Connie L. Hall1 Department of Biomedical Engineering, The College of New Jersey, 20 0 0 Pennington Road, Ewing, NJ 0862 8 e-m ail: chall@ tcnj.edu
Computational Modeling of Thrombotic Microparticle Deposition in Nonparallel Flow Regimes
Melissa Calt Department of Biomedical Engineering, The College of New Jersey, 20 0 0 Pennington Road, Ewing, NJ 08628 e-m ail: melissaacalt@ gmail.com
Thrombotic microparticles (MPs) released from cells and platelets in response to various stimuli are present in elevated numbers in various disease states that increase the risk for thrombotic events. In order to understand how particles of this size may localize in non parallel flow regimes and increase thrombotic risk, a computational analysis of flow and MP deposition was performed for 3 deg of stenosis at moderate Reynolds number (20 < Re < 80) and for recirculation zones at low Reynolds (~1) number. The results indicate that MP deposition results primarily from impaction and not by diffusive flux. [DOT: 10.1115/1.4028134]
In tro d u c tio n
In the past several years, the clinical importance of cell-derived vesicles that arise from membrane microdomains (lipid rafts) fol lowing cell stimulation or apoptosis have been extensively investi gated for their role in hemostasis and thrombosis [1], These so-named “MPs” are characterized by markers from the source cell as well as other active protein components, specifically those that localize to lipid rafts and have been reported to be 0.1-1.0 pm in diameter. They have been detected in normal subjects [2] and in elevated numbers in various disease states including cardiovas cular disease, cancer, sepsis, and diabetes [3-6]. They are impli cated in inflammation, thrombosis and vascular dysfunction [7,8]. Specifically, tissue factor (TF) bearing MPs from monocytes, a subset of all possible MPs, have been identified in elevated num bers in patients with diabetes [3], acute coronary syndromes [5], cancer [6], and following cardiopulmonary bypass [9,10]. They can transfer the proteins they retain from the source cell, such as TF, to other cells via membrane fusion, activate other cells, or facilitate cell-cell interactions [9,11,12]. The procoagulant nature of the particles stems from the negatively charged phospholipids [13], such as phosphatidylserine, present in the outer leaflet of the vesicle membrane that support the formation of coagulation com plexes and from the presence of TF [14,15]. Studies utilizing a mouse model of thrombosis found that monocyte-derived MPs containing TF also contain PSGL-1 and are incorporated into thrombi by binding to P-selectin on activated platelets [16-18], Their incorporation provides a source of TF for continued fibrin formation beyond that initiated by vessel wall TF [19], The significance of all circulating MPs to thrombotic events in vivo has not been completely elucidated. The importance of the physical forces involved in the transport and localization of par ticles in flowing blood cannot be emphasized enough. In order for MPs to contribute to the growth of thrombus, they must leave the bulk blood flow and localize to the site of thrombus formation. The flow and hansport factors that contribute to this process have not been examined in detail, experimentally, for MPs (O.l-l.OjUm) as has been done previously for leukocytes and pla telets [20-24]. The physical forces that influence delivery of MPs to the injured vessel wall or the site of mural thrombus must be examined in order to determine the conditions that facilitate or promote the localization of MPs into thrombi. The effect of size Corresponding author. Manuscript received January 9, 2014; final manuscript received July 26, 2014; accepted manuscript posted August 1, 2014; published online September 4, 2014. Assoc. Editor: Dalin Tang.
