An accurate, robust, and efficient finite element framework with applications to anisotropic, nearly and fully incompressible elasticity

A Modular Framework for Implicit 3D-0D Coupling in Cardiac Mechanics

In numerical simulations of cardiac mechanics, coupling the heart to a model of the circulatory system is essential for capturing physiological cardiac behavior. A popular and efficient technique is to use an electrical circuit analogy, known as a lumped parameter network or zero-dimensional (0D) fluid model, to represent blood flow throughout the cardiovascular system. Due to the strong physical interaction between the heart and the blood circulation, developing accurate and efficient numerical coupling methods remains an active area of research. In this work, we present a modular framework for implicitly coupling three-dimensional (3D) finite element simulations of cardiac mechanics to 0D models of blood circulation. The framework is modular in that the circulation model can be modified independently of the 3D finite element solver, and vice versa. The numerical scheme builds upon a previous work that combines 3D blood flow models with 0D circulation models (3D fluid - 0D fluid). Here, we extend it to couple 3D cardiac tissue mechanics models with 0D circulation models (3D structure - 0D fluid), showing that both mathematical problems can be solved within a unified coupling scheme. The effectiveness, temporal convergence, and computational cost of the algorithm are assessed through multiple examples relevant to the cardiovascular modeling community. Importantly, in an idealized left ventricle example, we show that the coupled model yields physiological pressure-volume loops and naturally recapitulates the isovolumic contraction and relaxation phases of the cardiac cycle without any additional numerical techniques. Furthermore, we provide a new derivation of the scheme inspired by the Approximate Newton Method of Chan (1985), explaining how the proposed numerical scheme combines the stability of monolithic approaches with the modularity and flexibility of partitioned approaches.

Efficient approximation of cardiac mechanics through reduced-order modeling with deep learning-based operator approximation

Reducing the computational time required by high-fidelity, full-order models (FOMs) for the solution of problems in cardiac mechanics is crucial to allow the translation of patient-specific simulations into clinical practice. Indeed, while FOMs, such as those based on the finite element method, provide valuable information on the cardiac mechanical function, accurate numerical results can be obtained at the price of very fine spatio-temporal discretizations. As a matter of fact, simulating even just a few heartbeats can require up to hours of wall time on high-performance computing architectures. In addition, cardiac models usually depend on a set of input parameters that are calibrated in order to explore multiple virtual scenarios. To compute reliable solutions at a greatly reduced computational cost, we rely on a reduced basis method empowered with a new deep learning-based operator approximation, which we refer to as Deep-HyROMnet technique. Our strategy combines a projection-based POD-Galerkin method with deep neural networks for the approximation of (reduced) nonlinear operators, overcoming the typical computational bottleneck associated with standard hyper-reduction techniques employed in reduced-order models (ROMs) for nonlinear parametrized systems. This method can provide extremely accurate approximations to parametrized cardiac mechanics problems, such as in the case of the complete cardiac cycle in a patient-specific left ventricle geometry. In this respect, a 3D model for tissue mechanics is coupled with a 0D model for external blood circulation; active force generation is provided through an adjustable parameter-dependent surrogate model as input to the tissue 3D model. The proposed strategy is shown to outperform classical projection-based ROMs, in terms of orders of magnitude of computational speed-up, and to return accurate pressure-volume loops in both physiological and pathological cases. Finally, an application to a forward uncertainty quantification analysis, unaffordable if relying on a FOM, is considered, involving output quantities of interest such as, for example, the ejection fraction or the maximal rate of change in pressure in the left ventricle.

A Doppler-exclusive non-invasive computational diagnostic framework for personalized transcatheter aortic valve replacement

Given the associated risks with transcatheter aortic valve replacement (TAVR), it is crucial to determine how the implant will affect the valve dynamics and cardiac function, and if TAVR will improve or worsen the outcome of the patient. Effective treatment strategies, indeed, rely heavily on the complete understanding of the valve dynamics. We developed an innovative Doppler-exclusive non-invasive computational framework that can function as a diagnostic tool to assess valve dynamics in patients with aortic stenosis in both pre- and post-TAVR status. Clinical Doppler pressure was reduced by TAVR (52.2 ± 20.4 vs. 17.3 ± 13.8 [mmHg], p < 0.001), but it was not always accompanied by improvements in valve dynamics and left ventricle (LV) hemodynamics metrics. TAVR had no effect on LV workload in 4 patients, and LV workload post-TAVR significantly rose in 4 other patients. Despite the group level improvements in maximum LV pressure (166.4 ± 32.2 vs 131.4 ± 16.9 [mmHg], p < 0.05), only 5 of the 12 patients (41%) had a decrease in LV pressure. Moreover, TAVR did not always improve valve dynamics. TAVR did not necessarily result in a decrease (in 9 out of 12 patients investigated in this study) in major principal stress on the aortic valve leaflets which is one of the main contributors in valve degeneration and, consequently, failure of heart valves. Diastolic stresses increased significantly post-TAVR (34%, 109% and 81%, p < 0.001) for each left, right and non-coronary leaflets respectively. Moreover, we quantified the stiffness and material properties of aortic valve leaflets which correspond with the reduced calcified region average stiffness among leaflets (66%, 74% and 62%; p < 0.001; N = 12). Valve dynamics post-intervention should be quantified and monitored to ensure the improvement of patient conditions and prevent any further complications. Improper evaluation of biomechanical valve features pre-intervention as well as post-intervention may result in harmful effects post-TAVR in patients including paravalvular leaks, valve degeneration, failure of TAVR and heart failure.

