Optical Flow Analysis of Left Ventricle Wall Motion with Real-Time Cardiac Magnetic Resonance Imaging in Healthy Subjects and Heart Failure Patients

Abstract

In cardiology, magnetic resonance imaging (MRI) provides a clinical standard for measuring ventricular volumes. Owing to their reliability, volumetric measurements with cardiac MRI have become an essential tool for quantitative assessment of ventricular function. However, as volumetric indices are indirectly related to myocardial motion that drives ventricular filling and ejection, cardiac MRI cannot provide comprehensive evaluation of ventricular performance. To overcome this limitation, the presented work sought to measure ventricular wall motion directly with optical flow analysis of real-time cardiac MRI. By modeling left ventricle (LV) walls in real-time images based on myocardial architecture, we developed an optical flow approach to analyzing LV radial and circumferential wall motion for improved quantitative assessment of ventricular function. For proof-of-concept, a cardiac MRI study was conducted with healthy volunteers and heart failure (HF) patients. It was found that, as real-time images provided sufficient temporal information for correlation analysis between different LV wall motion velocity components, optical flow assessment detected the difference of ventricular performance between the HF patients and the healthy volunteers more effectively than volumetric measurements. We expect that this model-based optical flow assessment with real-time cardiac MRI would offer intricate analysis of ventricular function beyond conventional volumetric measurements.

Appendix

The coefficient terms A_m(x,y,z,t-t') and B_mn(x,y,z,t) in Eq. 4 are given as below:

$$A_{0} \left( {x,y,z,t - t^{\prime}} \right) = \left[ {\frac{{\partial I\left( {x,y,z,t} \right)}}{\partial x}\cos \left( \theta \right) + \frac{{\partial I\left( {x,y,z,t} \right)}}{\partial y}\sin \left( \theta \right)} \right]s_{v} \left( {z,t - t^{\prime}} \right)$$

$$A_{1} \left( {x,y,z,t - t^{\prime}} \right) = \left[ {\frac{{\partial I\left( {x,y,z,t} \right)}}{\partial x}\cos \left( \theta \right)\sin \left( \theta \right) + \frac{{\partial I\left( {x,y,z,t} \right)}}{\partial y}\sin^{2} \left( \theta \right)} \right]s_{v} \left( {z,t - t^{\prime}} \right)$$

$$A_{2} \left( {x,y,z,t - t^{\prime}} \right) = \left[ {\frac{{\partial I\left( {x,y,z,t} \right)}}{\partial x}\sin^{2} \left( \theta \right) + \frac{{\partial I\left( {x,y,z,t} \right)}}{\partial y}\cos \left( \theta \right)\sin \left( \theta \right)} \right]s_{v} \left( {z,t - t^{\prime}} \right)$$

$$A_{3} \left( {x,y,z,t - t^{\prime}} \right) = \left[ { - \frac{{\partial I\left( {x,y,z,t} \right)}}{\partial x}r \sin \left( \theta \right) + \frac{{\partial I\left( {x,y,z,t} \right)}}{\partial y}r \cos \left( \theta \right)} \right]s_{v} \left( {z,t - t^{\prime}} \right)$$

$$A_{4} \left( {x,y,z,t - t^{\prime}} \right) = \left[ { - \frac{{\partial I\left( {x,y,z,t} \right)}}{\partial x}r^{2} \sin \left( \theta \right) + \frac{{\partial I\left( {x,y,z,t} \right)}}{\partial y}r^{2} \cos \left( \theta \right)} \right]s_{v} \left( {z,t - t^{\prime}} \right)$$

$$A_{5} \left( {x,y,z,t - t^{\prime}} \right) = \left[ { - \frac{{\partial I\left( {x,y,z,t} \right)}}{\partial x}rz \sin \left( \theta \right) + \frac{{\partial I\left( {x,y,z,t} \right)}}{\partial y}rz \cos \left( \theta \right)} \right]s_{v} \left( {z,t - t^{\prime}} \right)$$