Journal of Biomechanical Engineering
(0.5-10 pm) or gravitational effects on the transport and nonspe cific adhesion of fluorescent, polystyrene microspheres were studied in parallel streamlined flow [25,26]. Most studies demon strating the incorporation of MP into growing thrombi have been in animal models of thrombosis [17] where the flow conditions are not well defined. There is one published in vitro study examin ing the adhesion of platelet derived MPs under well characterized flow conditions. Keuren et al. investigated platelet derived MP ad hesion to collagen, von Willebrand factor and a monolayer of pla telets from parallel, streamlined flow of thrombocytopenic blood in a parallel plate chamber under wall shear rates of 100 and 1000 s-1 [27]. The study concluded that platelet MPs adhered to each surface equally at both shear rates; hence, the adhesion was not shear dependent at the conditions employed. To date, there have not been extensive computational studies on MP transport and deposition and no studies of MP transport and deposition in characterized nonparallel flow regimes (also referred to as “disturbed” flow) produced by stenosis or sudden expansions such as the reverse vertical step. Nanometer to milli meter diameter particle transport models have been applied exten sively to understanding particulate (drugs or ah contaminants) deposition in the pulmonary airways [28], Lagrangian tracking models are commonly employed and are better able to capture features of particle transport in nonparallel flow regimes in com parison to homogenous models [29]. Disturbed flow is of particu lar interest in thrombotic and early atherosclerotic events. Many investigators have used in vitro flow chambers that predictably create disturbed flow regimes and enable the examination of leu kocyte or platelet deposition [21,22,30,31] in addition to computa tional models of platelet transport and adhesion [32]. There are no computational models specific to thrombotic MP transport and deposition and limited experimental studies under controlled flow conditions. Therefore, a computational analysis of MP transport in disturbed flow regimes produced in these commonly employed in vitro devices was performed to predict MP localization under either venous or arterial disturbed flow conditions. The stenosis chamber was selected to reproduce arterial flow conditions and results in recirculation at Re = 40-80. The reverse, vertical step chamber was selected to produce recirculation at venous flow conditions (Re ~ 1). These devices also include sections in which well-developed flow with parallel streamlines is produced at a wide range of wall shear rates. The analysis utilizes Eulerian analysis of the flow field and a Lagrangian analysis of particle transport and deposition. The reverse, vertical step flow chamber [21] has been used to evaluate white blood cell or platelet deposition in recirculation
Copyright © 2014 by ASME
NOVEMBER 2014, Vol. 136 / 111002-1
extensively by Sakariassen, mimics disturbed flow in a stenosed artery and has primarily been used to study platelet deposition [30,31],
Methods Model Geometries and Flow Conditions. Three parallel plate perfusion chambers with 60%, 80%, and 89% occlusions (Fig. 1) representative of coronary artery stenoses [31,33] and one reverse step parallel plate device (Fig. 2) were created in 3D using cfd - geom (ESI Group Software v. 2011). The three-dimensional models were utilized rather than a 2D approximation as the mini mum width to height ratios of the devices fell just below 1:10. Figure 3 shows a magnified view of the grid at the upstream contraction zone of the 89% stenosis and for the vertical step.
Stenosis geometries (60%, 80%, and 89% ) and dimen sions. The y dim ension is scaled up by a factor of 3 to allow vis Fig. 1
ualization of the full length (scale x 1). The contraction appears sharp as a result of com pressing the figure. Refer to Fig. 3 for a magnified view of the contraction zone.
zones produced by the sudden step under venous shear conditions (400 s_1). The sudden expansion, although not a specific configu ration in the vasculature, reproduces recirculation under relatively low shear conditions. The stenosis chamber developed and used
Stenosis. The stenosis model dimensions are identical to the chambers developed by Sakariassen and used for many years [31]. The device is 5 mm wide with pre- and poststenotic heights of 0.7 mm, yielding a minimum width to height ratio of 1-7.14. The prestenotic portion is 9 mm long and the poststenotic region 18 mm long to allow for a fully developed flow profile well before the constriction and also prior to the flow outlet. At longitudinal position of 9 mm and spanning 0.5 mm, the cover declines in the shape of a cosine curve to one of three degrees of occlusion (60%, 80%, or 89%) at the height of 0.28, 0.14, and 0.08 mm, respec tively. The stenotic region proceeds for 18 mm before expanding to the full 0.7 mm height following the reverse of the declination. The structured grid was comprised of 211, 185 cells with a more refined grid in the transitions regions. The average inlet velocities, volumetric flow rates examined were assigned to achieve Reynolds numbers of 10, 20, 40, and 80. The values are given in Table 1. Reverse Vertical Step. The reverse vertical step design was constrained by the need to create recirculation downstream of the step while requiring relatively low volumetric flow rates. The height dimensions are similar to chambers used to examine adhe sion of white blood cells [21], The inlet height was 0.0254 cm and the outlet height was double that of the inlet at 0.0508 cm. The step was positioned 1.4 cm downstream of the inlet and 2.9 cm upstream of the outlet. The chamber width was 5 mm.
y
Fig. 2 The vertical step geometry and dimensions. The y dim ension is scaled by a factor of 3 to allow visualization of the full length (scale x 1).
