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# 1. Illustration relating various object-sets used by CHeart to solve physics problems. In this case, we consider the coupled fluid-solid interaction (see Results). The fluid is solved on a tetrahedral grid using P 2 − P 1 Taylor–Hood elements and the ALE Navier–Stokes formulation introduced in section 2.3. The solid is solved on a hexahedral curvilinear grid using Q 3 − Q 2 Taylor– Hood elements and the solid mechanics formulation from section 2.2. Object-set B is constructed using five basis object types with varying order shape and dimension. These are used with five topology objects comprising T , and the interface I object-set is composed of two identity and two injective interfaces. V is then built from eight variables used in the four core problems feeding into the monolithic solver.

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From basic science to translation, modern biomedical research demands computational models which integrate several interacting physical systems. This paper describes the infrastructural framework for a generic multi- physics integration implemented in the software CHeart, a finite-element code for biomedical research. To generalize the coupling of...

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## Citations

... FEBio provides rich capabilities for vascular flow simulations, differing from other CFD programs mainly by the use of fluid dilatation instead of pressure as a primary variable [52]. CHeart is a framework for multiphysics finite element simulations in biomedical research, including a CFD solver with advanced numerical features, an Arbitrary Lagrangian-Eulerian (ALE) formulation and the Streamline Upwind Petrov-Galerkin (SUPG) stabilization [53]. simVascular provides a complete computational framework, from the construction of an anatomic model to finite element simulation and postprocessing [42], with the possibility to solve the incompressible Navier-Stokes equations in an arbitrary domain and specifically designed for cardiovascular simulations. ...

Computational fluid dynamics (CFD) is an important tool for the simulation of the cardiovascular function and dysfunction. Due to the complexity of the anatomy, the transitional regime of blood flow in the heart, and the strong mutual influence between the flow and the physical processes involved in the heart function, the development of accurate and efficient CFD solvers for cardiovascular flows is still a challenging task. In this paper we present lifex-cfd: an open-source CFD solver for cardiovascular simulations based on the lifex finite element library, written in modern C++ and exploiting distributed memory parallelism. We model blood flow in both physiological and pathological conditions via the incompressible Navier-Stokes equations, accounting for moving cardiac valves, moving domains, and transition-to-turbulence regimes. In this paper, we provide an overview of the underlying mathematical formulation, numerical discretization, implementation details and instructions for use of lifex-cfd. The code has been verified through rigorous convergence analyses, and we show its almost ideal parallel speedup. We demonstrate the accuracy and reliability of the numerical methods implemented through a series of idealized and patient-specific vascular and cardiac simulations, in different physiological flow regimes. The lifex-cfd source code is available under the LGPLv3 license, to ensure its accessibility and transparency to the scientific community, and to facilitate collaboration and further developments.

... The computational simulations were performed with CHeart [82], a custom multiphysics finite element solver, and the solution procedure follows the Shamanskii-Newton-Raphson (SNR) method [83]. Noting the split of S n ⋆ in Eq. (29) into terms in iteration n and terms in iteration n − 1, it is convenient to define the residual function in two parts ...

Biomechanics plays an important role in the diagnosis and treatment of pathological conditions of the heart. Computational models are paving the way for personalized therapeutic treatment but they rely on accurate constitutive equations for predicting their biomechanical behavior. Even so, viscoelasticity remains under-explored in computational modeling despite experimental observations. To facilitate the viscoelastic modeling of cardiovascular soft tissues, we previously developed a fractional viscoelastic modeling approach, which extends existing hyperelastic models. This has comparable computational costs to the conventional hyperelastic model and only requires two additional material parameters for the viscoelastic response. This approach was demonstrated to be able to accurately capture the viscoelastic response of the human myocardium. However, the numerical properties of this fractional viscoelastic approach have not yet been examined. In this work, we present its implementation in Finite Element Analysis, examine its numerical properties in uniaxial extension and 2D inflation test examples, and examine its physiological implication in a computational model of an idealized left ventricle in a fully idealized circulatory system. Optimal convergence properties were observed and the importance of viscoelasticity during passive filling, ventricular motion, and regional fiber strain and stresses were explained.

