Benchmarking the Immersed Boundary Method for Viscoelastic Flows.

We present and analyze a series of benchmark tests regarding the application of the immersed boundary (IB) method to viscoelastic flows through and around non-trivial, stationary geometries. The IB method is widely used to simulate biological fluid dynamics and other modeling scenarios in which a structure is immersed in a fluid. Although the IB method has been most commonly used to model systems involving viscous incompressible fluids, it also can be applied to visoelastic fluids, and has enabled the study of a wide variety of dynamical problems including the settling of vesicles and the swimming of elastic filaments in fluids modeled by the Oldroyd-B constitutive equation. In the viscoelastic context, however, relatively little work has explored the accuracy or convergence properties of this numerical scheme. Herein, we present benchmarking results for an IB solver applied to viscoelastic flows in and around non-trivial geometries using either the idealized Oldroyd-B constitutive model or the more physcially realistic, polymer-entanglementbased Rolie-Poly constitutive equations. We use two-dimensional numerical test cases along with results from rheology experiments to benchmark the IB method and compare it to more complex finite element and finite volume viscoelastic flow solvers. Additionally, we analyze different choices of regularized delta function and relative Lagrangian grid spacings which allow us to identify and recommend the key choices of these numerical parameters depending on the present flow regime.

A stabilized linear finite element method for anisotropic poroelastodynamics with application to cardiac perfusion.

We propose a variational multiscale method stabilization of a linear finite element method for nonlinear poroelasticity. Our approach is suitable for the implicit time integration of poroelastic formulations in which the solid skeleton is anisotropic and incompressible. A detailed numerical methodology is presented for a monolithic formulation that includes both structural dynamics and Darcy flow. Our implementation of this methodology is verified using several hyperelastic and poroelastic benchmark cases, and excellent agreement is obtained with the literature. Grid convergence studies for both anisotropic hyperelastodynamics and poroelastodynamics demonstrate that the method is second-order accurate. The capabilities of our approach are demonstrated using a model of the left ventricle (LV) of the heart derived from human imaging data. Simulations using this model indicate that the anisotropicity of the myocardium has a substantial influence on the pore pressure. Furthermore, the temporal variations of the various components of the pore pressure (hydrostatic pressure and pressure resulting from changes in the volume of the pore fluid) are correlated with the variation of the added mass and dynamics of the LV, with maximum pore pressure being obtained at peak systole. The order of magnitude and the temporal variation of the pore pressure are in good agreement with the literature.

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 sharp interface Lagrangian-Eulerian method for flexible-body fluid-structure interaction.

This paper introduces a sharp-interface approach to simulating fluid-structure interaction (FSI) involving flexible bodies described by general nonlinear material models and across a broad range of mass density ratios. This new flexible-body immersed Lagrangian-Eulerian (ILE) scheme extends our prior work on integrating partitioned and immersed approaches to rigid-body FSI. Our numerical approach incorporates the geometrical and domain solution flexibility of the immersed boundary (IB) method with an accuracy comparable to body-fitted approaches that sharply resolve flows and stresses up to the fluid-structure interface. Unlike many IB methods, our ILE formulation uses distinct momentum equations for the fluid and solid subregions with a Dirichlet-Neumann coupling strategy that connects fluid and solid subproblems through simple interface conditions. As in earlier work, we use approximate Lagrange multiplier forces to treat the kinematic interface conditions along the fluid-structure interface. This penalty approach simplifies the linear solvers needed by our formulation by introducing two representations of the fluid-structure interface, one that moves with the fluid and another that moves with the structure, that are connected by stiff springs. This approach also enables the use of multi-rate time stepping, which allows us to use different time step sizes for the fluid and structure subproblems. Our fluid solver relies on an immersed interface method (IIM) for discrete surfaces to impose stress jump conditions along complex interfaces while enabling the use of fast structured-grid solvers for the incompressible Navier-Stokes equations. The dynamics of the volumetric structural mesh are determined using a standard finite element approach to large-deformation nonlinear elasticity via a nearly incompressible solid mechanics formulation. This formulation also readily accommodates compressible structures with a constant total volume, and it can handle fully compressible solid structures for cases in which at least part of the solid boundary does not contact the incompressible fluid. Selected grid convergence studies demonstrate second-order convergence in volume conservation and in the pointwise discrepancies between corresponding positions of the two interface representations as well as between first and second-order convergence in the structural displacements. The time stepping scheme is also demonstrated to yield second-order convergence. To assess and validate the robustness and accuracy of the new algorithm, comparisons are made with computational and experimental FSI benchmarks. Test cases include both smooth and sharp geometries in various flow conditions. We also demonstrate the capabilities of this methodology by applying it to model the transport and capture of a geometrically realistic, deformable blood clot in an inferior vena cava filter.

