Interactive CardioVascular and Thoracic Surgery Advance Access published April 22, 2014
A computational fluid dynamics simulation study of coronary blood flow affected by graft placement† Jason V. Lassalinea and Byung C. Moonb,* a b
Ryerson University, Toronto, Canada Southlake Regional Health Centre, Newmarket, Canada
* Corresponding author. Division of Cardiac Surgery, Southlake Regional Health Centre, 625 Davis Drive, Newmarket, ONT, Canada L3Y 2R2. Tel: +1-905-8688386; fax: +1-905-8300931; e-mail: [email protected] (B.C. Moon). Received 1 October 2013; received in revised form 19 January 2014; accepted 29 January 2014
Abstract OBJECTIVES: To determine the effect of graft placement and orientation on flow rates through a partially obstructed coronary artery. METHODS: A numerical, parametric study of blood flow in the human coronary artery was conducted using computational fluid dynamics simulation. A cylindrical approximation of the coronary artery with varying degrees of stenosis, with and without a bypass graft, was modelled to determine trends in volumetric flow rates. Steady and transient simulations were conducted for geometric variations of percentage of blockage, length and shape of stenosis, graft position relative to the coronary blockage and graft orientation. Accurate simulations were performed using a non-Newtonian fluid model and pressure-driven viscous flow. RESULTS: Simulations demonstrate, as expected, that total outlet flow rates of grafted arteries are consistently improved for upstream stenosis varying between 0 and 90% blockage. Grafts angled towards the artery provided increased total outflow. However, flow rates in the coronary artery upstream of the graft are substantially reduced in comparison with the non-grafted configuration due to competing flows. For some configurations (reduced blockage, graft placed close to long grafts), flow rates in the graft are below that of the flow rate through the stenosis. In general, a graft angled more towards the artery increased flow rates upstream of the graft. CONCLUSIONS: Placement and orientation of a graft may adversely affect upstream flow, with the degree of effect dependent on geometric factors of downstream position and graft angle. Keywords: Coronary stenosis • Computational fluid dynamics • Anastomosis
INTRODUCTION Recent fluid dynamic research on vascular anastomosis has focused on determining the relationship between haemodynamics and graft failures. Haemodynamic forces, in particular fluid shear stress at the vessel wall, have been shown to affect vessel remodelling and patency [1–4]. Studying blood flow patterns for grafted vessels may lead to an understanding of the relationship between patency and geometry. However, modelling blood flow at anastomotic junctions is difficult, due to the wide variety of vessel geometries and complex pulsatile flow. The aim of this study was to explore the effect of competing, mixing flows on flow rates for a partially obstructed coronary artery (CA) with different graft configurations. As this is a preliminary study, an approximation of the fluid geometry was created using a parametric model for ease of simulation. By varying geometry parameters, blood flow through various grafted arteries can be evaluated using computational fluid dynamics (CFD). The numerical studies presented in this work were completed with consideration of numerical accuracy and physical fidelity. † Presented at the 27th Annual Meeting of the European Association for CardioThoracic Surgery, Vienna, Austria, 5–9 October 2013.
MATERIALS AND METHODS Geometry Flow through a partially obstructed CA was simulated using CFD with a simplified geometric model (see Fig. 1). The stenosis was modelled as a single, centred and abrupt change in vessel diameter. An optional graft was modelled as a cylindrical volume attached at a downstream location at a given angle α with respect to the artery axis. A parametric geometric model allowed for the variation of: artery internal diameter, length of the stenosis, stenosis internal diameter, distance to graft attachment distal to stenosis exit, graft internal diameter and graft angle. As the graft vessel diameter may not necessarily match that of the artery, the crosssection of the graft cylinder was taken to be an ellipse with a minor-axis diameter matching the artery diameter. The area of this ellipse was set to be equivalent to that of a circle of diameter equal to the graft vessel internal diameter. Thus, the parametric geometry can accommodate grafts of diameter different from the artery diameter. Although this simplified geometry may appear to be a rough approximation of physical anatomy, a parametric definition is a useful tool for demonstrating relationships between flow parameters (e.g. flow rates) and geometry parameters.
