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Computational modeling of coupled blood-wall mass transport of LDL: effects of local wall shear stress

Laboratory of Thermodynamics in Emerging Technologies, Department of Mechanical and Process Engineering, ETH Zurich, Zurich, Switzerland

Submitted 18 September 2007 ; accepted in final form 7 December 2007

	ABSTRACT

TOP ABSTRACT Glossary Superscripts and Subscripts METHODS RESULTS DISCUSSION GRANTS REFERENCES

 
The work herein represents a novel approach for the modeling of low-density lipoprotein (LDL) transport from the artery lumen into the arterial wall, taking into account the effects of local wall shear stress (WSS) on the endothelial cell layer and its pathways of volume and solute flux. We have simulated LDL transport in an axisymmetric representation of a stenosed coronary artery, where the endothelium is represented by a three-pore model that takes into account the contributions of the vesicular pathway, normal junctions, and leaky junctions also employing the local WSS to yield the overall volume and solute flux. The fraction of leaky junctions is calculated as a function of the local WSS based on published experimental data and is used in conjunction with the pore theory to determine the transport properties of this pathway. We have found elevated levels of solute flux at low shear stress regions because of the presence of a larger number of leaky junctions compared with high shear stress regions. Accordingly, we were able to observe high LDL concentrations in the arterial wall in these low shear stress regions despite increased filtration velocity, indicating that the increase in filtration velocity is not sufficient for the convective removal of LDL.

low-density lipoprotein transport; lipid accumulation; atherosclerosis; leaky junction

Early experimental studies on LDL transport were performed in animals. The main method was systemic injection of LDL solution with subsequent harvesting and examination of the animal's arteries. Lin et al. (19) observed thoracic aortas of rats with this method and counted the number of leakage sites of LDL, as well as the portion of leakage sites associated with mitotic cells. Their results support the hypothesis that transiently opened leaky junctions provide pathways for LDL transport. Truskey et al. (38) used the same method on rabbits to quantify the apparent permeability of the endothelium to LDL. Tompkins et al. (37) measured LDL concentration at various sites in the circulatory system of monkeys. Meyer et al. (21) used a different method and excised aortic segments of rabbits to incubate them in vitro at various pressure levels with LDL solutions. With this method, they were able to assess the effect of intraluminal pressure on LDL transport. Recently, Cancel et al. (2) used endothelial cell cultures to study LDL transport under pressurized conditions and found that LDL transport through leaky junctions constitutes the major part of total LDL transport (90.9%), whereas only a small portion occurs via the vesicular pathway (9.1%). Because the occurrence of leaky junctions is governed by the adjacent endothelial cells, and because the behavior of the endothelial cells is in turn influenced by the shear stress acting on their surface, LDL transport in the arterial wall is in part governed by local blood flow characteristics that determine the level of wall shear stress. The effects of shear stress on endothelial cells have been investigated in various studies. Although there is no study that directly investigates the relationship between shear stress and the fraction of leaky junctions, there have been studies on the effect of shear stress on cell shape (4, 17, 27), proliferation rate (12, 41), and hydraulic conductivity and permeability (27, 28). Levesque et al. (17) observed a stenosed dog aorta and correlated the cell shape with levels of shear stress. Suciu (32), Chiu et al. (4), and Sakamoto et al. (27) used cell cultures to observe the cell alignment of endothelial cells in response to shear stress. These experimental findings are used in this study to obtain a single correlation between shear stress and cell shape. Chien (3) observed rabbit thoracic aortas and counted the number of mitotic cells in regions of specific cell shapes. Their findings are used in this study to correlate cell shape to the number of mitotic cells. Finally, the percentage of leaky cells associated with mitotic cells reported by Lin et al. (19) is used as a last step to obtain a correlation between shear stress and fraction of leaky junctions.

On the computational side, there have been several approaches to model LDL transport. In simple wall-free models, transport is solely calculated in the artery lumen, and flow characteristics are correlated with the solute distribution. Wada et al. (40) used a wall-free model to study the concentration polarization phenomenon, and Kaazempur-Mofrad and Ethier (13) calculated the mass transport in a realistic human right coronary artery. The main drawback of wall-free models is that they do not provide any information on the transmural flow and solute dynamics in the arterial wall. Lumen-wall models, on the other hand, account for both the arterial lumen and wall transport: Stangeby and Ethier (31) treated the wall as a homogeneous single-layer porous medium and solved the coupled luminal blood flow and transmural fluid flow using Brinkman's equations. Multilayer models have also been used to represent intima and media separately (1, 14, 25), which better represent the arterial wall, but are rather complex compared with the single-layer models.

