Near-Wall Deposition Probability of Blood Elements as ... - Springer Link

C 2006) pp. 958–970 Annals of Biomedical Engineering, Vol. 34, No. 6, June 2006 ( DOI: 10.1007/s10439-006-9096-6

Near-Wall Deposition Probability of Blood Elements as A New Hemodynamic Wall Parameter MIN-CHEOL KIM,1 JIN HYUN NAM,2 and CHONG-SUN LEE3 1 School of Mechanical and Aerospace Engineering, Seoul National University, Seoul, Korea; 2 School of Mechanical & Automotive Engineering, Kookmin University, Seoul, Korea; and 3 Department of Mechanical and Control Engineering, Handong Global University, Pohang, Korea

(Received 7 September 2005; accepted 22 February 2006; published online: 13 May 2006)

dp h

Abstract—The present study was performed to investigate deposition probability of blood particles on the vessel walls. To track dynamics of movement and adhesion of blood particles in the near wall region, two models such as the particle rolling model (PR-model) and the near wall force model (NWF-model) were employed in the present study. Simulations of the present models for the pre-activated platelets in the stagnated point flow chamber and for the pre-activated monocytes in the stenotic perfusion tube resulted in significant correlations with the experimental data. The proposed near wall deposition probability (NWDP) index exhibited good fits with the experimental data of the stagnation point flow chamber for the platelet. As for the monocyte, the NWDP index exhibited the best fit with the experimental data of the stenotic tube. The new hemodynamic index, NWDP, is different from the wall shear stress (WSS)-based hemodynamic parameters, such as MWSS (Mean Wall Shear Stress), AWSS (Amplitude of Wall Shear Stress), and OSI (Oscillatory Shear Index) in that it locates regions of both the high and low WSS. The proposed NWDP index needs to be tested and compared in real geometries for its effectiveness in locating regions of lesion-prone sites.

hp r rp R t ta tm T U y z Red α ν η ρ ρp ω

Keywords—NWDP (Near-wall deposition probability), Particle deposition model, Platelet, Monocyte, Particle trajectory, Computer simulation, Hemodynamics.

NOMENCLATURE

particle diameter (m) distance from the horizontal plate in a stagnation flow chamber (m) distance from the center of the particle to the wall surface (m) spatial coordinate in the radial direction (m) particle radius (m) radius (m) time (s) time scale for the adhesion of a particle (s) time scale for the movement of a particle (s) pulse period (s) average velocity (ms−1 ) relative coordinate in the wall normal direction (m) spatial coordinate in the axial direction (m) diameter based Reynolds √number (Ud/v) Wormersley number (R ω/ν) correlation coefficient kinematic viscosity (η/ρ) (m2 s−1 ) dynamic viscosity (kg m−1 s−1 ) fluid density (kg m−3 ) particle density (kg m−3 ) angular velocity (2π /T ) (s−1 ) Subscripts

a A Ap b1 b2 c1 c2 d

ratio of correction factors c1 and c2 in NWDP index area (m2 ) contact area of the particle on the wall (m2 ) constant in Eq. (8) constant in Eq. (8) correction factor of tm to fit with actual movement of the particle correction factor of ta to fit with actual adhesion of the particle diameter (m)

f p

INTRODUCTION Atherosclerosis is a progressive vascular disease characterized by the accumulation of lipids, smooth muscle cells, collagen, and platelets onto the arterial wall. All of these components form plaques which later develop into complex ulcerations at the luminal surface and gradually occlude blood vessels by the formation of a thrombus or blood clot, resulting in myocardial infarction or stroke.24 Blood particles are temporarily arrested at the vessel wall due to local fluid dynamics developing near bifurcated

Address correspondence to Chong-Sun Lee, Professor of Department of Mechanical and Control Engineering, CE2-104, Handong Global University, Heunghae-eup, Buk-gu, Pohang, Kyungbuk, Seoul 791-708, Korea. Electronic mail: [email protected]