An ultrasound-exclusive non-invasive computational diagnostic framework for personalized cardiology of aortic valve stenosis

Aortic stenosis (AS) is an acute and chronic cardiovascular disease and If left untreated, 50% of these patients will die within two years of developing symptoms. AS is characterized as the stiffening of the aortic valve leaflets which restricts their motion and prevents the proper opening under transvalvular pressure. Assessments of the valve dynamics, if available, would provide valuable information about the patient's state of cardiac deterioration as well as heart recovery and can have incredible impacts on patient care, planning interventions and making critical clinical decisions with life-threatening risks. Despite remarkable advancements in medical imaging, there are no clinical tools available to quantify valve dynamics invasively or noninvasively. In this study, we developed a highly innovative ultrasound-based non-invasive computational framework that can function as a diagnostic tool to assess valve dynamics (e.g. transient 3-D distribution of stress and displacement, 3-D deformed shape of leaflets, geometric orifice area and angular positions of leaflets) for patients with AS at no risk to the patients. Such a diagnostic tool considers the local valve dynamics and the global circulatory system to provide a platform for testing the intervention scenarios and evaluating their effects. We used clinical data of 12 patients with AS not only to validate the proposed framework but also to demonstrate its diagnostic abilities by providing novel analyses and interpretations of clinical data in both pre and post intervention states. We used transthoracic echocardiogram (TTE) data for the developments and transesophageal echocardiography (TEE) data for validation.

A cell-based framework for modeling cardiac mechanics

Cardiomyocytes are the functional building blocks of the heart-yet most models developed to simulate cardiac mechanics do not represent the individual cells and their surrounding matrix. Instead, they work on a homogenized tissue level, assuming that cellular and subcellular structures and processes scale uniformly. Here we present a mathematical and numerical framework for exploring tissue-level cardiac mechanics on a microscale given an explicit three-dimensional geometrical representation of cells embedded in a matrix. We defined a mathematical model over such a geometry and parametrized our model using publicly available data from tissue stretching and shearing experiments. We then used the model to explore mechanical differences between the extracellular and the intracellular space. Through sensitivity analysis, we found the stiffness in the extracellular matrix to be most important for the intracellular stress values under contraction. Strain and stress values were observed to follow a normal-tangential pattern concentrated along the membrane, with substantial spatial variations both under contraction and stretching. We also examined how it scales to larger size simulations, considering multicellular domains. Our work extends existing continuum models, providing a new geometrical-based framework for exploring complex cell-cell and cell-matrix interactions.

Robust and efficient fixed-point algorithm for the inverse elastostatic problem to identify myocardial passive material parameters and the unloaded reference configuration

Image-based computational models of the heart represent a powerful tool to shed new light on the mechanisms underlying physiological and pathological conditions in cardiac function and to improve diagnosis and therapy planning. However, in order to enable the clinical translation of such models, it is crucial to develop personalized models that are able to reproduce the physiological reality of a given patient. There have been numerous contributions in experimental and computational biomechanics to characterize the passive behavior of the myocardium. However, most of these studies suffer from severe limitations and are not applicable to high-resolution geometries. In this work, we present a novel methodology to perform an automated identification of in vivo properties of passive cardiac biomechanics. The highly-efficient algorithm fits material parameters against the shape of a patient-specific approximation of the end-diastolic pressure-volume relation (EDPVR). Simultaneously, an unloaded reference configuration is generated, where a novel line search strategy to improve convergence and robustness is implemented. Only clinical image data or previously generated meshes at one time point during diastole and one measured data point of the EDPVR are required as an input. The proposed method can be straightforwardly coupled to existing finite element (FE) software packages and is applicable to different constitutive laws and FE formulations. Sensitivity analysis demonstrates that the algorithm is robust with respect to initial input parameters.