Vol. 282, Issue 2, H576-H584, February 2002
Flow through internal elastic lamina affects shear stress on
smooth muscle cells (3D simulations)
Shigeru
Tada1 and
John
M.
Tarbell2
1 Energy Phenomena Laboratory, Department of Mechanical
Engineering and Science, Tokyo Institute of Technology, Tokyo 152-8859, Japan; and 2 Biomolecular Transport Dynamics Laboratory,
Chemical Engineering and Bioengineering Department, The
Pennsylvania State University, University Park, Pennsylvania
16802-4400
 |
ABSTRACT |
We describe a three-dimensional
numerical simulation of interstitial flow through the medial layer of
an artery accounting for the complex entrance condition associated with
fenestral pores in the internal elastic lamina (IEL) to investigate the
fluid mechanical environment around the smooth muscle cells (SMCs)
right beneath the IEL. The IEL was modeled as an impermeable barrier to
water flow except for the fenestral pores, which were assumed to be
uniformly distributed over the IEL. The medial layer was modeled as a
heterogeneous medium composed of a periodic array of cylindrical SMCs
embedded in a continuous porous medium representing the interstitial
proteoglycan and collagen matrix. Depending on the distance between the
IEL bottom surface and the upstream end of the proximal layer of SMCs,
the local shear stress on SMCs right beneath the fenestral pore could
be more than 10 times higher than that on the cells far removed from
the IEL under the conditions that the fenestral pore diameter and area
fraction of pores were kept constant at 1.4 µm and 0.05, respectively. Thus these proximal SMCs may experience shear stress
levels that are even higher than endothelial cells exposed to normal
blood flow (order of 10 dyn/cm2). Furthermore, entrance
flow through fenestral pores alters considerably the interstitial flow
field in the medial layer over a spatial length scale of the order of
the fenestral pore diameter. Thus the spatial gradient of shear stress
on the most superficial SMC is noticeably higher than computed for
endothelial cell surfaces.
computer simulation; fenestral pore
| |
INTRODUCTION |
IT HAS BEEN WIDELY
ACCEPTED that a confluent layer of endothelial cells forms a
mechanosensitive barrier separating blood and the underlying tissue
space and regulates many complex processes in the vasculature (6,
19). In contrast to extensive studies describing the response of
endothelial cells to fluid mechanical stimulation (shear stress), much
less attention has been focused on the biochemical responses of smooth
muscle cells (SMCs) to fluid flow forces. The level of shear stress on
SMC, in the presence of an intact endothelium, was thought to be too
low to influence SMC function due to very low interstitial flow
velocities (order of 10
6 cm/s) in the medial layer.
Recent studies, however, indicate that SMCs can be subjected to
significant levels of shear stress associated with interstitial flow
even though the superficial velocity of interstitial flow is very low
(27).
Wang and Tarbell (27) developed a two-dimensional (2D)
model describing the medial layer of an artery as a periodic array of
cylindrical SMCs residing in a uniform matrix composed of proteoglycan and collagen fibers and simulated interstitial flow in the media. Their
results showed that under typical physiological conditions, the level
of shear stress on SMCs could be of the order of 1 dyn/cm2,
a level that has been shown to affect SMC biology (1, 15, 28). Tada and Tarbell (22) subsequently extended
the 2D modeling of interstitial flow by incorporating the complex
entrance condition induced by the funneling of flow through leaky
fenestral pores in the internal elastic lamina (IEL). Their 2D results
indicated that the average shear stress on the most superficial SMCs,
immediately below the IEL, could be 10-60 times higher than on
cells located far from the IEL.
Because of the growing awareness of the importance of fluid shear
stress in modulating SMC function and the particular relevance of the
most superficial layer in atherosclerosis and intimal hyperplasia (12, 25), the present study describes a full
three-dimensional (3D) simulation of the flow field around the SMC
right beneath a fenestral pore. We find that previous 2D calculations
provide useful upper bounds on shear stress levels, but full 3D
calculations are required to determine the distribution of shear stress
along the cell axis.