958 0090-6964/06/0600-0958/0

C 2006

Biomedical Engineering Society

fluid particle

Near-Wall Deposition Probability of Blood Elements

or stenosed regions of arteries. Actual deposition of blood particles is believed to be completed by biochemical reaction, and cross-linking of the particles and vessel wall through binding agents. Activated platelets release their granules, containing cytokines and growth factors, and contribute to the migration and proliferation of smooth muscle cells and monocytes.4 In addition, thrombin-activated platelets readily bind to endothelial cells by a GPIIbIIIadependent bridging mechanism, which involves plateletbound adhesive proteins, such as fibrinogen, fibronectin, and vWF, and the endothelial cell receptors ICAM-1, α v β 3 integrin, and GPIbα.4 On the other hand, monocytes react with adhesion molecules, such as β2 integrin, VLA-4, and PCAM-1 at the damaged vessel wall.20 An early atherosclerotic lesion is characterized by the formation of foam cells. These cells are ultimate results of monocytes adhesion, migration of monocytes into intima, and a series of transformations of monocytes into intimal macrophage with a scavenger receptor and modified lipoprotein particle.20 Biochemistry experiments on the adhesion of blood element have shown that atherosclerotic lesion growth was promoted by endothelial P-selectin,8 cross-talk between platelet and leukocyte under blockade of GPIIbIIIa or P-selectin,11,19 and platelet adhesion to fibrinogen.5 Several studies indicated that hemodynamic effect plays an important role on the adhesion of blood elements on the arterial wall or atherogenesis. High shear stress was found to enhance the degree of activation of blood elements and the reactivity of arterial walls, as indicated by the in vitro experiment carried out in a parallel plate flow chamber.2 The adhesion and aggregation of platelets onto injured endothelium cells was promoted at elevated fluid shear stresses due to increased activation of platelets and surface reactivity. However, the concentration of blood elements near the arterial wall also governs the adhesion of blood elements, especially when shear stress is not large or the degree of activation is the same as surface reactivity. Examination on in vitro behaviors of pre-activated platelets17,18 and monocytes3,26 in recirculation flow fields has shown that the adhesion of blood elements is directly proportional to the particle residence time (PRT), which is equivalent to the concentration of the blood elements.28 The rate of platelet adhesion on collagen-coated glass in an annular expansion flow field was observed to be highest in the recirculation zone with higher PRTs and lowest at the reattachment point.17,18 Similarly, the rate of monocyte adhesion was promoted in low wall shear stress region where PRT was large.26 Numerical simulations have been carried out to visualize the characteristics of the transient motion of non-interacting blood elements,7,10,16,28 where particles were recognized as deposited when they entered the numerical wall boundary layer (deposition model). In case of the deposition model, because the particle was located within the numerical wall

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boundary layer, which is equal to the radius of the particle, the deformation and permanent deposition of the spherical particle must occur unless the particle pierced through the computational domain, and thus the calculation for the particle trajectory inevitably stopped.28 However, instead of direct depositing on the wall, blood elements move rather freely within the near wall region. Longest et al.22 considered detailed hydrodynamic particle-wall interaction to track the motion of particles near the wall, proposed the near wall residence time (NWRT) as a non-dimensional parameter for calculating the probabilities of particle deposition, and validated their proposed NWRT parameter for the both the platelet and the monocyte in two different CFD systems based on experiments of blood elements.1,15 In the present study, to track the dynamics of movement and adhesion of blood particles in the near wall region, two models, the particle rolling model (PR-model) and the near-wall force model (NWF-model), were employed. The PR-model was different from previous deposition models in that the numerical calculation continued to allow the particle to roll over the near-wall region because of the characteristics of pulsatile blood flow. That is, periodic responsive force due to pulsatile flow would act on the particle, inducing rolling of the particle on the front and back of the wall. Additionally, the rate of deposition of particles onto the activated or dysfunctional endothelial cells is assumed to be dependent on the time required for the particles to move to the wall and the time required to adhere. By integrating the contribution of all the particle trajectories with time, we propose the near-wall deposition probability (NWDP) as an index of particle deposition probability on the wall. By the scaling analysis of the two time scales required for movement and adhesion of particles, the weighting factor of NWDP that account for the particle-wall separation distance can be derived theoretically. The scaling analysis revealed that the deposition probability is dependent on the activating force on the particle exerted by the fluid velocity normal to the wall. Numerical simulations were conducted to obtain the proposed NWDP index for two different flow systems which were also done by Longest et al.,22 and our results were compared to previous experimental data and numerical results.1,15,22 The proposed NWDP index must be validated before application to real arteries. METHOD Flow Systems Continuity and Navier-Stokes (NS) equations for laminar incompressible flow are expressed as: ∇ · u f = 0 ∂ u f 1 + ( u f · ∇) u f = (−∇ P + η∇ 2 u f ) ∂t p

(1) (2)

