Computational and Mathematical Methods in Medicine

Volume 2015, Article ID 108274, 15 pages

http://dx.doi.org/10.1155/2015/108274

## A Reconstruction Method of Blood Flow Velocity in Left Ventricle Using Color Flow Ultrasound

^{1}Department of Computational Science and Engineering, Yonsei University, Seoul 120-749, Republic of Korea^{2}Division of Computational Mathematics, National Institute for Mathematical Sciences, Daejeon 305-811, Republic of Korea^{3}School of Mechanical Engineering, Yonsei University, Seoul 120-749, Republic of Korea

Received 29 October 2014; Accepted 25 December 2014

Academic Editor: Tianye Niu

Copyright © 2015 Jaeseong Jang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Vortex flow imaging is a relatively new medical imaging method for the dynamic visualization of intracardiac blood flow, a potentially useful index of cardiac dysfunction. A reconstruction method is proposed here to quantify the distribution of blood flow velocity fields inside the left ventricle from color flow images compiled from ultrasound measurements. In this paper, a 2D incompressible Navier-Stokes equation with a mass source term is proposed to utilize the measurable color flow ultrasound data in a plane along with the moving boundary condition. The proposed model reflects out-of-plane blood flows on the imaging plane through the mass source term. The boundary conditions to solve the system of equations are derived from the dimensions of the ventricle extracted from 2D echocardiography data. The performance of the proposed method is evaluated numerically using synthetic flow data acquired from simulating left ventricle flows. The numerical simulations show the feasibility and potential usefulness of the proposed method of reconstructing the intracardiac flow fields. Of particular note is the finding that the mass source term in the proposed model improves the reconstruction performance.

#### 1. Introduction

Vortex flow imaging has recently attracted much attention in the field of clinical cardiac assessment owing to reports of its feasibility for analyzing intraventricular vortex flows [1–3]. The vorticity of intraventricular blood flow describes a rotational flow pattern that offers possible clinical indices of cardiac functions such as sphericity, vortex depth, vortex length, and vortex pulsation correlation.

There are several methods to compute and visualize the velocity fields of blood flow inside the left ventricle (LV), with echo particle image velocimetry (E-PIV) being representative of the commonly used noninvasive methods [4]. It tracks the speckle patterns of blood flow to estimate blood motion within the imaging plane. Although it is generally unable to measure out-of-plane particle motion from 2D echocardiography data (called B-mode images), a recent study extending E-PIV to 3D volume data demonstrated the possibility of out-of-plane assessment [5]. However, E-PIV is not completely noninvasive because it requires the intravenous injection of a contrast agent to obtain images suitable for the speckle-tracking algorithm.

To develop less invasive techniques, methods to reconstruct blood flows from color flow images (also called C-mode images, color Doppler images, color Doppler data, or Doppler echocardiography) have been proposed. The color flow images reflect the projected velocity components in the direction of ultrasound beam propagation [6]. To compute the flow velocity from color flow images, Garcia et al. [7] assumed a 2D divergence-free condition on the velocity fields; they decomposed each 2D velocity vector into a radial component obtained from the color flow data and an unknown angular component, which was computed using their assumption of the 2D flow. Ohtsuki and Tanaka [8] also assumed 2D flows and recovered the 2D velocity fields from the color Doppler data using the concepts of stream function and streamline in 2D fluid flow. However, the assumption of a 2D divergence-free condition is an oversimplification that ignores out-of-plane flows.

Arigovindan et al. [9] proposed a velocity reconstruction method using color Doppler data acquired from beams in two different directions. To cope with the nonuniformly sampled data of multiple imaging planes, they used 2D B-spline on each of the velocity components to be estimated, and the unknown coefficients of the 2D B-spline were calculated from the measured color Doppler data using least squares. Similar to the 2D reconstruction, Gomez et al. [10] recovered 3D velocity fields from multiple registered color Doppler images using 3D B-spline and least squares. The registration of multiple imaging planes for the above two methods remains very challenging in a practical environment.

Recently, a new imaging modality (Doppler vortography) based on 2D color Doppler data was introduced by Mehregan et al. [11], who assumed that a vortex flow pattern has axisymmetric features in the neighborhood of its center. Their method employs a simple kernel filter designed to find the positions of axisymmetry in the 2D color Doppler images. The vortex flow was recovered using a color Doppler-variable vorticity function that directly computes vorticities from color Doppler values. However, the assumption of axisymmetry does not reflect detailed flow patterns, and it may lead to inaccurate vortex positions and vorticity values in patients with severe dysfunction where axisymmetry cannot be assumed at all.

In this paper, we propose a 2D Navier-Stokes model to reconstruct intraventricular flows using color flow images and LV boundaries extracted from echocardiography data. Although the use of the full 2D Navier-Stokes equations in this setting has already been proposed and evaluated for 2D flow field regularization [12], the originality of this work is the inclusion of a source-term to deal with the out-of-plane flow component. The proposed model considers both in-plane and out-of-plane blood flows for an imaging plane in apical long-axis three-chamber (A3CH) view. Particular attention is given to the appearance and disappearance of the out-of-plane components in the imaging plane, which is modeled as a mass source term of a source-sink distribution. Blood flows in the imaging domain are reconstructed through solving a system of equations, which include a 2D incompressible Navier-Stokes equation for the mass source term and the color flow data measurement equation describing the projected velocity component for the color flow data. The boundary conditions required to solve the system of equations are given by the LV borders extracted from echocardiography data.

The performance of the proposed method is evaluated numerically using synthetic flow data with LV motion. The proposed method is shown to be feasible and potentially valuable for reconstructing intracardiac flow fields.

#### 2. Materials and Methods

Commonly used ultrasound systems can provide not only 2D echo images but also color flow images, which represent the scanline directional components of the velocity fields using the phase-shift estimated by a standard autocorrelation algorithm [13]. Our flow reconstruction method is to reconstruct the intraventricular flows using color flow images and the LV boundaries extracted from the echo images. In this section, we describe the overall outline of our flow reconstruction method using those ultrasound measurements based on two assumptions as follows:(i)the time difference between sequential color flow imaging frames is very small;(ii)the echo and color flow images are acquired simultaneously and separately for the entire heart cycle.

##### 2.1. Mathematical Model on 2D Imaging Plane

We focus on the dominant vortex flow appearing in the A3CH view, which passes through the apex and the mitral and aortic valves as shown in Figure 1(b), and mathematical model for blood flows inside the LV on the imaging plane of the A3CH view.