In this study, we hypothesize that the dependence of fluid and LDL transport through the endothelium on local blood flow characteristics can be modeled more accurately using a three-pore model than with a single pathway approach (15, 25, 33, 34). To test this hypothesis, we use an axisymmetric representation of a stenosed coronary artery, where the arterial wall is treated as a single-layer porous medium. The volume flux and the solute flux across the interface between the fluid and the porous domains are governed by a three-pore model representing the vesicular pathway, normal junctions, and leaky junctions. The transport properties of the leaky junctions, i.e., hydraulic conductivity, diffusive permeability, and reflection coefficient, are calculated locally in dependence of the fraction of leaky junctions. The fraction of leaky junctions is correlated to wall shear stress using the available experimental data as outlined above.

	Glossary

TOP ABSTRACT Glossary Superscripts and Subscripts METHODS RESULTS DISCUSSION GRANTS REFERENCES

 
WSS

a

Radius of LDL

A_p

Total pore area

c

LDL concentration

C_o

LDL concentration at the inlet of the artery

D

Diffusion coefficient

d

Artery diameter

F

Function for the restricted diffusivity in a pore

J_v

Volume flux

J_s

Solute flux

k

Local mass transfer coefficient

K_lag

LDL lag coefficient

K_p

Darcy's permeability in the arterial wall

l_lj

Length of a leaky junction

L_p

Hydraulic conductivity

n

Local mass flux

p

Pressure

p_o

Pressure at the outlet of the artery

P

Diffusive permeability

P_app

Apparent permeability

Pe

Péclet number

r

Radial location

r_w

LDL degradation rate

R

Resistance

R_a

Radius of the artery

R_a,t

Radius of the artery at the throat section of the stenosis

R_cell

Radius of single endothelial cell

S

Surface area

Sc

Schmidt number

Sh

Sherwood number

SI

Shape index

t_a

Thickness of the arterial wall

u

Velocity

U_o

Mean velocity at the inlet of the artery

w

Half-width of a leaky junction

WSS

Wall shear stress

z

Axial location

z_b

Base point

Z

Fractional reduction factor in solute concentration gradient at the pore entrance

µ

Blood viscosity

µ_p

Plasma viscosity

Half-distance between leaky junctions

Blood density

_p

Plasma density

_d

Osmotic reflection coefficient

_f

Solvent-drag reflection coefficient

ø

Fraction of leaky junctions

Partition coefficient

#MC

Number of mitotic cells

#LC

Number of leaky cells

	Superscripts and Subscripts

TOP ABSTRACT Glossary Superscripts and Subscripts METHODS RESULTS DISCUSSION GRANTS REFERENCES

 
WSS.

end

Endothelium

adv

Adventitia

l

Artery lumen

w

Arterial wall

v

Vesicular pathway

nj

Normal junctions

lj

Leaky junctions

slj

Single leaky junction

i

Single pathway

	METHODS

TOP ABSTRACT Glossary Superscripts and Subscripts METHODS RESULTS DISCUSSION GRANTS REFERENCES

 
Mathematical model. A schematic of the mathematical model of LDL transport from the artery lumen into the arterial wall is shown in Fig. 1. A description of the model integration is given in the section Model flowchart. Although blood flow has a pulsatile nature, we assumed steady-state conditions in this study for simplicity. Although these will yield a different wall shear stress distribution than transient conditions, the fundamental properties of our model are independent of the type of flow. The current model could be modified to take into account transient conditions in cases where quantification of the results for transient conditions is desired. Under the assumption of steady-state conditions and incompressible blood with Newtonian rheology, the fluid dynamics in the artery lumen are governed by the Navier-Stokes and continuity equations:

(1)

(2)

where and denote the gradient and the Laplacian operator, respectively, u_l is the velocity vector field in the lumen, p_l is the lumen pressure, and µ is the blood viscosity. The transmural velocity vector field u_w in the arterial wall is calculated with Darcy's Law,

(3)

(4)

where _p, µ_p, and p_w are the permeability of the arterial wall, viscosity of the blood plasma, and pressure within the arterial wall, respectively. The viscosity and density are taken as µ = 0.0035 Pa·s and = 1,050 kg/m³ for blood and as µ_p = 0.001 Pa·s and _p = 1,000 kg/m³ for blood plasma (22).

