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As both fluid flow measurement techniques and computer simulation methods continue to improve, there is a growing need for numerical simulation approaches that can assimilate experimental data into the simulation in a flexible and mathematically consistent manner. The problem of interest here is the simulation of blood flow in the left ventricle with the assimilation of experimental data provided by ultrasound imaging of microbubbles in the blood. The weighted least-squares finite element method is used because it allows data to be assimilated in a very flexible manner so that accurate measurements are more closely matched with the numerical solution than less accurate data. This approach is applied to two different test problems: a flexible flap that is displaced by a jet of fluid and blood flow in the porcine left ventricle. By adjusting how closely the simulation matches the experimental data, one can observe potential inaccuracies in the model because the simulation without experimental data differs significantly from the simulation with the data. Additionally, the assimilation of experimental data can help the simulation capture certain small effects that are present in the experiment, but not modeled directly in the simulation.

The physics of blood flow in the left ventricle of the heart has traditionally been studied using either experimental measurement of flow properties (e.g., ultrasound or magnetic resonance imaging) or computational fluid dynamic models. Experimental approaches are generally limited to obtaining flow information at only few spatial locations and using time-averaged properties. Computational models require assumptions along the mathematical domain boundaries, and they include numerical approximation error and model error. In many cases, however, it is desirable to have the more comprehensive spatial and temporal data provided by computational fluid dynamics combined with the data provided by experimental measurement. The weighted least-square finite element method (WLSFEM) is a computational modeling approach that allows experimental data to be assimilated into the model in a flexible framework so that the numerical approximation matches the more accurate experimental data while, at the same time, not being contaminated by errors in the noisier experimental data. The application of this method to the simulation of blood flow in the left ventricle is examined here.

One approach that we have used previously for obtaining experimental blood flow data in the left ventricle is echocardiographic particle imaging velocimetry (echo PIV) [

A complementary approach to echo PIV that could allow the approximation of all 3 components of the 3D velocity field is to use computational fluid dynamics to simulate blood flow in the left ventricle. A number of computational models have been developed specifically to model blood flow in the heart (c.f., [

In an earlier paper, we developed the WLSFEM for the assimilation of data when solving partial differential equations, including the steady Navier-Stokes equations [

The physical phenomena of interest here are typically modeled by partial differential equations. In particular, incompressible, Newtonian fluids are modeled by the Navier-Stokes equations, which are generally considered appropriate for modeling blood flow in the heart [^{2}-norm on the 3D fluid domain and ^{2}-norm along the 2D boundary surfaces (^{2}-norm, used along the boundary surfaces, is an approximation of the H^{1/2}-norm that deemphasizes oscillatory components (i.e., noisy components) relative to the H^{1/2}-norm [

The boundary functional weights,

When modeling blood flow in the left ventricle, or any fluid-structure interaction problem, the shape of the fluid domain is continuously changing. There are a number of numerical strategies 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 WLSFEM has a number of computational and algorithmic advantages for the problem of solving the Navier-Stokes equations and pseudosolid domain mapping equations with assimilated data:

it provides tremendous flexibility in handling the additional conditions imposed by the experimental data, including the ability to weight data based on the accuracy of the experimental data, that is, accurate data can be weighted and matched more closely by the CFD solution while less accurate data is only loosely matched by the CFD approximation;

the mathematical framework of least-squares minimization leads to symmetric positive definite matrices, which generally allows for efficient algebraic multigrid solvers [

the functional itself provides a natural sharp local error estimator, which could enable effective adaptive refinement [

To solve the least-squares problem, the equations in the functional (

All simulations were performed using the ParaFOS code, written by the authors. The code imports hexahedral meshes from the Cubit mesh generation package (Sandia National Laboratory). The finite element meshes are then partitioned using the Metis graph partitioning library [

To test the WLSFEM on problems with moving domains and PIV data assimilation, two different test problems are examined. The first problem is a flexible flap that is displaced by a fluid jet, and optical PIV is used to obtain experimental data. The second problem is a simulation of blood flow in the left ventricle of a pig using echo PIV data. It is this second problem that was the primary motivation for the development of the numerical modeling approach described here.

