Simulations of blood flow in patient-specific models of heart ventricles is a rapidly developing field of research, showing promise to improve future treatment of heart diseases. Fluid-structure interaction simulation of the mitral valve, with its complex structure including leaflets, chordae tendineae, and papillary muscles, provides additional prospects as well as challenges to such models. In this study, we combine a patient-specific model of the left ventricle with an idealized unified continuum fluid-structure interaction model of the mitral valve, to simulate the intraventricular diastolic blood flow. To the best of our knowledge, no monolithic fluid-structure interaction model, without the need for remeshing, has ever been used before to simulate the native mitral valve within the left ventricle. The chordae tendineae are simulated as a region of porous medium, to partially hinder the flow. Simulation results from this model are compared to those of a model with the same patient-specific left ventricle, but with the mitral valve simply modeled as a time-variant inflow boundary condition. The blood flow is analyzed with the E-wave propagation index, and by use of the triple decomposition of the velocity gradient tensor, which decomposes the flow into rigid body rotational flow, shearing flow, and irrotational straining flow. The triple decomposition enables analysis of the formation of initially large dominant flow features, such as the E-wave jet and the vortex ring around it, and their subsequent decay into smaller turbulent flow structures. This analysis of the development of flow structures over the duration of diastole appears to be in general agreement with the theory of the stability of rotation, shear, and strain structures. Elevated shear levels are investigated, but are found only in limited amounts that do not indicate significant risks of thrombus formation or other blood damage, which is to be expected in this healthy ventricle. The highest shear levels are localized at the leaflets in the fluid-structure interaction model, and at the ventricle wall in the planar model. The computed E-wave propagation indices are 1.21 for the fluid-structure interaction model and 1.90 for the planar valve model, which indicates proper washout in the apical region with no significant risk of blood stasis that could lead to left ventricular thrombus formation.

Introduction: Aortic stiffness plays a critical role in the evolution of cardiovascular diseases, but the assessment requires specialized equipment. Photoplethysmography (PPG) and single-lead electrocardiogram (ECG) are readily available in healthcare and wearable devices. We studied whether a brief PPG registration, alone or in combination with single-lead ECG, could be used to reliably estimate aortic stiffness. Methods: A proof-of-concept study with simultaneous high-resolution index finger recordings of infrared PPG, single-lead ECG, and finger blood pressure (Finapres) was performed in 33 participants [median age 44 (range 21–66) years, 19 men] and repeated within 2 weeks. Carotid–femoral pulse wave velocity (cfPWV; two-site tonometry with SphygmoCor) was used as a reference. A brachial single-cuff oscillometric device assessed aortic pulse wave velocity (aoPWV; Arteriograph) for further comparisons. We extracted 136 established PPG waveform features and engineered 13 new with improved coupling to the finger blood pressure curve. Height-normalized pulse arrival time (NPAT) was derived using ECG. Machine learning methods were used to develop prediction models. Results: The best PPG-based models predicted cfPWV and aoPWV well (root-mean-square errors of 0.70 and 0.52 m/s, respectively), with minor improvements by adding NPAT. Repeatability and agreement were on par with the reference equipment. A new PPG feature, an amplitude ratio from the early phase of the waveform, was most important in modelling, showing strong correlations with cfPWV and aoPWV (r = −0.81 and −0.75, respectively, both P < 0.001). Conclusion: Using new features and machine learning methods, a brief finger PPG registration can estimate aortic stiffness without requiring additional information on age, anthropometry, or blood pressure. Repeatability and agreement were comparable to those obtained using non-invasive reference equipment. Provided further validation, this readily available simple method could improve cardiovascular risk evaluation, treatment, and prognosis.

The triple decomposition of a velocity gradient tensor provides an analysis tool in fluid mechanics by which the flow can be split into a sum of irrotational straining flow, shear flow, and rigid body rotational flow. In 2007, Kolar formulated an optimization problem to compute the triple decomposition [V. Kolar, "Vortex identification: New requirements and limitations, " Int. J. Heat Fluid Flow 28, 638-652 (2007)], and more recently, the triple decomposition has been connected to the Schur form of the associated matrix. We show that the standardized real Schur form, which can be computed by state of the art linear algebra routines, is a solution to the optimization problem posed by Kolar. We also demonstrate why using the standardized variant of the real Schur form makes computation of the triple decomposition more efficient. Furthermore, we illustrate why different structures of the real Schur form correspond to different alignments of the coordinate system with the fluid flow and may, therefore, lead to differences in the resulting triple decomposition. Based on these results, we propose a new, simplified algorithm for computing the triple decomposition, which guarantees consistent results.

In this work we propose a new stabilized approach for solving the incompressible Navier-Stokes equations on fixed overlapping grids. This new approach is based on the partition of unity finite element method, which defines the solution fields as weighted sums of local fields, supported by the different grids. Here, the discrete weak formulation of the problem is re-set in cG(1)cG(1) stabilized form, which has the dual benefit of lowering grid resolution requirements for convection dominated flows and allowing for the use of velocity and pressure discretizations which do not satisfy the inf-sup condition. Additionally, we provide an outline of our implementation within an existing distributed parallel application and identify four key options to improve the code efficiency namely: the use of cache to store mapped quadrature points and basis function gradients, the intersection volume splitting algorithm, the use of lower order quadrature schemes, and tuning the partition weight associated with the interface elements. The new method is shown to have comparable accuracy to the single mesh boundary-fitted version of the same stabilized solver based on three transient flow tests including both 2D and 3D settings, as well as low and moderate Reynolds number flow conditions. Moreover, we demonstrate how the four implementation options have a synergistic effect lowering the residual assembly time by an order of magnitude compared to a naive implementation, and showing good load balancing properties.