© The Author 2014. Published by Oxford University Press on behalf of the European Association for Cardio-Thoracic Surgery. All rights reserved.
ORIGINAL ARTICLE
ORIGINAL ARTICLE – ADULT CARDIAC
Interactive CardioVascular and Thoracic Surgery (2014) 1–5 doi:10.1093/icvts/ivu034
2
J.V. Lassaline and B.C. Moon / Interactive CardioVascular and Thoracic Surgery
Figure 1: The parametric model of fluid geometry for a simplified coronary artery including an abrupt stenosis and a bypass graft. Shown here are the parameters affecting the length of the stenosis, the attachment point of the graft and the angle of the graft. Parameters not shown include upstream and downstream lengths, taper ratio and artery and graft diameters.
Numerical simulation Simulation of blood flow through the modelled geometry was completed for a range of geometry parameters using a numerical simulation of pressure-driven flow for both steady and unsteady boundary conditions. Blood was approximated as an incompressible, non-Newtonian fluid with a density of 1.060 g/ml. To model the shear-thinning behaviour of blood, the Carreau–Yasuda model was used to determine viscosity, with model parameters η0 = 0.1600 Pa s, η∞ = 0.0035 Pa s, λ = 8.2 s, a = 0.64 and n = 0.2128 [5, 6]. Although blood is a complex multiphase mixture, past research has shown the above model of fluid properties to be accurate for this flow regime [7]. All simulations used a second-order finite-volume spatial discretization with pressure boundary conditions set at the inlets and outlets of the geometry, and a no-slip condition at solid boundaries. Turbulence was simulated using the shear stress transport k-Ω eddy-viscosity model, with inlet values of turbulent kinetic energy k = 0.24 m2s−2 and specific dissipation Ω = 1.78 s−1. For the given problem, this turbulence model provides an economic balance between computational accuracy and simulation costs. For steady-state simulations, pressure at the inlets was set to 120 mmHg with a pressure of 80 mmHg at the outlet. Steady-state simulations were iterated to at least five orders of convergence using the semi-implicit method for pressure linked equations scheme. For unsteady simulations, inlet pressure was set to vary from 80 to 120 mmHg according to a pulsatile waveform (see Fig. 2) approximating aortic pressure over a heartbeat cycle while the outlet back pressure was held at 80 mmHg [8]. Unsteady simulations were time-marched with second-order accuracy using the merged pressure-implicit split-operator scheme. Simulations were postprocessed to determine the mass flow rate at the inlets and outlets of the geometry. For steady-state simulations, the mass flow rate was determined from the converged solution. Unsteady flow simulations were initialized from a steady-state solution, with mass flow rates calculated as the time average over a simulation of four consecutive heartbeat cycles (e.g. see Fig. 3).
Numerical studies Numerical simulations were completed using a geometry representing a 2-mm inner-diameter CA with various degrees of blockage. Blockage b was defined by the percentage of artery cross-sectional area occluded by the stenosis. Geometry parameters shown in
Figure 2: Inlet pressure boundary condition modelled as a pressure pulse under normal conditions and 70 bpm.
Figure 3: Flow rate history for a 2-mm internal diameter coronary artery (CA) with 50% blockage by area, with and without a 2-mm internal diameter graft. Total outflow (dashed line) is the sum of the CA inflow (short dash) and the graft inflow (dotted). In this example, the stenosis is 10-mm long and the graft is centred at 20-mm downstream from the end of the stenosis.
Table 1 were combined for all combinations to produce multiple configurations.
Study A A total of 720 simulations were completed over a range of percentage of blockages with and without a 2-mm-diameter graft, for both steady and unsteady flow. The geometry was further varied by the length of the stenosis LS (5, 10 and 20-mm), the distance from the stenosis to the graft attachment LSG (5, 10 and 20-mm) and the angle of the graft α (0°, 67° and 45°).