... Each component of Virchow's triad was modelled by finite-element simulations in CHeart [4]. LA models were generated from temporally varying Cine MRI data from two patients [5]. ...

Atrial fibrillation (AF) is associated with a significantly increased risk of stroke due to the presence of three pro-thrombotic mechanisms known as Virchow’s triad – blood stasis, endothelial damage and hypercoagulability – which primarily occur in the left atrial appendage (LAA). In-silico evaluation of each factor can improve upon the current empirical stroke risk stratification for AF patients.
Computational fluid dynamics simulations were performed on two patient-specific models of the left atrium, one in sinus rhythm (SR) and one in AF to quantify blood stasis and metrics of endothelial damage. Hypercoagulability was assessed by solving reaction- diffusion-convection equations for thrombin, fibrinogen and fibrin – three key clotting proteins, and varying initial concentrations of fibrinogen in accordance with clinical literature. An original grading system is proposed (A = low, B = moderate, C = high risk) for each component of the triad to form a patient-specific risk profile.
The SR patient had a risk profile of [A, B, A] showing a low-moderate risk of thrombus formation, while the AF patient had [C, B, C], indicating a very high risk of thrombus formation and increased potential for stroke.
This novel modelling approach encapsulates all fundamental mechanisms of thrombus formation and may be used to improve stroke risk assessment for AF patients.

... Each component of Virchow's triad was modelled by finite-element simulations in CHeart [4]. LA models were generated from temporally varying Cine MRI data from two patients [5]. ...

Atrial fibrillation (AF) is associated with a significantly increased risk of stroke due to the presence of three pro-thrombotic mechanisms known as Virchow's triad-blood stasis, endothelial damage and hypercoagulability-which primarily occur in the left atrial appendage (LAA). In-silico evaluation of each factor can improve upon the current empirical stroke risk stratification for AF patients. Computational fluid dynamics simulations were performed on two patient-specific models of the left atrium, one in sinus rhythm (SR) and one in AF to quantify blood stasis and metrics of endothelial damage. Hypercoagulability was assessed by solving reaction-diffusion-convection equations for thrombin, fibrinogen and fibrin-three key clotting proteins, and varying initial concentrations of fibrinogen in accordance with clinical literature. An original grading system is proposed (A = low, B = moderate, C = high risk) for each component of the triad to form a patient-specific risk profile. The SR patient had a risk profile of [A, B, A] showing a low-moderate risk of thrombus formation, while the AF patient had [C, C, C], indicating a very high risk of thrombus formation and increased potential for stroke. This novel modelling approach encapsulates all fundamental mechanisms of thrombus formation and may be used to improve stroke risk assessment for AF patients.

... Overall, the required implementation overhead to enable parallel-in-time integration through MGRIT is relatively low. For example, the finite element package CHeart [45,46] has about 98500 lines of Fortran code (includes comments), whereas the wrapper routines only amount to about 2050 lines of Fortran code (includes comments). 3 ...

... To solve the spatial problem described above, we employ the finite element solver CHeart [45,47], which is based on the matrix solver MUMPS [48]. For sequential time-stepping runs, CHeart steers the simulation by itself and it is set up to run 10 subsequent cardiac cycles. ...

... The flow is modeled by a stabilized general Galerkin scheme (instead of using, e.g., an inf-sup stable Taylor-Hood finite element discretization scheme) for the incompressible Navier-Stokes equations; namely the cG(1)cG(1) scheme as given in the study of Hoffman et al. [57]. The scheme was implemented in CHeart [45,46], and validated in a previous work [47]. ...

In this paper, a time-periodic MGRIT algorithm is proposed as a means to reduce the time-to-solution of numerical algorithms by exploiting the time periodicity inherent to many applications in science and engineering. The time-periodic MGRIT algorithm is applied to a variety of linear and nonlinear single- and multiphysics problems that are periodic-in-time. It is demonstrated that the proposed parallel-in-time algorithm can obtain the same time-periodic steady-state solution as sequential time-stepping. It is shown that the required number of MGRIT iterations can be estimated a priori and that the new MGRIT variant can significantly and consistently reduce the time-to-solution compared to sequential time-stepping, irrespective of the number of dimensions, linear or nonlinear PDE models, single-physics or coupled problems and the employed computing resources. The numerical experiments demonstrate that the time-periodic MGRIT algorithm enables a greater level of parallelism yielding faster turnaround, and thus, facilitating more complex and more realistic problems to be solved.

