deterioration mediated by fatigue and tissue mineralization. Improving BHV design remains an important clinical goal and represents a unique cardiovascular engineering challenge. Transcatheter heart valves (THVs) have emerged as a minimally invasive alternative to surgical bioprosthetic heart valves therapy. THVs offer advantages such as less postoperative pain, faster rehabilitation, and better pressure gradients. However, issues such as paravalvular leakage, leaflet fatigue, and valve migration limit the widespread use of THV in the younger population, especially due to the lack of data concerning its long-term performance and durability. The friction force and the radial force between THV frames and the surrounding anatomy are important indicators for the safe anchoring. Thus, in-vitro measurement of these forces is vital for pre-operative planning of transcatheter aortic valve replacement (TAVR) procedures. There is a profound need to develop a general understanding of heart valve mechanism through novel simulation technologies that take advantage of fluid–structure interactions (FSI). In this work, a framework for modeling BHVs using recently proposed isogeometric analysis based parametric design platform and immersogeometric FSI analysis is presented. Due to the complex motion of the heart valve leaflets, the blood flow domain undergoes large deformations, including changes of topology. The FSI simulations are carried out using our hybrid arbitrary Lagrangian--Eulerian/immersogeometric methodology, which allows us to efficiently perform a computation that combines a boundary-fitted, deforming-mesh treatment of the artery with a non-boundary-fitted treatment of the leaflets. The development of modeling and simulation of full THV is integrated with the immersogeometric FSI analysis. With an effective material model considering the collagen fibers network of heart valve leaflets, and a novel method for the THV frame isogeometric design and simulation, a biomechanically rigorous and physiologically realistic computational FSI framework is carried out to study the interaction between THVs and aortic wall. From the computed friction force analysis, the anchoring ability of THVs is estimated, which is a valuable information for clinical planning and decision making of TAVR

Heart valve isogeometric sequentially-coupled FSI analysis with the space–time topology change method

Heart valve fluid–structure interaction (FSI) analysis is one of the computationally challenging cases in cardiovascular fluid mechanics. The challenges include unsteady flow through a complex geometry, solid surfaces with large motion, and contact between the valve leaflets. We introduce here an isogeometric sequentially-coupled FSI (SCFSI) method that can address the challenges with an outcome of high-fidelity flow solutions. The SCFSI analysis enables dealing with the fluid and structure parts individually at different steps of the solutions sequence, and also enables using different methods or different mesh resolution levels at different steps. In the isogeometric SCFSI analysis here, the first step is a previously computed (fully) coupled Immersogeometric Analysis FSI of the heart valve with a reasonable flow solution. With the valve leaflet and arterial surface motion coming from that, we perform a new, higher-fidelity fluid mechanics computation with the space–time topology change method and isogeometric discretization. Both the immersogeometric and space–time methods are variational multiscale methods. The computation presented for a bioprosthetic heart valve demonstrates the power of the method introduced

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

Isogeometric Kirchhoff–Love shell formulations for general hyperelastic materials

We present formulations for compressible and incompressible hyperelastic thin shells which can use general 3D constitutive models. The necessary plane stress condition is enforced analytically for incompressible materials and iteratively for compressible materials. The thickness stretch is statically condensed and the shell kinematics are completely described by the first and second fundamental forms of the midsurface. We use C1-continuous isogeometric discretizations to build the numerical models. Numerical tests, including structural dynamics simulations of a bioprosthetic heart valve, show the good performance and applicability of the presented methods

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