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... depending on the coupling between the left ventricle and the systemic arterial circulation-the basic (extended) model without (with) coupling to the systemic arterial circulation. Our models consist of 9 or 12 ordinary differential equations, and thus require significantly less computational resources and time compared with more advanced models [19] governed by partial differential equations. In particular, [19] simulated left ventricular fluidsolid mechanics through the cardiac cycle under LVAD support. ...

... Our models consist of 9 or 12 ordinary differential equations, and thus require significantly less computational resources and time compared with more advanced models [19] governed by partial differential equations. In particular, [19] simulated left ventricular fluidsolid mechanics through the cardiac cycle under LVAD support. CircAdapt [20] is another example of advanced models incorporating electric and sarcomere dynamics, but a systematic study with LAVD seems lacking. ...

... Finally, we note that the direct comparison of our results with those from more advanced (distributed) models with the LVAD support [19] is difficult. if not impossible, because of the use of different variables or parameters and the lack of a systematic study on these advanced models (due to high computational cost). ...

The breakdown of cardiac self-organization leads to heart diseases and failure, the number one cause of death worldwide. The left ventricular pressure-volume relation plays a key role in the diagnosis and treatment of heart diseases. Lumped-parameter models combined with pressure-volume loop analysis are very effective in simulating clinical scenarios with a view to treatment optimization and outcome prediction. Unfortunately, often invoked in this analysis is the traditional, time-varying elastance concept, in which the ratio of the ventricular pressure to its volume is prescribed by a periodic function of time, instead of being calculated consistently according to the change in feedback mechanisms (e.g., the lack or breakdown of self-organization) in heart diseases. Therefore, the application of the time-varying elastance for the analysis of left ventricular assist device (LVAD)-heart interactions has been questioned. We propose a paradigm shift from the time-varying elastance concept to a synergistic model of cardiac function by integrating the mechanical, electric, and chemical activity on microscale sarcomere and macroscale heart levels and investigating the effect of an axial rotary pump on a failing heart. We show that our synergistic model works better than the time-varying elastance model in reproducing LVAD-heart interactions with sufficient accuracy to describe the left ventricular pressure-volume relation.

... They demonstrated that co-pulse and counter-pulse modes are better than constant speed mode in view aspect of achieving unloading performance. To simulate the LVAD through the complete cardiac cycle, a non-conforming finite element fluid-solid mechanics approach has been developed recently by McCormic et al. [11,12]. They also investigated a patient supported by LVAD through simulation in six cases: Constant flow, sinusoidal in-sync and sinusoidal counter-sync according to the cardiac cycle at two different flow rates [13]. ...

... In order to pump the blood flow with the strong power through the balloon, the wall of the balloon must move toward the center of the balloon. In this study, twelve different displacement factor curves (different from the six curves in the previous study) which are applied to the balloon's wall are presented in Fig. 4 and denoted by numbers from (1) to (12). As seen from this figure, from number (1) to (5), the time in which the balloon is in maximum pressure is increased (maximum inflation time period (MITP)); the time duration of the deflation is increased from number (6) to (9) (deflation time period (DTP)); the time duration of the inflation is increased from number (10) to (12) (inflation time period (ITP)). ...

... Figs. 6(1)- (12) show the difference between patterns of exerted force associated with the twelve displacement factor curves on the aortic valve for three different boundary conditions during one heart cycle. ...

AVICENA is a new cardiac assist device which helps the patients suffering from left ventricular failure. This study aims to investigate the performance of this device and its effect on the aortic valve within different patterns of inflation and deflation of the balloon. Twelve different displacement factor curves which are applied to the balloon part of AVICENA and required for the performing of balloon inflation and deflation are selected and examined in three cases named low-pressure, normal-pressure, and high-pressure subjects. The results of generated energy, aortic valve force and pressure required for the balloon’s inflation and deflation are presented within a heart cycle. The outcomes of the present study can be utilized in designing of the AVICENA with more power generation and less aortic valve force for different subjects.

... These processes enable partitioned coupling, monolithic coupling or a mixed approach for different components of the physical systems allowing for a multitude of possible solution strategies. The infrastructural organisation outlined here has allowed us to avoid a burdensome re-engineering of the code when interlinking a variety of model combinations, including fluid-solid coupling [82,83,85,70, 22,71,72], vascular-porous flow [62], and electromechanics [104] to mention a few. ...

... For problems with monolithic integration, their implicit solution requires the formulation of a Jacobian matrix (or some approximation to it) which may be inverted or iteratively solved to give the solution or a solution update (as in Newton-Raphson methods [71,41]). Similar to other structures, we construct an object-set M = {M k } of matrices. ...

... The application of updates to the subgroup may either be done once or repeated using a fixed point criteria. The criteria used to terminate the fixed point iteration may be defined to depend on the size of computed updates, residual error or some combination of the two (for more details, see [71]). In addition, when linked with a fixed point procedure, the subgroup may also use line search [71], which effectively scales all updates by a single scalar α ∈ (0, 1] to minimize residual error. ...

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 physics systems, we introduce a framework in which the geometric and operator relationships between the constituent systems are rigorously defined. We then introduce the notion of topological interfaces and define specific operators encompassing many common model coupling requirements. These interfaces enable the evaluation of weak form integrals between mesh subregions of arbitrary finite element bases orders, types, and spatial dimensions. Equation maps are introduced which provide abstract representations of the individual physics systems that can be automatically combined to permit a monolithic matrix assembly. Flexible solution strategies for the resulting coupled systems are implemented, permitting fine–tuning of solution updates during fixed point iterations, and sub-grouping where several problems are being solved together. Partitioning of coupled mesh domains for optimal load balancing is also supported, taking into account the per-processor cost of the entire coupled problem within the graph problem. The demonstration of the performance is illustrated through important real-world multi-physics problems relevant to cardiac physiology.

... A nonconforming monolithic FSI method and model [182,183] were used to simulate passive / active cardiac mechanics on patient-specific geometries using the Costa constitutive equation [43]. This model was later applied to study congenital heart diseases [49,50] and assisted left ventricles [155,156,157] (see Figure 2b). Krittian [128] presented some of the first validation work, developing an experimental heart setup and modeled using both FSI and boundary driven flow modeling, illustrating good qualitative agreement between data and model particularly for the boundary driven flow model. ...

... Figure 5a illustrates the use of an electromechanical model used to access the efficacy of a mitral valve annuloplasty device that aims to reduce mitral regurgitation. Fluid-structure interaction models have been applied to assess left ventricular assist devices [155,156,157], examining how alterations in device settings influence myocardial unloading as well as the potential for LV thrombus formation. Electromechanical modeling has been used to assess the AdjuCor 5 extravascular ventricular assist device, whereby pneumatic cushions are used to improve ventricular stroke volume while unloading the heart [66]. ...

... Even the most efficient data assimilation methods become significantly more challenging and computationally intensive as the number of personalized parameters grow or the computational problem itself increases in demand. As a consequence, simplified models are often being used to parameterize components of more complex models; as was the case in [156] where a solid mechanics-Windkessel model was used to tune model parameters prior to simulating the full FSI-Windkessel model. Moreover, very complex models encompassing the widest range of physiological knowledge through integration many smaller models are not necessarily more predictive or reliable. ...

With heart and cardiovascular diseases continually challenging healthcare systems worldwide, translating basic research on cardiac (patho)physiology into clinical care is essential. Exacerbating this already extensive challenge is the complexity of the heart, relying on its hierarchical structure and function to maintain car- diovascular flow. Computational modeling has been proposed and actively pursued as a tool for accelerating research and translation. Allowing exploration of the relationships between physics, multiscale mechanisms and function, computational modeling provides a platform for improving our understanding of the heart. Further integration of experimental and clinical data through data assimilation and parameter estimation techniques is bringing computational models closer to use in routine clinical practice. This article reviews developments in computational cardiac modeling and how their integration with medical imaging data is providing new pathways for translational cardiac modeling.

... Finally, the WK model is suitable for the modeling of the systemic and pulmonary arterial systems (15). A lumped parameter model is based on differential equations expressed in terms of hydraulic or electrical networks making it suitable to study heart failure and ventricular unloading by an assist device (16)(17)(18)(19). The analogy with electric circuits facilitates the formulation of the necessary equations as shown in Table 1. ...

... A Lagrange multiplier coupling approach (74) can be applied to LVAD modeling (75) using fictitious domain methods (76,77) to address the interactions between the LVAD cannula and the ventricle at the expense of significant increase in computational time. These challenging aspects may be overcome with a 0-D WK (18,(78)(79)(80). A WK model and a time-varying resistance have been used more recently for dynamic modeling of pulsatile pumps (81). ...

... The three-element WK describes input impedance and system behaviour as a whole (23), while the reservoir-wave approach separates wave from reservoir component (93). Given the successful application of a 0-D WK in LVAD modeling (18,(78)(79)(80), the reservoir-wave approach may have a role to play. ...

Modeling of the cardiovascular system is challenging, but it has the potential to further advance our understanding of normal and pathological conditions. Morphology and function are closely related. The arterial system provides steady blood flow to each organ and damps out wave fluctuations as a consequence of intermittent ventricular ejection. These actions can be approached separately in terms of mathematical relationships between pressure and flow. Lumped parameter models are helpful for the study of the interactions between the heart and the arterial system. The arterial windkessel model still plays a significant role despite its limitations. This review aims to discuss the model and its modifications and derive the fundamental equations by applying electric circuits theory. In addition, its role during left ventricular assist device assistance is explored and discussed in relation to rotary blood pumps.
Copyright © 2015 International Center for Artificial Organs and Transplantation and Wiley Periodicals, Inc.

... These models have been used to investigate blood flow within the ventricular cavities and the efficiency of the heart as a pump from diastole [3] through to systole [11,22,32,33]. Recently, we [15,16] have extended a nonconforming finite element fluid-solid mechanics scheme [21] to facilitate the simulation of LVAD supported LVs through the full cardiac cycle. Using a fictitious domain (FD) [31] method to prescribe the LVAD cannula, the application of this approach enables the interaction between the cannula and the myocardial wall to be captured, facilitating the simulation of the full range of cardiac behaviour. ...

... Derived from the principles of conservation of mass and momentum, and as outlined in detail in our previous publications [16,22], we have developed a model that provides a physiological description of the myocardium and ventricular blood flow. In brief, the model was solved using a non-conforming Galerkin finite element scheme to enable varying degrees of refinement to adequately resolve the blood and myocardial spatial domains. ...

... This was achieved by integrating the 3D FSI model with a 0D Windkessel representation of systemic circulation. In this work, we coupled the Shi and Korakianitis 0D Windkessel model [28] using a fixed point prescribed flow rate technique [16]. Using this technique, flow was prescribed according to the pressure gradient across the valve using Bernoulli's equation for the conservation of energy along the same streamline. ...