A Nodal Immersed Finite Element-Finite Difference Method.

The immersed finite element-finite difference (IFED) method is a computational approach to modeling interactions between a fluid and an immersed structure. The IFED method uses a finite element (FE) method to approximate the stresses, forces, and structural deformations on a structural mesh and a finite difference (FD) method to approximate the momentum and enforce incompressibility of the entire fluid-structure system on a Cartesian grid. The fundamental approach used by this method follows the immersed boundary framework for modeling fluid-structure interaction (FSI), in which a force spreading operator prolongs structural forces to a Cartesian grid, and a velocity interpolation operator restricts a velocity field defined on that grid back onto the structural mesh. With an FE structural mechanics framework, force spreading first requires that the force itself be projected onto the finite element space. Similarly, velocity interpolation requires projecting velocity data onto the FE basis functions. Consequently, evaluating either coupling operator requires solving a matrix equation at every time step. Mass lumping, in which the projection matrices are replaced by diagonal approximations, has the potential to accelerate this method considerably. This paper provides both numerical and computational analyses of the effects of this replacement for evaluating the force projection and for the IFED coupling operators. Constructing the coupling operators also requires determining the locations on the structure mesh where the forces and velocities are sampled. Here we show that sampling the forces and velocities at the nodes of the structural mesh is equivalent to using lumped mass matrices in the IFED coupling operators. A key theoretical result of our analysis is that if both of these approaches are used together, the IFED method permits the use of lumped mass matrices derived from nodal quadrature rules for any standard interpolatory element. This is different from standard FE methods, which require specialized treatments to accommodate mass lumping with higher-order shape functions. Our theoretical results are confirmed by numerical benchmarks, including standard solid mechanics tests and examination of a dynamic model of a bioprosthetic heart valve.

A model of fluid-structure and biochemical interactions for applications to subclinical leaflet thrombosis.

Subclinical leaflet thrombosis (SLT) is a potentially serious complication of aortic valve replacement with a bioprosthetic valve in which blood clots form on the replacement valve. SLT is associated with increased risk of transient ischemic attacks and strokes and can progress to clinical leaflet thrombosis. SLT following aortic valve replacement also may be related to subsequent structural valve deterioration, which can impair the durability of the valve replacement. Because of the difficulty in clinical imaging of SLT, models are needed to determine the mechanisms of SLT and could eventually predict which patients will develop SLT. To this end, we develop methods to simulate leaflet thrombosis that combine fluid-structure interaction and a simplified thrombosis model that allows for deposition along the moving leaflets. Additionally, this model can be adapted to model deposition or absorption along other moving boundaries. We present convergence results and quantify the model's ability to realize changes in valve opening and pressures. These new approaches are an important advancement in our tools for modeling thrombosis because they incorporate both adhesion to the surface of the moving leaflets and feedback to the fluid-structure interaction.

Microphysiological model of PIK3CA-driven vascular malformations reveals a role of dysregulated Rac1 and mTORC1/2 in lesion formation.