Glossary
| a |
Distance between IEL and upstream end of SMC
|
| D |
SMC diameter
|
| d |
Fenestral pore diameter
|
| F |
Volume fraction of SMC
|
| f |
Area fraction of fenestral pore
|
| Kp |
Permeability of media
|
| L |
Distance between the center of neighboring SMC
|
| l |
Fenestral pore spacing
|
| P |
Pressure
|
| Re |
Reynolds number
|
| r |
Radial coordinate
|
| U |
Superficial flow velocity
|
| U0 |
Flow velocity at fenestral pores
|
| u |
Velocity vector
|
| ux |
x-component of velocity
|
| ur |
r-component of velocity
|
u |
 -component of velocity
|
| x |
Positional vector
|
| x |
x-axis of Cartesian coordinate system
|
| y |
y-axis of Cartesian coordinate system
|
| |
Angular coordinate
|
| µ |
Viscosity
|
 |
Fluid density
|
 |
Shear stress
|
 |
Boundary layer thickness
|
 |
Dimensionless permeability
|
| wall |
Value at the SMC surface
|
| * |
Dimensionless value
|
| |
METHODS |
Numerical analysis.
In the present study, under realistic physiological flow conditions, 3D
numerical simulations of the interstitial flow in the tunica media were
performed to estimate the magnitude and spatial distribution of shear
stress on SMCs taking into account the influence of the IEL in
distributing flow to the media.
As described in detail elsewhere (23), but not
shown here, transmural volume (water) flow driven by the transvascular
pressure gradient enters the intima after passing through
interendothelial junctions and then spreads laterally in the
subendothelial intima that is separated from the media by the IEL. The
elastin-rich IEL is relatively impermeable to water flow except for its
fenestral pores, which are filled with extracellular matrix material
that is permeable to water (9).
A schematic illustration of the medial layer right beneath the
endothelium is shown in Fig. 1. The IEL
separates the subendothelial intimal from the medial layer and provides
a complex entrance flow condition through the fenestral pores. In the
present 3D computational model, the medial layer consists of a periodic
array of cylindrical SMCs embedded in a continuous, porous interstitial phase. Transmural flow is distributed into the medial layer from the
intimal layer through the fenestral pores in the IEL. The fenestral
pores are modeled as circular openings distributed in a square array
over the IEL. The IEL is assumed to be an impermeable rigid wall except
for its pore openings. Justification for this assumption has been
discussed extensively in previous studies (9, 23). SMCs
are treated as circular cylinders impermeable to fluid because of the
low hydraulic conductance of the cell membrane relative to that of
interstitium (27). Under the assumptions described above,
flow in the interstitial phase filled with fiber matrix is treated as
Newtonian fluid flow (viscosity µ) through a homogeneous porous
medium having a Darcy permeability coefficient, Kp.
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Fig. 1.
Schematic illustration of arterial medial layer and
subendothelial intima. Internal elastic lamina (IEL), an impermeable
barrier to water flow except for its fenestral openings, is depicted as
a plane sheet having circular fenestral pores. Smooth muscle cells
(SMCs) residing in the medial layer are modeled as circular cylinders
arranged in a square array configuration. Between SMCs in the medial
layer is interstitial phase containing proteoglycan and collagen
fibers. Transmural flow from lumen is concentrated at the fenestral
pores in the IEL before spreading into the medial layer.
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Mathematical formulation.
The governing equations for fluid flow are Brinkman's equation
(4), a generalization of Darcy's law for flow in
saturated porous medium, which allows for satisfaction of no-slip
boundary conditions on the surfaces of SMCs and the IEL
|
(1)
|
and the equation of continuity
|
(2)
|
In the above, 
is the Nabla operator, 
is the
Laplacian, u is the local flow velocity vector, P is the
pressure, µ is the viscosity of the fluid, and
Kp is the Darcy permeability of the fiber
matrix. The term on the left-hand side of Eq.1 is the
pressure gradient that drives the flow, the first term on the
right-hand side of Eq.1 represents the viscous force that allows satisfaction of the no-slip condition, and the last term on the
right-hand side, the Darcy-Forchheimer term, characterizes flow in the
porous medium away from the solid boundaries. After defining
dimensionless variables
|
(3)
|
where D is the diameter of the SMC, U is
the superficial velocity of the fully developed interstitial flow, is the fluid density, and x is the position vector in
Cartesian coordinates, Eqs. 1 and 2 are rewritten
in dimensionless form as
|
(4)
|
|
(5)
|
Re is Reynolds number defined as
|
(6)
|
Physiological parameters.