... In this generalized form, with an appropriate choice of the projection operator, we may choose to strengthen or weaken the constraint resulting in the PL, LM or penalty formulations. Using this generalization, we derive an estimate detailing the error convergence of these methods (in a linear setting) and introduce modifications to a Newton-Raphson scheme [54,55] to significantly improve nonlinear convergence properties for standard and weakly penalized formulations (particularly for high bulk modulus). The scheme is further augmented to take advantage of a Shamanskii-type Newton scheme [54,55] boosting computational performance by enabling re-use of the Jacobian matrix (and its inverses or preconditioners) estimated at previous time/loading steps. ...

... Using this generalization, we derive an estimate detailing the error convergence of these methods (in a linear setting) and introduce modifications to a Newton-Raphson scheme [54,55] to significantly improve nonlinear convergence properties for standard and weakly penalized formulations (particularly for high bulk modulus). The scheme is further augmented to take advantage of a Shamanskii-type Newton scheme [54,55] boosting computational performance by enabling re-use of the Jacobian matrix (and its inverses or preconditioners) estimated at previous time/loading steps. As this re-use is particularly sensitive to stiffness, we modify the scheme to effectively maintain nonlinear convergence behavior. ...

... In order to solve the mechanical system introduced in Eq. (30) (as well as the others discussed in the Appendix A), we look to use the global Shamanskii-Newton-Raphson (SNR) method [54]. This method has been shown to be effective for problems in fluid-structure interaction [55], enabling faster computation by re-using the Jacobian matrix over multiple time/load steps. Following the procedure outlined in [55], on the n th SNR iteration we update each subsequent guess of the solution, ...

The Lagrange Multiplier (LM) and penalty methods are commonly used to enforce incompressibility and compressibility in models of cardiac mechanics. In this paper we show how both formulations may be equivalently thought of as a weakly penalized system derived from the statically condensed Perturbed Lagrangian formulation, which may be directly discretized maintaining the simplicity of penalty formulations with the convergence characteristics of LM techniques. A modified Shamanskii–Newton–Raphson scheme is introduced to enhance the nonlinear convergence of the weakly penalized system and, exploiting its equivalence, modifications are developed for the penalty form. Focusing on accuracy, we proceed to study the convergence behavior of these approaches using different interpolation schemes for both a simple test problem and more complex models of cardiac mechanics. Our results illustrate the well-known influence of locking phenomena on the penalty approach (particularly for lower order schemes) and its effect on accuracy for whole-cycle mechanics. Additionally, we verify that direct discretization of the weakly penalized form produces similar convergence behavior to mixed formulations while avoiding the use of an additional variable. Combining a simple structure which allows the solution of computationally challenging problems with good convergence characteristics, the weakly penalized form provides an accurate and efficient alternative to incompressibility and compressibility in cardiac mechanics.

... fluids and (often deformable) solid structures or particles, for example, in biomedical or aerospace engineering applications [1,2,3]. Well-established parallelization techniques, such as spatial domain decomposition methods, provide a straightforward and scalable approach for reducing the wall clock time for many FSI algorithms. ...

... with density ρ, Cauchy stress tensor σ(u, p) = µ(F − I) − pI, material stiffness parameter µ, the hydrostatic pressure variable p and initial velocityv 0 . Equation 1 can be transformed to a system of first-order equations, ...

This paper presents some recent advances for parallel-in-time methods applied to linear elasticity. With recent computer architecture changes leading to stagnant clock speeds, but ever increasing numbers of cores, future speedups will be available through increased concurrency. Thus, sequential algorithms, such as time stepping, will suffer a bottleneck. This paper explores multigrid reduction in time (MGRIT) for an important application area, linear elasticity. Previously, efforts at parallel-in-time for elasticity have experienced difficulties, for example, the beating phenomenon. As a result, practical parallel-in-time algorithms for this application area currently do not exist. This paper proposes some solutions made possible by MGRIT (e.g., slow temporal coarsening and FCF-relaxation) and more importantly, a different formulation of the problem that is more amenable to parallel-in-time methods. Using a recently developed convergence theory for MGRIT and Parareal, we show that the changed formulation of the problem avoids the instability issues and allows reduction of the error using two temporal grids. We then extend our approach to the multilevel case, where we demonstrate how slow temporal coarsening improves convergence. The paper ends with supporting numerical results showing a practical algorithm enjoying speedup benefits over the sequential algorithm.

... In this paper, we consider a monolithic ALE FSI technique that is able to use non-conforming meshes at the interface, 5,[15][16][17] such that meshes can be designed based on the requirements of the physics of the coupled subsystems leading to improved accuracy and to decreased computational cost (by avoiding underrefinement and overrefinement, respectively). Coupling of subdomain equations is achieved via introduction of an additional coupling domain and enforcing interface constraints by means of a Lagrange multiplier variable. ...

... It has been extended recently to enable modeling of turbulent flow phenomena by a stabilized cG(1)cG(1) scheme 21 to extend the use of the method over a larger range of Reynolds numbers. Besides the various biomedical engineering applications, 5,17,20 the method has been assessed and verified using test problems; however, it was not validated in any previous work. Thus, validation of the method will be the focus of this work as well as validation of using the cG(1)cG(1) scheme within the Lagrange multiplier-based coupling method. ...

This paper details the validation of a non-conforming arbitrary Lagrangian-Eulerian fluid-structure interaction (FSI) technique using a recently developed experimental 3D FSI benchmark problem. Numerical experiments for steady and transient test cases of the benchmark were conducted employing an inf-sup stable and a general Galerkin scheme. The performance of both schemes is assessed. Spatial refinement with three mesh refinement levels and fluid domain truncation with two fluid domain lengths are studied as well as employing a sequence of increasing time step sizes for steady-state cases. How quickly an approximate steady-state or periodic steady-state is reached is investigated and quantified based on error norm computations. Comparison of numerical results with experimental phase-contrast magnetic resonance imaging data shows very good overall agreement including governing of flow patterns observed in the experiment.

... The general solution procedure to solve equation 32 is outlined in algorithm 1. Briefly, we follow a Shamanskii-Newton Raphson procedure introduced previously in [51] and used in [39]. Here, the Jacobian matrix, K, and its inverse are initially computed and only recomputed when residual convergence slows. ...

... Endo-volume 21,22 [2,46] Applied using Λe = R and using data-derived V lv . Methods vary in how V (u) is computed: SVI: V calculated using eqn 24 VI: V calculated using eqn 28 Endo-volumerate 26 [39,26] Applied using Λe = R and using data-derived ∂tV lv . VRI: V calculated using eqn 26 Table 2: Summary of some common and proposed endocardial boundary conditions for LV mechanics models. ...

Supported by the wide range of medical data available, cardiac biomechanical modeling has exhibited sig- nificant potential in improving our understanding of the complex heart function and assisting in patient diagnosis and treatment. A critical step towards the development of accurate patient-specific models is the deployment of boundary conditions capable of integrating data into the model to enhance model fi- delity. This step is often hindered by sparse or noisy data that, if applied directly, commonly introduces non-physiological forces and artifacts into the model. To address these issues, in this paper we propose novel boundary conditions which aim to balance the accurate use of the data with physiological boundary forces and model outcomes though the use of data-derived boundary energies. The introduced techniques employ Lagrange multipliers, penalty methods and moment-based constraints to achieve robustness to data of varying quality and quantity. The proposed methods are compared with commonly used boundary con- ditions over an idealized left ventricle as well as over in vivo models, exhibiting significant improvement in model accuracy. The boundary conditions are also employed in in vivo full-cycle models of healthy and diseased hearts, demonstrating the ability of the proposed approach to reproduce data-derived deformation and physiological boundary forces over a varied range of cardiac function.

... represents the residual function. The nonlinear mechanics system is subsequently solved using the Newton-Raphson scheme with line search and Jacobian reuse outlined in [51]. ...

... To accelerate the simulation of cardiac mechanics, we examined the first-order effects, which influence the compute time of the whole cycle. Because of the system matrix reuse strategy which significantly improves compute times [51] by reducing the number of matrix builds and factorizations, residual evaluations consume most of the compute time. As a result, our initial focus was to port both residual evaluations and Jacobian computations to the GPU. ...

In this paper, we look at the acceleration of weakly coupled electromechanics using the graphics processing unit (GPU). Specifically, we port to the GPU a number of components of CHeart-a CPU-based finite element code developed for simulating multi-physics problems. On the basis of a criterion of computational cost, we implemented on the GPU the ODE and PDE solution steps for the electrophysiology problem and the Jacobian and residual evaluation for the mechanics problem. Performance of the GPU implementation is then compared with single core CPU (SC) execution as well as multi-core CPU (MC) computations with equivalent theoretical performance. Results show that for a human scale left ventricle mesh, GPU acceleration of the electrophysiology problem provided speedups of 164 × compared with SC and 5.5 times compared with MC for the solution of the ODE model. Speedup of up to 72 × compared with SC and 2.6 × compared with MC was also observed for the PDE solve. Using the same human geometry, the GPU implementation of mechanics residual/Jacobian computation provided speedups of up to 44 × compared with SC and 2.0 × compared with MC. © 2013 The Authors. International Journal for Numerical Methods in Biomedical Engineering published by John Wiley & Sons, Ltd.

... So far, a few finite element-based computational models that account for the coupling between a cardiac assist device and the heart have been developed. The existing work presents simulations for the ventricular pumps that are coupled with a univentricular (26) or biventricular heart models (27) via a cannula. A computational model for the innovative support system from AdjuCor GmbH is presented in Hirschvogel et al. (28). ...

The present computational study investigates the effects of an epicardial support pressure mimicking a heart support system without direct blood contact. We chose restrictive cardiomyopathy as a model for a diseased heart. By changing one parameter representing the amount of fibrosis, this model allows us to investigate the impairment in a diseased left ventricle, both during diastole and systole. The aim of the study is to determine the temporal course and value of the support pressure that leads to a normalization of the cardiac parameters in diseased hearts. These are quantified via the end-diastolic pressure, end-diastolic volume, end-systolic volume, and ejection fraction. First, the amount of fibrosis is increased to model diseased hearts at different stages. Second, we determine the difference in the left ventricular pressure between a healthy and diseased heart during a cardiac cycle and apply for the epicardial support as the respective pressure difference. Third, an epicardial support pressure is applied in form of a piecewise constant step function. The support is provided only during diastole, only during systole, or during both phases. Finally, the support pressure is adjusted to reach the corresponding parameters in a healthy rat. Parameter normalization is not possible to achieve with solely diastolic or solely systolic support; for the modeled case with 50% fibrosis, the ejection fraction can be increased by 5% with purely diastolic support and 14% with purely systolic support. However, the ejection fraction reaches the value of the modeled healthy left ventricle (65.6%) using a combination of diastolic and systolic support. The end-diastolic pressure of 13.5 mmHg cannot be decreased with purely systolic support. However, the end-diastolic pressure reaches the value of the modeled healthy left ventricle (7.5 mmHg) with diastolic support as well as with the combination of the diastolic and systolic support. The resulting negative diastolic support pressure is −4.5 mmHg, and the positive systolic support pressure is 90 mmHg. We, thereby, conclude that ventricular support during both diastole and systole is beneficial for normalizing the left ventricular ejection fraction and the end-diastolic pressure, and thus it is a potentially interesting therapy for cardiac insufficiency.