Somatic activating mutations of PIK3CA are associated with development of vascular malformations (VMs). Here, we describe a microfluidic model of PIK3CA-driven VMs consisting of human umbilical vein endothelial cells expressing PIK3CA activating mutations embedded in three-dimensional hydrogels. We observed enlarged, irregular vessel phenotypes and the formation of cyst-like structures consistent with clinical signatures and not previously observed in cell culture models. Pathologic morphologies occurred concomitant with up-regulation of Rac1/p21-activated kinase (PAK), mitogen-activated protein kinase cascades (MEK/ERK), and mammalian target of rapamycin (mTORC1/2) signaling networks. We observed differential effects between alpelisib, a PIK3CA inhibitor, and rapamycin, an mTORC1 inhibitor, in mitigating matrix degradation and network topology. While both were effective in preventing vessel enlargement, rapamycin failed to reduce MEK/ERK and mTORC2 activity and resulted in hyperbranching, while inhibiting PAK, MEK1/2, and mTORC1/2 mitigates abnormal growth and vascular dilation. Collectively, these findings demonstrate an in vitro platform for VMs and establish a role of dysregulated Rac1/PAK and mTORC1/2 signaling in PIK3CA-driven VMs.

Semi-Automated Construction of Patient-Specific Aortic Valves from Computed Tomography Images.

This paper presents a semi-automatic method for the construction of volumetric models of the aortic valve using computed tomography angiography images. Although the aortic valve typically cannot be segmented directly from a computed tomography angiography image, the method described herein uses manually selected samples of an aortic segmentation derived from this image to inform the construction. These samples capture certain physiologic landmarks and are used to construct a volumetric valve model. As a demonstration of the capabilities of this method, valve models for 25 pediatric patients are created. A selected valve anatomy is used to perform fluid-structure interaction simulations using the immersed finite element/difference method with physiologic driving and loading conditions. Simulation results demonstrate this method creates a functional valve that opens and closes normally and generates pressure and flow waveforms that are similar to those observed clinically.

Patient-Specific Immersed Finite Element-Difference Model of Transcatheter Aortic Valve Replacement.

Transcatheter aortic valve replacement (TAVR) first received FDA approval for high-risk surgical patients in 2011 and has been approved for low-risk surgical patients since 2019. It is now the most common type of aortic valve replacement, and its use continues to accelerate. Computer modeling and simulation (CM&S) is a tool to aid in TAVR device design, regulatory approval, and indication in patient-specific care. This study introduces a computational fluid-structure interaction (FSI) model of TAVR with Medtronic's CoreValve Evolut R device using the immersed finite element-difference (IFED) method. We perform dynamic simulations of crimping and deployment of the Evolut R, as well as device behavior across the cardiac cycle in a patient-specific aortic root anatomy reconstructed from computed tomography (CT) image data. These IFED simulations, which incorporate biomechanics models fit to experimental tensile test data, automatically capture the contact within the device and between the self-expanding stent and native anatomy. Further, we apply realistic driving and loading conditions based on clinical measurements of human ventricular and aortic pressures and flow rates to demonstrate that our Evolut R model supports a physiological diastolic pressure load and provides informative clinical performance predictions.

Rule-based definition of muscle bundles in patient-specific models of the left atrium.

Atrial fibrillation (AF) is the most common arrhythmia encountered clinically, and as the population ages, its prevalence is increasing. Although the CHA2DS2- VASc score is the most used risk-stratification system for stroke risk in AF, it lacks personalization. Patient-specific computer models of the atria can facilitate personalized risk assessment and treatment planning. However, a challenge faced in creating such models is the complexity of the atrial muscle arrangement and its influence on the atrial fiber architecture. This work proposes a semi-automated rule-based algorithm to generate the local fiber orientation in the left atrium (LA). We use the solutions of several harmonic equations to decompose the LA anatomy into subregions. Solution gradients define a two-layer fiber field in each subregion. The robustness of our approach is demonstrated by recreating the fiber orientation on nine models of the LA obtained from AF patients who underwent WATCHMAN device implantation. This cohort of patients encompasses a variety of morphology variants of the left atrium, both in terms of the left atrial appendages (LAAs) and the number of pulmonary veins (PVs). We test the fiber construction algorithm by performing electrophysiology (EP) simulations. Furthermore, this study is the first to compare its results with other rule-based algorithms for the LA fiber architecture definition available in the literature. This analysis suggests that a multi-layer fiber architecture is important to capture complex electrical activation patterns. A notable advantage of our approach is the ability to reconstruct the main LA fiber bundles in a variety of morphologies while solving for a small number of harmonic fields, leading to a comparatively straightforward and reproducible approach.