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coating (Fig. 2(b)). U937 cells express sLex , β 1 - and β 2 integrins; thus, they can adhere to endothelial cells in a manner similar to monocytes.14 The density and viscosity of culture media were 1.0 g cm−3 and 0.008 dynes cm−2 respectively. Figure 2(a) shows a pulsatile flow waveform in both the experiment and the simulation during one pulse period (T = 10 s). In Fig. 2(a), t/T = 0.6 corresponds to the minimum volumetric flow rate of the periodic cycle. Pulsatile flow waveforms had Remean (mean Reynolds number) of 117 and Wormersley number α of 3. Based on the geometry of tube model shown in Fig. 2(b), the diameters at the throat and sudden expansion regions were 3.96 mm and 6.35 mm, respectively. Figure 2(c) depicts the velocity vectors and the contours of velocity magnitude obtained by the CFD analysis at the moment of minimum flow. Recirculation zone is noted in the expansion region right after the throat. Particle Hemodynamics

FIGURE 1. Configuration of stagnation point flow chamber1 ; (a) chamber geometry, and (b) velocity vectors and contours of velocity magnitude.

where u f is velocity vector of fluid, and P is pressure. A commercial CFD tool, STAR-CD (version 3.15a, CD-adpaco), was employed to solve the flow field. Two different flow systems were selected to simulate and test our proposed NWDP index because Longest et al.22 used these flow systems to validate the near wall residence time (NWRT) parameter. Firstly, as shown in Fig. 1(a), a 3D axisymmetric stagnation point flow system, which had been experimented by Affeld et al.,1 was selected to measure platelet (dp = 2.3 µm, ρ p = 1.068 g cm−3 ) deposition along the wall of the chamber. Platelet-rich plasma (PRP: ρ = 1.03 g cm−3 , η = 0.014 dynes cm−2 ) was supplied with a fully developed inlet velocity profile at a uniform flow rate of 20 ml/h. The diameter of the circular cylinder was 0.67 mm, and the distance to the horizontal plate was 0.4 mm. The velocity magnitude shown in Fig. 1(b) decreased along the radial direction because the cross-sectional area of the outer cylinder increased with the radial distance. Secondly, a 3D axisymmetric flow tube with a stenosis and a sudden expansion was selected to evaluate monocytes deposition. Hinds et al.15 studied monocytic cell deposition using pre-activated U937 cells (dp = 2 µm, ρ p = 1.076 g cm−3 ) in the same stenotic tube with E-selectin

Blood elements were modeled as spherical particles in a Newtonian fluid. Deformation of blood particles and interaction between the particles were not considered in this study. Transient motion of every single particle was solved by the Lagrangian approach including spatial and temporal interpolations of the flow velocity at the particle’s position using the pre-computed steady or transient flow field from NS Eqs. (1) and (2). Because of the hemodynamic condition where blood elements are much smaller than the vessel and are in sufficiently low concentration (platelet; 3 × 105 mm3 , monocyte; 400 mm3 ), solutions of the particle trajectory and the flow field can be uncoupled. Then, each particle’s trajectory was determined by integrating simultaneous ordinary differential equations for particle velocity ( u p ) and location vector ( x p ) as given by mp

d u p = FD + FP + FL dt d x p = u p dt

(3) (4)

where mp denotes the mass of the particle, and the righthand side terms of Eq. (3) are the forces due to Stokes drag ( FD ), fluid pressure gradient ( FP ), and the near-wall lift ( FL ). The integration was carried out by using the fourth order Rosenbrock method, which is more efficient for solving steep ordinary differential equations than the Runge-Kutta method. The Stokes drag force is given by FD = C D

πρd 2p 8

| u p − u f |( u f − u p )Ciw

(5)

The drag coefficient CD was chosen from Stokes flow condition (Re p 1); CD =

24 Re p

(6)


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FIGURE 2. Configuration of stenotic tube15 ; (a) pulsatile flow waveform, (b) tube geometry, and (c) velocity vectors and contours of velocity magnitude at the moment of minimum flow rate (t/T = 0.6).

where Re p (= | u p − u f |d p /v) is the particle Reynolds number. In Eq. (5), Ciw represents a wall correction factor for the drag force, which includes the effect of the presence of the wall. For the directions parallel to the wall, the wall correction factor becomes:12 1 dp 3 9 dp + C1w = C3w = 1 − 16 2h p 8 2h p 4 45 d p 1 d p 5 −1 − − . (7) 256 2h p 16 2h p For the direction normal to the wall, the wall correction factor13 was given by 1 dp 3 9 dp + C2w = 1 − 8 2h p 2 2h p −1 2h p 45 −b1 − b2 (8) × 1 − exp 256 dp where hp denotes the distance between the center of the particle and the wall. Subscripts 1 and 3 represent directions parallel to the wall and subscript 2 represents normal direction. These wall correction factors were based on Wakiya’s

approximate solution and Brenner’s exact solution.6 The constants b1 and b2 in Eq. (8) were set to 2.686 and 0.9999, respectively, in the present simulation.13 The wall correction factors can be used under assumption of small Rep . Force due to fluid pressure gradient is given as:22 π d 3p u1f − u0f D u f ≈ρ Fp = m f Dt 6 dt