Study B A second study was completed to examine the influence of graft angle α on flow rates within the artery and graft. A total of 240
J.V. Lassaline and B.C. Moon / Interactive CardioVascular and Thoracic Surgery
3
Study A Percentage of blockage Stenosis length Distance to the graft attachment Graft angle Study B Percentage of blockage Stenosis length Distance to the graft attachment Graft angle Study C Percentage of blockage Stenosis length Distance to the graft attachment Graft angle
Parameter
Values
b LS LSG
10, 20, …, 90% 5, 10, 20-mm 5, 10, 20-mm
α
0°, 67° and 45°
b LS LSG
10, 20, …, 90% 5, 10, 20-mm 20-mm
α
90°, 80°, …, 30°
b LS LSG
10, 20, …, 90% 5, 10, 20-mm 5, 10, 15, …, 30, 40, 50, 100-mm 45°
α
simulations were completed over a range of percentage of blockage with a 2-mm-diameter graft located 20-mm downstream from stenoses of varied length.
ORIGINAL ARTICLE
Table 1: Geometric parameters used in combination for each numerical study
Figure 4: Steady-state flow rate versus percentage of blockage for a steady simulation of a 2-mm-diameter coronary artery (CA) with a 2-mm-diameter graft centred 20.0-mm downstream from a 10-mm long stenosis. Grafted artery outflow (long dash) is improved in comparison with the non-grafted artery (solid). Artery inlet flow rate (short dash) plus graft flow rate (dotted) sum to the total outflow.
Study C A third study was performed to examine the sensitivity of flow rates to the graft axial location. A total of 540 simulations were completed over a range of percentage of blockages with a 2-mm-diameter graft attached at a 45° angle.
RESULTS Study A Steady and unsteady simulations demonstrate similar flow rate sensitivities to percentage of blockage, both for the grafted and the non-grafted artery. For example, steady-state results shown in Fig. 4 demonstrate that the total outflow of the grafted artery is significantly improved in comparison with the non-grafted artery over the entire range of blockage. Note that the outflow of the grafted artery is equal to the sum of the inflow of the upstream artery and graft as required for the conservation of mass. However, the total outflow of the grafted artery is not simply double the value of the non-grafted artery due to viscous losses. These losses occur primarily due to the mixing of competing flows at the graft junction. It should be noted that the flow rate of the artery upstream of the graft junction is consistently reduced in comparison with that of the non-grafted artery. This effect is more pronounced as the percentage of blockage is increased. This is a natural consequence of the mixing of pressure-driven flows with a constrained outlet area. Similar results were found for all other simulated configurations. The imbalance of flow rates within the graft and within the upstream portion of the artery are a function of the length of the stenosis and the position of the graft. For grafts where the distance
Figure 5: Time-averaged flow rate versus percentage of blockage for an unsteady simulation (four heart beats) of a 2-mm-diameter coronary artery (CA) with a 2-mm-diameter graft centred 20.0-mm downstream from a 10-mm long stenosis.
between the junction and the stenosis is similar to the length of the stenosis itself, the flow rates in each vessel are closer in magnitude. For some simulations, the flow rate within the graft was less than that of the partially occluded artery; however, this only occurred at low percentage of blockage. Flow rates within the graft increase with increased artery blockage; however, at low degrees of blockage, graft flow rates are consistently less than that of the original non-grafted artery.
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J.V. Lassaline and B.C. Moon / Interactive CardioVascular and Thoracic Surgery
Figure 6: Steady-state flow rates versus graft angle α for a 2-mm-diameter artery with a 2-mm-diameter graft attached at a distance of 20-mm downstream from a 10-mm long stenosis. Curves are labelled with respect to the percentage of blockage.
Time-averaged unsteady results (see Fig. 5) show similar flow rate sensitivity to percentage of blockage. Unsteady results, driven by a physically realistic transient boundary condition, provide a more accurate simulation of actual flow rates, unlike steady-state results that do not match circulatory behaviour. Given that steady-state simulations require significantly less computational resources, this is a distinct advantage in exploring flow rate sensitivity to variations in geometry.