... A classical occurrence in nature is represented by the interaction of the blood flow in the human circulatory system, which has attracted the interest of scientists for decades [13,14,39,64,157]. In particular, the understanding of aneurysm in arteries constitutes an active area of research [15,44,110,156], as well as the comprehension of the pumping motion of the human heart [105,161,165]. FSI is also omnipresent in a vast range on engineering applications. ...

The interest in the simulation of fluid-structure interaction (FSI) phenomena has increased significantly over the years. Despite the constant growth in available computing resources, the demand for more robust and efficient computational methods does not cease.
This thesis proposes novel schemes for the solution of FSI problems with weakly compressible flows. Special attention is devoted to the spatial discretization of the fluid problem by means of the hybridizable discontinuous GALERKIN (HDG) method and to the coupling of the fluid field with the structural one, discretized by means of the continuous GALERKIN (CG) method.

... To account for differences in constitutive behavior between the two phases, a non-conforming monolithic framework [360] was used to build patient-specific models able to simulate both active and passive mechanics [361]. Later adaptations of this model were used to study the effect of left ventricular assist devices [316,318,317] as well as diastolic function in hypoplastic left heart patients [89,486]. More recently, Gao et al. [143] demonstrated the feasibility of merging the LV and mitral valve into a monolithic anatomically accurate FSI model. ...

The hierarchical construction of the myocardium plays a pivotal role in the biomechanics of the heart muscle and the resulting flow of blood. In disease, the construction of the heart remodels, altering the structure of the tissue from the subcellular level all the way to the whole organ. Elucidating the impact of these fundamental alterations on the biomechanics of the heart presents challenges to diagnosis, therapy planning and treatment. Computational modeling provides an innovative tool, enabling the simulation of complex biomechanics that capture the complexity of tissue, its growth and remodeling and the resulting blood flow. In this chapter, we review the key ways that computational models can address challenging biomechanical questions in the heart and how these tools can change the way treatment is approached across a range of heart diseases.

... Implementing hemodynamic models in vitro involves many technical challenges, so computational models were developed to predict the hemodynamics of the heart. Hemodynamic models predict blood flow within the heart under a repeated expansion-contraction cycle, coupled with cardiac electromechanics [226,232]. In particular, computational fluid dynamics (CFD) models based on the Navier-Stokes equation were used to simulate the pumping of the heart [233]. ...

Cardiac tissue engineering aims to generate in vivo-like functional tissue for the study of cardiac development, homeostasis, and regeneration. Since the heart is composed of various types of cells and extracellular matrix with a specific microenvironment, the fabrication of cardiac tissue in vitro requires integrating technologies of cardiac cells, biomaterials, fabrication, and computational modeling to model the complexity of heart tissue. Here, we review the recent progress of engineering techniques from simple to complex for fabricating matured cardiac tissue in vitro. Advancements in cardiomyocytes, extracellular matrix, geometry, and computational modeling will be discussed based on a technology perspective and their use for preparation of functional cardiac tissue. Since the heart is a very complex system at multiscale levels, an understanding of each technique and their interactions would be highly beneficial to the development of a fully functional heart in cardiac tissue engineering.

... Our current understanding of flow through an LVAD is largely based on ex vivo and in vivo modeling using a combination of artificial circuits, computer simulations and animal models (3)(4)(5)(6)(7)(8). Few studies have examined the flow patterns in humans supported with LVADs. ...

Our current understanding of flow through the circuit of left ventricular assist device (LVAD), left ventricle and ascending aorta remains incompletely understood. Computational fluid dynamics, which allow for analysis of flow in the cardiovascular system, have been used for this purpose, although current simulation models have failed to fully incorporate the interplay between the pulsatile left ventricle and continuous-flow generated by the LVAD. Flow-through the LVAD is dependent on the interaction between device and patient-specific factors with suboptimal flow patterns evoking increased risk of LVAD-related complications. Computational fluid dynamics can be used to analyze how different pump and patient factors affect flow patterns in the left ventricle and the aorta. Computational fluid dynamics simulations were carried out on a patient with a HeartMate II. Simulations were also conducted for theoretical scenarios substituting HeartWare HVAD, HeartMate 3 (HM3) in continuous mode and HM3 with Artificial Pulse. An anatomical model of the patient was reconstructed from computed tomography (CT) images, and the LVAD outflow was used as the inflow boundary condition. The LVAD outflow was calculated separately using a lumped-parameter-model of the systemic circulation, which was calibrated to the patient based on the patient-specific ventricular volume change reconstructed from 4 dimensional computed tomography and pulmonary capillary wedge pressure tracings. The LVADs were implemented in the lumped-parameter-model via published pressure head versus flow (H-Q) curves. To quantify the flushing effect, virtual contrast agent was released in the ascending aorta and its flushing over the cycles was quantified. Shear stress acting on the aortic endothelium and shear rate in the bloodstream were also quantified as indicators of normal/abnormal blood flow, especially the latter being a biomarker of platelet activation and hemolysis. LVAD speeds for the HVAD and HM3 were selected to match flow rates for the patient's HMII (9,000 RPM for HMII, 5,500 RPM for HM3, and 2,200 RPM for HVAD), the cardiac outputs were 5.81 L/min, 5.83 L/min, and 5.92 L/min, respectively. The velocity of blood flow in the outflow cannula was higher in the HVAD than in the two HeartMate pumps with a cycle average (range) of 0.92 m/s (0.78-1.19 m/s), 0.91 m/s (0.86-1.00 m/s), and 1.74 m/s (1.40-2.24 m/s) for HMII, HM3, and HVAD, respectively. Artificial pulse increased the peak flow rate to 9.84 L/min for the HM3 but the overall cardiac output was 5.96 L/min, which was similar to the continuous mode. Artificial pulse markedly decreased blood stagnation in the ascending aorta; after six cardiac cycles, 48% of the blood was flushed out from the ascending aorta under the continuous operation mode while 60% was flushed under artificial pulse. Shear stress and shear rate in the aortic arch were higher with the HVAD compared to the HMII and HM3, respectively (shear stress: 1.76 vs. 1.33 vs. 1.33 Pa, shear rate: 136 vs. 91.5 vs. 89.4 s-1). Pump-specific factors such as LVAD type and programmed flow algorithms lead to unique flow patterns which influence blood stagnation, shear stress, and platelet activation. The pump-patient interaction can be studied using a novel computational fluid dynamics model to better understand and potentially mitigate the risk of downstream LVAD complications.

... In industry, we have examples such as the design of parachutes [1][2][3] and wind-turbines [4,5]. In cardiovascular research [6][7][8], the simulation of valves [9,10] and implanted devices [11][12][13][14] holds a great potential for better understanding and treatment of a number of pathologies such as valve stenosis, regurgitation, heart failure and outflow obstruction. Building such models, however, remains challenging due to the need to balance computational costs and solution accuracy. ...

For problems involving large deformations of thin structures, simulating fluid-structure interaction (FSI) remains a computationally expensive endeavour which continues to drive interest in the development of novel approaches. Overlapping domain techniques have been introduced as a way to combine the fluid-solid mesh conformity, seen in moving-mesh methods, without the need for mesh smoothing or re-meshing, which is a core characteristic of fixed mesh approaches. In this work, we introduce a novel overlapping domain method based on a partition of unity approach. Unified function spaces are defined as a weighted sum of fields given on two overlapping meshes. The method is shown to achieve optimal convergence rates and to be stable for steady-state Stokes, Navier-Stokes, and ALE Navier-Stokes problems. Finally, we present results for FSI in the case of 2D flow past an elastic beam simulation. These initial results point to the potential applicability of the method to a wide range of FSI applications, enabling boundary layer refinement and large deformations without the need for re-meshing or user-defined stabilization.

... Significant recent improvements have been made in determination of diastolic (Sack et al.) and systolic (Sack et al., 2016) myocardial material properties in FE models of normal human LV (Genet et al., 2014). Relatively few computational studies have investigated the effects of mechanical circulatory support devices on cardiac function using realistic geometries (Lim et al., 2012;McCormick et al., 2013). A recent study of the effects of an LV assist device on acute left heart failure (Sack et al., 2016) presented stress results in both ventricles. ...

... To our knowledge, previous finite element models of heart-LVAD interaction did not consider the biventricular interaction. For example, the study by McCormick et al. (2013) expands the LV FSI model of Nordsletten et al. (2011) by adding an immersed cannula structure, but does not implement any electrophysiology physics. Other studies by Sack et al. (2016) and Heikhmakhtiar et al. (2017) simply modeled the LVAD contribution by adding a constant flow from the heart and into the aortic circuit of the four-chamber Living Heart Model (Baillargeon et al., 2014) and the Gurev et al. (2011) biventricular model, respectively. ...

Computational models have become essential in predicting medical device efficacy prior to clinical studies. To investigate the performance of a left-ventricular assist device (LVAD), a fully-coupled cardiac fluid-electromechanics finite element model was developed, incorporating electrical activation, passive and active myocardial mechanics, as well as blood hemodynamics solved simultaneously in an idealized biventricular geometry. Electrical activation was initiated using a simplified Purkinje network with one-way coupling to the surrounding myocardium. Phenomenological action potential and excitation-contraction equations were adapted to trigger myocardial contraction. Action potential propagation was formulated within a material frame to emulate gap junction-controlled propagation, such that the activation sequence was independent of myocardial deformation. Passive cardiac mechanics were governed by a transverse isotropic hyperelastic constitutive formulation. Blood velocity and pressure were determined by the incompressible Navier-Stokes formulations with a closed-loop Windkessel circuit governing the circulatory load. To investigate heart-LVAD interaction, we reduced the left ventricular (LV) contraction stress to mimic a failing heart, and inserted a LVAD cannula at the LV apex with continuous flow governing the outflow rate. A proportional controller was implemented to determine the pump motor voltage whilst maintaining pump motor speed. Following LVAD insertion, the model revealed a change in the LV pressure-volume loop shape from rectangular to triangular. At higher pump speeds, aortic ejection ceased and the LV decompressed to smaller end diastolic volumes. After multiple cycles, the LV cavity gradually collapsed along with a drop in pump motor current. The model was therefore able to predict ventricular collapse, indicating its utility for future development of control algorithms and pre-clinical testing of LVADs to avoid LV collapse in recipients.