On the Lagrangian-Eulerian Coupling in the Immersed Finite Element/Difference Method.

The immersed boundary (IB) method is a non-body conforming approach to fluid-structure interaction (FSI) that uses an Eulerian description of the momentum, viscosity, and incompressibility of a coupled fluid-structure system and a Lagrangian description of the deformations, stresses, and resultant forces of the immersed structure. Integral transforms with Dirac delta function kernels couple the Eulerian and Lagrangian variables, and in practice, discretizations of these integral transforms use regularized delta function kernels. Many different kernel functions have been proposed, but prior numerical work investigating the impact of the choice of kernel function on the accuracy of the methodology has often been limited to simplified test cases or Stokes flow conditions that may not reflect the method's performance in applications, particularly at intermediate-to-high Reynolds numbers, or under different loading conditions. This work systematically studies the effect of the choice of regularized delta function in several fluid-structure interaction benchmark tests using the immersed finite element/difference (IFED) method, which is an extension of the IB method that uses a finite element structural discretization combined with a Cartesian grid finite difference method for the incompressible Navier-Stokes equations. Whereas the conventional IB method spreads forces from the nodes of the structural mesh and interpolates velocities to those nodes, the IFED formulation evaluates the regularized delta function on a collection of interaction points that can be chosen to be denser than the nodes of the Lagrangian mesh. This opens the possibility of using structural discretizations with wide node spacings that would produce gaps in the Eulerian force in nodally coupled schemes (e.g., if the node spacing is comparable to or broader than the support of the regularized delta functions). Earlier work with this methodology suggested that such coarse structural meshes can yield improved accuracy for shear-dominated cases and, further, found that accuracy improves when the structural mesh spacing is increased. However, these results were limited to simple test cases that did not include substantial pressure loading on the structure. This study investigates the effect of varying the relative mesh widths of the Lagrangian and Eulerian discretizations in a broader range of tests. Our results indicate that kernels satisfying a commonly imposed even-odd condition require higher resolution to achieve similar accuracy as kernels that do not satisfy this condition. We also find that narrower kernels are more robust, in the sense that they yield results that are less sensitive to relative changes in the Eulerian and Lagrangian mesh spacings, and that structural meshes that are substantially coarser than the Cartesian grid can yield high accuracy for shear-dominated cases but not for cases with large normal forces. We verify our results in a large-scale FSI model of a bovine pericardial bioprosthetic heart valve in a pulse duplicator.

A Hybrid Semi-Lagrangian Cut Cell Method for Advection-Diffusion Problems with Robin Boundary Conditions in Moving Domains.

We present a new discretization approach to advection-diffusion problems with Robin boundary conditions on complex, time-dependent domains. The method is based on second order cut cell finite volume methods introduced by Bochkov et al. [8] to discretize the Laplace operator and Robin boundary condition. To overcome the small cell problem, we use a splitting scheme along with a semi-Lagrangian method to treat advection. We demonstrate second order accuracy in the L 1, L 2, and L ∞ norms for both analytic test problems and numerical convergence studies. We also demonstrate the ability of the scheme to convert one chemical species to another across a moving boundary.

Adherens junction engagement regulates functional patterning of the cardiac pacemaker cell lineage.