All of the physiological parameters used in the present study were
defined and justified in the earlier 2D study of Tada and Tarbell
(22). The SMC diameter (D) was 4.0 µm based
on data for the rabbit thoracic aorta. The volume fraction of SMC
(F) was always set at 0.4. For the fiber matrix in the
interstitial phase, the Kp was set at 1.432 × 1014 cm2, which is consistent with the
value used by Wang and Tarbell (27) based on the data of
Tedgui and Lever (24) for the rabbit aorta. The
superficial velocity of the fully developed interstitial flow, U
= 5.8 × 106 cm/s, was obtained by applying
Darcy's law to physiological data (10). The Reynolds
number (Eq. 6) was 3.4 × 107. In
addition to the constants mentioned above, the area fraction of
fenestral pores (f), the diameter of fenestral pores
(d), and the distance between the IEL and the upstream end
of the most proximal SMC (a) are important parameters
defining the flow field. According to recent experimental data (rat
thoracic aorta at 0-150 mmHg pressure) presented by Huang et al.
(8), f and d are set at 0.05 and
1.40 µm, respectively. A range of values for each of these parameters
was considered in our earlier 2D study (22), but we are
restricted here by the computational demands of 3D simulations. The
parameter a was estimated to be 0.36 µm (rabbit thoracic
aorta) (22). However, because the distance a is
thought to be the most critical parameter affecting the distribution of flow between the IEL and the first layer of SMC, the value of a was varied in the range from 0.2 to 0.8 µm to examine
its influence on the flow field and associated shear stress on the SMCs
in the immediate vicinity of the IEL. All of the constants and
parameters described above are listed in Table
1.
Numerical analysis.
The FIDAP software package (8.52, parallel computing version; FLUENT
Lebanon, NH, http://www.fluent.com) was used for the numerical
simulations. The set of governing equations was integrated using a
direct Gaussian elimination method with a structured, boundary-fitted
coordinate system. The boundary-fitted coordinate was generated by
using FI-GEN, a submodule of the FIDAP solver. The global matrix
arising from the finite element method discretization was decomposed
into smaller submatrices to save memory space. Velocity grid points
were arranged on each nodal point of the coordinate system, whereas the
pressure was solved at the centroid of every finite element hexahedral
element. In every finite element, biquadratic interpolation functions
were used to approximate the velocities. For the pressure variable, a
piece-wise constant approximation was applied. Numerical simulations
were carried out on the parallel supercomputing machine, Silicon
Graphics Origin-2000, at the National Center for Supercomputing
Applications (NCSA, Champaign, IL).
The geometric configuration of the present 3D flow model is shown in
Fig. 2. The fenestral pores were assumed
to be aligned with SMC centers thus generating the highest possible
shear stress on SMC surfaces. SMCs were arranged in a square array
configuration leading to a relatively simple computational domain.
Therefore, symmetry boundary conditions could be applied on the lateral
surfaces of the rectangular computational domain the width of which was equal to the half-spacing of the fenestral pores. At the downstream end, a gradient-free (
u*/z* = 0) outlet
boundary condition was applied. The effect of the position of the
outlet boundary on the resulting flow field was examined by increasing
the number of columns of SMC parallel to the IEL surface (e.g., moving
the boundary A to boundary B as shown in Fig. 2).
The distribution of the shear stress over the last column of SMCs was
used as an indicator of the fully developed outlet flow condition.
Results were not affected by the exit length when two or more layers of SMCs were present. The value of the shear stress distribution over the
second SMC layer, calculated by applying the outlet boundary described
as boundary A in Fig. 2, showed less than a 2% maximum difference from that of the fully developed interstitial flow field.