(9)

where mf denotes the fluid mass of spherical volume, and D/Dt means the material derivative. u0f denotes the fluid velocity vector at the previous time and position x0p while u1f denotes the velocity vector at the present time and predicted position ( x 1p = x0p + dt u0p ). Pressure gradient term is approximated by the material derivative of the acceleration of a fluid element, neglecting the viscous terms in the NS equation. McLaughlin25 considered the effect of the presence of the wall on the lift force acting on a spherical particle and tabulated the effect for particles in shear field (not too close to the wall). The lift force acting on a particle near the wall (hp ∼ rp ) was derived by Cherukak and McLaughlin9 and

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FIGURE 3. Comparison of particle trajectory between experiment and computational simulations; (a) experiment of a hardened red blood cell,18 b) computational simulation of a spherical red blood cell,21 and (c) computational simulation of a spherical red blood cell (Current research).

was expressed for a freely rotating sphere as: FL = ρr 2p Us2 I (κ, )

Validation of Particle Trajectory Code (10)

where rp is the radius of the particle, and I (κ, ) is the tabulated function of κ(= r p / h p ) and non-dimensional shear,

(= γ˙ r p /Us ) where γ˙ is shear rate, and Us is wall tangential slip velocity, which is expressed by ( u p − u f ) · tˆ, where tˆ is the wall-tangential unit vector in the direction of particle motion. Negative non-dimensional shear results in a positive lift force (motion away from the wall), while positive non-dimensional shear results in a negative lift force (motion toward the wall).9,21 Trajectories of blood particles were calculated with two numerical models, the particle rolling model (PR-model) and the near-wall forces model (NWF-model). The PRmodel included the Stokes drag and pressure gradient terms in the equation of particle motion and assumed contact with the vessel wall. However, the NWF-model included the lift force additionally and assumed no contact in the near-wall region.

The result of our particle tracking algorithm, containing Stokes drag and pressure gradient force terms, i.e. PRmodel, was compared to the experimental result of annular expansion obtained by Karino and Goldsmith.17 Under the sinusoidal flow waveform, hardened red blood cells were seeded in annular expansion. The current numerical simulation was in good agreement with the experimental result of Karino and Goldsmith, as shown in Fig. 3(a). Our result was also close to the numerical result of Longest21 as shown in Fig. 3(b). Near-Wall Deposition Probability (NWDP) The deposition probability of a blood particle in the near wall region is assumed to be related to the distance from the center of the particle to the wall surface (hp ) and the inward vertical fluid-phase velocity (Vh ). Vh is defined as follows: | u f · nw | if u f · nw < 0 (11) Vh = 0 else


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Adhesion time is inversely proportional to the pressure between the particle and the wall (Pp−w ) exerted by the vertical fluid-phase velocity. Because the force acting on the particle is proportional to the vertical fluid-phase velocity, Pp−w is expressed as Pp−w ∼

FIGURE 4. Schematic view of movement and activation of a particle; tm is the time scale for particle movement and ta is the time scale for adhesion of the particle.

where nw is an inward pointing normal vector of the wall surface. With respect to the particle-wall separation distance, the deposition probability will increase as the particle is placed closer to the wall. In other words, time of flight will get shorter as the particle approaches the wall. With respect to the inward vertical fluid-phase velocity, higher value of Vh will increase the pressure on the particle, and thereby, the particle will experience more activation and strongly adhere to the lesion prone site. Figure 4 depicts the time scales for the movement and adhesion of a particle; the left panel shows the time scale for the movement (tm ) of a particle located at hp , and the right panel shows the time scale for the adhesion (ta ). The deposition time scale can be expressed by the summation of the two time scales (td = tm + ta ). The deposition probability of a blood particle in the near wall region is thought to be inversely proportional to the deposition time scale (td ). Therefore, probability of the particle deposition can be expressed by the following equation: Nd ∼

1 1 = . td tm + ta

(12)