Fig. 3 Magnified view of computational grids, (a) The com putational grid from the prestenotic zone through the contraction zone and into the stenosis and (b) the com putational grid in the vicinity of the vertical step.
111002-2 / Vol. 136, NOVEMBER 2014
Transactions of the ASME
T a b le 1
Reynolds No.
motion [34], The solution to Eq. (3) is decoupled from the flow solution
F lo w c o n d itio n s fo r s te n o s is m o d el
MP mass flow rate (ng s-1)
Average inlet velocity (cm s-1)
Volumetric flow rate (ml min-1)
30.4 60.7 121.6 243.1
2.36 4.72 9.43 18.86
5 10 20 40
10 20 40 80
« p “ ^ = CsPp| V - v | ^ + F b
The minimum width to height ratio in the device was 1-8.6 down stream of the step. The structured grid was comprised of 99,997 cells. The inlet velocities were calculated in order to achieve wall shear rates of 50, 100, and 400 s-1 in the fully developed flow re gime upstream of the step. The corresponding Reynolds number, volumetric flow rates and mass flow rates are provided in Table 2.
(3)
where mp is particle mass, v = ui + vj + wk is Cartesian velocity vector, Cs is slip drag coefficient (see Eq. (8)), pp is particle den sity, Ap is particle frontal area, V is velocity of the surrounding medium, and FB is Force due to Brownian motion. The particle position vector, r, is found by numerical integra tion of the velocity equation (3)
Flow Solution. The flow fields were computed with cfd-ace+ multiphysics software (ESI Group Software v. 2011) for laminar flow conditions. The three-dimensional governing equations for laminar, steady, incompressible flow were solved iteratively using the finite volume technique to obtain the velocity fields. The governing equations of fluid flow are Continuity Steady state momentum
V •V = 0
(1)
p(V ■VV) = \ p + p \/2\
(2)
where V is velocity of the fluid, p is pressure of the surrounding medium, p is dynamic viscosity of blood, and p is density of the fluid. The fluid was considered Newtonian with the density and dynamic viscosity of blood, specified as 1060 kg m-3 and 0.0035 kg m-1 s~ , respectively. A subset of flow conditions were examined using the Casson model for blood rheology. This included Re =10, 20, and 40 in the stenosis model and Re = 0.046, 0.1625, and 0.325 in the vertical step chamber. The no-slip condition was applied at all walls and the inlet velocity specified based on desired Reynolds number (stenosis model) or the wall shear rate (reverse step chamber) in the fully developed flow regime preceding the step. A fixed pressure (atmospheric) was set at the outlet. The flow was laminar in all cases as the Reynolds numbers ranged from less than 1-80. MP Tracking Model. Parcels of particles were tracked using the Lagrangian equation of motion and the solution decoupled from the solution of the flow field. The volume/volume ratio of suspending fluid volume to particle volume is 1 x 107. The suspension was dilute and therefore the particles did not influence the flow solution. MPs were considered neutrally buoyant (density = 1060 kg m-3) with the suspending fluid, the same vis cosity as the fluid (3.5 cp), 0.5 pm in diameter, the midpoint of the 0.1-1.0 pm range in the published literature, and released into the inlet flow. Equation (3) is used to calculate individual particles in
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Stenosis Model. The velocity profiles were fully developed from 0.34 mm to 0.75 mm of the chamber inlet for Re = 1 0 to Re = 80. Upon entering the stenotic zone, the flow redevelops within 0.18 mm-0.36 mm for Re = 10-80, respectively. At all Reynolds numbers (10-80), the contraction zone results in flow acceleration and an increasing velocity component directed downward into the lower wall. The extent of stenosis having the greatest impact, and the increase in Reynolds number influencing the magnitude of the velocity component directed to ward the wall to a lesser extent. The strain rate on the bottom wall increases proportionally to Reynolds number and with the decrease in height squared resulting in exceptionally large values in the stenotic zone. The change in strain rate along the centerline of the bottom wall from just upstream of the contraction (x = 0.8 cm) to 0.5 mm into the stenosis zone are shown in Fig. 4. The strain rate rises rapidly through the contraction leveling off at the new value for Reynolds numbers 10-40. For the highest Re = 80, the contraction results in a peak strain rate at the end of the contraction zone that is 14—18% greater than the value in the fully developed flow region of the stenosis. The contraction results in an increase in the magnitude of the y velocity compo nent directed toward the lower wall. There was no recirculation zone formed at a Re = 10 and 20 in the downstream expansion zone, irrespective of the degree of ste nosis (60%, 80%, or 89%). A recirculation zone formed poststeno sis at Reynolds number of 40 and 80, for all degrees of stenosis. The size of the zone increased with degree of stenosis and with increasing Reynolds numbers from 40 to 80 with flow separation