... This model is extensively applied in nonlinear biomechanics-for example in simulations of the heart (Wang et al. 2009, Nordsletten et al. 2011a, McCormick et al. 2013, 2014, Hadjicharalambous et al. 2014, breast (Rajagopal et al. 2010, Reynolds et al. 2011, Gamage et al. 2011, arterial wall (Holzapfel 2000, Gasser et al. 2006, Hariton et al. 2007, etc.-where specific material response is usually defined through biorheological experiments. The simplest material models are purely elastic, based on spring rheological elements. ...

Characterisation of soft tissue mechanical properties is a topic of increasing interest in translational and clinical research. Magnetic resonance elastography (MRE) has been used in this context to assess the mechanical properties of tissues in vivo noninvasively. Typically, these analyses rely on linear viscoelastic wave equations to assess material properties from measured wave dynamics. However, deformations that occur in some tissues (e.g. liver during respiration, heart during the cardiac cycle, or external compression during a breast exam) can yield loading bias, complicating the interpretation of tissue stiffness from MRE measurements. In this paper, it is shown how combined knowledge of a material’s rheology and loading state can be used to eliminate loading bias and enable interpretation of intrinsic (unloaded) stiffness properties. Equations are derived utilising perturbation theory and Cauchy’s equations of motion to demonstrate the impact of loading state on periodic steady-state wave behaviour in nonlinear viscoelastic materials. These equations demonstrate how loading bias yields apparent material stiffening, softening and anisotropy. MRE sensitivity to deformation is demonstrated in an experimental phantom, showing a loading bias of up to twofold. From an unbiased stiffness of \(4910.4 \pm 635.8\) Pa in unloaded state, the biased stiffness increases to 9767.5 \(\pm \,\)1949.9 Pa under a load of \(\approx \) 34% uniaxial compression. Integrating knowledge of phantom loading and rheology into a novel MRE reconstruction, it is shown that it is possible to characterise intrinsic material characteristics, eliminating the loading bias from MRE data. The framework introduced and demonstrated in phantoms illustrates a pathway that can be translated and applied to MRE in complex deforming tissues. This would contribute to a better assessment of material properties in soft tissues employing elastography.

... Alternatively, a fluid-solid left ventricular model coupled with a lumped parameter model [38][39][40][41][42] can be optimized with high order interpolation at the fluid-solid boundary and obtain simulations of fluid-solid interaction over a complete cardiac cycle during circulatory assistance. [43] Here we present a heart failure patient discussed during MDT meeting whose outcome was compared to the results carried out with CARDIOSIM c software to investigate the role of this approach as a planning strategy to guide intervention and predict outcome. ...

Clinical practice heavily relies on results from randomized controlled trials, which may not reflect completely individual patients. Patient-specific modelling has received increasing attention in recent years. Although still far from clinical application on a daily basis, the potential of this approach is significant. The treatment of advanced heart failure may benefit from a modelling framework to guide device treatment and predict outcome. The role of mechanical circulatory support as a long-term solution is increasing in view of the evolving technology and worsening heart failure patient population. Therefore, a preoperative strategy with the ability to predict the course of events in a simulation setting may be justified. Here we present a heart failure patient discussed at a multidisciplinary team meeting whose outcome was compared with simulations carried out with CARDIOSIM$^{©}$ software to investigate the role of this approach as a planning strategy to guide intervention and predict outcome. The clinical decision process is complex and many factors are involved. Patient-specific modelling may have a role to play as part of a preoperative planning strategy with more quantitative evaluation to smooth decision-making.

... A Lagrange multiplier coupling approach [49] can be applied to LVAD modelling [50] using fictitious domain methods [51,52] to address the interactions between the LVAD cannula and the ventricle at the expense of significant increase in computational time and instability in specific regions of interface between the fluid and solid meshes. To overcome these limitations, a fluid-solid left ventricular model coupled with a 0-D Windkessel model [53,54] already successfully applied to different types of LVAD [55][56][57] can be used where optimization with high order interpolation at the fluid-solid boundary allows simulations of fluid-solid interaction over a complete cardiac cycle during LVAD support [58]. ...

Background
Modelling and simulation may become clinically applicable tools for detailed evaluation of the cardiovascular system and clinical decision-making to guide therapeutic intervention. Models based on pressure–volume relationship and zero-dimensional representation of the cardiovascular system may be a suitable choice given their simplicity and versatility. This approach has great potential for application in heart failure where the impact of left ventricular assist devices has played a significant role as a bridge to transplant and more recently as a long-term solution for non eligible candidates. ResultsWe sought to investigate the value of simulation in the context of three heart failure patients with a view to predict or guide further management. CARDIOSIM© was the software used for this purpose. The study was based on retrospective analysis of haemodynamic data previously discussed at a multidisciplinary meeting. The outcome of the simulations addressed the value of a more quantitative approach in the clinical decision process. Conclusions
Although previous experience, co-morbidities and the risk of potentially fatal complications play a role in clinical decision-making, patient-specific modelling may become a daily approach for selection and optimisation of device-based treatment for heart failure patients. Willingness to adopt this integrated approach may be the key to further progress.

... Here, the system under consideration models the interaction between fluids and (often deformable) solid structures or particles, for example, in biomedical or aerospace engineering applications. [1][2][3] Well-established parallelization techniques, such as spatial domain decomposition methods, provide a straightforward and scalable approach for reducing the wall-clock time for many FSI algorithms. However, spatial parallelism saturates when communication tasks become dominant over computation tasks. ...

This paper presents some recent advances for parallel-in-time methods applied to linear elasticity. With recent computer architecture changes leading to stagnant clock speeds, but ever increasing numbers of cores, future speedups will be available through increased concurrency. Thus, sequential algorithms, such as time stepping, will suffer a bottleneck. This paper explores multigrid reduction in time (MGRIT) for an important application area, linear elasticity. Previously, efforts at parallel-in-time for elasticity have experienced difficulties, for example, the beating phenomenon. As a result, practical parallel-in-time algorithms for this application area currently do not exist. This paper proposes some solutions made possible by MGRIT (e.g., slow temporal coarsening and FCF-relaxation) and, more importantly, a different formulation of the problem that is more amenable to parallel-in-time methods. Using a recently developed convergence theory for MGRIT and Parareal, we show that the changed formulation of the problem avoids the instability issues and allows the reduction of the error using two temporal grids. We then extend our approach to the multilevel case, where we demonstrate how slow temporal coarsening improves convergence. The paper ends with supporting numerical results showing a practical algorithm enjoying speedup benefits over the sequential algorithm.

... A Lagrange multiplier coupling approach [79] can be applied to LVAD modelling [80] using fictitious domain methods [81][82] to address the interactions between the LVAD cannula and the ventricle at the expense of significant increase in computational time and instability in specific regions of interface between the fluid and solid meshes. To overcome these limitations, a fluid-solid left ventricular model coupled with a 0D Windkessel model [83][84] already successfully applied to different types of LVAD [85][86][87] can be used where optimization with high order interpolation at the fluid-solid boundary allows simulations of fluid-solid interaction over a complete cardiac cycle during LVAD support [88]. ...

The impact of ventricular assist devices (VADs) for the treatment of advanced heart failure has played a significant role as a bridge to transplant and more recently as a long-term solution for non eligible candidates. Continuous flow rotary blood pumps are currently the most popular devices in view of their smaller size, increased reliability and higher durability compared to pulsatile flow devices. The trend towards their use is increasing. Mathematical modelling and computer simulation are invaluable tools to investigate the interactions between VADs and the cardiovascular system. In view of its complexity, assumptions and simplifications have to be made in order to achieve a balance between computational time and reliability of the modelling approach. The time-varying elastance theory and lumped parameter models have been widely used for this purpose. Dynamic modelling with time-varying resistance and nonlinear time-varying lumped parameter models have been applied with promising results. Speed modulation and a Starling-like controller for continuous flow VADs are the subject of intense modelling and simulation to optimize loading conditions and achieve a more physiological response pattern. Simulations for continuous flow VADs are usually performed under steady flow conditions and optimization of the device is based on these results. A simulation-based approach in the context of patient-specific modelling as a potential preoperative strategy may be an additional tool to obtain more accurate predictions of the performance of these devices in a clinical setting. Besides, it would be a valuable training opportunity for medical and nursing staff involved with the care of this complex and challenging group of patients. The aim of this review is to give an overview of the different approaches currently used and attempt to link current thinking to future developments.

... Raised afterload and RV dilatation are associated with fiber reorientation toward the circumferential direction (32), and our data indicate that such adverse structural remodeling independently contributes to survival. (24,25). Atlas-based analysis in the heart has been applied to describing shape variation among asymptomatic adults, identifying persisting effects of preterm delivery on ventricular geometry, and demonstrating patterns of remodeling after myocardial infarction (9,26,27). ...

Purpose To determine if patient survival and mechanisms of right ventricular failure in pulmonary hypertension could be predicted by using supervised machine learning of three-dimensional patterns of systolic cardiac motion. Materials and Methods The study was approved by a research ethics committee, and participants gave written informed consent. Two hundred fifty-six patients (143 women; mean age ± standard deviation, 63 years ± 17) with newly diagnosed pulmonary hypertension underwent cardiac magnetic resonance (MR) imaging, right-sided heart catheterization, and 6-minute walk testing with a median follow-up of 4.0 years. Semiautomated segmentation of short-axis cine images was used to create a three-dimensional model of right ventricular motion. Supervised principal components analysis was used to identify patterns of systolic motion that were most strongly predictive of survival. Survival prediction was assessed by using difference in median survival time and area under the curve with time-dependent receiver operating characteristic analysis for 1-year survival. Results At the end of follow-up, 36% of patients (93 of 256) died, and one underwent lung transplantation. Poor outcome was predicted by a loss of effective contraction in the septum and free wall, coupled with reduced basal longitudinal motion. When added to conventional imaging and hemodynamic, functional, and clinical markers, three-dimensional cardiac motion improved survival prediction (area under the receiver operating characteristic curve, 0.73 vs 0.60, respectively; P < .001) and provided greater differentiation according to difference in median survival time between high- and low-risk groups (13.8 vs 10.7 years, respectively; P < .001). Conclusion A machine-learning survival model that uses three-dimensional cardiac motion predicts outcome independent of conventional risk factors in patients with newly diagnosed pulmonary hypertension. Online supplemental material is available for this article.