In case of the platelet, which is small, the vertical fluid-phase velocity profile can be modeled as linear, i.e. v{y} = (y/ h)Vh . Then, tm required for a particle located at hp to move to the wall is obtained by integrating the linear velocity profile from rp to hp as hp hp hp hp hp dy = c1 tm = c1 dy = c1 ln Vh rp r p v(y) r p y Vh (13a) where c1 represents a correction factor of tm to fit with actual movement of the particle. However, in case of the monocyte, whose diameter is larger than that of the platelet, the assumption of a linear velocity profile needs to be replaced by that of parabolic profile, i.e. v{y} = (y 2 / h 2 )Vh . Then, tm required for a particle located at hp to move to the wall is obtained by integrating the parabolic velocity profile as hp hp h 2p dy = c1 dy tm = c1 2 r p v{y} r p y Vh h 2p 1 1 . (13b) = c1 ln − Vh rp hp

∼

6π ηr p ν ∗f {r p } ην ∗f {r p } FW N ∼ ∼ Ap πr 2p rp ηVh r p / h p ηVh ∼ rp hp

(14)

where FWN represents the wall-normal drag force, Ap is the contact area of the particle on the wall, and v ∗f {r p } means the wall-normal fluid velocity component as the particle contacts the wall (i.e. y = rp ). Then, the adhesion time scale is ta ∼

η Pp−w

ta = c2

∼

hp Vh

hp . Vh

(15a)

(15b)

where c1 represents the correction factor of ta to fit with actual adhesion of the particle. The number of deposition probability for platelets can be written by superposing Eqs. (13a) and (15b) and the result is: N d ∼ K td = K

h c1 Vhp

1

hp h ln r p + c2 Vhp

(16a)

where K is defined as the constant. Eq. (16a) can be modified to Eq. (16b) by introducing the proportionality constant K (= K /c1 ) and a: Nd = K

Vh 1

hp h p ln +a rp

(16b)

where parameter a is defined as the ratio of the correction factors c2 and c1 . In case of monocytes, the number of deposition probability is evaluated in a similar manner to that of the platelets by superposing Eqs. (13b) and (15b) and the result is: Nd ∼ K

1 1 = K h2

h td c1 Vhp r1p − h1p + c2 Vhp = K

Vh 1

h p hp + a − 1 rp

(17)

Inward normal fluid velocity component, Vh , in Eqs. (16b) and (17) is considered to be zero when the particle moves away from the wall. In order to obtain Nd in the throat and the recirculation region of the stenotic tube, Eq. (17) was modified to include a critical rolling velocity (Vh,roll ) of 5 µm s−1 because the inward normal velocity lay within 30◦ from the wall. The algorithm for obtaining Nd for monocytes in the stenotic tube is graphically described in Fig. 5.

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KIM et al. TABLE 1. Values of 2 from linear regression between NWDP distribution calculated in the rolling model (PRmodel) and normalized platelet density distribution measured by Affeld et al.1 in the stagnation point flow chamber with the variation of the parameter ‘a’. Parameter ‘a’

2

Parameter ‘a’

2

0.01 0.05 0.1 0.2 0.3

0.420 0.603 0.661 0.709 0.728

0.4 0.5 0.6 0.7 1.0

0.733 0.730 0.727 0.719 0.693

RESULTS

FIGURE 5. Algorithm for evaluating number of deposition probability (Nd ) for monocyte in the stenotic tube; critical rolling velocity, Vh,roll , was set to 5 µm/s; (a) flowchart for determining Nd , and (b) three cases for directions of fluid velocity.

The deposition probability (DPi ) on the wall at the i-th element must consider all seeded particles visiting the element during the whole flow period as described in Eq. (18a) and (18b): all P ∞ j Hi (t)Nd dt (18a) D Pi = j

0

j

where Hi (t) is a conditional function which equals to one if a particle j is within a cell i; otherwise, it equals to 0. Therefore, the proposed non-dimensional near-wall deposition probability parameter, NWDP, is expressed by the following equation: Reactive Surface

Ak DPi k (18b) NWDPi =

Surface Ai Reactive DPk k

The reactive local surface area, Ai , is included to show that the mesh size is independent of particle seeding. The cutoff height for the near-wall region was set to 6rp . To validate our theory with the experimental results, parameter ‘a’ in Eq. (16b) was set to 0.4 for platelets, and parameter ‘a’ in Eq. (17) was set to more than one for monocytes.