Therefore, the boundary A in Fig. 2 (two columns of SMC present in the computational domain) was taken as the outlet
boundary throughout the remaining numerical analysis.
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Fig. 2.
Geometrical configuration of three-dimensional (3D) interstitial
flow model. Calculation domain is indicated with dotted lines. Because
fenestral pores and SMCs are assumed to be organized in spatially
periodic square arrays for computational convenience, symmetry
conditions are imposed on lateral surfaces of the rectangular
parallelepiped the width of which is equal to the half-spacing of the
fenestral pores. In the transmural flow direction of this calculation
domain, the "exit" boundary is moved outward by successive SMC rows
(i.e., boundary A to boundary B) with the values
of shear stress along the periphery of the distal SMC being matched to
the fully developed flow values.
|
|
The complete computational domain used in the present study is shown in
Fig. 3. A uniform velocity profile was
imposed on the interstitial flow at the fenestral pore inlet. On all
surfaces, i.e., the bottom surface of IEL and the periphery of SMCs, a
no-slip boundary condition was applied. Because thin Brinkman boundary layers arise in the vicinity of no-slip surfaces, a finer mesh of grid
points taken perpendicular to surfaces was employed to ensure
sufficiently high resolution within the boundary layer of the
interstitial flow. In the present problem, the Brinkman boundary layer
thickness () is of the order 
, or
in dimensionless form, * = O(
). The order of
magnitude of * is estimated to be 1 × 103;
hence, a minimum mesh size of 1 × 105 was
taken in the proximity of both the SMC and IEL surfaces. After it was
established that the numerical results were independent of mesh
density, a computational mesh consisting of ~48,000 finite elements
was applied. As the convergence criteria for implicit Gaussian
elimination iterations, a relative error of velocity
|
(7)
|
was applied to the value of the nth iteration.
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Fig. 3.
3D computational domain. Continuous fiber matrix between
SMCs is assumed to be a homogeneous porous medium with a hydraulic
permeability of Kp = 1.432 × 1014 cm2. A uniform interstitial flow
velocity profile (U0) is specified at the exit
of the fenestral pore. Bottom of IEL and peripheral boundaries of SMCs
are treated as "no-slip" surfaces. Symmetry boundary conditions are
imposed on the lateral surfaces of the computational domain (see Fig.
2).
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|
Numerical validation.
To validate the present interstitial flow simulation method, the wall
shear stress on a SMC () was computed under the physiological condition that the SMC volume fraction (F) for an
"infinite" (without IEL) periodic square array of SMC was 0.4. The
computed results were compared with the analytic solution of Wang and
Tarbell (26), and the 2D numerical simulation result of
Tada and Tarbell (22). In the present 3D calculations,
perfect 2D profiles of the shear stress distribution (no variation of
shear stress profile along the axes of the SMC) were obtained on the
second SMC for each value of the distance a. The maximum
value of the dimensionless shear stress on the second SMC obtained in
the present 3D calculation was 2.59, whereas the analytically obtained
value for fully developed interstitial flow (26) was 2.57, about a 0.7% difference. On the other hand, the average value of shear
stress around the second SMC was 1.333, which is identical to that of
the 2D simulation result (22) and only 0.3% greater than
the analytic value (1.329) obtained by Wang and Tarbell
(26). Both comparisons show favorable agreement and
confirm the validity of the present numerical simulation procedure.
| |
RESULTS |
The mathematical model and computational methods described in the
previous sections were applied in analyzing dynamics of flow through
the medial layer near the IEL for different values of a, the
distance between the IEL and the first layer of SMCs. The incoming flow
rate at the fenestral pore (U0) used in the study was derived from the relation
|
(8)
|
where U is the superficial velocity of the interstitial
flow far removed from the IEL, l is the distance between
neighboring fenestral pores, and d is the fenestral pore
diameter. Equation 8 ensures that the volumetric flow rate
distal to the IEL is equal to that supplied from the fenestral pores
(see Fig. 3). The velocity profile at the fenestral pore was assumed to
be a uniform flow distribution. This assumption is reasonable because
the flow velocity at the pore exit has a flat distribution except in
the immediate vicinity of the pore edge within the Brinkman boundary
layer (22). The distance between neighboring SMCs
(L) was determined from the volume fraction of the SMC
(F) and the SMC diameter (D) as
|
(9)
|
L = 5.6 µm was determined by using the physiological
values F = 0.4 and D = 4.0 µm. The
fenestral pore spacing (l) was taken to be the same as
L. The use of this assumption and a fenestral pore diameter
of 1.4 µm leads to a calculated fenestral pore area fraction of
f = 0.049, which is physiological (8).