View larger version (16K): [in this window] [in a new window]   Fig. 1. Illustration of the mathematical domains corresponding to the artery lumen and the arterial wall with the respective governing equations. Jv and Js are the volume and solute flux, respectively, from the artery lumen into the arterial wall across the endothelium.

(5)

where c_l and D₁ are the LDL concentration and diffusion coefficient in the lumen, respectively. The transport of LDL in the arterial wall is modeled similarly with an additional term that represents LDL degradation, i.e.,

(6)

where c_w and D_w are the LDL concentration and diffusion coefficient in the arterial wall, respectively, K_lag is the solute lag coefficient, and r_w is the LDL degradation rate in the arterial wall. The diffusivity of LDL in the blood and in the wall are taken as D_l = 5.0 x 10^–12 m/s and D_w = 8.0 x 10^–13 m/s. D_l is calculated with the Stokes-Einstein Equation as described previously (6), and D_w is the value reported by Prosi et al. (25) and the references therein. The solute lag coefficient K_lag is taken as 0.1486, the value proposed by Sun et al. (33). Published LDL degradation rates are 1.4 – 6.05 x 10^–14 s^–1 (1, 25, 33, 34). We have chosen here an average value of r_w = 3.0 x 10^–4 s^–1.

The coupling of flow and solute dynamics in the artery lumen with that in the arterial wall is achieved by a three-pore model that takes into account the vesicular pathway, normal endothelial junctions, and leaky junctions. Following Curry (6), the volume flux J_v,i through a single pathway is given by

(7)

where L_p,i and _d,i are the hydraulic conductivity and the osmotic reflection coefficient of the single pathway, respectively, and p and are the hydraulic and osmotic pressure differences across the endothelial layer, respectively. In the present study, the osmotic pressure difference is neglected to decouple the fluid dynamics from the solute dynamics. Therefore, Eq. 7 can be written as

(8)

Again following Ref. 6, the solute flux J_s,s through a single pathway is given by

(9)

where c and c are the luminal and arterial wall side LDL concentrations at the endothelium, respectively, P_i and _f,i are the diffusive permeability and the solvent-drag reflection coefficient of the single pathway, respectively, and Pe_i is a modified Péclet number that is defined as the ratio of convective to diffusive fluxes through the pathway:

(10)

The endothelium has a high resistance to LDL transport (23, 31, 35) and, as a result, c is much smaller compared with c. Neglecting c, Eq. 9 can be arranged as

(11)

where P_app,i is the apparent permeability coefficient given as

(12)

and Z_i is a fractional reduction factor in solute concentration gradient at the pore entrance given as

(13)

In the absence of flow through the pore, i.e., when convection vanishes, Pe_i 0 and Z_i 1, which represents the pure diffusion case. As Pe_i , i.e., convection dominates over diffusion, Z_i 0.

In our three-pore model, the total volume flux J_v and the total solute flux J_s can be obtained by summing the fluxes over each single pathway, i.e.,

(14)

(15)

where J_v,v, J_v,nj, and J_v,lj are the volume flux through vesicular pathway, normal junctions, and leaky junctions, respectively. P_app is the total apparent permeability of the endothelial layer given as the sum of apparent permeabilities of each single pathway:

(16)

where P_app,nj = P_njZ_nj + J_v,nj(1 – _f,nj) and P_app,lj = P_ljZ_lj + J_v,lj(1 – _f,lj) are the apparent permeabilities of the normal junctions and the leaky junctions, respectively, and P_v is the permeability of the vesicular pathway.

Volume flux occurs through normal and leaky junctions (23, 35). Therefore, with J_v,v = 0, Eq. 14 becomes

(17)

LDL flux, herein referred to as solute flux, occurs by vesicular transport and through leaky junctions, but not through normal junctions since they block the passage of solutes with a radius larger than 2 nm (LDL has a radius of 11 nm) (23, 35). Consequently, the diffusive permeability and the solvent-drag reflection coefficient become P_nj = 0 and _f,nj = 1, respectively, resulting in P_app,nj = 0. Accordingly, Eq. 16 can be rewritten as

(18)

The endothelium and the arterial wall exhibit a combined resistance against inflow from the artery lumen. We use an electrical analogy for the fluid dynamics as shown in Fig. 2. The flow, which is analogous to current in an electric circuit, is driven by the pressure difference, which is analogous to potential difference. The flow resistances for the normal and leaky junction pathways are derived from Eq. 8 as R_nj = 1/L_p,nj and R_ij = 1/L_p,lj, respectively. Similarly, the flow resistance for the arterial wall is derived from Darcy's Law, Eq. 3, as R_wall = µ_pt_a/_p, where t_a is the thickness of the artery. Therefore, the filtration velocity can be expressed as