... The finite element simulations are performed using Cheart [115] , a generic multi-physics finite-element software for biomedical research. A simple cylindrical mesh was created in gmesh [116] using quadratic tetrahedral elements ( n = 5 , 551 ) for velocity and displacement, and linear interpolation for pressure. ...

Residual stress is thought to play a critical role in modulating stress distributions in soft biological tissues and in maintaining the mechanobiological stress environment of cells. Residual stresses in arteries and other tissues are classically assessed through opening angle experiments, which demonstrate the continuous release of residual stresses over hours. These results are then assessed through nonlinear biomechanical models to provide estimates of the residual stresses in the intact state. Although well studied, these analyses typically focus on hyperelastic material models despite significant evidence of viscoelastic phenomena over both short and long timescales. In this work, we extended the state-of-the-art structural tensor model for arterial tissues from Holzapfel and Ogden for fractional viscoelasticity. Models were tuned to capture consistent levels of hysteresis observed in biaxial experiments, while also minimizing the fractional viscoelastic weighting and opening angles to correctly capture opening angle dynamics. Results suggest that a substantial portion of the human abdominal aorta is viscoelastic, but exhibits a low fractional order (i.e. more elastically). Additionally, a significantly larger opening angle in the fully unloaded state is necessary to produce comparable hysteresis in biaxial testing. As a consequence, conventional estimates of residual stress using hyperelastic approaches over-estimate their viscoelastic counterparts by a factor of 2. Thus, a viscoelastic approach, such as the one illustrated in this study, in combination with an additional source of rate-controlled viscoelastic data is necessary to accurately analyze the residual stress distribution in soft biological tissues.
Statement of significance
Residual stress plays a crucial role in achieving a homeostatic stress environment in soft biological tissues. However, the analysis of residual stress typically focuses on hyperelastic material models despite evidence of viscoelastic behavior. This work is the first attempt at analyzing the effects of viscoelasticity on residual stress. The application of viscoelasticity was crucial for producing realistic opening dynamics in arteries. The overall residual stresses were estimated to be 50% less than those from using hyperelastic material models, while the opening angles were projected to be 25% more than those measured after 16 hours, suggesting underestimated residual strain. This study highlights the importance viscoelasticity in the analysis of residual stress even in weakly dissipative materials like the human aorta.

... The multiphysics finite-element solver, CHeart, was used to perform simulations of flow and coagulation [8]. A pulsatile blood flow velocity of 7.5 / for SR and 5.0 / in AF was applied through the PVs. ...

Atrial fibrillation (AF) is a major cause of stroke and there has been much interest in the underlying mechanisms leading to this higher risk of thrombus formation. The latter risk correlates with four morphologies of the left atrial appendage (LAA), i.e. chicken wing (CW), broccoli (BR), cactus (CA) and windsock (WS). We present a mechanistic study of coagulation dynamics in blood flow in a series of 2D models of the left atrium (LA) to dissect the impact of LAA shape on thrombus formation. Interactions between blood flow, viscosity and key clotting proteins (thrombin, fibrinogen and fibrin) were modelled during 1 minute of pulsatile LA blood flow to simulate the blood gelification process leading to thrombus formation. Simulations were performed in sinus rhythm (SR) and AF by varying the active contraction of the LAA and pulmonary vein inflow velocities. In the CW morphology, fibrin inside the LAA was almost completely washed out after 28 seconds in SR, while in AF the gelification process was slow, suggesting the CW has the lowest risk of thrombus formation. Conversely, the BR morphology had the highest risk of thrombus formation due to a region of sustained flow stasis which prevented fibrin washout during SR and facilitated the shortest time to thrombus formation in AF.