Ten thousand pre-activated spherical platelets of 4.6 µm in diameter were seeded near the centerline of the cylindrical stagnation point flow chamber, as shown in Fig. 1(a), to obtain convergent NWDP values. The trend of the NWDP distribution calculated in the PR-model of platelet was similar to that of the normalized platelet density distribution measured by Affeld et al. in the stagnation point flow chamber.1 Table 1 indicates that the linear regression factor between the two distributions is much influenced by the parameter ‘a’. As shown in Fig. 6, for ‘a’ = 0.01, a poor correlation was obtained between the two distributions (2 = 0.420), showing a strong deviation beyond the region of r/R = 1. A far better correlation was obtained for ‘a’ = 0.1 (2 = 0.661) because of the decreased NWDP for r/R > 1. NWDP distribution for ‘a’ = 0.4 was in good agreement with the measured platelet density distribution for the entire range of the radial distance, resulting in the best correlation (2 = 0.733) among the cases considered in the present study. The good agreement of NWDP with the measured values in the region of r/R > 1 demonstrated that, for larger values of the parameter ‘a’, the velocity component in the wall-normal direction increases in the PR-model and that this increase hinders the particle deposition in the same region. NWDP values for ‘a’ = 1.0 also showed good correlation. As the parameter ‘a’ was raised from 0.01 to 1.0, the peak value of NWDP was slightly shifted to the left and NWDP values around r/R = 0.25 was notably increased. Figure 7 compares NWRT and NWDP distributions calculated in the NWF-model of the platelet to the normalized platelet density distribution measured by Affeld et al.1 In case of Fig. 7(a), NWDP index was modified by replacing the NWDP probability term, (v 0f / h p )(1/(ln(h p /r p ) + a)), with the Longest et al.’s NWRT term, (rp /hp )2 , for comparison. The correlation (2 = 0.540) showed moderately good agreement. Compared to the results of the PRmodel, a significant deviation is observed in the region of r/R > 1. This deviation indicates the existence of a fluid velocity component that presses the particle towards the


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FIGURE 6. Comparison of NWDP distribution along radial distance calculated in the rolling model (PR-model) of platelet with normalized platelet density distribution measured by Affeld et al.1 in the stagnation point flow chamber; (a) NWDP (‘a’ = 0.01) vs. normalized platelet density, (b) linear regression for ‘a’ = 0.01 (2 = 0.420), (c) NWDP (‘a’ = 0.1) vs. normalized platelet density, (d) linear regression for ‘a’ = 0.1 (2 = 0.661), (e) NWDP (‘a’ = 0.4) vs. normalized platelet density, (f) linear regression for ‘a’ = 0.4 (2 = 0.733), (g) NWDP (‘a’ = 1.0) vs. normalized platelet density, and (h) linear regression for ‘a’ = 1.0 (2 = 0.693).

near-wall region in the NWRT index.23 Figure 7(c) shows that a stronger correlation between NWDP (2 = 0.759) and the Affeld et al.’s data can be obtained.1 The deviation observed in the NWRT index was much reduced in the region r/R > 1 by selection of an appropriate value of the parameter ‘a’ in the NWDP index. The good corre-

lation, shown in Fig. 7(c), was obtained with the parameter ‘a’ = 0.4. Figure 8 demonstrates the changes of the weighting term, 1/(ln(h p /r p ) + a), of NWDP index and the weighting term, (rp /hp )2 , of NWRT index along the dimensionless wall-normal distance (hp /rp ). The weighting term was increased as the particle approached the wall. A

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FIGURE 7. Comparison of NWDP distribution along radial distance calculated in the near-wall force model (NWF-model) of platelet with normalized platelet density distribution measured by Affeld et al.1 in the stagnation point flow chamber; (a) simulated NWRT vs. normalized platelet density, (b) linear regression for (a) (2 = 0.540), (c) NWDP (‘a’ = 0.4) vs. normalized platelet density, and (d) linear regression for (c) (2 = 0.759).

steeper increase was noted for smaller values of the parameter ‘a’. For h p /r p > 3, the weighting term approached a saturated value. The magnitude of the weighting function of NWDP with ‘a’ = 1 is observed to be similar to that of NWRT with s = 2. Therefore, the NWDP index proposed in the current study is approximately the same as the NWRT index suggested by Longest et al. at a specific value of the parameter ‘a’. Sixty pre-activated spherical monocytes of 20 µm in diameter were seeded at every time step(0.05 s) for a pulse period; that is, 12,000 particles were seeded for one period. To get a saturated NWDP value, all the trajectories for one period were recycled for more than 500 periods. The trajectories of the monocytes were calculated only with the PR-model. Table 2 indicates that the trend of NWDP distribution calculated in the PR-model is similar to that of adherent U937 cells measured by Hinds et al. in a stenotic geometry under the pulsatile flow condition.15 As shown in Fig. 9, for ‘a’ = 0.01, a poor correlation is observed in the stenosis, the throat, and the expansion region (2 = 0.376). For ‘a’ = 0.1, correlation is slightly enhanced at the stenosis region (2 = 0.494). As the parameter ‘a’ is increased, the

