The left ventricle (LV) pumps oxygenated blood from the lungs to the rest of the body through systemic circulation. The efficiency of such a pumping function is dependent on blood flow within the LV chamber. It is therefore crucial to accurately characterize LV hemodynamics. Improved understanding of LV hemodynamics is expected to provide important clinical diagnostic and prognostic information. We review the recent advances in numerical and experimental methods for characterizing LV flows and focus on analysis of intraventricular flow fields by echocardiographic particle image velocimetry (echo-PIV), due to its potential for broad and practical utility. Future research directions to advance patient-specific LV simulations include development of methods capable of resolving heart valves, higher temporal resolution, automated generation of three-dimensional (3D) geometry, and incorporating actual flow measurements into the numerical solution of the 3D cardiovascular fluid dynamics.
The key function of the heart is to maintain blood circulation. This is accomplished through repetitive cycles of systolic contraction and diastolic relaxation. Systole and diastole consist, however, of several component phases that impact intraventricular flow. In a normally functioning heart, following isovolumic contraction (during which myocardial tension increases, but both aortic and mitral valves are closed), the aortic valve opens, and the LV ejects blood into the aorta. As a result of the helical architecture of its fibers [
Thus, the LV is not a simple positive displacement pump [
Ventricular flow can be studied using clinical imaging techniques and image-based computational fluid dynamics (CFD). Ventricular flow measurements using clinical imaging have been summarized elsewhere [
Echo particle image velocimetry study of normal heart. (a) Epicardial echo scan. Notice the well-defined LV boundary. Anatomic structures are labeled. After injection of microbubbles, digital transfer, and offline tracking, (b) early and (c) late diastolic inflows and vortex are mapped by velocity vectors and isovelocity streamlines
Ventricular flow is mainly driven by motion of the myocardium, a dynamically evolving surface boundary inside the LV. A major challenge in the numerical simulation is to satisfy the boundary conditions on the LV surface that is continuously changing in time. The methods for handling large deformations/movements of the boundary can be broadly categorized into two classes: (a) boundary-conforming techniques; and (b) fixed grid techniques. In boundary-conforming techniques the grid moves with the moving boundary, whereas, in fixed grid techniques, the grid is not moving and effects of the moving boundaries are transferred onto the closest fixed grid nodes that are in the vicinity of the moving boundary. The boundary-conforming techniques can retain good grid resolution near the moving boundaries. Consequently, they typically require fewer grid points relative to the fixed grid techniques to achieve the same level of resolution near the moving boundaries. Large boundary movements, however, in the boundary-conforming approaches create highly skewed grids that reduce the convergence and accuracy of the numerical scheme [
In boundary-conforming methods the computational grid is fitted to and moves/deforms with, the moving boundary. The grid movement is taken into account in the formulation of the Navier-Stokes equation by incorporating the grid velocity terms, which is referred to as the Arbitrary Lagrangian-Eulerian (ALE) formulation [
The ALE method works well for problems with relatively simple geometries and moderate deformations, such as those encountered in compliant blood vessels. However, obtaining smooth computational meshes at every time step for problems with significant structural deformations is difficult, and frequent remeshing may be the only option [
The fixed grid techniques are of particular interest in problems involving large deformations and movements of the boundary. Many fixed grid techniques have been developed in the past decades, including the immersed boundary (IB) and immersed interface methods [
From the different fixed grid methods developed, the immersed boundary, pioneered by Peskin [
Patient-specific data on cardiac tissue motion and flow measurements is essential for image-based CFD simulations to provide realistic boundary conditions. This is due to the fact that the accuracy and realism of such CFD simulations strongly depend on the specifications along the heart wall boundary, including the location, geometry, and velocity of the heart wall. Such boundary conditions can only be provided by the experimental measurements or a mathematical model of the LV wall. However, mathematical models of the heart wall are still not sufficiently accurate to be used for specific patients [
Determination of boundary conditions over the duration of the cardiac cycle requires tracking of the inner (endocardial) border of the muscle defining the analyzed cardiac cavity, such as the LV. Manual delineation of the endocardial border is currently involved in most studies, at least as an initial, user-determined estimate of the boundary at a given time point. Software tools, such as Omega Flow (Siemens) [
Echo-PIV is minimally invasive (injection or infusion of diluted microbubbles), relatively inexpensive, does not involve ionizing radiation, and offers high temporal resolution combined with suitable 2D spatial resolution, when compared to existing techniques. Information regarding several commonly used intracardiac flow visualization techniques is detailed by Sengupta et al. [