Study B As may be expected, simulations show that flow rates in grafted arteries are improved by reducing the angle between the artery and graft (see Fig. 6), regardless of the percentage of blockage. For a high degree of blockage (e.g. 90%), total outflow of the grafted artery remains high, decreasing with angle α. A similar result holds for the flow rate within the grafted artery; however, flow rates upstream of the graft junction remain relatively constant (significantly lower values compared with those of the non-grafted state) at high degrees of blockage.
Study C Simulations demonstrate that, while it is desirable to place a graft as close to the stenosis as possible, the net improvement in artery outflow is small. As can be seen in Fig. 7, grafted artery outflow remains nearly constant regardless of the distance from the stenosis to the graft junction at a higher percentage of blockage. Similar flow rate sensitivity to graft placement can be seen for the graft itself and the artery upstream of the junction. For graft placement close to the stenosis, there is an interaction between a jet flow exiting the stenosis and the incoming graft flow, producing
Figure 7: Steady-state flow rates versus graft position LSG downstream from a 10-mm long stenosis within a 2-mm-diameter artery with a 2-mm-diameter graft attached at a 45° angle. Curves are labelled with respect to the percentage of blockage.
strong vortices. For graft placement further away from the stenosis, this interaction is reduced as the jet expands downstream.
DISCUSSION In this study, we have shown the effect of graft position and orientation on flow rates within a CA using an idealized geometric model. Steady-state simulations provide sufficient detail regarding the nature of the flow within these vessels at a substantial reduced computational cost in comparison with unsteady simulations. Flow rates within a partially obstructed artery upstream of the graft junction are significantly impeded by the addition of a graft, although the total outflow of the grafted vessel is improved. The balance of flow rates between the graft and grafted vessel are primarily dependent on the level of obstruction. In general, graft placement and orientation that minimizes strong vortices and mixing produces the highest outflow rates.
Funding Computations were performed on the GPC supercomputer at the SciNet HPC Consortium. SciNet is funded by: the Canada Foundation for Innovation under the auspices of Compute Canada; the Government of Ontario; Ontario Research Fund Research Excellence and the University of Toronto. Conflict of interest: none declared.
REFERENCES [1] Ku DN, Giddens DP, Zarins CK, Glagov S. Pulsatile flow and atherosclerosis in the human carotid bifurcation. Positive correlation between plaque location and low oscillating shear stress. J Am Heart Assoc 1985;5:293–302.
J.V. Lassaline and B.C. Moon / Interactive CardioVascular and Thoracic Surgery
[8] Conlon M, Russell D, Mussivand T. Development of a mathematical model of the human circulatory system. Ann Biomed Eng 2006;34:1400–13.
APPENDIX: CONFERENCE DISCUSSION Dr K. Vural (Ankara, Turkey): Your presentation was very enlightening. Have you ever studied haemodynamics in sequential grafts? If so, what would be the clinical implications for daily use? Dr Moon: I don’t really have the answer at the moment. Thank you for the question, because that’s exactly what we are studying right now, and I’m hoping that we should be able to have some results within the next couple of months. But we are actually in the process of doing all the simulations at this stage.
ORIGINAL ARTICLE
[2] Bertolotti C, Deplano V. Three-dimensional numerical simulations of flow through a stenosed coronary bypass. J Biomech 2000;33:1011–22. [3] Sherwin SJ, Doorly DJ. Flow dynamics within model distal arterial bypass grafts. Adv Fluid Mech 2003;34:327–74. [4] Loth F, Fischer PF, Bassiouny HS. Blood flow in end-to-side anastomoses. Annu Rev Fluid Mech 2008;40:367–93. [5] Abraham F, Behr M, Heinkenschloss M. Shape optimization in steady blood flow: a numerical study of non-Newtonian effects. Comput Methods Biomech Biomed Engin 2005;8:127–37. [6] Boyd J, Buick JM, Green S. Analysis of the Casson and Carreau-Yasuda non-Newtonian blood models in steady and oscillatory flows using the lattice Boltzmann method. Phys Fluids 2007;19:093103. [7] Cho Y, Kensey K. Effects of the non-Newtonian viscosity of blood on flows in a diseased arterial vessel. Part 1: steady flows. Biorheology 1991;28: 241–62.