correlation became much more enhanced in the throat and in the recirculation region as well (2 = 0.696, 2 = 0.750 for ‘a’ = 1 and 10 respectively). Two conspicuous peaks are, however, observed: one occurred at about 1.0 diameter proximal to the throat and the other at about 1.5 diameter distal to the throat. The first one is attributed to the particle impact to the wall as the flow orifice area is reduced towards the throat, and the second one is related to the reattachment points following the sudden expansion. DISCUSSION Hemodynamics forces have been known to transport platelets and monocytes to the dysfunction-prone sites of vascular endothelium. Adhesion of platelets is a key event in thrombosis and inflammation,11 and increased monocytic adhesion to vascular endothelium results in greater monocyte deformation and frequent transmigration of moncytes into the intima, where they proliferate, differentiate into macrophages and take up the lipoproteins, forming foam cells at the initiation of atherosclerosis.24,27 Simulations of our models for the pre-activated platelets in the stagnated point flow chamber and for the pre-activated


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TABLE 2. Values of 2 from the linear regression between NWDP distribution calculated in the rolling model of monocyte and normalized monocyte distribution measured by Hinds et al.15 in the stenotic perfusion tube with the variation of the parameter ‘a’. Parameter ‘a’ 0.01 0.1 0.5 1

FIGURE 8. Comparison of weighting terms of NWDP index in the PR-model of platelets, 1/(ln(h/r p ) + a), with weighting term of NWRT index, (r p /h)2 23 with variation of ‘a’; (a) full view of weighting terms, and (b) magnified view of the circle marked in (a).

monocytes in the stenotic perfusion tube resulted in significant correlations with the experimental results.1,15 The proposed NWDP index for the platelet obtained from linear velocity profile exhibited good fits with the experimental data of the stagnation point flow chamber when the value of the parameter ‘a’ was set larger than 0.1 (2 > 0.661). As for the monocyte, the NWDP index obtained from parabolic velocity profile exhibited the best fit with the experimental data of the stenotic tube when the value of the parameter ‘a’ was set larger than 1.0 (2 > 0.696). Smaller values of the parameter ‘a’ used in platelets imply that platelets adhere faster than monocytes due to smaller mass. As for the monocyte, NWDP index was modified to allow the particle to roll on the throat and recirculation zone in the Hinds et al.’s stenotic tube.15 Number of deposition probability was modified depending on the direction of the fluid velocity with the introduction of the critical rolling velocity (See Fig. 5). As the result, stronger correlation was obtained with the experimental data even at the stenotic throat if the parameter ‘a’ exceeded 1.

2 0.376 0.493 0.642 0.696

Parameter ‘a’

2

5 10 100 1000

0.746 0.750 0.752 0.752

Table 3 indicates the physical significance of the parameter ‘a’ in the two time scales. The parameter ‘a’ is defined as the ratio of correction factors c2 and c1 in Eqs. (16a) and (17). Firstly, when the correction factor equals unity, the parameter ‘a’ is related to the time taken by the particle to roll on the wall until it adheres to the wall. That is, the parameter ‘a’ is positively proportional to the correction factor c2 , which means that as the value of the correction factor c2 becomes smaller, a rolling particle on the wall adheres more quickly. Thus, in case of small value of the parameter ‘a’ for the platelet (‘a’ = 0.4), time scale for the adhesion of the particle (ta in Fig. 4) in the near wall region contribute to the deposition probability (Nd ) due to the inverse proportion of the deposition time scale (td ). However, in case of the monocyte, a higher value of the correction factor c2 , which is more than unity, results in longer rolling motion. Additionally, a zero value of the parameter ‘a’ is a special case, in which the time for the adhesion of the particle is negligible compared with the time for the movement, that is, the particle does not adhere to the wall. As shown in Fig. 8, for ‘a’ = 0, the weighting function becomes infinitely large when the particle contacts the wall. Therefore, the zero value of the parameter ‘a’ needs to be avoided to guarantee the convergence of the numerical computation. Secondly, when the correction factor c2 equals unity, the parameter ‘a’ is related to the time it takes for the particle to move toward the wall until it contacts to the wall. That is, the parameter ‘a’ and the correction factor c1 are in inverse proportion. Hence, the time it takes for the particle to move becomes negligible (very fast) in the near wall region when the parameter ‘a’ becomes larger than unity. In this case, the weight function reaches a constant value regardless of the distance of the particle from the wall. Therefore, with respect to large values of the parameter ‘a’, our particle rolling model with a constant weighting term becomes similar to the general residence model,28 which do not consider the particle rolling event along the wall. The appropriate value of the parameter ‘a’ can be selected by comparing the numerical simulation with the experimental data. Comparison of the NWDP index with ‘a = 1’ and the NWRT index of Longest and Kleinstreuer with ‘s = 2’ in Fig. 8 shows that weighting terms for both