X Position (cm) A 4 0 0 1 /$
©yi Recirculation trace in the expansion zone past the verti cal step. R e c ir c u la tio n w a s e v id e n t f o r a ll R e y n o ld s n u m b e r s .
F ig . 6
T h e t r a c e in d ic a te s a m a s s le s s flu id p a r tic le t r a je c to r y in d ic a tin g a g r o w in g r e c ir c u la tio n z o n e w ith in c r e a s in g R e y n o ld s n u m b e r a n d w a ll s h e a r ra te . F r o m t o p to b o tto m , th e p a n e ls d is p la y t r a c e s fo r p r e s te p w a ll s h e a r r a te s o f 1 2 .5 , 5 0 , 4 0 0 , a n d 1 6 0 0 s ' 1.
Journal of Biomechanical Engineering
0 1 6 0 0 1 /$
The y velocity component at a position 0.25pm from the lower wall. T h e v e lo c it y c o m p o n e n t p e r p e n d ic u la r t o th e
F ig . 7
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Computational Modeling of Thrombotic Microparticle Deposition in Nonparallel Flow Regimes
Melissa Calt Department of Biomedical Engineering, The College of New Jersey, 20 0 0 Pennington Road, Ewing, NJ 08628 e-m ail: melissaacalt@ gmail.com
Thrombotic microparticles (MPs) released from cells and platelets in response to various stimuli are present in elevated numbers in various disease states that increase the risk for thrombotic events. In order to understand how particles of this size may localize in non parallel flow regimes and increase thrombotic risk, a computational analysis of flow and MP deposition was performed for 3 deg of stenosis at moderate Reynolds number (20 < Re < 80) and for recirculation zones at low Reynolds (~1) number. The results indicate that MP deposition results primarily from impaction and not by diffusive flux. [DOT: 10.1115/1.4028134]
In tro d u c tio n
In the past several years, the clinical importance of cell-derived vesicles that arise from membrane microdomains (lipid rafts) fol lowing cell stimulation or apoptosis have been extensively investi gated for their role in hemostasis and thrombosis [1], These so-named “MPs” are characterized by markers from the source cell as well as other active protein components, specifically those that localize to lipid rafts and have been reported to be 0.1-1.0 pm in diameter. They have been detected in normal subjects [2] and in elevated numbers in various disease states including cardiovas cular disease, cancer, sepsis, and diabetes [3-6]. They are impli cated in inflammation, thrombosis and vascular dysfunction [7,8]. Specifically, tissue factor (TF) bearing MPs from monocytes, a subset of all possible MPs, have been identified in elevated num bers in patients with diabetes [3], acute coronary syndromes [5], cancer [6], and following cardiopulmonary bypass [9,10]. They can transfer the proteins they retain from the source cell, such as TF, to other cells via membrane fusion, activate other cells, or facilitate cell-cell interactions [9,11,12]. The procoagulant nature of the particles stems from the negatively charged phospholipids [13], such as phosphatidylserine, present in the outer leaflet of the vesicle membrane that support the formation of coagulation com plexes and from the presence of TF [14,15]. Studies utilizing a mouse model of thrombosis found that monocyte-derived MPs containing TF also contain PSGL-1 and are incorporated into thrombi by binding to P-selectin on activated platelets [16-18], Their incorporation provides a source of TF for continued fibrin formation beyond that initiated by vessel wall TF [19], The significance of all circulating MPs to thrombotic events in vivo has not been completely elucidated. The importance of the physical forces involved in the transport and localization of par ticles in flowing blood cannot be emphasized enough. In order for MPs to contribute to the growth of thrombus, they must leave the bulk blood flow and localize to the site of thrombus formation. The flow and hansport factors that contribute to this process have not been examined in detail, experimentally, for MPs (O.l-l.OjUm) as has been done previously for leukocytes and pla telets [20-24]. The physical forces that influence delivery of MPs to the injured vessel wall or the site of mural thrombus must be examined in order to determine the conditions that facilitate or promote the localization of MPs into thrombi. The effect of size Corresponding author. Manuscript received January 9, 2014; final manuscript received July 26, 2014; accepted manuscript posted August 1, 2014; published online September 4, 2014. Assoc. Editor: Dalin Tang.
