Introduction 3 21 38 7 8 7 22 4 23 18 24 32 33 37 28 40 41 28 41 38 17 12 Methods Computational Model 1 d R mean 9 15 16 2 Figure 1. A schematic model of the left anterior descending coronary artery (LAD).  Blood Flow Boundary Conditions 2 22 q o Re κ Figure 2. The LAD inlet flow and curvature waveform.  1 \documentclass[12pt]{minimal}
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$$ \kappa = \left( {\frac{d} {{2R}}} \right)^{1/2} Re $$\end{document} T μ ρ 3 16 29 Wall Motion Boundary Conditions 32 ε 2 \documentclass[12pt]{minimal}
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$$ \varepsilon = \frac{{\left( {R_{\max } - R_{\min } } \right)}} {{2R_{{\text{mean}}} }} $$\end{document} R 15 2 3 \documentclass[12pt]{minimal}
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$$ R\left( t \right) = R_{{\text{mean}}} \left[ {1 + \varepsilon \cdot \cos (\omega t)} \right] $$\end{document} Mass Transfer Boundary Conditions 4 \documentclass[12pt]{minimal}
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$$ \frac{{\partial C}} {{\partial t}} + {\text{u}} \cdot \nabla C = - D\nabla ^2 C $$\end{document} C D 10 K −8 K −03 36 5 11 39 C o 36 39 5 \documentclass[12pt]{minimal}
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$$ V_{\text{w}} C_{\text{w}} - D\left. {\frac{{\partial C}} {{\partial n}}} \right|_{{\text{wall}}} = KC_{\text{w}} \quad {\text{at}}\;{\text{the}}\;{\text{wall}} $$\end{document} V w −6 36 C w D −8 2 39 K 5 6 \documentclass[12pt]{minimal}
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$$ - D\left. {\frac{{\partial C}} {{\partial n}}} \right|_{{\text{wall}}} = (K - V_{\text{w}} )C_{\text{w}} $$\end{document} 31 1 31 20 31 7 \documentclass[12pt]{minimal}
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$$ - D\left. {\frac{{\partial C}} {{\partial n}}} \right|_{{\text{wall}}} = KC_{\text{w}} $$\end{document} −5 2 1 Numerical Method 3 t Modelling Approach q o R R q o Case 4: Pulsatile flow inlet with dynamic geometry, to study the combined effects of pulsatility and wall motion. Model Validation 32 3 C w C o μ ρD 5 dv D 7 Figure 3. Comparison of the time-averaged normalised LDL wall flux predicted with three different meshes and time step sizes to demonstrate mesh (top panel) and time step (bottom panel) independence of the computational solutions.  Results 32 Velocity Patterns 4 4 4 4 4 4 4 4 4 Figure 4. R q o R q o  WSS Patterns 5 Figure 5. Temporal axial inner and outer WSS distribution at 5 diameters (top panel) and 9 diameters (bottom panel) from the LAD inlet.  The pulsatility of blood flow caused time varying WSS distribution along the outer and inner walls. The WSS increased with the pulsatile flow rate and overall this increase was greater along the outer wall. Hence peak WSS was noted along the outer wall at maximum flow rate. Together, wall motion and flow pulsatility produced instantaneous variations in the outer and inner WSS which were somewhat similar to the WSS distribution in the pulsatile flow static model. Although the gross distribution of WSS primarily followed the flow rate curve, the dynamic curvature effects were evident in the central section where the location and magnitude of peak WSS were different from that of the pulsatile flow static model. 6 Figure 6. Time-averaged axial WSS distribution along the inner and outer walls of the LAD.  The combined effects of wall motion and pulsatility caused an increase in the outer WSS up to its peak value and a subsequent decrease in the WSS in comparison with that of the pulsatile flow static model, and an overall increase in the outer WSS when compared to the steady flow models. Along the inner wall the flow pulsatility and the wall motion produced contrasting effects whereby the former decreased the WSS and the later increased the WSS. Due to this opposing effect of wall motion and flow pulsatility their combination had only caused smaller variations in the WSS distribution. Oxygen Transport KC w KC o 7 7 Figure 7. Temporal normalised oxygen flux to the inner and outer walls at 5 diameters (top panel) and 9 diameters (bottom panel) from the LAD inlet.  8 Figure 8. Time-averaged normalised oxygen wall flux distribution along the inner and outer walls of the LAD.  Along the outer wall after an initial sharp drop the flow pulsation caused a gradual increase and then a gradual decrease in the oxygen flux. The wall motion produced a net reduction in the oxygen flux to the outer wall from the inlet. When both the wall motion and flow pulsatility were included the oxygen flux decreased sharply closer to the inlet followed by a gradual increase in the medial region and a gradual decrease thereafter. Nevertheless, in all the models the time-averaged outer wall oxygen flux was more than that to the inner wall. LDL Transport KC w KC o 9 9 Figure 9. Temporal normalised LDL flux to the inner and outer walls at 5 diameters (top panel) and 9 diameters (bottom panel) from the LAD inlet.  10 Figure 10. Time-averaged normalised LDL flux distribution along the inner and outer walls of the LAD.  Discussion 4 16 27 29 32 4 4 4 1 4 32 t 4 41 16 27 29 6 28 41 5 11 39 5 35 5 25 KC w KC o 1 17 38 31 KC w KC o 39 17 17 39 7 9 8 10 11 19 34 34 19 11 7 9 8 10 7 9 8 10 8 10 39 6 8 10 14 21 5 17 17 39 6 10 1 38 26 36 13 21 30 36 1 27 The results predicted in this study are specific to the flow and wall motion used in our model and therefore would be expected to change if a different flow or wall motion were imposed. For instance, if the flow and wall motion were more out of phase it may be expected that this would reduce the instantaneous Dean number and therefore one might expect the impact of the dynamic wall motion to be decreased from what was predicted. Although the wall motion effects were secondary to the pulsatile flow effects this does not mean that the wall motion is unimportant. Given the considerable variability in coronary artery motion, geometry and flow patterns in vivo there may be instances where wall motion could play a more significant role and therefore its effects should be considered on a case by case basis. 6 Conclusion In this study, we have incorporated the effects of idealised motion into a simulation of oxygen and LDL transport in the LAD. The LDL and oxygen transport were modelled differently due to their differences in in-vivo transport mechanisms. We carried out simulations under both the steady and pulsatile flow conditions in the static and dynamic models to quantify the relative importance of pulsatility and wall motion on mass transport patterns. This study showed that the effects of wall motion on mass transport in the coronary arteries could be modelled in the commercial CFD code Fluent. Our results predicted elevation of LDL flux, impaired oxygen flux and low wall shear stress (WSS) along the inner wall of curvature, a region known for its predilection to atherosclerosis. The temporal variations in velocity and WSS patterns in the dynamic model did not influence time dependent mass transport and were only secondary to the pulsatile flow effects. However, wall motion may influence mass transport if shear dependent variations in species permeability and other factors are considered in the calculations. Nevertheless, the wall motion did alter the time-averaged mass transfer in the medial and distal regions in the order of 26% and 12% for oxygen and LDL transport when compared to the corresponding static models. Taken together, these results suggest that wall motion may play an important role in coronary arterial transport processes. However, future studies on more physiologic models are warranted to gain a fundamental understanding of the role of wall motion on mass transport and atherosclerosis.