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FIGURE 9. Comparison of NWDP distribution along tube axis calculated in the rolling model (PR-model) of monocyte with normalized monocyte distribution measured by Hinds15 in the stenotic tube; (a) NWDP (‘a’ = 0.01) vs. number of adherent cells, (b) linear regression for ‘a’ = 0.01 (2 = 0.376), (c) NWDP (‘a’ = 0.1) vs. number of adherent cells, (d) linear regression for ‘a’ = 0.1 (2 = 0.493), (e) NWDP (‘a’ = 1) vs. number of adherent cells, (f) linear regression for ‘a’ = 1 (2 = 0.696), (g) NWDP (‘a’ = 10) vs. number of adherent cells, and (h) linear regression for ‘a’ = 10 (2 = 0.750).

cases are similar and approach 1 as the platelet moves towards the wall and contacts the wall. As for the monocyte, the weighting term, (1/(h p /r p + a − 1)), of the NWDP index with ‘a = 1’ is mathematically equivalent to the weighting term, ((r p / h p )s ), of the NWRT index with ‘s = 1’. Therefore, the NWRT index for the platelet and monocyte can be regarded as another form of the NWDP index with

the parameter ‘a’ = 1. Therefore, the NWDP index accounts for both the movement time and the adhesion time, and the weight between the two time scales can be adjusted with the magnitude of the parameter ‘a’. For both the PR-model and NWF-model in the platelet simulation, the NWDP index showed good correlations with the experimental data when the parameter ‘a’ was set

Near-Wall Deposition Probability of Blood Elements TABLE 3. Physical significance of the parameter ‘a’ in two time scales (tm ,ta ). ‘a’ = c2 /c1

tm (movement; when c2 = 1)

ta (adhesion; when c1 = 1)

‘a’ = 0 (special) c2 = ∞ (very slow) c2 = 0 (very fast) 0 < ‘a’ < 1 (platelet) 1 < c1 < ∞ (slow) 0 < c2 < 1 (fast) ‘a ≥ 1 (monocyte) c1 ≤ 1 (fast) c2 ≥ 1 (slow)

to 0.4. The PR-model showed a better correlation than the NWF-model at the downstream site of the stagnation point flow chamber (r/R > 0.75). Comparison of the NWDP and NWRT indices indicates that inclusion of the inward normal velocity component in the calculation of the number of deposition probability significantly reduced the NWDP values at the downstream site of the flow chamber and yielded a better correlation with of the NWDP index with the experimental data than the NWRT index with scale factor, ‘s’ = 2 did.23 As the value of the parameter ‘a’ increased, the first peak of the NWDP distribution near r/R = 0.25 became increased, while the second peak slightly shifted to the left. This trend can be explained as follows. A large value of the parameter ‘a’ means increased time it takes for the particle to roll on the wall. Therefore, rolling distance is increased. Longer rolling distance results in higher platelet adhesion flux near r/R = 0.25 due to the higher overlapping of the seeded particles. Our numerical results showed a strong adhesion of the monocytes at the reattachment site of the stenotic perfusion tube while this trend was not observed in the experimental data obtained by Hind et al.15 This difference is attributed to the vertical model used in the Hind’s experiment where the gravitational force reduced the cell adhesion at the reattachment point. The current study investigated the newly proposed NWDP index for platelets and monocytes. This new hemodynamic index is different from the WSS-based hemodynamic parameters, such as MWSS (Mean Wall Shear Stress), AWSS (Amplitude of Wall Shear Stress), and OSI (Oscillatory Shear Index) in that it locates regions of both high and low WSS. The proposed NWDP index needs to be tested and compared in real geometries to evaluate its effectiveness in locating regions of lesion-prone sites. ACKNOWLEDGEMENTS This work was conducted with the supports of Korean Ministry of Health and Welfare, 02-PJ3-PG3-31403-0004. REFERENCES 1

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