... In the field of biomedical engineering, fluid-structure interaction (FSI) modeling is playing an increasingly important role due to the coupling of fluid flow and tissue mechanics vital to many physical phenomena. Use of FSI models for the assessment of medical devices as well as clinical evaluation [6,7,8] is becoming increasingly common. In silico testing of devices using FSI models can help to expedite and augment pre-production development as well as assist in understanding the implications of an implant and its interaction in the human body. ...

In this paper a fluid-structure interaction (FSI) experiment is presented. The aim of this experiment is to provide a challenging yet easy-to-setup FSI test case that addresses the need for rigorous testing of FSI algorithms and modeling frameworks. Steady-state and periodic steady-state test cases with constant and periodic inflow were established. Focus of the experiment is on biomedical engineering applications with flow being in the laminar regime with Reynolds numbers 1283 and 651. Flow and solid domains were defined using CAD tools. The experimental design aimed at providing a straight-forward boundary condition definition. Material parameters and mechanical response of a moderately viscous Newtonian fluid and a nonlinear incompressible solid were experimentally determined. A comprehensive data set was acquired by employing magnetic resonance imaging to record the interaction between the fluid and the solid, quantifying flow and solid motion.

... In this novel computational study concerned with the normal human heart, acute left HF, and LVAD therapy, we sought to detail significant improvements in the systolic material properties of the Dassault Systèmes Living Heart Model (LHM) and use it to compute left and right ventricular myofiber stress distributions under the following 4 conditions: (1) normal cardiac function; (2) acute left HF with an LV ejection fraction (EF) of 28%; (3) acute left HF using an LVAD flow rate of 2 L/min; and (4) acute left HF treated using an LVAD flow rate of 4.5 L/min. Relatively few computational studies have investigated the effect of LVADs on cardiac function utilizing realistic geometries (17,18) due to the complexities involved, and to the best of our knowledge, our study is the first to present stress and strain results in both ventricles. ...

Purpose
Heart failure is a worldwide epidemic that is unlikely to change as the population ages and life expectancy increases. We sought to detail significant recent improvements to the Dassault Systèmes Living Heart Model (LHM) and use the LHM to compute left ventricular (LV) and right ventricular (RV) myofiber stress distributions under the following 4 conditions: (1) normal cardiac function; (2) acute left heart failure (ALHF); (3) ALHF treated using an LV assist device (LVAD) flow rate of 2 L/min; and (4) ALHF treated using an LVAD flow rate of 4.5 L/min.
Methods and Results
Incorporating improved systolic myocardial material properties in the LHM resulted in its ability to simulate the Frank-Starling law of the heart. We decreased myocardial contractility in the LV myocardium so that LV ejection fraction decreased from 56% to 28%. This caused mean LV end diastolic (ED) stress to increase to 508% of normal, mean LV end systolic (ES) stress to increase to 113% of normal, mean RV ED stress to decrease to 94% of normal and RV ES to increase to 570% of normal. When ALHF in the model was treated with an LVAD flow rate of 4.5 L/min, most stress results normalized. Mean LV ED stress became 85% of normal, mean LV ES stress became 109% of normal and mean RV ED stress became 95% of normal. However, mean RV ES stress improved less dramatically (to 342% of normal values).
Conclusions
These simulations strongly suggest that an LVAD is effective in normalizing LV stresses but not RV stresses that become elevated as a result of ALHF.

... In this context, a synergy between clinical imaging and computer modeling has the potential to provide accurate patient-specific information to assist the clinical decision-making process. Recently, computational modeling has reached a stage of development with capability to simulate cardiac function realistically and to augment the traditional clinical approaches (McCormick et al., 2013;Tang et al., 2010). However, this potential is currently not fully exploited, as model personalization often relies on invasive pressure data acquired through catheterization, which in complex pathologies is limited by technical difficulties and risks for the patient, and on detailed information on the orientation of the myocardial fiber architecture. ...

Current state-of-the-art imaging techniques can provide quantitative information to characterize ventricular function within the limits of the spatiotemporal resolution achievable in a realistic acquisition time. These imaging data can be used to personalize computer models, which in turn can help treatment planning by quantifying biomarkers that cannot be directly imaged, such as flow energy, shear stress and pressure gradients. To date, computer models have typically relied on invasive pressure measurements to be made patient-specific. When these data are not available, the scope and validity of the models are limited. To address this problem, we propose a new methodology for modeling patient-specific hemodynamics based exclusively on noninvasive velocity and anatomical data from 3D+t echocardiography or Magnetic Resonance Imaging (MRI). Numerical simulations of the cardiac cycle are driven by the image-derived velocities prescribed at the model boundaries using a penalty method that recovers a physical solution by minimizing the energy imparted to the system. This numerical approach circumvents the mathematical challenges due to the poor conditioning that arises from the imposition of boundary conditions on velocity only. We demonstrate that through this technique we are able to reconstruct given flow fields using Dirichlet only conditions. We also perform a sensitivity analysis to investigate the accuracy of this approach for different images with varying spatiotemporal resolution. Finally, we examine the influence of noise on the computed result, showing robustness to unbiased noise with an average error in the simulated velocity approximately 7% for a typical voxel size of 2mm3 and temporal resolution of 30ms. The methodology is eventually applied to a patient case to highlight the potential for a direct clinical translation.

... The remeshing method authorises, at given predefined time step, a complete remesh the deformed geometry. The ALE method has been previously used in biomechanical problems, specifically in cardiovascular problems where fluid-structure interactions between blood flow and natural tissues [17][18][19][20][21][22][23] or devices [24][25][26] are modelled, or in deformation problems of soft organs such as breasts and lungs [27,28], or in car safety simulation such as airbag deployment [29]. However, to the best of the authors' knowledge, it has not been used with dentistry-related models. ...

In finite element simulations of orthodontic tooth movement, one of the challenges is to represent long term tooth movement. Large deformation of the periodontal ligament and large tooth displacement due to bone remodelling lead to large distortions of the finite element mesh when a Lagrangian formalism is used. We propose in this work to use an Arbitrary Lagrangian Eulerian (ALE) formalism to delay remeshing operations. A large tooth displacement is obtained including effect of remodelling without the need of remeshing steps but keeping a good-quality mesh. Very large deformations in soft tissues such as the periodontal ligament is obtained using a combination of the ALE formalism used continuously and a remeshing algorithm used when needed. This work demonstrates that the ALE formalism is a very efficient way to delay remeshing operations.

... The advantage of such approaches are that the interactions between the cardiac electromechanics and cardiac flow can be investigated via first-principles. For example, McCormick et al. [36,37] investigated the effect of LVAD on the LV structure and hemodynamics using the coupled FSI simulations (see Fig. 4). Another recent example of a one-way coupled model is that of Choi et al. (2015) [35] which coupled a detailed multiscale model for cardiac electromechanics with a Navier-Stokes model for the hemodynamics, and used it to examine the effect of heart failure and dyssynchony on LV hemodynamics. ...

The proliferation of four-dimensional imaging technologies, increasing computational speeds, improved simulation algorithms, and the widespread availability of powerful computing platforms is enabling simulations of cardiac hemodynamics with unprecedented speed and fidelity. Since cardiovascular disease is intimately linked to cardiovascular hemodynamics, accurate assessment of the patient's hemodynamic state is critical for the diagnosis and treatment of heart disease. Unfortunately, while a variety of invasive and non-invasive approaches for measuring cardiac hemodynamics are in widespread use, they still only provide an incomplete picture of the hemodynamic state of a patient. In this context, computational modeling of cardiac hemodynamics presents as a powerful non-invasive modality that can fill this information gap, and significantly impact the diagnosis as well as the treatment of cardiac disease. This article reviews the current status of this field as well as the emerging trends and challenges in cardiovascular health, computing, modeling and simulation and that are expected to play a key role in its future development. Some recent advances in modeling and simulations of cardiac flow are described by using examples from our own work as well as the research of other groups.

... Next, we describe the in vivo datasets processed and present pipeline outputs for these cases. All simulations were performed using CHeart (McCormick et al. 2011(McCormick et al. , 2013, 8 a multiphysics solver package developed at the Biomedical Engineering Department, King's College London. ...

Advances in medical imaging and image processing are paving the way for personalised cardiac biomechanical modelling. However, for clinical utility to be achieved, model-based analyses mandate robust model selection and parameterisation. In this paper we introduce a patient-specific biomechanical model for the left ventricle aiming to balance model fidelity with parameter identifiability. Using non-invasive data and common clinical surrogates we illustrate unique identifiability of passive and active parameters over the full cardiac cycle. Identifiability and accuracy of the estimates in the presence of controlled noise are verified with a number of in silico datasets. Unique parametrisation is then obtained for a dataset acquired in vivo from a healthy volunteer. The model predictions show good agreement with the data extracted from the images providing a pipeline for personalised biomechanical analysis.

... The partitioned approach governs structure and fluid separately and updates information iteratively at the interface until convergence criteria are satisfied. The partitioned approach is more popular, owing to its capability of utilizing sophisticated structure and fluid solvers [6,[9][10][11]. During a numerical modeling, grids have to cope with the large leaflet rotation in a cardiac cycle, and two distinct methods have been developed: fixed grid and moving grid [1]. ...

This work presents a numerical simulation of intraventricular flow after the implantation of a bileaflet mechanical heart valve at the mitral position. The left ventricle was simplified conceptually as a truncated prolate spheroid and its motion was prescribed based on that of a healthy subject. The rigid leaflet rotation was driven by the transmitral flow and hence the leaflet dynamics were solved using fluid-structure interaction approach. The simulation results showed that the bileaflet mechanical heart valve at the mitral position behaved similarly to that at the aortic position. Sudden area expansion near the aortic root initiated a clockwise anterior vortex, and the continuous injection of flow through the orifice resulted in further growth of the anterior vortex during diastole, which dominated the intraventricular flow. This flow feature is beneficial to preserving the flow momentum and redirecting the blood flow towards the aortic valve. To the best of our knowledge, this is the first attempt to numerically model intraventricular flow with the mechanical heart valve incorporated at the mitral position using a fluid-structure interaction approach. This study facilitates future patient-specific studies. © 2015 Su et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

... Here, image-derived information is used for building, constraining and validating the model, which in turn provides a complete representation of ventricular functions with very high spatial and temporal resolution. The ability of the numerical models to characterise cardiac fluidand solid-mechanics in a patient-specific context has been demonstrated by numerous studies (Cheng et al., 2005;De Vecchi et al., 2012;McCormick et al., 2013;McQueen and Peskin, 2000;Watanabe et al., 2004). All of these models have the potential to be applied to clinical practice. ...

... The finite element method is used to solve all the models presented in this paper, all of which are implemented within an inhouse parallelised, multi-physics code, CHeart McCormick et al., 2013;Nordsletten et al., 2010). A brief description for each model component follows. ...