Journal of Biomechanical Engineering
(0.5-10 pm) or gravitational effects on the transport and nonspe cific adhesion of fluorescent, polystyrene microspheres were studied in parallel streamlined flow [25,26]. Most studies demon strating the incorporation of MP into growing thrombi have been in animal models of thrombosis [17] where the flow conditions are not well defined. There is one published in vitro study examin ing the adhesion of platelet derived MPs under well characterized flow conditions. Keuren et al. investigated platelet derived MP ad hesion to collagen, von Willebrand factor and a monolayer of pla telets from parallel, streamlined flow of thrombocytopenic blood in a parallel plate chamber under wall shear rates of 100 and 1000 s-1 [27]. The study concluded that platelet MPs adhered to each surface equally at both shear rates; hence, the adhesion was not shear dependent at the conditions employed. To date, there have not been extensive computational studies on MP transport and deposition and no studies of MP transport and deposition in characterized nonparallel flow regimes (also referred to as “disturbed” flow) produced by stenosis or sudden expansions such as the reverse vertical step. Nanometer to milli meter diameter particle transport models have been applied exten sively to understanding particulate (drugs or ah contaminants) deposition in the pulmonary airways [28], Lagrangian tracking models are commonly employed and are better able to capture features of particle transport in nonparallel flow regimes in com parison to homogenous models [29]. Disturbed flow is of particu lar interest in thrombotic and early atherosclerotic events. Many investigators have used in vitro flow chambers that predictably create disturbed flow regimes and enable the examination of leu kocyte or platelet deposition [21,22,30,31] in addition to computa tional models of platelet transport and adhesion [32]. There are no computational models specific to thrombotic MP transport and deposition and limited experimental studies under controlled flow conditions. Therefore, a computational analysis of MP transport in disturbed flow regimes produced in these commonly employed in vitro devices was performed to predict MP localization under either venous or arterial disturbed flow conditions. The stenosis chamber was selected to reproduce arterial flow conditions and results in recirculation at Re = 40-80. The reverse, vertical step chamber was selected to produce recirculation at venous flow conditions (Re ~ 1). These devices also include sections in which well-developed flow with parallel streamlines is produced at a wide range of wall shear rates. The analysis utilizes Eulerian analysis of the flow field and a Lagrangian analysis of particle transport and deposition. The reverse, vertical step flow chamber [21] has been used to evaluate white blood cell or platelet deposition in recirculation
Copyright © 2014 by ASME
NOVEMBER 2014, Vol. 136 / 111002-1
extensively by Sakariassen, mimics disturbed flow in a stenosed artery and has primarily been used to study platelet deposition [30,31],
Methods Model Geometries and Flow Conditions. Three parallel plate perfusion chambers with 60%, 80%, and 89% occlusions (Fig. 1) representative of coronary artery stenoses [31,33] and one reverse step parallel plate device (Fig. 2) were created in 3D using cfd - geom (ESI Group Software v. 2011). The three-dimensional models were utilized rather than a 2D approximation as the mini mum width to height ratios of the devices fell just below 1:10. Figure 3 shows a magnified view of the grid at the upstream contraction zone of the 89% stenosis and for the vertical step.