(19)

where R_T = R_end + R_wall is the total flow resistance formed by the endothelial resistance R_end and wall resistance R_wall connected in series, and, p and p_adv are the lumen side pressure over the endothelium and the pressure at the media-adventitia interface, respectively. The normal and leaky junctions are two parallel flow pathways with a combined flow resistance across the endothelium, R_end, given as

(20)

Accordingly, the volume fluxes through normal junctions and leaky junctions are

(21)

and

(22)

View larger version (7K): [in this window] [in a new window]   Fig. 2. Electrical analogy for the filtration velocity. Rnj and Rlj are the flow resistances of the normal junctions and leaky junctions, respectively, connected in series between the two sides of the endothelium, and Rwall is the flow resistance of the arterial wall. p, p, and padv are the pressures at the lumen side of the endothelium, at the wall side of the endothelium, and at the media-adventitia interface, respectively.

(23)

where R_cell is the endothelial cell radius taken to be 15 µm (10, 42, 43) (Fig. 3).

View larger version (22K): [in this window] [in a new window]   Fig. 3. Illustration of randomly distributed leaky cells. A leaky cell is present at the center of each periodic circular unit of radius represented by dashed lines. The gray background represents the normal cells. Cells have a radius of Rcell, and leaky junctions have a half-width of w.

(24)

where w and l_lj are the half-width and the length of the leaky junction taken to be 20 nm and 2 µm, respectively (10, 42, 43). The diffusive permeability of a single leaky junction is given by

(25)

where D_lj is the diffusion coefficient in a leaky junction, which is related to the free diffusion coefficient in the lumen, D_l, by

(26)

where _lj = a/w is the ratio of the radius a of the transported molecule to the half-width of the leaky junction, and F(_lj) is a function describing the restricted diffusivity in a pore. The solvent drag coefficient _f,lj for the leaky junctions is given by

(27)

The hydraulic conductivity and diffusive permeability of the leaky junction pathway are given by

(28)

(29)

respectively, where = 1 – _lj is the partition coefficient, i.e., the ratio of the pore area available for solute transport to the total pore area. A_p/S represents the fraction of the surface area S occupied by the leaky junctions. Referring to Fig. 3, A_p/S is A_lj/² in the periodic unit circle of radius , where A_lj = (2R_cell)(2w) is the area of a single leaky junction. Using Eq. 23 for the fraction of leaky junctions ø, A_p/S is related to ø as

(30)

Therefore, once ø is known, the transport properties of the leaky junctions are calculated accordingly. In our model, ø is assigned locally at the endothelium as a function of WSS. The correlation between WSS and ø is obtained using published experimental data in several steps as described in the following paragraphs. To the best of our knowledge, no direct correlations between WSS and ø have been published to date.

Levesque et al. (17) studied stenosed dog aortas and correlated the cell shape with levels of shear stress. Cell shape is generally expressed in terms of a shape index (SI) defined as

(31)

where area and perimeter are the area and perimeter of a single cell. SI is one for a circle and approaches zero for a line. Suciu (32), Chiu et al. (4), and Sakamoto et al. (27) used cell cultures to observe the cell alignment of endothelial cells as a response to shear stress. As shown in Fig. 4, we have used the data from these studies to establish an empirical correlation between wall shear stress and endothelial cell shape index, SI:

(32)

View larger version (12K): [in this window] [in a new window]   Fig. 4. Experimental data on shape response of endothelial cells to shear stress and the correlation (Eq. 32) used in the simulations at hand.

(33)

View larger version (9K): [in this window] [in a new window]   Fig. 5. Experimental data on the no. of mitotic cells/unit area of 0.64 mm2 in specific regions of shape index and the correlation (Eq. 33) used in the simulations at hand.

(34)

Because the fraction of leaky junctions ø is defined as the ratio of the area of leaky cells to the area of all cells (10), next to the form given in Eq. 23, ø can also be written as

(35)

Equations 32–35 constitute the correlation between wall shear stress and fraction of leaky junctions used in our simulations.