Shear stress distributions on SMC.
Shear stress values are nondimensionalized with respect to the
reference value introduced by Wang and Tarbell (26)
|
(10)
|
where µ, U, and Kp are water
viscosity, the superficial flow velocity, and hydraulic permeability of
the media, respectively. The nondimensional local shear stress on the
surface of the SMC is defined as
|
(11)
|
where and
x are defined as
|
(12)
|
where the asterisk indicates that the value is
nondimensional, and the subscript "Wall" represents the value taken
on the SMC surface (r* = 1). u is
the angular component of the velocity (see Fig. 3), and
ur is the radial component of the velocity when
the origin of the polar coordinate system is taken at the center of the
SMC. 
is calculated to first-order
discretization accuracy using the numerically obtained velocity
variables uy and uz.
Moreover, the wall shear stress averaged over the entire SMC surface is
defined as
|
(13)
|
where the integration is taken with respect to the unit length of
cylindrical SMC surface (0 
2
, 0 x 2.8 µm, r = 2 µm).
Figure 4 shows a perspective rendering of
the shear stress distributions over a SMC for three different values of
the distance between the IEL and the upstream end of the first SMC,
a. The color scale represents the magnitude of the local
shear stress () over the cell
surface. For each value of a, a focal elevation of shear
stress occurs on the upstream portion of the first SMC, because this
surface is exposed to high-velocity flow issuing from the fenestral
pore (fenestral pores are indicated with thick solid lines drawn on the
IEL). Because the point on the surface of the SMC right beneath the
fenestral pore center is a stagnation point of the interstitial flow,
the wall shear stress is zero at this point. Moving away from the
stagnation point, wall shear stress increases steeply along the
direction of flow but then decreases as fluid spreads into the media.
The maximum value of the shear stress on the surface of the first SMC
(above * = 18 for a = 0.2 µm) appears within the
projection of the fenestral pore periphery onto the SMC surface, and
this value decreases markedly with an increase in the distance
a. Hence, the influence of the IEL on the shear stress
distribution over the SMC extinguishes within a spatial length scale of
the order of the fenestral pore diameter (
1 µm in this case). The
shear stress on the second SMC is much lower than on the first and
exhibits the same magnitude and distribution as that on SMCs exposed to fully developed interstitial flow. Thus both the shear stress level and
spatial distribution of shear stress on the first SMC are greatly
altered due to the presence of the IEL.
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Fig. 4.
Perspective rendering of shear
stress distributions over SMCs for 3 different values of the distance
between IEL and upstream end of the first SMC, a. Color
scale represents the magnitude of the dimensionless local shear stress
(*) over the SMC surface. For reference, the magnitude of
the average shear stress on SMC exposed to fully developed interstitial
flow is * = 1.3 ( 0.7 dyn/cm2). Maximum
values of * are 18, 10, and 4 for a = 0.20 (A), 0.36 (B), and 0.80 µm (C),
respectively. Elevation of shear stress occurs on the upstream surface
of the first SMC where the cell is exposed to concentrated flow issuing
from the fenestral pore.
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Figure 5 shows the distribution of
around the periphery of the first
and second SMCs at x* = 0 (the fenestral pore center axis)
for three different values of the distance between the bottom of IEL
and upstream end of the first SMC. The circumferential coordinate is taken in the counterclockwise direction from the downstream end of
the SMC as indicated in Fig. 5. The flow direction is also indicated
with an arrow in the figure. To obtain a sense of the dimensional
magnitude of , the normalization
factor in Eq. 10,
µU/ takes on a value of
0.52 dyn/cm2 based on data for the rabbit thoracic aorta
(24). For each value of a, the maximum shear
stress occurs at 7/8, on the upstream side of the first SMC.