5
A computational fluid dynamics simulation study of coronary blood flow affected by graft placement† Jason V. Lassalinea and Byung C. Moonb,* a b
Ryerson University, Toronto, Canada Southlake Regional Health Centre, Newmarket, Canada
* Corresponding author. Division of Cardiac Surgery, Southlake Regional Health Centre, 625 Davis Drive, Newmarket, ONT, Canada L3Y 2R2. Tel: +1-905-8688386; fax: +1-905-8300931; e-mail: [email protected] (B.C. Moon). Received 1 October 2013; received in revised form 19 January 2014; accepted 29 January 2014
Abstract OBJECTIVES: To determine the effect of graft placement and orientation on flow rates through a partially obstructed coronary artery. METHODS: A numerical, parametric study of blood flow in the human coronary artery was conducted using computational fluid dynamics simulation. A cylindrical approximation of the coronary artery with varying degrees of stenosis, with and without a bypass graft, was modelled to determine trends in volumetric flow rates. Steady and transient simulations were conducted for geometric variations of percentage of blockage, length and shape of stenosis, graft position relative to the coronary blockage and graft orientation. Accurate simulations were performed using a non-Newtonian fluid model and pressure-driven viscous flow. RESULTS: Simulations demonstrate, as expected, that total outlet flow rates of grafted arteries are consistently improved for upstream stenosis varying between 0 and 90% blockage. Grafts angled towards the artery provided increased total outflow. However, flow rates in the coronary artery upstream of the graft are substantially reduced in comparison with the non-grafted configuration due to competing flows. For some configurations (reduced blockage, graft placed close to long grafts), flow rates in the graft are below that of the flow rate through the stenosis. In general, a graft angled more towards the artery increased flow rates upstream of the graft. CONCLUSIONS: Placement and orientation of a graft may adversely affect upstream flow, with the degree of effect dependent on geometric factors of downstream position and graft angle. Keywords: Coronary stenosis • Computational fluid dynamics • Anastomosis
INTRODUCTION Recent fluid dynamic research on vascular anastomosis has focused on determining the relationship between haemodynamics and graft failures. Haemodynamic forces, in particular fluid shear stress at the vessel wall, have been shown to affect vessel remodelling and patency [1–4]. Studying blood flow patterns for grafted vessels may lead to an understanding of the relationship between patency and geometry. However, modelling blood flow at anastomotic junctions is difficult, due to the wide variety of vessel geometries and complex pulsatile flow. The aim of this study was to explore the effect of competing, mixing flows on flow rates for a partially obstructed coronary artery (CA) with different graft configurations. As this is a preliminary study, an approximation of the fluid geometry was created using a parametric model for ease of simulation. By varying geometry parameters, blood flow through various grafted arteries can be evaluated using computational fluid dynamics (CFD). The numerical studies presented in this work were completed with consideration of numerical accuracy and physical fidelity. † Presented at the 27th Annual Meeting of the European Association for CardioThoracic Surgery, Vienna, Austria, 5–9 October 2013.
MATERIALS AND METHODS Geometry Flow through a partially obstructed CA was simulated using CFD with a simplified geometric model (see Fig. 1). The stenosis was modelled as a single, centred and abrupt change in vessel diameter. An optional graft was modelled as a cylindrical volume attached at a downstream location at a given angle α with respect to the artery axis. A parametric geometric model allowed for the variation of: artery internal diameter, length of the stenosis, stenosis internal diameter, distance to graft attachment distal to stenosis exit, graft internal diameter and graft angle. As the graft vessel diameter may not necessarily match that of the artery, the crosssection of the graft cylinder was taken to be an ellipse with a minor-axis diameter matching the artery diameter. The area of this ellipse was set to be equivalent to that of a circle of diameter equal to the graft vessel internal diameter. Thus, the parametric geometry can accommodate grafts of diameter different from the artery diameter. Although this simplified geometry may appear to be a rough approximation of physical anatomy, a parametric definition is a useful tool for demonstrating relationships between flow parameters (e.g. flow rates) and geometry parameters.