Among the three transport properties of normal junctions, only the hydraulic conductivity has to be determined, since the diffusive permeability P_nj = 0 and the solvent reflection coefficient _f,nj = 1 as described in the previous section. To set the hydraulic conductivity of the normal junctions, we use the electrical analogy as introduced in the previous section at the base point z*_b and set the filtration velocity to 1.78 x 10^–8 m/s as reported by Meyer et al. (21). There are three unknowns in Eq. 19 for the filtration velocity (L_p,nj, L_p,lj, and _p). L_p,lj is calculated with the pore theory as described earlier based on the fraction of leaky junctions at z*_b. Darcy's permeability of the arterial wall is estimated using the electrical analogy by employing the equation for filtration velocity between the wall side of the endothelium and adventitia, i.e.,

(36)

where p is the wall side pressure at the endothelium, which is set using the pressure drop of p– p= 18 mmHg given by Tedgui and Lever (36), since p is known from fluid dynamics calculations. The remaining unknown in Eq. 36 is Darcy's permeability of the arterial wall, and it is estimated to be K_p = 1.2 x 10^–18 m², which agrees well with the literature values of 1.0 – 2.0 x 10^–18 m² (10, 25, 31, 34, 39). Once the permeability of the wall is fixed, the last remaining unknown, L_p,nj, is estimated with Eq. 19 as L_p,nj = 1.16 x 10^–9 m/(s·mmHg), which is within the range of estimated values of L_p,nj by Yuan et al. (43) and close to the value of L_p,nj = 1.58 x 10^–9 m/(s·mmHg) by Tedgui and Lever (36) and L_p,nj = 1.19 x 10^–9 m/(s·mmHg) by Vargas et al. (39).

The only transport property to be determined for the vesicular pathway is the vesicular permeability P_v. To find out P_v, P_app,lj in Eq. 18 is calculated at z*_b. P_lj and _f,lj are calculated from the pore theory as described earlier, and J_v,lj is calculated from the electrical analogy for the fluid dynamics with Eq. 22. Accordingly, P_app,lj assumes a value of 1.9 x 10^–10 m/s. Applying the proportion between the leaky junction pathway and vesicular transport reported by Cancel et al. (2), P_v is calculated to be 1.92 x 10^–11 m/s.

Model flowchart. A general flowchart of the model and its modules is shown in Fig. 6 and described in this section. The two modeled domains (artery lumen and arterial wall) are separated by an interface, the endothelium. The gray-colored modules in Fig. 6 labeled with lowercase letters and connected with solid lines are used over the entire artery, whereas the white-colored modules labeled with Roman numerals and connected with dashed lines are used at a single point z*_b. The governing Navier-Stokes, continuity, and convection-diffusion equations (Fig. 6b) in artery lumen are solved with proper boundary conditions and material properties (Fig. 6a) to obtain the blood flow velocity, pressure, and LDL concentration fields (Fig. 6c). WSS (Fig. 6d) is calculated from the velocity field, and a base point z*_b (I in Fig. 6) is chosen far downstream of the stenosis, where local WSS reaches a constant value after the disturbances introduced by the stenosis. The WSS value at this point is used with the WSS-SI and SI-#MC correlations and #MC-#LC proportion in the module (II in Fig. 6) to calculate the fraction of leaky junctions associated with mitotic and nonmitotic cells. The fraction associated with nonmitotic cells is kept constant with respect to WSS in the overall correlation between WSS and fraction of leaky junctions (Fig. 6e). Values of the fraction of leaky junctions (Fig. 6f) are assigned at the endothelium depending on the local WSS using the overall correlation between WSS and the fraction of leaky junctions (Fig. 6e). These local ø values are fed into the pore theory (Fig. 6g) to calculate the local transport properties of the leaky junction pathway. The calculated transport properties are fed into the three-pore model (Fig. 6h), and their values at the base point are used to determine Darcy's permeability of the wall, hydraulic conductivity of normal junctions, and vesicular permeability. The experimental values of Meyer et al. (21) for the filtration velocity and of Tedgui and Lever (36) for the pressure drop across the endothelium (III in Fig. 6) are used at the base point, and an electrical analogy for the fluid dynamics in the wall is adapted to calculate Darcy's permeability of the wall and the hydraulic conductivity of the normal junctions. In addition, the ratio between the leaky junction pathway and vesicular transcytosis for LDL transport reported by Cancel et al. (2) (IV in Fig. 6) is used to calculate the vesicular permeability. These parameters are fed into the three-pore model (Fig. 6h), which uses the luminal side pressure and concentration at the endothelium from the module (Fig. 6c) to calculate the solute flux J_s and, with the help of an electrical analogy, the volume flux J_v and its distribution among the normal and leaky junction pathways. Volume and solute flux are used as boundary conditions for the solution of the governing equations in the artery lumen (Fig. 6b) and the arterial wall (Fig. 6j). Darcy's law and the convection-diffusion equations (Fig. 6j) governing in the arterial wall are solved with proper boundary conditions and material properties (Fig. 6i) to obtain the transmural velocity, pressure, and LDL concentration fields in the artery wall (Fig. 6k).