$$B_{0n} \left( {x,y,z,t} \right) = \left[ {\frac{{\partial I\left( {x,y,z,t} \right)}}{\partial x}\cos \left( \theta \right) + \frac{{\partial I\left( {x,y,z,t} \right)}}{\partial y}\sin \left( \theta \right)} \right]\mathop \sum \limits_{{t^{\prime} = 0}}^{{t_{a} }} s_{w} \left( {z,t - t^{\prime}} \right)T_{n} \left( {\frac{{t^{\prime}}}{{t_{a} }}} \right)$$

$$B_{1n} \left( {x,y,z,t} \right) = \left[ {\frac{{\partial I\left( {x,y,z,t} \right)}}{\partial x}x\cos \left( \theta \right) + \frac{{\partial I\left( {x,y,z,t} \right)}}{\partial y}x \sin \left( \theta \right)} \right]\mathop \sum \limits_{{t^{\prime} = 0}}^{{t_{a} }} s_{w} \left( {z,t - t^{\prime}} \right)T_{n} \left( {\frac{{t^{\prime}}}{{t_{a} }}} \right)$$

$$B_{2n} \left( {x,y,z,t} \right) = \left[ {\frac{{\partial I\left( {x,y,z,t} \right)}}{\partial x}y\cos \left( \theta \right) + \frac{{\partial I\left( {x,y,z,t} \right)}}{\partial y}y \sin \left( \theta \right)} \right]\mathop \sum \limits_{{t^{\prime} = 0}}^{{t_{a} }} s_{w} \left( {z,t - t^{\prime}} \right)T_{n} \left( {\frac{{t^{\prime}}}{{t_{a} }}} \right)$$

$$B_{3n} \left( {x,y,z,t} \right) = \left[ { - \frac{{\partial I\left( {x,y,z,t} \right)}}{\partial x}\sin \left( \theta \right) + \frac{{\partial I\left( {x,y,z,t} \right)}}{\partial y}\cos \left( \theta \right)} \right]\mathop \sum \limits_{{t^{\prime} = 0}}^{{t_{a} }} s_{w} \left( {z,t - t^{\prime}} \right)T_{n} \left( {\frac{{t^{\prime}}}{{t_{a} }}} \right)$$

$$B_{4n} \left( {x,y,z,t} \right) = \left[ { - \frac{{\partial I\left( {x,y,z,t} \right)}}{\partial x}x \sin \left( \theta \right) + \frac{{\partial I\left( {x,y,z,t} \right)}}{\partial y}x \cos \left( \theta \right)} \right]\mathop \sum \limits_{{t^{\prime} = 0}}^{{t_{a} }} s_{w} \left( {z,t - t^{\prime}} \right)T_{n} \left( {\frac{{t^{\prime}}}{{t_{a} }}} \right)$$

$$B_{5n} \left( {x,y,z,t} \right) = \left[ { - \frac{{\partial I\left( {x,y,z,t} \right)}}{\partial x}y \sin \left( \theta \right) + \frac{{\partial I\left( {x,y,z,t} \right)}}{\partial y}y \cos \left( \theta \right)} \right]\mathop \sum \limits_{{t^{\prime} = 0}}^{{t_{a} }} s_{w} \left( {z,t - t^{\prime}} \right)T_{n} \left( {\frac{{t^{\prime}}}{{t_{a} }}} \right)$$

where n={0, 1, 2,…, N} is the order of the Chebyshev polynomial T_n(t), I(x,y.z,t) represents a real-time image voxel (x,y) in the slice z and at the time t, (r,θ) is the radial coordinates of the image voxel (x,y) in the slice z, ∂I(x,y,z,t)/∂x represents the partial derivative of the real-time image along the x direction and ∂I(x,y,z,t)/∂y along the y direction, s_v(z,t) and s_w(z,t) are the systole/diastole and expiration/inspiration binary indicator functions (Fig. 2(c)), and [0,t_a] is the data acquisition window.