The maximum value predicted in the present study is ~9.0
dyn/cm2 for a = 0.2 µm. This order of
magnitude corresponds to shear stress on endothelial cell surfaces
exposed to blood flow. On the downstream side, for 5/8,
the shear stress profiles are independent of a and identical
to the distribution on the second SMC, indicating that the entrance
effect associated with the fenestral pore has been dissipated
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Fig. 5.
Local shear stress distribution
() around the periphery of the first
and second SMCs at x* = 0 for three different values of the
distance between the bottom of IEL and upstream end of the first SMC,
a. Circumferential coordinate is taken in a
counterclockwise direction from the downstream end of SMC. Flow
direction is indicated with an arrow. d, Fenestral pore
diamter; f, area fraction of fenestral pore; F,
volume fraction of SMC.
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The shear stress distribution transverse to the primary flow direction
is of particular interest because this could not be estimated in our
previous 2D calculations (22). For exactly the same
interstitial flow conditions, the shear stress distribution along the
leading edge (meridian at = ) of the first SMC is shown in
Fig. 6. As expected, the shear stress
gradient is steeper for smaller values of a, and the maximum
shear stress is higher for smaller a. When Figs. 5 and
6 are compared, it is clear that the fenestral pore has a similar
effect on shear stress distribution over the first SMC surface in both
the axial and circumferential direction, but the maximum shear stress
in the direction of the SMC axis is slightly lower than in the
circumferential direction
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Fig. 6.
Local shear stress distribution along the leading edge
(meridian) of the SMC for 3 different values of a.
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Figure 7 shows the variation of the
average (Eq. 13) and maximum shear stress on the first SMC
as the distance a changes. Elevation of the local shear
stress on the first SMC due to concentrated flow through the fenestral
pore does not alter the average shear stress very much because there
are also regions of low shear stress (e.g., near stagnation points).
The average shear stress, therefore, attains a low value (order of 1 dyn/cm2) regardless of the distance a. The
results in Fig. 7 demonstrate that the influence of the IEL on the
shear stress distribution over the first SMC is extinguished at
distances beyond about 1 µm from the IEL. For spacing greater than
a 1 µm, the magnitude of the maximum and average shear
stress are on the order of 1 dyn/cm2, corresponding to
fully developed interstitial flow.
Comparison with 2D simulations.
A 3D calculation requires far greater memory capacity and computation
time than a 2D calculation, even for the idealized 3D geometry
demonstrated in the present study. Therefore, comparisons between
results obtained using the 2D model previously proposed by Tada and
Tarbell (22) and those of the present 3D model are shown
in Figs. 8 and 9 to assess the accuracy of 2D predictions. The
physiological parameters U, d, f, and
F applied in the 2D calculation by Tada and Tarbell
(22) are 5.8 × 106 cm/s, 0.8 µm,
0.016, and 0.4, respectively. These parameters again correspond to a
case where the fenestral pores are centered on the SMC axes. The same
value of U0 (Eq. 8) was used in both the 2D and 3D calculations.
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Fig. 8.
Comparison of the 3D and 2D shear stress distributions
around the periphery of first SMC at x* = 0 for a
= 0.36 µm. Circumferential coordinate follows the convention
defined in Fig. 5.
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Fig. 9.
Comparison of maximum shear stress predicted by 3D and 2D
computations for different values of a.
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Figure 8 compares the 2D and 3D
distribution of around the
periphery of the first SMC at x* = 0 for the case a
= 0.36 µm. The 2D and 3D shear stress profiles are of similar
shape; both profiles display a maximum at 7/8 and another
local maximum at /2 before monotonically decreasing to zero
at 0. The level of the shear stress predicted in the 3D model,
however, is lower than that of the 2D model. Because fluid is not
allowed to flow out of the SMC cross-sectional plane in the 2D model, a
higher level of fluid velocity and shear stress are induced over the
SMC surface relative to the 3D model, which allows for the spreading of
fluid along the SMC axis.