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... The partitioned approach governs structure and fluid separately and updates information iteratively at the interface until convergence criteria are satisfied. The partitioned approach is more popular, owing to its capability of utilizing sophisticated structure and fluid solvers [6,[9][10][11]. During a numerical modeling, grids have to cope with the large leaflet rotation in a cardiac cycle, and two distinct methods have been developed: fixed grid and moving grid [1]. ...

In this study, an ideal elliptical left ventricle was simulated with a prosthetic heart valve placed at the mitral orifice. The deformation of the left ventricle was predefined, and fluid-structure interaction (FSI) approach was applied to the prosthetic mitral heart valve. The simulation results have shown that the anterior and posterior leaflets behaved differently due to the non-uniform ventricular flow at downstream.

... Versatility of the meshing solution, illustrated with four examples of meshes automatically personalized. (a) LV of a dilated cardiomyopathy patient from SSFP MRI; (b) BiV truncated mesh from a sparse cine-MRI dataset; (c) LV with a hole in the apex from SSFP MRI, needed for the simulation of an assisted device [25] and (d) LV of a mouse with detailed papillary and trabecular anatomy smoothed by the mesh. Images show the mesh (white smooth geometry) overlapping with the isosurface of the binary mask (red geometry) provided as input. ...

Computational cardiac physiology has great potential to improve the management of cardiovascular diseases. One of the main bottlenecks in this field is the customization of the computational model to the anatomical and physiological status of the patient. We present a fully automatic service for the geometrical personalization of cardiac ventricular meshes with high-order interpolation from segmented images. The method is versatile (able to work with different species and disease conditions) and robust (fully automatic results fulfilling accuracy and quality requirements in 87% of 255 cases). Results also illustrate the capability to minimize the impact of segmentation errors, to overcome the sparse resolution of dynamic studies and to remove the sometimes unnecessary anatomical detail of papillary and trabecular structures. The smooth meshes produced can be used to simulate cardiac function, and in particular mechanics, or can be used as diagnostic descriptors of anatomical shape by cardiologists. This fully automatic service is deployed in a cloud infrastructure, and has been made available and accessible to the scientific community.

... Stijnen et al. [16] also simulated the dynamic behavior of a two-dimensional moving rigid heart valve using the FD method. Lastly, McCormick et al. formulated a FSI model of the left ventricle with LVAD using a modification of the Newton± Raphson/line algorithm and optimizing the interpolation scheme at the fluid±solid boundaries [4,17]. ...

The support of a failing heart with pump devices has been an essential element in cardiac health care for several decades. It is therefore important to understand the left ventricular response to the pumping action of these devices when connected to the native heart. Furthermore, monitoring of aortic valve opening and closure is important in avoiding valve stenosis and thrombogenesis during pump support. This paper reports the first steps in simulating the effects of outlet pump pressure on aortic valve closure of the heart assisted by an implantable blood pump. A two-dimensional fluid structure interaction aortic valve model is presented with blood flow in left ventricular chamber using the Arbitrary Lagrangian-Eulerian Finite Element Method formulation to predict the AV closure during outflow of blood from the left ventricle into the left ventricular assist device (LVAD).

... The anatomy was captured by CT images, and a robust model-based segmentation strategy [5] was used to identify the extents of the left ventricle. These meshes have been used for the optimization of the treatment of heart failure with LVAD [11,12]. ...

Sharing data between scientists and with clinicians in cardiac research has been facilitated significantly by the use of web technologies. The potential of this technology has meant that information sharing has been routinely promoted through databases that have encouraged stakeholder participation in communities around these services. In this paper we discuss the Anatomical Model Database (AMDB) (Gianni et al. Functional imaging and modeling of the heart. Springer, Heidelberg, 2009; Gianni et al. Phil Trans Ser A Math Phys Eng Sci 368:3039-3056, 2010) which both facilitate a database-centric approach to collaboration, and also extends this framework with new capabilities for creating new mesh data. AMDB currently stores cardiac geometric models described in Gianni et al. (Functional imaging and modelling of the heart. Springer, Heidelberg, 2009), a number of additional cardiac models describing geometry and functional properties, and most recently models generated using a web service. The functional models represent data from simulations in geometric form, such as electrophysiology or mechanics, many of which are present in AMDB as part of a benchmark study. Finally, the heartgen service has been added for producing left or bi-ventricle models derived from binary image data using the methods described in Lamata et al. (Med Image Anal 15:801-813, 2011). The results can optionally be hosted on AMDB alongside other community-provided anatomical models. AMDB is, therefore, a unique database storing geometric data (rather than abstract models or image data) combined with a powerful web service for generating new geometric models.

Development of cardiac multiphysics models has progressed significantly over the decades and simulations combining multiple physics interactions have become increasingly common. In this review, we summarise the progress in this field focusing on various approaches of integrating ventricular structures. electrophysiological properties, myocardial mechanics, as well as incorporating blood hemodynamics and the circulatory system. Common coupling approaches are discussed and compared, including the advantages and shortcomings of each. Currently used strategies for patient-specific implementations are highlighted and potential future improvements considered.

The objective of the current study was to propose a sensitivity analysis of a 3D left ventricle model in order to assess the influence of parameters on myocardial mechanical dispersion. A finite element model of LV electro-mechanical activity was proposed and a screening method was used to evaluate the sensitivity of model parameters on the standard deviation of time to peak strain. Results highlight the importance of propagation parameters associated with septal and lateral segments activation. Simulated curves were compared to myocardial strains, obtained from echocardiography of one healthy subject and one patient diagnosed with intraventricular dyssynchrony and coronary artery disease. Results show a close match between simulation and clinical strains and illustrate the model ability to reproduce myocardial strains in the context of intraventricular dyssynchrony.

Mechanical Circulatory Support (MCS) therapy is increasingly considered for patients with advanced heart failure unresponsive to optimal medical treatments. In this context, we: 1) presented an overview of clinical issues raised by MCS implantation, 2) designed a novel computer-assisted approach for planning the implantation, 3) implemented a CFD model to understand the ventricle hemodynamics induced by the inflow cannula pose. With the aim of decreasing complications and morbidity, quantitative criteria for optimizing ventricle unloading could be determined through CFD, and the planning approach may provide valuable information for choosing the device and adapting the clinical strategy.

Accelerated methods for acquiring phase contrast (PC) MRI allow the acquisition of 4D flow data of the whole heart in clinically acceptable times. These datasets are becoming interesting both for clinicians --- to better stratify diagnosis --- and in the modeling community --- to constrain patient-specific models. One of the difficulties related to PC data is a limited accuracy in the regions of low flow such as close to the myocardial wall, where the velocity field may even produce observed blood motion across the endocardial surface. To address this issue we propose to constrain the motion of blood in cavity during the analysis by using cine MRI and replacing the PC velocity in the peri-myocardial zone by neighboring tissue velocity obtained by analysis of 3D tagged MRI. We demonstrate the effect of these corrections on 2 healthy volunteer datasets and on one patient with a hypoplastic left ventricle.

This series of notes was prepared for the COMMAS summer school in 2013. The primary object of these notes is to provide additional material and references to augment the seminar slides introduced in class. Over the three part lecture series, the slides as well as these notes aim to expose students to the issues facing the cardiac clinical community and show the potential for modeling to address these issues. These notes focus on:
◦ Understanding Cardiac Multi-Scale / Multi-Physics Issues
◦ Understanding the Anisotropic Hyperelastic Behavior of Cardiac Tissue
◦ Understanding Cauchy’s first Law, Solution with Finite Element Method
◦ Understanding the Basic issues of Incompressibility
This material is by no means comprehensive, but instead seeks to provide an introduction to the field of cardiac biomechanics, and stress its relevance as both an important clinical research area as well as a challenging system of study.
Dr. David Nordsletten
Dept of Biomedical Engineering, KCL

A variation of the Shamanskii method is used to obtain a superlinearly convergent method for a class of nonlinear equations having singular Fréchet derivative at the root. The cost of a superlinear step is one derivative evaluation and two function evaluations.

In this paper we propose a method for coupling distributed and lumped models for the blood circulation. Lumped parameter models,
based on an analogy between the circulatory system and an electric or a hydraulic network are widely employed in the literature
to investigate different systemic responses in physiologic and pathologic situations (see e.g. [13, 24, 30, 15, 4, 27, 11,
14]). From the mathematical viewpoint these models are represented by ordinary differential equations. On the other hand,
for the accurate description of local phenomena, the Navier–Stokes equations for incompressible fluids are considered. In
the multiscale perspective, lumped models have been adopted (see e.g. [16]) as a numerical preprocessor to provide a quantitative
estimate of the boundary conditions at the interfaces. However, the two solvers (i.e. the lumped and the distributed one)
have been used separately. In the present work, we introduce a genuinely heterogeneous multiscale approach where the local
model and the systemic one are coupled at a mathematical and numerical level and solved together. In this perspective, we
have no longer boundary conditions on the artificial sections, but interface conditions matching the two submodels. The mathematical model and its numerical approximation are carefully addressed and several test
cases are considered.

An implicit function theorem and a resulting modified Newton-Raphson method for roots of functions between finite dimensional
spaces, without assuming non-singularity of the Jacobian at the initial approximation.

We derive a class of globally and quadratically converging algorithms for a system of nonlinear equations,g(u)=0, whereg is a sufficiently smooth homeomorphism. Particular attention is directed to key parameters which control the iteration. Several examples are given that have been successful in solving the coupled nonlinear PDEs which arise in semiconductor device modelling.

The loss of cardiac pump function accounts for a significant increase in both mortality and morbidity in Western society, where there is currently a one in four lifetime risk, and costs associated with acute and long-term hospital treatments are accelerating. The significance of cardiac disease has motivated the application of state-of-the-art clinical imaging techniques and functional signal analysis to aid diagnosis and clinical planning. Measurements of cardiac function currently provide high-resolution datasets for characterizing cardiac patients. However, the clinical practice of using population-based metrics derived from separate image or signal-based datasets often indicates contradictory treatments plans owing to inter-individual variability in pathophysiology. To address this issue, the goal of our work, demonstrated in this study through four specific clinical applications, is to integrate multiple types of functional data into a consistent framework using multi-scale computational modelling.

We compare the relative performance of monolithic and segregated (partitioned) solvers for large- displacement fluid–structure
interaction (FSI) problems within the framework of oomph-lib, the object-oriented multi-physics finite-element library, available as open-source software at http://www.oomph-lib.org. Monolithic solvers are widely acknowledged to be more robust than their segregated counterparts, but are believed to be
too expensive for use in large-scale problems. We demonstrate that monolithic solvers are competitive even for problems in
which the fluid–solid coupling is weak and, hence, the segregated solvers converge within a moderate number of iterations.
The efficient monolithic solution of large-scale FSI problems requires the development of preconditioners for the iterative
solution of the linear systems that arise during the solution of the monolithically coupled fluid and solid equations by Newton’s
method. We demonstrate that recent improvements to oomph-lib’s FSI preconditioner result in mesh-independent convergence rates under uniform and non-uniform (adaptive) mesh refinement,
and explore its performance in a number of two- and three-dimensional test problems involving the interaction of finite-Reynolds-number
flows with shell and beam structures, as well as finite-thickness solids.