Stenosis geometries (60%, 80%, and 89% ) and dimen sions. The y dim ension is scaled up by a factor of 3 to allow vis Fig. 1
ualization of the full length (scale x 1). The contraction appears sharp as a result of com pressing the figure. Refer to Fig. 3 for a magnified view of the contraction zone.
zones produced by the sudden step under venous shear conditions (400 s_1). The sudden expansion, although not a specific configu ration in the vasculature, reproduces recirculation under relatively low shear conditions. The stenosis chamber developed and used
Stenosis. The stenosis model dimensions are identical to the chambers developed by Sakariassen and used for many years [31]. The device is 5 mm wide with pre- and poststenotic heights of 0.7 mm, yielding a minimum width to height ratio of 1-7.14. The prestenotic portion is 9 mm long and the poststenotic region 18 mm long to allow for a fully developed flow profile well before the constriction and also prior to the flow outlet. At longitudinal position of 9 mm and spanning 0.5 mm, the cover declines in the shape of a cosine curve to one of three degrees of occlusion (60%, 80%, or 89%) at the height of 0.28, 0.14, and 0.08 mm, respec tively. The stenotic region proceeds for 18 mm before expanding to the full 0.7 mm height following the reverse of the declination. The structured grid was comprised of 211, 185 cells with a more refined grid in the transitions regions. The average inlet velocities, volumetric flow rates examined were assigned to achieve Reynolds numbers of 10, 20, 40, and 80. The values are given in Table 1. Reverse Vertical Step. The reverse vertical step design was constrained by the need to create recirculation downstream of the step while requiring relatively low volumetric flow rates. The height dimensions are similar to chambers used to examine adhe sion of white blood cells [21], The inlet height was 0.0254 cm and the outlet height was double that of the inlet at 0.0508 cm. The step was positioned 1.4 cm downstream of the inlet and 2.9 cm upstream of the outlet. The chamber width was 5 mm.
y
Fig. 2 The vertical step geometry and dimensions. The y dim ension is scaled by a factor of 3 to allow visualization of the full length (scale x 1).
Fig. 3 Magnified view of computational grids, (a) The com putational grid from the prestenotic zone through the contraction zone and into the stenosis and (b) the com putational grid in the vicinity of the vertical step.
111002-2 / Vol. 136, NOVEMBER 2014
Transactions of the ASME
T a b le 1
Reynolds No.
motion [34], The solution to Eq. (3) is decoupled from the flow solution
F lo w c o n d itio n s fo r s te n o s is m o d el
MP mass flow rate (ng s-1)
Average inlet velocity (cm s-1)
Volumetric flow rate (ml min-1)
30.4 60.7 121.6 243.1
2.36 4.72 9.43 18.86
5 10 20 40
10 20 40 80
« p “ ^ = CsPp| V - v | ^ + F b
The minimum width to height ratio in the device was 1-8.6 down stream of the step. The structured grid was comprised of 99,997 cells. The inlet velocities were calculated in order to achieve wall shear rates of 50, 100, and 400 s-1 in the fully developed flow re gime upstream of the step. The corresponding Reynolds number, volumetric flow rates and mass flow rates are provided in Table 2.