View larger version (34K): [in this window] [in a new window]   Fig. 6. Flowchart of the model. The two treated domains (artery lumen and arterial wall) are separated by the endothelium. The gray modules connected with solid lines are used over the entire computational domain, and the white modules connected with dashed lines are used at the base point z*b. Volume flux Jv and solute flux Js are calculated locally in dependence of WSS and are specified as boundary conditions for both domains.

View larger version (3K): [in this window] [in a new window]   Fig. 7. Computational, axisymmetric geometry representing a coronary artery with 40% reduction of inner diameter at the location of stenosis. The dashed center line represents the axis of symmetry.

(37)

where r(z) is the radius of the artery at the axial location z, R_a,t is the radius of the artery at the throat of the stenosis, and z₁ = 8R_a and z₂ = 10R_a are the start and end points of the stenosis, respectively.

For the flow calculations in the lumen, a parabolic velocity profile is used at the inlet:

(38)

where U₀ is the mean inlet velocity taken to be 24 cm/s as the average of the mean diastolic and systolic velocities in the LAD (11). To match the experimental conditions of Meyer et al. (21), a constant pressure boundary condition of 70 mmHg is used at the outlet. The volume flux J_v, calculated with Eq. 19, is specified in the normal direction to the endothelium for both the artery lumen and the arterial wall.

For the calculation of transmural flow, a constant pressure boundary condition is used at the media-adventitia interface with the value of p_adv = 17.5 mmHg [Ai and Vafai (1) assumed p_adv = 30 mmHg for an intraluminal pressure of 100 mmHg]. At both axial ends of the wall, zero flux boundary conditions are applied in the axial direction.

For the mass transport calculations at the lumen, a constant concentration boundary condition of C_o = 3.12 mol/m³ is used at the inlet in accordance with the common human blood LDL concentration level (8). To ensure that LDL is convected out of the domain with flow, at the lumen outlet, a convective flux condition is used. Because the solute dynamics in the artery lumen are convection dominated, the diffusive flux is set to zero:

(39)

The solute flux, J_s, calculated with Eq. 15 is specified in the normal direction to the endothelium for both the artery lumen and the arterial wall. At the media-adventitia interface, a constant LDL concentration value of c_adv/C_o = 0.005 as reported by Meyer et al. (21) is used. At both axial ends of the wall, insulation boundary conditions are applied:

(40)

A commercial finite element code, Comsol Multiphysics, Version 3.3 (COMSOL), was used to implement the model. The artery lumen was meshed with 29,100 quadrilateral elements and the arterial wall was meshed with 25,220 quadrilateral elements for the solution of fluid dynamics. Second-order elements for velocity and linear elements for pressure were used in the lumen. Linear elements were used for flow in the arterial wall. For the LDL transport calculations, the artery lumen was meshed with 86,400 quadrilateral elements and the arterial wall was meshed with 144,000 quadrilateral elements. Linear elements were used to solve the convection-diffusion equations in both domains. The flow calculations were performed first. Next, the obtained velocity field was used for the mass transport calculations. Grid independence tests were performed successfully for both fluid flow and LDL transport calculations. The maximum relative change in velocity, pressure, and LDL concentration values was under 0.1% compared with a coarser grid with one-half the number of elements.

	RESULTS

TOP ABSTRACT Glossary Superscripts and Subscripts METHODS RESULTS DISCUSSION GRANTS REFERENCES

 
To ensure that the computational mesh used in the simulations is adequate, the wall shear stress distribution and Sherwood number (Sh) along the endothelium in the axial direction are compared with the corresponding analytical solution to the Graetz-Nusselt problem for a straight tube in Figs. 8 and 9, respectively. Leveque's approximate solution (16) to the Graetz-Nusselt problem is used, since the Schmidt number is large (Sc = µ/D = 666,666). The Sherwood number is a dimensionless mass transfer coefficient defined as Sh = kd/D, where k = n/(C_o – c) is a local mass transfer coefficient and n is the local mass flux. The simulation results for the wall shear stress and Sherwood number agree very well with the analytical solution at the inlet section. Downstream of the stenosis, the simulation results again converge to the analytical solution after having deviated from them due to the disturbances induced by the stenosis. This is an indication that the computational mesh is sufficiently fine.