© The Author 2014. Published by Oxford University Press on behalf of the European Association for Cardio-Thoracic Surgery. All rights reserved.
ORIGINAL ARTICLE
ORIGINAL ARTICLE – ADULT CARDIAC
Interactive CardioVascular and Thoracic Surgery (2014) 1–5 doi:10.1093/icvts/ivu034
2
J.V. Lassaline and B.C. Moon / Interactive CardioVascular and Thoracic Surgery
Figure 1: The parametric model of fluid geometry for a simplified coronary artery including an abrupt stenosis and a bypass graft. Shown here are the parameters affecting the length of the stenosis, the attachment point of the graft and the angle of the graft. Parameters not shown include upstream and downstream lengths, taper ratio and artery and graft diameters.
Numerical simulation Simulation of blood flow through the modelled geometry was completed for a range of geometry parameters using a numerical simulation of pressure-driven flow for both steady and unsteady boundary conditions. Blood was approximated as an incompressible, non-Newtonian fluid with a density of 1.060 g/ml. To model the shear-thinning behaviour of blood, the Carreau–Yasuda model was used to determine viscosity, with model parameters η0 = 0.1600 Pa s, η∞ = 0.0035 Pa s, λ = 8.2 s, a = 0.64 and n = 0.2128 [5, 6]. Although blood is a complex multiphase mixture, past research has shown the above model of fluid properties to be accurate for this flow regime [7]. All simulations used a second-order finite-volume spatial discretization with pressure boundary conditions set at the inlets and outlets of the geometry, and a no-slip condition at solid boundaries. Turbulence was simulated using the shear stress transport k-Ω eddy-viscosity model, with inlet values of turbulent kinetic energy k = 0.24 m2s−2 and specific dissipation Ω = 1.78 s−1. For the given problem, this turbulence model provides an economic balance between computational accuracy and simulation costs. For steady-state simulations, pressure at the inlets was set to 120 mmHg with a pressure of 80 mmHg at the outlet. Steady-state simulations were iterated to at least five orders of convergence using the semi-implicit method for pressure linked equations scheme. For unsteady simulations, inlet pressure was set to vary from 80 to 120 mmHg according to a pulsatile waveform (see Fig. 2) approximating aortic pressure over a heartbeat cycle while the outlet back pressure was held at 80 mmHg [8]. Unsteady simulations were time-marched with second-order accuracy using the merged pressure-implicit split-operator scheme. Simulations were postprocessed to determine the mass flow rate at the inlets and outlets of the geometry. For steady-state simulations, the mass flow rate was determined from the converged solution. Unsteady flow simulations were initialized from a steady-state solution, with mass flow rates calculated as the time average over a simulation of four consecutive heartbeat cycles (e.g. see Fig. 3).
Numerical studies Numerical simulations were completed using a geometry representing a 2-mm inner-diameter CA with various degrees of blockage. Blockage b was defined by the percentage of artery cross-sectional area occluded by the stenosis. Geometry parameters shown in
Figure 2: Inlet pressure boundary condition modelled as a pressure pulse under normal conditions and 70 bpm.
Figure 3: Flow rate history for a 2-mm internal diameter coronary artery (CA) with 50% blockage by area, with and without a 2-mm internal diameter graft. Total outflow (dashed line) is the sum of the CA inflow (short dash) and the graft inflow (dotted). In this example, the stenosis is 10-mm long and the graft is centred at 20-mm downstream from the end of the stenosis.
Table 1 were combined for all combinations to produce multiple configurations.
Study A A total of 720 simulations were completed over a range of percentage of blockages with and without a 2-mm-diameter graft, for both steady and unsteady flow. The geometry was further varied by the length of the stenosis LS (5, 10 and 20-mm), the distance from the stenosis to the graft attachment LSG (5, 10 and 20-mm) and the angle of the graft α (0°, 67° and 45°).