(3)
where mp is particle mass, v = ui + vj + wk is Cartesian velocity vector, Cs is slip drag coefficient (see Eq. (8)), pp is particle den sity, Ap is particle frontal area, V is velocity of the surrounding medium, and FB is Force due to Brownian motion. The particle position vector, r, is found by numerical integra tion of the velocity equation (3)
Flow Solution. The flow fields were computed with cfd-ace+ multiphysics software (ESI Group Software v. 2011) for laminar flow conditions. The three-dimensional governing equations for laminar, steady, incompressible flow were solved iteratively using the finite volume technique to obtain the velocity fields. The governing equations of fluid flow are Continuity Steady state momentum
V •V = 0
(1)
p(V ■VV) = \ p + p \/2\
(2)
where V is velocity of the fluid, p is pressure of the surrounding medium, p is dynamic viscosity of blood, and p is density of the fluid. The fluid was considered Newtonian with the density and dynamic viscosity of blood, specified as 1060 kg m-3 and 0.0035 kg m-1 s~ , respectively. A subset of flow conditions were examined using the Casson model for blood rheology. This included Re =10, 20, and 40 in the stenosis model and Re = 0.046, 0.1625, and 0.325 in the vertical step chamber. The no-slip condition was applied at all walls and the inlet velocity specified based on desired Reynolds number (stenosis model) or the wall shear rate (reverse step chamber) in the fully developed flow regime preceding the step. A fixed pressure (atmospheric) was set at the outlet. The flow was laminar in all cases as the Reynolds numbers ranged from less than 1-80. MP Tracking Model. Parcels of particles were tracked using the Lagrangian equation of motion and the solution decoupled from the solution of the flow field. The volume/volume ratio of suspending fluid volume to particle volume is 1 x 107. The suspension was dilute and therefore the particles did not influence the flow solution. MPs were considered neutrally buoyant (density = 1060 kg m-3) with the suspending fluid, the same vis cosity as the fluid (3.5 cp), 0.5 pm in diameter, the midpoint of the 0.1-1.0 pm range in the published literature, and released into the inlet flow. Equation (3) is used to calculate individual particles in
I 1
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&
520G00
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A
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Wall shear rate (s ')
A
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ooooo
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T able 2
A
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40 and for all flow condi tions examined in the vertical step configuration.
Stenosis Model. The velocity profiles were fully developed from 0.34 mm to 0.75 mm of the chamber inlet for Re = 1 0 to Re = 80. Upon entering the stenotic zone, the flow redevelops within 0.18 mm-0.36 mm for Re = 10-80, respectively. At all Reynolds numbers (10-80), the contraction zone results in flow acceleration and an increasing velocity component directed downward into the lower wall. The extent of stenosis having the greatest impact, and the increase in Reynolds number influencing the magnitude of the velocity component directed to ward the wall to a lesser extent. The strain rate on the bottom wall increases proportionally to Reynolds number and with the decrease in height squared resulting in exceptionally large values in the stenotic zone. The change in strain rate along the centerline of the bottom wall from just upstream of the contraction (x = 0.8 cm) to 0.5 mm into the stenosis zone are shown in Fig. 4. The strain rate rises rapidly through the contraction leveling off at the new value for Reynolds numbers 10-40. For the highest Re = 80, the contraction results in a peak strain rate at the end of the contraction zone that is 14—18% greater than the value in the fully developed flow region of the stenosis. The contraction results in an increase in the magnitude of the y velocity compo nent directed toward the lower wall. There was no recirculation zone formed at a Re = 10 and 20 in the downstream expansion zone, irrespective of the degree of ste nosis (60%, 80%, or 89%). A recirculation zone formed poststeno sis at Reynolds number of 40 and 80, for all degrees of stenosis. The size of the zone increased with degree of stenosis and with increasing Reynolds numbers from 40 to 80 with flow separation
X Position (cm) A 4 0 0 1 /$
©yi Recirculation trace in the expansion zone past the verti cal step. R e c ir c u la tio n w a s e v id e n t f o r a ll R e y n o ld s n u m b e r s .
F ig . 6
T h e t r a c e in d ic a te s a m a s s le s s flu id p a r tic le t r a je c to r y in d ic a tin g a g r o w in g r e c ir c u la tio n z o n e w ith in c r e a s in g R e y n o ld s n u m b e r a n d w a ll s h e a r ra te . F r o m t o p to b o tto m , th e p a n e ls d is p la y t r a c e s fo r p r e s te p w a ll s h e a r r a te s o f 1 2 .5 , 5 0 , 4 0 0 , a n d 1 6 0 0 s ' 1.
Journal of Biomechanical Engineering
0 1 6 0 0 1 /$
The y velocity component at a position 0.25pm from the lower wall. T h e v e lo c it y c o m p o n e n t p e r p e n d ic u la r t o th e
F ig . 7
b o tto m
w a ll
(1 .4 c m
