Image collection
16 normal coronary arteries and 26 abnormal coronary arteries CT images were collected based on a 128-slice Siemens (SOMATOM Definition Flash) dual-source CT scanner (80 kV, 140 kV). The tube voltage was selected according to the Body Mass Index (BMI) as follows: if BMI≤18, tube voltage is 80 kV; if 18<BMI≤22, tube voltage is 100 kV; if 22<BMI≤27, tube voltage is 120 kV and if BMI>27, tube voltage is 140 kV. The contrast injection rate is 5-5.5 ml/s, and its dose is 60-80 ml. All patients did not use any drugs to control heart rate before scanning. They took 0.5 ml of nitroglycerin under the tongue before scanning, and they were scanned after breath training in calm state. Scan settings are listed in Table 1.
Artery models
The realistic coronary arterial models were reconstructed based on the collected samples with software Mimics (v9.0, Materialise, Ann Arbor, MI, USA). Geomagic Studio 2013 (3D Systems, Morrisville, NC, USA) was used to smooth the rough surfaces of the reconstructed models and generate solid surfaces subsequently. To make the numerical simulation easier, the small coronary branches of the coronary arterial models were removed. The main branch of right coronary artery was retained, and left anterior descending coronary artery, left main coronary artery and left circumflex coronary artery were retained for the left coronary artery. The ascending aorta was cut off about 0.55 centimeters away from the coronary sinus which is taken as the inlet of blood flow. The surface of reconstructed model was offset 0.5mm along the normal direction to get the outer wall of the vascular. A 3D CAD software, Solidworks ( Solidworks Corporation, Boston, MA, USA), was used to rebuild the vessel wall of the model. The vessel wall was created by the reconstruction of the blood solid and the offset solid. The blood model is shown in Fig. 1.
Assumption and governing equations
Blood was considered as a homogenous, incompressible and Newtonian fluid[24]. Blood was assumed to be isothermal. The blood flow was described by the three-dimensional incompressible Navier-Stokes equations and continuity equation [43]:
where u and σ respectively represent the fluid velocity vector and stress tensor, and ρ is the density. σ is defined by
where η and denote the viscosity of blood and shear rate, respectively. p is the pressure. D is the rate of deformation tensor which is defined by
where μ is the blood viscosity.
The vessel wall is assumed to be an isotropic, non-linear elastic material with no infiltration. The equation governing the solid domain:
where σs is the stress tensor, ρs is the density, and as is the acceleration.
Mesh generation
Meshes were generated using ICEM software (ANSYS, Inc., Canonsburg, PA, USA). The geometric models were meshed using unstructured tetrahedral volume meshes. The minimum and maximum sizes of the mesh were 0.06 mm and 1 mm, respectively, for the fluid part of the model. Five layers of fine mesh with a height ratio of 1.2 were used. The size of the mesh was 0.1 mm for the solid part of the model.
Boundary conditions
In this study, the inlet flow velocity is set as a constant, which is a simple way to compare the different effects of normal right coronary arteries and abnormal right coronary arteries on haemodynamics. The maximum exit velocity of the left ventricle was set as the entry boundary condition for the numerical simulation, as denoted by the red dot in Fig. 2 [44].
Finally, the boundary conditions are set as follows:
(1) On the entry of the aorta, the tangential velocity is set to 1 m/s, and the normal velocity is set to 0 m/s.
Vt=1 m/s;
Vn=0 m/s.
where the subscripts t and n are angential and normalt directions, respectively.
(2) On the exits of the aorta, the normal and tangential outlet pressure is 0 Pa.
pn=pt=0 Pa;
The wall of blood domain is assumed to has no slip, vn=vt=0 m/s. Fluid-solid coupling follows the most basic conservation principle, so the conservation of fluid and solid stress, displacement and flow should be met at the fluid-solid coupling interface.
where d is the displacement vectors, σ is the stress tensors, n is the boundary normal, and the subscripts f and s represent fluids and solids, respectively.
Numerical simulation
In the paper, a one-way fluid-solid coupling simulation was executed with ANSYS workbench. The finite volume method (FVM) mixed with the finite element method (FEM) is used to solve the governing equations. The structural analysis of vessel wall was solved with ANSYS Mechanical, and a CFD software based on the finite element method, ANSYS CFX (ANSYS CFX 19.0, Canonsburg, USA), was used for the fluid analysis of blood. The semi-implicit method for pressure linked equations consistent (SIMPLEC) algorithm was used to couple the outflow velocity term, and all equations were solved by the separation solution method. The convergence criterion was set to 1×10-4.
According to the Reynolds number, for Re<2300, the flow of blood is set to laminar. Re is defined by:
where v and ρ are the velocity and density of the fluid, respectively. d is the characteristic length, and μ is the viscosity of the fluid.
Statistical analysis
To eliminate the differences in patient specificity, multiple cases were analysed by statistical analysis. Statistical analysis was performed using GraphPad Prism (GraphPad Software 8.0, CA, USA). Correlation analysis was used to evaluate the relationship between area and flow rate, pressure and WSS. The difference in cross-sectional area of the inlet, volumetric flow, pressure and WSS between the two groups was studied by t-tests of two independent samples. The number in pressure and WSS was collected by means of the five-point sampling method. If correlation coefficient r was close to +1, 0 and -1, the results were positive correlation, negative correlation and no correlation, respectively. A correlation less than 0.5 was described as weak, whereas a correlation more than 0.8 was described as strong. The significance level was set at P<0.05.