This paper presents an efficient numerical solver for the finite element approximation of the incompressible Navier-Stokes equations within a moving three-dimensional domain. The moving domain is modeled using the arbitrary Lagrangian-Eulerian (ALE) formulation. Applying a finite element approximation leads to the solution of a large sparse system of equations. In this work we look at the application of the $Fp$ preconditioner within GMRES for efficiently solving such systems. Numerical results are presented for tetrahedral and hexahedral elements, and both structured and unstructured meshes. In all cases GMRES convergence rates are seen to be independent of mesh size. Finally, we show an application of this problem for modeling fluid flow within the heart.

We present a new method for solving the sparse linear system of equations arising from the discretization of the linearized steady-state Navier–Stokes equations (also known as the Oseen equations). The solver is an iterative method of Krylov subspace type for which we devise a preconditioner through a heuristic argument based on the fundamental solution tensor for the Oseen operator. The preconditioner may also be conceived through a weaker heuristic argument involving differential operators. Computations indicate that convergence for the preconditioned discrete Oseen problem is only mildly dependent on the viscosity (inverse Reynolds number) and, most importantly, that the number of iterations does not grow as the mesh size is reduced. Indeed, since the preconditioner is motivated through analysis of continuous operators, the number of iterations decreases for smaller mesh size which accords with better approximation of these operators.

Cardiac resynchronization therapy (CRT) has emerged as one of the few effective and safe treatments for heart failure. However, identifying patients that will benefit from CRT remains controversial. The dependence of CRT efficacy on organ and cellular scale mechanisms was investigated in a patient-specific computer model to identify novel patient selection criteria.
A biophysically based patient-specific coupled electromechanics heart model has been developed which links the cellular and sub-cellular mechanisms which regulate cardiac function to the whole organ function observed clinically before and after CRT. A sensitivity analysis of the model identified lack of length dependence of tension regulation within the sarcomere as a significant contributor to the efficacy of CRT. Further simulation analysis demonstrated that in the whole heart, length-dependent tension development is key not only for the beat-to-beat regulation of stroke volume (Frank-Starling mechanism), but also the homogenization of tension development and strain.
In individuals with effective Frank-Starling mechanism, the length dependence of tension facilitates the homogenization of stress and strain. This can result in synchronous contraction despite asynchronous electrical activation. In these individuals, synchronizing electrical activation through CRT may have minimal benefit.

This book deals with the numerical approximation of partial differential equations. Its scope is to provide a thorough illustration of numerical methods, carry out their stability and convergence analysis, derive error bounds, and discuss the algorithmic aspects relative to their implementation. A sound balancing of theoretical analysis, description of algorithms and discussion of applications is one of its main features. Many kinds of problems are addressed. A comprehensive theory of Galerkin method and its variants, as well as that of collocation methods, are developed for the spatial discretization. These theories are then specified to two numerical subspace realizations of remarkable interest: the finite element method and the spectral method.

In this paper, we first of all review the morphology and structure of the myocardium and discuss the main features of the mechanical response of passive myocardium tissue, which is an orthotropic material. Locally within the architecture of the myocardium three mutually orthogonal directions can be identified, forming planes with distinct material responses. We treat the left ventricular myocardium as a non-homogeneous, thick-walled, nonlinearly elastic and incompressible material and develop a general theoretical framework based on invariants associated with the three directions. Within this framework we review existing constitutive models and then develop a structurally based model that accounts for the muscle fibre direction and the myocyte sheet structure. The model is applied to simple shear and biaxial deformations and a specific form fitted to the existing (and somewhat limited) experimental data, emphasizing the orthotropy and the limitations of biaxial tests. The need for additional data is highlighted. A brief discussion of issues of convexity of the model and related matters concludes the paper.

Aortic flow and pressure result from the interactions between the heart and arterial system. In this work, we considered these interactions by utilizing a lumped parameter heart model as an inflow boundary condition for three-dimensional finite element simulations of aortic blood flow and vessel wall dynamics. The ventricular pressure-volume behavior of the lumped parameter heart model is approximated using a time varying elastance function scaled from a normalized elastance function. When the aortic valve is open, the coupled multidomain method is used to strongly couple the lumped parameter heart model and three-dimensional arterial models and compute ventricular volume, ventricular pressure, aortic flow, and aortic pressure. The shape of the velocity profiles of the inlet boundary and the outlet boundaries that experience retrograde flow are constrained to achieve a robust algorithm. When the aortic valve is closed, the inflow boundary condition is switched to a zero velocity Dirichlet condition. With this method, we obtain physiologically realistic aortic flow and pressure waveforms. We demonstrate this method in a patient-specific model of a normal human thoracic aorta under rest and exercise conditions and an aortic coarctation model under pre- and post-interventions.

A new method for the computational analysis of fluid–structure interaction of a Newtonian fluid with slender bodies is developed. It combines ideas of the fictitious domain and the mortar element method by imposing continuity of the velocity field along an interface by means of Lagrange multipliers. The key advantage of the method is that it circumvents the need for complicated mesh movement strategies common in arbitrary Lagrangian–Eulerian (ALE) methods, usually used for this purpose. Copyright © 2001 John Wiley & Sons, Ltd.

A new method for the computational analysis of fluid-structure interaction of a Newtonian fluid with slender bodies is developed. It combines ideas of the fictitious domain and the mortar element method by imposing continuity of the velocity field along an interface by means of Lagrange multipliers. The key advantage of the method is that it circumvents the need for complicated mesh movement strategies common in arbitrary Lagrangian-Eulerian (ALE) methods, usually used for this purpose. Copyright (C) 2001 John Wiley & Sons, Ltd.

We investigate several ways to improve the performance of sparse LU factorization with partial pivoting, as used to solve unsymmetric linear systems. We introduce the notion of unsymmetric supernodes to perform most of the numerical computation in dense matrix kernels. We introduce unsymmetric supernode-panel updates and two-dimensional data partitioning to better exploit the memory hierarchy. We use Gilbert and Peierls's depth-first search with Eisenstat and Liu's symmetric structural reductions to speed up symbolic factorization. We have developed a sparse LU code using all these ideas. We present experiments demonstrating that it is significantly faster than earlier partial pivoting codes. We also compare its performance with UMFPACK, which uses a multifrontal approach; our code is very competitive in time and storage requirements, especially for large problems.

The paper is devoted to a general finite element approximation of the solution of the Stokes equations for an incompressible viscous fluid. Both conforming and nonconforming finite-element methods are studied and various examples of simplistic elements well suited for the numerical treatment of the incompressibility condition are given. Optimal error estimates are derived in the energy norm and in the L**2-norm.

The sixth editions of these seminal books deliver the most up to date and comprehensive reference yet on the finite element method for all engineers and mathematicians. Renowned for their scope, range and authority, the new editions have been significantly developed in terms of both contents and scope. Each book is now complete in its own right and provides self-contained reference; used together they provide a formidable resource covering the theory and the application of the universally used FEM. Written by the leading professors in their fields, the three books cover the basis of the method, its application to solid mechanics and to fluid dynamics.* This is THE classic finite element method set, by two the subject's leading authors * FEM is a constantly developing subject, and any professional or student of engineering involved in understanding the computational modelling of physical systems will inevitably use the techniques in these books * Fully up-to-date; ideal for teaching and reference

The central problem in modelling the multi–dimensional mechanics of the heart is in identifying functional forms and parameter of the constitutive equations, which describe the material properties of the resting and active, normal and diseased myocardium.
The constitutive properties of myocardium are three dimensional, anisotropic, nonlinear and time dependent. Formulating usefu constitutive laws requires a combination of multi–axial tissue testing in vitro, microstructural modelling based on quantitative morphology, statistical parameter estimation, and validation with measurement from intact hearts. Recent models capture some important properties of healthy and diseased myocardium including: the nonlinea interactions between the responses to different loading patterns; the influence of the laminar myofibre sheet architecture the effects of transverse stresses developed by the myocytes; and the relationship between collagen fibre architecture an mechanical properties in healing scar tissue after myocardial infarction.

A finite dimensional stability test for checking velocity/pressure finite element trial spaces is presented. Applications are made to a new class of element pairs proposed in this paper as well as to existing spaces.

In this paper, we provide a priori and a posteriori error analyses of an augmented mixed finite element method with Lagrange multipliers applied to elliptic equations in divergence form with mixed boundary conditions. The augmented scheme is obtained by including the Galerkin least-squares terms arising from the constitutive and equilibrium equations. We use the classical Babuska-Brezzi theory to show that the resulting dual-mixed variational formulation and its Galerkin scheme defined with Raviart-Thomas spaces are well posed, and also to derive the corresponding a priori error estimates and rates of convergence. Then, we develop a reliable and efficient residual-based a posteriori error estimate and a reliable and quasi-efficient Ritz projection-based one, as well. Finally, several numerical results illustrating the performance of the augmented scheme and the associated adaptive algorithms are reported.

To simulate fluid-structure interaction involved in the contraction of a human left ventricle, a 3D finite element based simulation program incorporating the propagation of excitation and excitation-contraction coupling mechanisms was developed. An ALE finite element method with automatic mesh updating was formulated for large domain changes, and a strong coupling strategy was taken. Under the assumption that the inertias of both fluid and structure are negligible and fluid-structure interaction is restricted to the pressure on the interface, the fluid dynamics part was eliminated from the FSI program, and a static structural FEM code corresponding to the cardiac muscles was also developed. The simulations of the contraction of the left ventricle in normal excitation and arrhythmia demonstrated the capability of the proposed method. Also, the results obtained by the two methods are compared. These simulators can be powerful tools in the clinical practice of heart disease.

Inexact Newton methods for finding a zero of F : R(n) --> R(n) are variations of Newton's method in which each step only approximately satisfies the linear Newton equation but still reduces the norm of the local linear model of F. Here, inexact Newton methods are formulated that incorporate features designed to improve convergence from arbitrary starting points. For each method, a basic global convergence result is established to the effect that, under reasonable assumptions, if a sequence of iterates has a limit point at which F' is invertible, then that limit point is a solution and the sequence converges to it. When appropriate, it is shown that initial inexact Newton steps are taken near the solution, and so the convergence can ultimately be made as fast as desired, up to the rate of Newton's method, by forcing the initial linear residuals to be appropriately small. The primary goal is to introduce and analyze new inexact Newton methods, but consideration is also given to ''globalizations'' of (exact) Newton's method that can naturally be viewed as inexact Newton methods.