The experimental apparatus consisted of a cellulose acetate flap (taken from an overhead transparency) that was fixed on one end and placed in a 15.4 cm cube filled with water and contrast agent particles (Figure

Image from the flap displacement PIV experiment. The flap (grey) is displaced by a jet of fluid, and the flow velocity is determined using PIV. The optical PIV particles appear as speckles, and the velocities are shown using arrows.

A sample image from the moving flap experiment is shown in Figure

WLSFEM simulation of the moving flap experiment at (a) 0.0 sec and (b) 0.2 sec. The finite element grid deforms in response to the moving flap, and the PIV data is assimilated into the simulation with a boundary functional weight of 1.0.

The WLSFEM algorithm was based on implicit time stepping, so from a numerical stability standpoint, any time step size could be used in the simulation. Here we used the same time step size in the simulation as was available from the PIV data, 20 msec. This means that PIV data was available at every time point in the simulation. If a simulation uses more time steps than are available from PIV data, then some simulation time steps cannot use assimilated PIV data or they must use interpolated PIV data. Based on error estimated by the experimentalists and the PIV software, a boundary functional weight of

It is difficult to quantitatively compare the simulation predictions with and without PIV data included, but one measure is to calculate the magnitude of the velocity at every node along a surface (in this case the PIV surface) and sum those magnitudes. Figure

The sum of the L^{2}-norm of every velocity vector along the PIV plane in the simulation versus the weight on the PIV data term in the functional. Slightly lower velocities are predicted by the simulation if the PIV data are not assimilated (

The PIV data for the left ventricle simulation was obtained in previous studies using an open-chest pig [

An ultrasound image showing microbubbles inside the left ventricle of the heart of a pig (left). The ultrasound probe is on the external side of the heart wall near the apex. The PIV data corresponding to the bubble motion is shown on the right. The data clearly contains some errors (e.g., the circled vector).

The WLSFEM simulation of the left ventricle required that the location of the heart walls be specified. The left ventricle was assumed to have a half-ellipsoid geometry, which is a common geometric approximation [

The simulation begins at the start of diastole, the filling of the ventricle, and the velocity along a single plane (the PIV plane) during early diastole is shown in Figure

WLSFEM simulations of blood flow in the left ventricle at

The PIV data has a larger impact on the simulation at later time points. Figure

WLSFEM simulation of flow in the left ventricle at

The challenge of assimilating experimental data into a computational simulation is very widespread. The general problem of interest here is solving the Navier-Stokes equations on a moving domain with additional experimental data provided by PIV experiments. In particular, the goal is the simulation of blood flow in the left ventricle with the assimilation and inclusion of 2-dimensional echo PIV data obtained using microbubbles. The WLSFEM used here is particularly well suited for this assimilation of data problem because of the flexibility in incorporating experimental data that are weighted based on accuracy. Accurate data can be closely matched with the simulation result, and less accurate data is not closely matched. The WLSFEM approach is demonstrated on two different test problems: (1) a flap that is displaced by a jet of fluid, and (2) blood flow in the left ventricle of the pig. By applying different weights to the PIV data, one can observe quantitative differences between the simulation without PIV data and the simulation that matches the PIV data more or less accurately. These comparisons can reveal inaccuracies in the model such as inaccurate boundary conditions or missing physics. The incorporation of PIV data can assist the simulation by capturing some effects that are not directly modeled. For example, in the left ventricle model presented here, the mitral valve was not modeled directly, but the effects of the valve on the blood flow could be partially captured by the simulation through the incorporated PIV data. There are other methods for incorporating experimental data into a numerical simulation, but the WLSFEM method is a flexible and efficient option for this class of problems.

This work was supported by NSF Grant DMS-0811275 and the Flight Attendant Medical Research Institute.