Which diameter and angle rule provides optimal flow patterns in a coronary bifurcation?

Abstract

The branching angle and diameter ratio in epicardial coronary artery bifurcations are two important determinants of atherogenesis. Murray's cubed diameter law and bifurcation angle have been assumed to yield optimal flows through a bifurcation. In contrast, we have recently shown a 7/3 diameter law (HK diameter model), based on minimum energy hypothesis in an entire tree structure. Here, we derive a bifurcation angle rule corresponding to the HK diameter model and critically evaluate the streamline flow through HK and Murray-type bifurcations. The bifurcations from coronary casts were found to obey the HK diameter model and angle rule much more than Murray's model. A finite element model was used to investigate flow patterns for coronary artery bifurcations of various types. The inlet velocity and pressure boundary conditions were measured by ComboWire. Y-bifurcation of Murray type decreased wall shear stress-WSS (10%-40%) and created an increased oscillatory shear index-OSI in atherosclerosis-prone regions as compared with HK-type bifurcations. The HK-type bifurcations were found to have more optimal flow patterns (i.e., higher WSS and lower OSI) than Murray-type bifurcations which have been traditionally believed to be optimized. This study has implications for changes in bifurcation angles and diameters in percutaneous coronary intervention.

Figure 1

(a–b) Coronary bifurcations with angle α of (a) 44° (Y bifurcation) and (b) 71° (T bifurcation) obtained from casts of porcine LAD arterial tree. The bifurcation angles α, β, and γ refer to the angles between daughter vessels, between mother and larger daughter vessels, and between mother and smaller daughter vessels, respectively. Bifurcation angles obey the angle rule (Eq. 1) and diameters comply with the HK diameter model as ${\text{D}}_{\text{m}}^{2\frac{1}{3}}={\text{D}}_{\text{l}}^{2\frac{1}{3}}+{\text{D}}_{\text{s}}^{2\frac{1}{3}}$, where D_m, D_l, and D_s are the mother, large and small daughter vessel diameters, respectively. (c) In vivo pulsatile flow velocity waveform measured at the inlet of porcine LAD arterial tree, which serves as inlet boundary condition for the flow simulation.

Figure 2

(a–b) Time-averaged (over a cardiac cycle) WSS (Unit: Dynes·cm⁻²) at Y bifurcations of (a) HK and (b) Murray corresponding to Table 1. Symbol A refers to the surface region in the smaller daughter vessel opposite to the carina of daughter vessels. Symbol B refers to the surface region in the larger daughter vessel opposite to the carina of daughter vessels. Symbol C refers to the joint surface regions of mother vessel and two daughter vessels lateral to the carina of daughter vessels, which penetrates into the carina of daughter vessels. (c–d) OSI at Y bifurcations of (c) HK and (d) Murray. In Y bifurcations, HK type is similar to Finet type.

Figure 3

(a–b) Time-averaged (over a cardiac cycle) WSS (Unit: Dynes·cm⁻²) at T bifurcations of (a) HK and (b) Finet corresponding to Table 1. Symbol A refers to the region in the smaller daughter vessel opposite to the carina of daughter vessels and Symbol B refers to the region in the larger daughter vessel lateral to the carina of daughter vessels. (c–d) OSI at T bifurcations of (c) HK and (d) Finet. In T bifurcations, HK type is similar to Murray type.

Figure 4

(a–c) Mean ± SD WSS (averaged over all nodes in the corresponding region) at (a) region A, (b) region B, and (c) region C in Y bifurcations of HK and Murray. (d–f) Mean ± SD OSI (averaged over all nodes in the corresponding region) at (d) region A, (e) region B, and (f) region C in Y bifurcations of HK and Murray. Regions A–C correspond to regions marked by symbols A–C in Fig. 2b. Regions A–C have surface areas of 0.94, 1.45, and 0.76 mm², respectively, where surface area in Region C only corresponds to one of two joint surface regions (anterior and posterior) of mother and two daughter vessels lateral to the carina of daughter vessels. OSI in regions B and C of HK-type bifurcations is zero.

Figure 5

(a–b) Mean ± SD WSS (averaged over all nodes in the corresponding region) at (a) region A and (b) region B in T bifurcations of HK and Finet. (c–d) Mean ± SD OSI (averaged over all nodes in the corresponding region) at (c) region A and (d) region B in T bifurcations of HK and Finet. Regions A and B correspond to regions marked by symbols A and B in Fig. 3b. Regions A and B have surface areas of 0.43 and 0.68 mm², respectively, where surface area in Region B only corresponds to one of two joint surface regions (anterior and posterior) of mother and two daughter vessels lateral to the carina of daughter vessels.

Figure A1

(a) Schematic representation of a bifurcation with an infinitesimal perturbation of mother vessel length; (b) Relationship between bifurcation angle α (the angle between two daughter vessels) and diameter ratio D_s/D_l determined by HK, Murray, and Finet angle rules and measurements of CT scans.