The stability of a higher-order Hood–Taylor method for the approximation of the stationary Stokes equations using continuous piecewise polynomials of degree 3 to approximate velocities and continuous piecewise polynomials of degree 2 to approximate the pressure is proved. This result implies that the standard finite element method using these spaces satisfies a quasi-optimal error estimate. The technique used may also be applied to prove the stability of Hood–Taylor rectangular elements of arbitrary degree k for velocities and $k - 1$ for pressure in each variable.

The MAC (marker and cell) discretization of fluid flow is analysed for the stationary Stokes equations. It is proved that the discrete approximations do in fact converge to the exact solutions of the flow equations. Estimates using mesh dependent norms analogous to the standard H 1 and L 2 norms are given for the velocity and pressure, respectively.

SUMMARY A new method for the computational analysis of fluid-structure interaction of a Newtonian fluid with slender bodies is developed. It combines ideas of the fictitious domain and the mortar element method by imposing continuity of the velocity field along an interface by means of Lagrange multipliers. The key advantage of the method is that it circumvents the need for complicated mesh movement strategies common in arbitrary Lagrangian-Eulerian (ALE) methods, usually used for this purpose. Copyright © 2001 John Wiley & Sons, Ltd.

Numerical simulations provide a unique approach for investigating the impact of left ventricular assist devices (LVADs) on the pump function of the heart. To this end, the fictitious domain (FD) method was incorporated within a non-conforming coupled fluid–solid mechanics solver creating a model capable of capturing the full range of cardiac motion under LVAD support, including contact of the cannula with the ventricular wall. To demonstrate and verify the properties of the applied method, convergence studies were performed showing linear convergence for both a fluid only problem with an immersed rigid body, as well as a fluid–solid coupled problem with a fluid immersed rigid body interacting with the coupled elastic solid. The model was implemented on a left ventricular (LV) geometry constructed from human MRI data. Simulations were performed to compare LV function with an FD prescribed LVAD cannula and with a cannula applied as a Dirichlet boundary. Good agreement was observed between the simulations for myocardial deformation, major flow features and rates of energy transfer. Finally, an LV simulation was performed to bring the ventricular wall into contact with the cannula, demonstrating the applicability of the model to investigating LV function under LVAD support. Copyright © 2011 John Wiley & Sons, Ltd.

Understanding the underlying feedback mechanisms of fluid/solid coupling and the role it plays in heart function is crucial for characterizing normal heart function and its behavior in disease. To improve this understanding, an anatomically accurate computational model of fluid–solid mechanics in the left ventricle is presented which assesses both the passive diastolic and active systolic phases of the heart. Integrating multiple data which characterize the hemodynamical and tissue mechanical properties of the heart, a numerical approach was applied which allows non-conformity in an optimal finite element scheme (J. Comp. Phys., submitted; Fluid–solid coupling for the simulation of left ventricular mechanics, University of Oxford, 2009). This approach is applied to look specifically at left ventricular fluid/solid coupling, allowing quantitative assessment of blood flow through the left ventricle, pressure distributions, activation and the loss of mechanical energy due to viscous dissipation. Copyright © 2010 John Wiley & Sons, Ltd.

We analyze the error of a fictitious-domain method with boundary Lagrange multiplier. It is applied to solve a non-homogeneous
steady incompressible Navier-Stokes problem in a domain with a multiply-connected boundary. The interior mesh in the fictitious
domain and the boundary mesh are independent, up to a mesh-length ratio.

Finite elasticity theory combined with finite element analysis provides the framework for analysing ventricular mechanics
during the filling phase of the cardiac cycle, when cardiac cells are not actively contracting. The orthotropic properties
of the passive tissue are described here by a “pole–zero” constitutive law, whose parameters are derived in part from a model
of the underlying distributions of collagen fibres. These distributions are based on our observations of the fibrous-sheet
laminar architecture of myocardial tissue. We illustrate the use of high order (cubic Hermite) basis functions in solving
the Galerkin finite element stress equilibrium equations based on this orthotropic constitutive law and for incorporating
the observed regional distributions of fibre and sheet orientations. Pressure–volume relations and 3D principal strains predicted
by the model are compared with experimental observations. A model of active tissue properties, based on isolated muscle experiments,
is also introduced in order to predict transmural distributions of 3D principal strains at the end of the contraction phase
of the cardiac cycle. We end by offering a critique of the current model of ventricular mechanics and propose new challenges
for future modellers.

Heart transplantation (HTx) improves symptoms and prolongs life in advanced heart failure (HF), but organ supply is limited. In recent years, mechanical circulatory support and specifically implantable left ventricular assist devices (LVADs) have undergone technical improvements, and outcomes have improved dramatically. Left ventricular assist devices are now viable options for patients with severe HF as bridge to transplantation, destination therapy, or as bridge to recovery. Many believe that LVADs may soon provide outcomes similar to, or better than, HTx, launching a new era of end-stage HF management. The key to improving outcomes is patient selection, but the field is changing rapidly and guidelines and consensus are limited. This review summarizes recent reports of predictors of poor outcomes and provides an overview of selection for LVAD therapy.

This paper is the second in a series that describes the development of a 3-dimensional computer model of the heart. The problem studied here is that of a contractile fiber-wound toroidal tube immersed in a viscous incompressible fluid. A wave of contraction propagates around the tube, and this results in peristaltic pumping of the internal fluid in the direction of the wave. When the contraction is sufficiently strong, there is a small region of entrained fluid that is convected along at the speed of the wave.

In this study, a Lagrange multiplier technique is developed to solve problems of coupled mechanics and is applied to the case of a Newtonian fluid coupled to a quasi-static hyperelastic solid. Based on theoretical developments in [57], an additional Lagrange multiplier is used to weakly impose displacement/velocity continuity as well as equal, but opposite, force. Through this approach, both mesh conformity and kinematic variable interpolation may be selected independently within each mechanical body, allowing for the selection of grid size and interpolation most appropriate for the underlying physics. In addition, the transfer of mechanical energy in the coupled system is proven to be conserved. The fidelity of the technique for coupled fluid–solid mechanics is demonstrated through a series of numerical experiments which examine the construction of the Lagrange multiplier space, stability of the scheme, and show optimal convergence rates. The benefits of non-conformity in multi-physics problems is also highlighted. Finally, the method is applied to a simplified elliptical model of the cardiac left ventricle.

A method is presented for modelling fluid–solid interaction with large transformations of a slender solid body. The fluid flow is described by the unsteady Navier–Stokes equation, and the solid deformation is described by an incompressible hyperelastic Neo-Hookean model. Although the fluid and solid mesh are non-conformal with respect to each other, both domains can be coupled using a Lagrange multiplier. Accuracy and robustness are improved by a computationally inexpensive adaptive meshing scheme which is applied to the fluid mesh at the position of the solid interface. To illustrate the applicability of this method, 2D and 3D model problems are presented that are closely related to dynamical heart-valve computations. To cite this article: R. van Loon et al., C. R. Mecanique 333 (2005).RésuméDans cet article, on présente une méthode pour la modélisation numérique de couplages fluide–solide lorsque le solide est un corps mince. L'écoulement est décrit par les équations de Navier–Stokes instationnaires, la déformation du solide l'étant par un modèle du type Neo-Hookien hyper élastique incompressible. Bien que les maillages fluide et solide ne soient pas en conformité, l'un par rapport à d'autres on peut coupler les regions respectives via un multiplicateur de Lagrange. Par rapport à d'autres approches de ce type, on améliore précision et robustesse par l'utilisation d'une méthode de maillage adaptative, peu coûteuse, appliquée au maillage fluide au voisinage de l'interface avec le solide. Pour évaluer les possibilités de la méthode, on l'applique à la résolution de problèmes modèles, bi et tri-dimensionnels, tous étroitement liés à la simulation numé rique du mouvement des valves cardiaques en régîme dynamique. Pour citer cet article : R. van Loon et al., C. R. Mecanique 333 (2005).

A computational method is proposed for problems where fluid–structure interaction is combined with solid-rigid contact. This combination is particularly important for the dynamics and impact of heart valves. The Navier–Stokes equation in an Eulerian setting is coupled to a Neo-Hookean solid model using a Lagrangian description. A fictitious domain method extended with a local mesh adaptation algorithm provides the required flexibility with respect to the motion and deformation of the valve. In addition, it ensures the solids ability of sustaining pressures present in the fluid. Making use of the fact that the fluid and solid mesh are not required to be connected conformingly, it is shown that the model can be extended with a contact algorithm without introducing meshing complications near the contact surfaces.

In this work we will introduce and analyze the Arbitrary Lagrangian Eulerian formulation for a model problem of a scalar advection–diffusion equation defined on a moving domain. Moving from the results illustrated in our previous work [J. Num. Math. 7 (1999) 105], we will consider first and second-order time advancing schemes and analyze how the movement of the domain might affect accuracy and stability properties of the numerical schemes with respect to their counterpart on fixed domains. Theoretical and numerical results will be presented, showing that stability properties are not, in general, preserved, while accuracy is maintained.

A new numerical technique is presented that has many advantages for obtaining solutions to a wide variety of time-dependent multidimensional fluid dynamics problems. The method uses a finite difference mesh with vertices that may be moved with the fluid (Lagrangian), be held fixed (Eulerian), or be moved in any other prescribed manner, as in the Arbitrary-Lagrangian-Eulerian (ALE) technique. In addition, it employs an implicit formulation similar to that of the Implicit Continuous-Fluid Eulerian (ICE) technique, making it applicable to flows at all speeds.This paper describes the basic methodology, presents finite difference approximations, and discusses such matters as stability, accuracy, and zoning. In addition, illustrations are included from a number of representative calculations.

The flow pattern of blood in the heart is intimately connected with the performance of the heart valves. This paper extends previous work on the solution of the Navier-Stokes equations in the presence of moving immersed boundaries which interact with the fluid. The boundary representation now includes the muscular heart wall. The fixed topology of the boundary representation is exploited in the solution of the nonlinear equations which implicitly define the boundary forces. An improved numerical representation of the δ-function is introduced. A fast Laplace-solver is used. The results of calculations with a natural valve and with a prosthetic valve are presented.

The approximation of incompressible, viscous, Newtonian fluids governed by Navier Stokes equations are considered. The upwinding techniques in finite elements are reviewed and the various formulations available to deal with the incompressibility conditions are presented. The different techniques related to automatic mesh generation and solution of systems of linear or nonlinear equations are surveyed. Bibtex entry for this abstract Preferred format for this abstract (see Preferences) Find Similar Abstracts: Use: Authors Title Keywords (in text query field) Abstract Text Return: Query Results Return items starting with number Query Form Database: Astronomy Physics arXiv e-prints