Cardiac pacemaker cells (CPCs) rhythmically initiate the electrical impulses that drive heart contraction. CPCs display the highest rate of spontaneous depolarization in the heart despite being subjected to inhibitory electrochemical conditions that should theoretically suppress their activity. While several models have been proposed to explain this apparent paradox, the actual molecular mechanisms that allow CPCs to overcome electrogenic barriers to their function remain poorly understood. Here, we have traced CPC development at single-cell resolution and uncovered a series of cytoarchitectural patterning events that are critical for proper pacemaking. Specifically, our data reveal that CPCs dynamically modulate adherens junction (AJ) engagement to control characteristics including surface area, volume, and gap junctional coupling. This allows CPCs to adopt a structural configuration that supports their overall excitability. Thus, our data have identified a direct role for local cellular mechanics in patterning critical morphological features that are necessary for CPC electrical activity.

Models for plasma kinetics during simultaneous therapeutic plasma exchange and extracorporeal membrane oxygenation.

This paper focuses on the derivation and simulation of mathematical models describing new plasma fraction in blood for patients undergoing simultaneous extracorporeal membrane oxygenation and therapeutic plasma exchange. Models for plasma exchange with either veno-arterial or veno-venous extracorporeal membrane oxygenation are considered. Two classes of models are derived for each case, one in the form of an algebraic delay equation and another in the form of a system of delay differential equations. In special cases, our models reduce to single compartment ones for plasma exchange that have been validated with experimental data (Randerson et al., 1982, Artif. Organs, 6, 43-49). We also show that the algebraic differential equations are forward Euler discretizations of the delay differential equations, with timesteps equal to transit times through model compartments. Numerical simulations are performed to compare different model types, to investigate the impact of plasma device port switching on the efficiency of the exchange process, and to study the sensitivity of the models to their parameters.

A poroelastic immersed finite element framework for modelling cardiac perfusion and fluid-structure interaction.

Modern approaches to modelling cardiac perfusion now commonly describe the myocardium using the framework of poroelasticity. Cardiac tissue can be described as a saturated porous medium composed of the pore fluid (blood) and the skeleton (myocytes and collagen scaffold). In previous studies fluid-structure interaction in the heart has been treated in a variety of ways, but in most cases, the myocardium is assumed to be a hyperelastic fibre-reinforced material. Conversely, models that treat the myocardium as a poroelastic material typically neglect interactions between the myocardium and intracardiac blood flow. This work presents a poroelastic immersed finite element framework to model left ventricular dynamics in a three-phase poroelastic system composed of the pore blood fluid, the skeleton, and the chamber fluid. We benchmark our approach by examining a pair of prototypical poroelastic formations using a simple cubic geometry considered in the prior work by Chapelle et al (2010). This cubic model also enables us to compare the differences between system behaviour when using isotropic and anisotropic material models for the skeleton. With this framework, we also simulate the poroelastic dynamics of a three-dimensional left ventricle, in which the myocardium is described by the Holzapfel-Ogden law. Results obtained using the poroelastic model are compared to those of a corresponding hyperelastic model studied previously. We find that the poroelastic LV behaves differently from the hyperelastic LV model. For example, accounting for perfusion results in a smaller diastolic chamber volume, agreeing well with the well-known wall-stiffening effect under perfusion reported previously. Meanwhile differences in systolic function, such as fibre strain in the basal and middle ventricle, are found to be comparatively minor.

Bioprosthetic aortic valve diameter and thickness are directly related to leaflet fluttering: Results from a combined experimental and computational modeling study.

OBJECTIVE: Bioprosthetic heart valves (BHVs) are commonly used in surgical and percutaneous valve replacement. The durability of percutaneous valve replacement is unknown, but surgical valves have been shown to require reintervention after 10 to 15 years. Further, smaller-diameter surgical BHVs generally experience higher rates of prosthesis-patient mismatch, which leads to higher rates of failure. Bioprosthetic aortic valves can flutter in systole, and fluttering is associated with fatigue and failure in flexible structures. The determinants of flutter in BHVs have not been well characterized, despite their potential to influence durability. METHODS: We use an experimental pulse duplicator and a computational fluid-structure interaction model of this system to study the role of device geometry on BHV dynamics. The experimental system mimics physiological conditions, and the computational model enables precise control of leaflet biomechanics and flow conditions to isolate the effects of variations in BHV geometry on leaflet dynamics. RESULTS: Both experimental and computational models demonstrate that smaller-diameter BHVs yield markedly higher leaflet fluttering frequencies across a range of conditions. The computational model also predicts that fluttering frequency is directly related to leaflet thickness. A scaling model is introduced that rationalizes these findings. CONCLUSIONS: We systematically characterize the influence of BHV diameter and leaflet thickness on fluttering dynamics. Although this study does not determine how flutter influences device durability, increased flutter in smaller-diameter BHVs may explain how prosthesis-patient mismatch could induce BHV leaflet fatigue and failure. Ultimately, understanding the effects of device geometry on leaflet kinematics may lead to more durable valve replacements.