View larger version (10K): [in this window] [in a new window]   Fig. 8. Absolute values of wall shear stress (WSS) at the endothelium as a function of the normalized axial location z* from the domain inlet. WSS is the same for the shear-dependent transport properties case and the constant transport properties case. z*b = 25 is the base point where the local WSS reaches a constant value after the disturbances introduced by the stenosis. The dashed line represents the analytical solution to the Graetz-Nusselt problem for a straight tube.

View larger version (13K): [in this window] [in a new window]   Fig. 9. Sherwood number at the endothelium as a function of the normalized axial location z* from the domain inlet. The dashed line represents the analytical solution to the Graetz-Nusselt problem for a straight tube.

View larger version (8K): [in this window] [in a new window]   Fig. 10. Velocity streamlines in the vicinity of the stenosis showing the separation point, reattachment point, and the recirculation zone.

View larger version (15K): [in this window] [in a new window]   Fig. 11. Total filtration velocity and the filtration velocities through the normal and leaky junction pathways in the vicinity of the stenosis as a function of the normalized axial location z*.

View larger version (9K): [in this window] [in a new window]   Fig. 12. Solute flux through the endothelium in the vicinity of the stenosis as a function of the normalized axial location z*.

View larger version (8K): [in this window] [in a new window]   Fig. 13. Wall side solute concentration at the endothelium normalized by the inlet concentration Co in the vicinity of the stenosis as a function of the normalized axial location z*.

View larger version (11K): [in this window] [in a new window]   Fig. 14. Comparison of the computed solute concentration profile across the artery wall at the base point z*b with the experimental results of Meyer et al. (21) at 70 mmHg. c*w is the arterial wall concentration normalized by the inlet concentration Co, and r* is the distance from the endothelium normalized by the thickness of the arterial wall.

	DISCUSSION

TOP ABSTRACT Glossary Superscripts and Subscripts METHODS RESULTS DISCUSSION GRANTS REFERENCES

The electrical analogy of the endothelium and the wall serves well in determining the model parameters and explaining the velocity distribution through the normal and leaky junctions. As can be seen in Fig. 11, volume flux through normal junctions decreases at the spots where volume flux through leaky junctions increases. This is most pronounced around the separation and reattachment points, where WSS is low. The increase in volume flux through leaky junctions is the result of the increased number of leaky junctions and accordingly decreased flow resistance through the entire leaky junction pathway. Because normal junctions and leaky junctions are parallel pathways, the decreased flow resistance of the leaky junction pathway leads to a decrease in the volume flux through the normal junction pathway as can be explained with the electrical analogy. This decrease in volume flux through normal junctions was shown experimentally by Sill et al. (28). They performed experiments on endothelial cell cultures investigating the effect of shear stress on the hydraulic conductivity of the cell layer. In these experiments, cells grew under no shear conditions. When the cell layer reached confluence, it was first exposed to a pressure gradient and, after stability in filtration velocity was reached, to flow for one to three hours. It was observed that the filtration velocity through the cell layer increased up to two- to fourfold with the flow. Sill et al. (28) hypothesized that a change in the number of leaky junctions may be responsible for the shear-induced solute permeability, whereas alterations of normal junctions may be responsible for the shear-induced hydraulic conductivity. Because the experiment duration of one to three hours was not long enough for the cells to respond in a proliferative manner, we believe that this increase in filtration velocity with WSS is only due to the alteration of the structure of the normal junctions. One simple explanation could be that, with shear stress, the cells are elongated and the total length of contact between the cells is increased, which can lead to a higher available normal junction area for the volume flux. According to the findings of Sill et al., one might expect low volume flux in low WSS regions. Our simulations indicate that their findings reflect decreased volume flux through normal junctions in these low WSS regions, whereas the total volume flux is observed to be increased.