Study B A second study was completed to examine the influence of graft angle α on flow rates within the artery and graft. A total of 240
J.V. Lassaline and B.C. Moon / Interactive CardioVascular and Thoracic Surgery
3
Study A Percentage of blockage Stenosis length Distance to the graft attachment Graft angle Study B Percentage of blockage Stenosis length Distance to the graft attachment Graft angle Study C Percentage of blockage Stenosis length Distance to the graft attachment Graft angle
Parameter
Values
b LS LSG
10, 20, …, 90% 5, 10, 20-mm 5, 10, 20-mm
α
0°, 67° and 45°
b LS LSG
10, 20, …, 90% 5, 10, 20-mm 20-mm
α
90°, 80°, …, 30°
b LS LSG
10, 20, …, 90% 5, 10, 20-mm 5, 10, 15, …, 30, 40, 50, 100-mm 45°
α
simulations were completed over a range of percentage of blockage with a 2-mm-diameter graft located 20-mm downstream from stenoses of varied length.
ORIGINAL ARTICLE
Table 1: Geometric parameters used in combination for each numerical study
Figure 4: Steady-state flow rate versus percentage of blockage for a steady simulation of a 2-mm-diameter coronary artery (CA) with a 2-mm-diameter graft centred 20.0-mm downstream from a 10-mm long stenosis. Grafted artery outflow (long dash) is improved in comparison with the non-grafted artery (solid). Artery inlet flow rate (short dash) plus graft flow rate (dotted) sum to the total outflow.
Study C A third study was performed to examine the sensitivity of flow rates to the graft axial location. A total of 540 simulations were completed over a range of percentage of blockages with a 2-mm-diameter graft attached at a 45° angle.
RESULTS Study A Steady and unsteady simulations demonstrate similar flow rate sensitivities to percentage of blockage, both for the grafted and the non-grafted artery. For example, steady-state results shown in Fig. 4 demonstrate that the total outflow of the grafted artery is significantly improved in comparison with the non-grafted artery over the entire range of blockage. Note that the outflow of the grafted artery is equal to the sum of the inflow of the upstream artery and graft as required for the conservation of mass. However, the total outflow of the grafted artery is not simply double the value of the non-grafted artery due to viscous losses. These losses occur primarily due to the mixing of competing flows at the graft junction. It should be noted that the flow rate of the artery upstream of the graft junction is consistently reduced in comparison with that of the non-grafted artery. This effect is more pronounced as the percentage of blockage is increased. This is a natural consequence of the mixing of pressure-driven flows with a constrained outlet area. Similar results were found for all other simulated configurations. The imbalance of flow rates within the graft and within the upstream portion of the artery are a function of the length of the stenosis and the position of the graft. For grafts where the distance
Figure 5: Time-averaged flow rate versus percentage of blockage for an unsteady simulation (four heart beats) of a 2-mm-diameter coronary artery (CA) with a 2-mm-diameter graft centred 20.0-mm downstream from a 10-mm long stenosis.
between the junction and the stenosis is similar to the length of the stenosis itself, the flow rates in each vessel are closer in magnitude. For some simulations, the flow rate within the graft was less than that of the partially occluded artery; however, this only occurred at low percentage of blockage. Flow rates within the graft increase with increased artery blockage; however, at low degrees of blockage, graft flow rates are consistently less than that of the original non-grafted artery.
4
J.V. Lassaline and B.C. Moon / Interactive CardioVascular and Thoracic Surgery
Figure 6: Steady-state flow rates versus graft angle α for a 2-mm-diameter artery with a 2-mm-diameter graft attached at a distance of 20-mm downstream from a 10-mm long stenosis. Curves are labelled with respect to the percentage of blockage.
Time-averaged unsteady results (see Fig. 5) show similar flow rate sensitivity to percentage of blockage. Unsteady results, driven by a physically realistic transient boundary condition, provide a more accurate simulation of actual flow rates, unlike steady-state results that do not match circulatory behaviour. Given that steady-state simulations require significantly less computational resources, this is a distinct advantage in exploring flow rate sensitivity to variations in geometry.