Mathematical modeling of the impact of recirculation on exchange kinetics in tandem extracorporeal membrane oxygenation and therapeutic plasma exchange.

Vascular access connection configurations during tandem extracorporeal membrane oxygenation (ECMO) and therapeutic plasma exchange (TPE) may impact exchange kinetics. In these tandem procedures, typically the TPE inlet line is proximal to the TPE return line with respect to blood flow in the ECMO device, maximizing the opportunity for replacement fluid homogenization within the ECMO circuit. However, if TPE inlet and return line connections are switched, recirculation-a phenomenon in which replacement fluid leaving the TPE return line is prematurely drawn into the TPE inlet line prior to satisfactory homogenization within the ECMO circuit-will occur. Such recirculation could diminish TPE efficacy in patients on ECMO and mitigate therapeutic benefits. Using a mathematical model of recirculation in tandem ECMO and TPE, we demonstrate that the predicted impact of recirculation is negligible and vascular access connection positioning does not appear to be a point of clinical concern with regard to TPE kinetics.

Immersed Methods for Fluid-Structure Interaction.

Fluid-structure interaction is ubiquitous in nature and occurs at all biological scales. Immersed methods provide mathematical and computational frameworks for modeling fluid-structure systems. These methods, which typically use an Eulerian description of the fluid and a Lagrangian description of the structure, can treat thin immersed boundaries and volumetric bodies, and they can model structures that are flexible or rigid or that move with prescribed deformational kinematics. Immersed formulations do not require body-fitted discretizations and thereby avoid the frequent grid regeneration that can otherwise be required for models involving large deformations and displacements. This article reviews immersed methods for both elastic structures and structures with prescribed kinematics. It considers formulations using integral operators to connect the Eulerian and Lagrangian frames and methods that directly apply jump conditions along fluid-structure interfaces. Benchmark problems demonstrate the effectiveness of these methods, and selected applications at Reynolds numbers up to approximately 20,000 highlight their impact in biological and biomedical modeling and simulation.

Stabilization approaches for the hyperelastic immersed boundary method for problems of large-deformation incompressible elasticity.

The immersed boundary method is a mathematical framework for modeling fluid-structure interaction. This formulation describes the momentum, viscosity, and incompressibility of the fluid-structure system in Eulerian form, and it uses Lagrangian coordinates to describe the structural deformations, stresses, and resultant forces. Integral transforms with Dirac delta function kernels connect the Eulerian and Lagrangian frames. The fluid and the structure are both typically treated as incompressible materials. Upon discretization, however, the incompressibility of the structure is only maintained approximately. To obtain an immersed method for incompressible hyperelastic structures that is robust under large structural deformations, we introduce a volumetric energy in the solid region that stabilizes the formulation and improves the accuracy of the numerical scheme. This formulation augments the discrete Lagrange multiplier for the incompressibility constraint, thereby improving the original method's accuracy. This volumetric energy is incorporated by decomposing the strain energy into isochoric and dilatational components, as in standard solid mechanics formulations of nearly incompressible elasticity. We study the performance of the stabilized method using several quasi-static solid mechanics benchmarks, a dynamic fluid-structure interaction benchmark, and a detailed three-dimensional model of esophageal transport. The accuracy achieved by the stabilized immersed formulation is comparable to that of a stabilized finite element method for incompressible elasticity using similar numbers of structural degrees of freedom.