Cardiac disease and clinical intervention may both lead to an increased risk for thrombosis events due to a modified blood flow in the heart, and thereby a change in the mechanical stimuli of blood cells passing through the chambers of the heart. Specifically, the degree of platelet activation is influenced by the level and type of mechanical stresses in the blood flow. In this article we analyze the blood flow in the left ventricle of the heart through a computational model constructed from patient-specific data. The blood flow in the ventricle is modelled by the Navier-Stokes equations, and the flow through the mitral valve by a parameterized model which represents the projected opening of the valve. A finite element method is used to solve the equations, from which a simulation of the velocity and pressure of the blood flow is constructed. The intraventricular blood flow is complex, in particular in diastole when the inflow jet from the atrium breaks down into turbulent flow on a range of scales. A triple decomposition of the velocity gradient tensor is then used to distinguish between rigid body rotational flow, irrotational straining flow, and shear flow. The triple decomposition enables the separation of three fundamentally different flow structures, that each generates a distinct type of mechanical stimulus on the blood cells in the flow. We compare the results in a simulation where a mitral valve clip intervention is modelled, which leads to a significant modification of the intraventricular flow. Further, we perform a sensitivity study of the results with respect to the positioning of the clip. It was found that the shear in the simulation cases treated with clips increased more compared to the untreated case than the rotation and strain did. A decrease in valve opening area of 64% in one of the cases led to a 90% increase in rotation and strain, but a 150% increase in shear. The computational analysis opens up for improvements in models of shear-induced platelet activation, by offering an algorithm to distinguish shear from other modalities in intraventricular blood flow.

Mathematical models of the cardiovascular system have come a long way since they were first introduced in the early 19th century. Driven by a rapid development of experimental techniques, numerical methods, and computer hardware, detailed models that describe physical scales from the molecular level up to organs and organ systems have been derived and used for physiological research. Mathematical and computational models can be seen as condensed and quantitative formulations of extensive physiological knowledge and are used for formulating and testing hypotheses, interpreting and directing experimental research, and have contributed substantially to our understanding of cardiovascular physiology. However, in spite of the strengths of mathematics to precisely describe complex relationships and the obvious need for the mathematical and computational models to be informed by experimental data, there still exist considerable barriers between experimental and computational physiological research. In this review, we present a historical overview of the development of mathematical and computational models in cardiovascular physiology, including the current state of the art. We further argue why a tighter integration is needed between experimental and computational scientists in physiology, and point out important obstacles and challenges that must be overcome in order to fully realize the synergy of experimental and computational physiological research.

An interface tracking finite element methodology is presented for 3D turbulent flow fluid-structure interaction, including full-friction contact and topology changes, with specific focus on heart valve simulations. The methodology is based on a unified continuum fluid-structure interaction model, which is a monolithic approach, where the fundamental conservation laws are formulated for the combined fluid-structure continuum. Contact is modeled by local phase changes in the unified continuum, and computational results show the promise of the approach. The core algorithms are all based on the solution of partial differential equations with standard finite element methods, and hence any general purpose finite element library which can leverage state of the art hardware platforms can be used for the implementation of the methodology.

In the context of flow visualization, a triple decomposition of the velocity gradient into irrotational straining flow, shear flow, and rigid body rotational flow was proposed by Kolar in 2007 [V. Kolar, "Vortex identification: New requirements and limitations," Int. J. Heat Fluid Flow, 28, 638-652 (2007)], which has recently received renewed interest. The triple decomposition opens for a refined energy stability analysis of the Navier-Stokes equations, with implications for the mathematical analysis of the structure, computability, and regularity of turbulent flow. We here perform an energy stability analysis of turbulent incompressible flow, which suggests a scenario where at macroscopic scales, any exponentially unstable irrotational straining flow structures rapidly evolve toward linearly unstable shear flow and stable rigid body rotational flow. This scenario does not rule out irrotational straining flow close to the Kolmogorov microscales, since there viscous dissipation stabilizes the unstable flow structures. In contrast to worst case energy stability estimates, this refined stability analysis reflects the existence of stable flow structures in turbulence over extended time.

We present a high performance computing framework for finite element simulation of blood flow in the left ventricle of the human heart. The mathematical model is described together with the discretization method and the parallel implementation in Unicorn which is part of the open source software framework FEniCS-HPC. We show results based on patient-specific data that capture essential features observed with other computational models and imaging techniques, and thus indicate that our framework possesses the potential to provide relevant clinical information for diagnosis and medical treatment. Several other studies have been conducted to simulate the three dimensional blood flow in the left ventricle of the human heart with prescribed wall movement. Our contribution to the field of cardiac research lies in establishing an open source framework modular both in modelling and numerical algorithms.