The wall side concentration profile at the endothelium in Fig. 13 reveals four high concentration locations: the two end points of the stenosis, the separation point, and the reattachment point. High concentration values are reached especially at the separation and the reattachment points, which is in agreement with the finding of Smedby (29) in an angiographic study showing that atherosclerotic plaques grow in the downstream direction of the stenoses significantly more frequently than in the upstream direction. The highest wall side concentration at the endothelium is observed at the reattachment point and is 26% of the luminal concentration. Tompkins et al. (37) measured LDL concentrations in four squirrel monkeys at a total of 280 sites. They observed elevated concentrations at 15 sites, with the highest one being 9.8% of the luminal concentration at 10% relative depth from the endothelium. At the reattachment point, we observed 14% of the luminal concentration at this depth. The discrepancy between our simulation results and the experimental observations could be explained in three ways. First, Tompkins et al. waited only for 30 min after the systemic injection of labeled LDL into the monkeys to measure the wall side concentrations. This may not be enough time for a stable concentration profile to be established at the wall. Therefore, the in vivo values of LDL may be higher than what the experiments suggest. Second, we have neglected the effects of pulsatile flow, which expands and contracts the flow recirculation zone downstream of the stenosis. This leads to a cyclic motion of the reattachment point, which would soften the local LDL peak observed at this point. Additionally, the effects of oscillatory shear stress on the formation of leaky junctions remain to be investigated. Finally, the correlation between WSS and fraction of leaky junctions in our model is based on several experimental results on various animals and cell cultures, since no direct correlation has been published to date. Especially, the correlation between the shape index and the number of mitotic cells is only based on the observations of Chien (3). The sharp increase in the number of mitotic cells with respect to shape index reported therein leads to a sharp increase of the fraction of leaky junctions with respect to WSS. This might be the reason for overestimated values of concentration at the high permeability and low WSS locations. To investigate this, we performed a sensitivity analysis by scaling the right-hand side (RHS) of Eq. 32, i.e., the correlation between wall shear stress and shape index and the RHS of Eq. 33, i.e., the correlation between shape index and number of mitotic cells by ±10%. Equation 34 was also updated accordingly as described in Calculation of transport properties. As can be seen in Fig. 15, the wall side concentration at the endothelium did not change qualitatively, but quantitatively. The maximum relative change was observed at the base point z*_b with 66% for +10% scaling of the RHS of both Eqs. 32 and 33, and 36% for –10% scaling of the RHS of both Eqs. 32 and 33. The relative change at the reattachment point was 56 and 46% for the +10 and –10% scaling, respectively. The sensitivity analysis shows that accurate experimental data on human endothelial cells or real artery segments are necessary to quantify the fraction of leaky junctions at different shear stress levels. The analysis also shows clearly that the qualitative behavior of our model is not sensitive to the fraction of leaky junctions as a function of shear stress.

View larger version (12K): [in this window] [in a new window]   Fig. 15. Sensitivity analysis for the wall side solute concentration at the endothelium normalized by the inlet concentration Co. The solid line is the result obtained with the original Eqs. 32 and 33. The dotted and the dashed lines are the results obtained with +10% and –10% scaling of the right-hand side of both Eqs. 32 and 33, respectively. Note that Eq. 34 is also updated accordingly in both cases.

One previous study attempting to incorporate the effects of local WSS on LDL transport from the blood to the arterial wall was performed by Sun et al. (34), where the data from Sill et al. (28) were used to define local hydraulic conductivity with respect to shear stress. As mentioned before, this approach only reflects the effects of WSS on normal junctions, and the filtration velocity profile obtained by Sun et al. is similar to our results for volume flux through normal junctions in the sense that it decreases at low shear stress regions. However, in our model, to fully cover the effects of local WSS on volume and solute flux, different pathways for the fluxes are defined, and transport properties for each pathway are calculated using the pore theory. Therefore, the effects of WSS on leaky junctions are also included, and the total filtration velocity is found to be increased in low shear stress regions. Sun et al. (34) calculated high concentration values in the low shear stress regions where their filtration velocity is decreased and explained this as a result of weaker local convective clearance effects of the transmural flow. According to Sun's argument about convective clearance effects, one would expect low concentration values in our results at the reattachment point where the total filtration velocity J_v is increased. However, this is not the case because of the elevated values of solute flux resulting from the leaky junction pathway. The relatively low increase in total filtration velocity is not enough to clear away the molecules, and high concentration values are reached at these low shear stress regions.

In conclusion, the work at hand constitutes a novel way of modeling LDL transport from the artery lumen into the arterial wall that incorporates for the first time the effect of WSS on the endothelial cells and the pathways of volume and solute flux. Further development of the model could be achieved with the incorporation of a direct relationship between WSS and the fraction of leaky junctions that could be obtained from appropriate experiments not available currently.

	GRANTS

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The financial support of the Swiss National Science Foundation through the National Center of Competence in Research in Computer-Aided and Image-Guided Medical Interventions is kindly acknowledged.

	FOOTNOTES

 

Address for reprint requests and other correspondence: D. Poulikakos, ETH Zurich, Sonneggstr. 3, ML J 36, 8092 Zurich, Switzerland (e-mail: dimos.poulikakos{at}ethz.ch)

The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.

	REFERENCES

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