MR Image-Based Geometric and Hemodynamic Investigation of the Right Coronary Artery with Dynamic Vessel Motion

Abstract

The aim of this study was to develop a fully subject-specific model of the right coronary artery (RCA), including dynamic vessel motion, for computational analysis to assess the effects of cardiac-induced motion on hemodynamics and resulting wall shear stress (WSS). Vascular geometries were acquired in the right coronary artery (RCA) of a healthy volunteer using a navigator-gated interleaved spiral sequence at 14 time points during the cardiac cycle. A high temporal resolution velocity waveform was also acquired in the proximal region. Cardiac-induced dynamic vessel motion was calculated by interpolating the geometries with an active contour model and a computational fluid dynamic (CFD) simulation with fully subject-specific information was carried out using this model. The results showed the expected variation of vessel radius and curvature throughout the cardiac cycle, and also revealed that dynamic motion of the right coronary artery consequent to cardiac motion had significant effects on instantaneous WSS and oscillatory shear index. Subject-specific MRI-based CFD is feasible and, if scan duration could be shortened, this method may have potential as a non-invasive tool to investigate the physiological and pathological role of hemodynamics in human coronary arteries.

Appendix

The strain energy due to stretching and bending is

$$ U = \int\limits_{l} {\frac{1}{2}\left[ {EA\left( {{\frac{{\partial \user2{u}}}{\partial s}}} \right)^{2} + EI\left( {{\frac{{\partial^{2} \user2{v}}}{{\partial s^{2} }}}} \right)^{2} + U_{\text{ext}} (\user2{X})} \right]} ds , $$

(1)

where u and v are displacement vectors tangential and normal to the centerline, s is coordinate along the centerline, E is elastic modulus, A is cross-sectional area, and I is second moment of area. The Cartesian coordinate vector is defined as X. U _ext is energy field in reference to image intensity that works as a source of external force attracting the centerline to that at the consecutive time point. It is given as

$$ U_{\text{ext}} (\user2{X}) = - \left| {\nabla G(I(\user2{X}))} \right|^{2} . $$

(2)

Here the energy U _ext at spatial position X is defined as a square of the image intensity gradient. A unit image intensity U ₀ is given along the “target” centerline, i.e., the centerline at the consecutive time-point, and the intensity field is smoothed by Gaussian filter function G. In the case shown in Fig. 5, the image intensity I(X) was set to 1.00 (=U ₀) for the centerline at time 725 ms to which the centerline at time 600 ms was attracted. The constants in the equations were set based on approximate physiological values for coronary arteries; E = 1 MPa, A = 1.26 × 10⁻⁵ m², I = 1.26 × 10⁻¹¹ m⁴ assuming constant radius r = 0.002 m. It has been found that the results are insensitive to the actual values of these parameters. Displacement that minimizes U (Eq. 1) must satisfy the Euler equation

$$ EA{\frac{{\partial^{2} \user2{u}}}{{\partial s^{2} }}} - EI{\frac{{\partial^{4} \user2{u}}}{{\partial s^{4} }}} - \nabla U_{\text{ext}} (\user2{X}) = 0. $$

(3)

Equation (3) was solved as a time-dependent problem by firstly treating nodal coordinate x as a function of coordinate along the line and time, i.e., x (s, t), and then equating the temporal derivative of the nodal coordinate to Eq. (3) to give the following:

$$ {\frac{{\partial \user2{x}}}{\partial t}} = EA{\frac{{\partial^{2} \user2{u}}}{{\partial s^{2} }}} - EI{\frac{{\partial^{4} \user2{u}}}{{\partial s^{4} }}} - \nabla U_{\text{ext}} (\user2{X}), $$

(4)

where t is time. The first and second terms on the right hand side represent stretching and bending, respectively. Contribution of torsion is not taken into account in this study assuming that it is not as significant as stretching and bending. Motion of a reference point was controlled by Eq. (4) discretized in time:

$$ \user2{x}^{t + \Updelta t} = \user2{x}^{t} + \Updelta t\left[ {EA{\frac{{\partial^{2} \user2{u}}}{{\partial s^{2} }}} - EI{\frac{{\partial^{4} \user2{u}}}{{\partial s^{4} }}} - \nabla U_{\text{ext}} (\user2{X})} \right] $$

which allowed the reference points to be mapped onto the centerline at the next time point.