... For further details regarding the fibre distributions, see the Supplementary Materials. All in silico experiments were simulated in CHeart (Lee et al., 2016). ...

Elastography has become widely used clinically for characterising changes in soft tissue mechanics that are associated with altered tissue structure and composition. However, some soft tissues, such as muscle, are not isotropic as is assumed in clinical elastog-raphy implementations. This limits the ability of these methods to capture changes in anisotropic tissues associated with disease. The objective of this study was to develop and validate a novel elastography reconstruction technique suitable for estimating the linear viscoelastic mechanical properties of transversely isotropic soft tissues. We derived a divergence-free formulation of the governing equations for acoustic wave propagation through a linearly transversely isotropic viscoelastic material, and transformed this into a weak form. This was then implemented into a finite element framework, enabling the analysis of wave input data and tissue structural fibre orientations, in this case based on diffusion tensor imaging. To validate the material constants obtained with this method, numerous in silico phantom experiments were run which encompassed a range of variations in wave input directions, material properties, fibre structure and noise. The method was also tested on ex vivo muscle and in vivo human volunteer calf muscles, and compared with a previous curl-based inversion method. The new method robustly extracted the transversely isotropic shear moduli (G ⊥ , G , G) from the in silico phantom tests with minimal bias, including in the presence of experimentally realistic levels of noise in either fibre orientation or wave data. This new method performed better than the previous method in the presence of noise. Anisotropy estimates from the ex vivo muscle phantom agreed well with rheological tests. In vivo experiments on human calf muscles were able to detect increases in muscle shear moduli with passive muscle stretch. This new reconstruction method can be applied to quantify tissue mechanical properties of anisotropic soft tissues, such as muscle, in health and disease.

... We note that throughout this paper, the solid may be referred to as either the skeleton or the myocardium, the pore fluid is the coronary flow while the chamber flow is the cavity blood fluid. Techniques for explicitly modelling the coronary vascular network have been refined over several decades 2,3,4,5,6 , but models accounting for interactions within the micro vascular network 1 , where clinical perfusion is typically assessed, 7 remain scarce. Further, despite continued improvement in the spatial resolution of clinical imaging, a fully detailed representation of the coronary vessels within the ventricular wall remains unachievable. ...

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.
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... In initial experiments, we used a single pencil beam positioned parallel to the direction of shear wave propagation (i.e., along the length of the LV septum). Numerical simulations using a realistic heart model 46 indicate that waves do propagate preferentially from base to apex. However, for in vivo applications, this approach was not robust enough due to partial volume effects. ...

Changes in myocardial stiffness may represent a valuable biomarker for early tissue injury or adverse remodeling. In this study, we developed and validated a novel transducer-free magnetic resonance elastography (MRE) approach for quantifying myocardial biomechanics using aortic valve closure-induced shear waves. Using motion-sensitized two-dimensional pencil beams, septal shear waves were imaged at high temporal resolution. Shear wave speed was measured using time-of-flight of waves travelling between two pencil beams and corrected for geometrical biases. After validation in phantoms, results from twelve healthy volunteers and five cardiac patients (two left ventricular hypertrophy, two myocardial infarcts, and one without confirmed pathology) were obtained. Torsional shear wave speed in the phantom was 3.0 ± 0.1 m/s, corresponding with reference speeds of 2.8 ± 0.1 m/s. Geometrically-biased flexural shear wave speed was 1.9 ± 0.1 m/s, corresponding with simulation values of 2.0 m/s. Corrected septal shear wave speeds were significantly higher in patients than healthy volunteers [14.1 (11.0–15.8) m/s versus 3.6 (2.7–4.3) m/s, p = 0.001]. The interobserver 95%-limits-of-agreement in healthy volunteers were ± 1.3 m/s and interstudy 95%-limits-of-agreement − 0.7 to 1.2 m/s. In conclusion, myocardial shear wave speed can be measured using aortic valve closure-induced shear waves, with cardiac patients showing significantly higher shear wave speeds than healthy volunteers. This non-invasive measure may provide valuable insights into the pathophysiology of heart failure.