Study B As may be expected, simulations show that flow rates in grafted arteries are improved by reducing the angle between the artery and graft (see Fig. 6), regardless of the percentage of blockage. For a high degree of blockage (e.g. 90%), total outflow of the grafted artery remains high, decreasing with angle α. A similar result holds for the flow rate within the grafted artery; however, flow rates upstream of the graft junction remain relatively constant (significantly lower values compared with those of the non-grafted state) at high degrees of blockage.
Study C Simulations demonstrate that, while it is desirable to place a graft as close to the stenosis as possible, the net improvement in artery outflow is small. As can be seen in Fig. 7, grafted artery outflow remains nearly constant regardless of the distance from the stenosis to the graft junction at a higher percentage of blockage. Similar flow rate sensitivity to graft placement can be seen for the graft itself and the artery upstream of the junction. For graft placement close to the stenosis, there is an interaction between a jet flow exiting the stenosis and the incoming graft flow, producing
Figure 7: Steady-state flow rates versus graft position LSG downstream from a 10-mm long stenosis within a 2-mm-diameter artery with a 2-mm-diameter graft attached at a 45° angle. Curves are labelled with respect to the percentage of blockage.
strong vortices. For graft placement further away from the stenosis, this interaction is reduced as the jet expands downstream.
DISCUSSION In this study, we have shown the effect of graft position and orientation on flow rates within a CA using an idealized geometric model. Steady-state simulations provide sufficient detail regarding the nature of the flow within these vessels at a substantial reduced computational cost in comparison with unsteady simulations. Flow rates within a partially obstructed artery upstream of the graft junction are significantly impeded by the addition of a graft, although the total outflow of the grafted vessel is improved. The balance of flow rates between the graft and grafted vessel are primarily dependent on the level of obstruction. In general, graft placement and orientation that minimizes strong vortices and mixing produces the highest outflow rates.
Funding Computations were performed on the GPC supercomputer at the SciNet HPC Consortium. SciNet is funded by: the Canada Foundation for Innovation under the auspices of Compute Canada; the Government of Ontario; Ontario Research Fund Research Excellence and the University of Toronto. Conflict of interest: none declared.
REFERENCES [1] Ku DN, Giddens DP, Zarins CK, Glagov S. Pulsatile flow and atherosclerosis in the human carotid bifurcation. Positive correlation between plaque location and low oscillating shear stress. J Am Heart Assoc 1985;5:293–302.
J.V. Lassaline and B.C. Moon / Interactive CardioVascular and Thoracic Surgery
[8] Conlon M, Russell D, Mussivand T. Development of a mathematical model of the human circulatory system. Ann Biomed Eng 2006;34:1400–13.
APPENDIX: CONFERENCE DISCUSSION Dr K. Vural (Ankara, Turkey): Your presentation was very enlightening. Have you ever studied haemodynamics in sequential grafts? If so, what would be the clinical implications for daily use? Dr Moon: I don’t really have the answer at the moment. Thank you for the question, because that’s exactly what we are studying right now, and I’m hoping that we should be able to have some results within the next couple of months. But we are actually in the process of doing all the simulations at this stage.
ORIGINAL ARTICLE
[2] Bertolotti C, Deplano V. Three-dimensional numerical simulations of flow through a stenosed coronary bypass. J Biomech 2000;33:1011–22. [3] Sherwin SJ, Doorly DJ. Flow dynamics within model distal arterial bypass grafts. Adv Fluid Mech 2003;34:327–74. [4] Loth F, Fischer PF, Bassiouny HS. Blood flow in end-to-side anastomoses. Annu Rev Fluid Mech 2008;40:367–93. [5] Abraham F, Behr M, Heinkenschloss M. Shape optimization in steady blood flow: a numerical study of non-Newtonian effects. Comput Methods Biomech Biomed Engin 2005;8:127–37. [6] Boyd J, Buick JM, Green S. Analysis of the Casson and Carreau-Yasuda non-Newtonian blood models in steady and oscillatory flows using the lattice Boltzmann method. Phys Fluids 2007;19:093103. [7] Cho Y, Kensey K. Effects of the non-Newtonian viscosity of blood on flows in a diseased arterial vessel. Part 1: steady flows. Biorheology 1991;28: 241–62.
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