The primary constraint of current echo-PIV is the inability to measure out-of-plane particle motion; thus, only 2D velocity fields can be generated. However, recent advancements in 3D ultrasound equipment or the use of multiplanar acquisition techniques may represent a step towards resolving this problem [
Extensive validation of echo-PIV has been conducted both
Another approach for intracardiac flow vortex analysis, which does not require the use of PIV, has been introduced by Gharib and his colleagues [
As reviewed above, a number of numerical approaches have been developed and applied to the modeling of blood flow in the LV. An obvious question then is why should another method be considered? The weighted least-squares finite element method (WLSFEM) has three relatively unique attributes that make the approach particularly appealing for simulating the complex flows inside the moving LV, especially in settings where patient-specific blood flow data are available. First, with most existing methods, as the computational mesh is refined, the computational time increases roughly quadratically with the number of grid points. The WLSFEM, however, has the potential to offer optimal scalability, that is, the computational time is directly proportional to the number of grid points for any number of points [
Second, in the WLSFEM, the approximation problem is written as an optimization problem: the goal is to minimize the value of a functional for a finite element approximation space. The value of the functional is a sharp measure of the error everywhere in the domain [
The third advantage of the WLSFEM approach is that it enables a straightforward and flexible platform for assimilating experimental data into the process of solving the governing equations [
The WLSFEM requires defining new variables and rewriting the Navier-Stokes equations as a system of first-order equations. A number of different first-order systems have been derived by others [
The following first-order system is now used to replace the Navier-Stokes equations:
The WLSFEM approach uses a functional that includes boundary weights:
The boundary functional weights,
When modeling blood flow in the LV or any fluid-structure interaction problem, the shape of the fluid domain is continuously changing. Many numerical strategies exist for addressing the changing domain shape, including the generation of a new mesh every time step or grid mapping using equations such as the Winslow generator [
The use of the WLSFEM for the simulation of flow in the LV is illustrated in Figure
Simulation of a filling flow jet entering the left ventricle (top panels) and flow vortex in the inflow region along with an ejection jet in the left ventricular outflow tract (bottom panels) by using the weighted least-squares finite element method (WLSFEM), without the assimilation of particle image velocimetry (PIV) data (left panels) and with assimilated data (right panels). Blue, green, and red colors indicate low, mid, and high flow velocities, respectively.
The development of any mathematical model requires that assumptions are made. In the case of the LV model shown in Figure
The earliest simulation of ventricular flow dates back to the late 1970s with the pioneering work of Peskin using the immersed boundary method [
Saber et al. [
Domenichini et al. [
Recently, Schenkel et al. [
As can be observed from the previous section, in all of the LV simulations published to date, the heart valves are not resolved; that is, they are replaced with simplified inflow/outflow conditions. However, several studies have argued the importance and sensitivity of the solution to the inflow condition [
Simulation of the left ventricle and a bioprosthetic heart valve in the aortic position. The 3D vortical structures downstream of the valve are visualized using isosurfaces of q-criteria, whereas the flow inside the LV is visualized using velocity vectors.
In the simulation pictured in Figure
Another issue that requires attention is automating the generation of a 3D geometry from experimental data. Usually a few cross-sections are available from imaging data, which requires interpolation for the 3D geometry reconstruction from the 2D slices. Currently, most of this process is done manually or semiautomatically in computer-aided design software, which is quite time consuming. Interpolation strategies are required to reconstruct the geometry in time instants, when experimental image frames are not available. Automating this process would save time and minimize or eliminate subjectivity of the currently required user interaction. Finally, the uncertainty in the reconstruction of the 3D geometry from imaging data including cycle-to-cycle LV motion variability, subject motion during the imaging process, and uncertainty in the location of the imaging planes, needs to be quantified to support clinical decision making in the future.
The simulations to date have used boundary conditions at the LV wall. However, in many cases, simultaneous flow measurements in a few 2D planes are also available. Incorporating the sparse flow measurements into the 3D simulations to augment boundary data will increase the accuracy and realism of the simulations, as suggested in Figure
In summary, we propose the following developments to advance the image-based LV simulations. Methods capable of simulating the LV with heart valves resolved. Higher temporal resolution for experimental and clinical imaging data. Automated generation of 3D geometry for CFD analysis from imaging data and quantifying the uncertainty. Incorporating actual (even if sparse) flow measurements into the numerical solution of the 3D cardiovascular fluid dynamics.
The authors were supported by the NSF Grant CBET 1249950, Center for Computational Research of the University at Buffalo, and Arizona State University/Mayo Clinic Seed Grant. They thank Gillian Murphy for secretarial assistance in preparation of the paper.