The difference between the 2D and 3D predictions depends strongly on
the distance between the IEL and the upstream end of the first SMC
(a). Figure 9 shows a
comparison of the maximum local shear stress as a function of
a for the 2D and 3D models. For both the 2D and the 3D
calculations, the maximum shear stress decreases with increasing
a, showing a similar trend. However, the discrepancy between
the 2D and 3D predictions of 
becomes
larger with increasing a. For instance, when
a = 0.2 µm, the 2D model predicts
55, which is about 1.5 times
greater than the value predicted by the 3D model. When
a = 0.8 µm, the 2D model predicts
17, which is about fourfold
greater than the 3D prediction. With these comparisons in mind, it
should be possible to estimate the influence of fenestral pore diameter
(d) and area fraction (f) on
maximum shear stress values by referring to the 2D simulation results
in (22).
It should be pointed out that in another 2D study of flow distribution
to a network of fenestral pores (23), we modeled the pores
as rectangular slits having the same area as the circular pores. That
model is equivalent to assuming the flow has been spread over the
length of the SMC upon entrance to the media. This model results in
lower shear stresses than the 3D predictions but will not be considered here.
| |
DISCUSSION |
The presence of IEL with fenestral pores can greatly alter the
flow field around the SMCs bordering the subendothelial intima. Because
the area fraction of fenestral pores is in the range 0.002-0.01, and the mean diameter of the fenestral pores ranges from 0.4 to 2.1 µm (8, 17), the velocity of water issuing from an
individual pore could be more than 100-fold greater than the
superficial flow velocity in the medial layer, resulting in a
significant change in the flow and shear stress distribution on the
surface of SMCs lying in the immediate vicinity of the IEL. The
elevated shear stresses on the cells in the most proximal layer may
affect SMC biology and could play a role in SMC proliferation and
migration that occur in atherosclerosis and intimal hyperplasia
(20).
In this paper, particular emphasis was placed on determination of the
shear stress distribution on the 3D SMC, which differs significantly
from that of the 2D model reported previously by Tada and Tarbell
(22). The detailed 3D simulation provides more definitive
data on the distribution and magnitude of shear stress on the surface
of SMCs residing right beneath the fenestral pore entrance.
Depending on the distance between the IEL bottom surface and the
upstream end of this proximal layer of SMCs (a), local shear stress on the most proximal layer of cells could be 3-14 times higher than on the cells far removed from the IEL, which have been
estimated to experience an average shear stress on the order of 1 dyn/cm2 (23). The 2D results in Ref.
22 indicate even higher levels of shear stress elevation
on the first SMC for fenestral pore diameters, and area fractions
smaller than were considered in the present study (d = 1.4 µm; f = 0.05). Thus the first layer of SMCs may experience
shear stress levels that are even higher than endothelial cells exposed
to normal blood flow (order of 10 dyn/cm2)
(13). On the other hand, the simulations show that the
effect of the IEL on shear stress levels is reduced sharply with
increasing values of a. For a greater than 1
µm, both the shear stress level and the shear stress distribution on
the first SMC approach the characteristics of SMCs far removed from the
IEL. In contrast to the significantly elevated SMC shear stress in the
focal area right beneath the fenestral pore, the shear stress averaged
with respect to the whole cell surface remains on the order of 1 dyn/cm2. This indicates the presence of steep spatial
gradients of shear stress on the SMC surface due to the jet issuing
from the fenestral pore. The calculated maximum shear stress gradient
on the first layer of SMCs is 200
dyn · cm2 · µm1, whereas
for the distal layers the maximum gradient is only 1 dyn · cm2 · µm1. It is
interesting to note that the maximum shear stress gradient over the
surface of endothelial cells in blood flow has been computed to be on
the order of 5 dyn · cm2 · µm1
(2). Although there is no direct evidence available to
indicate that shear stress gradients affect SMC biology, analogies to
endothelial cells in which the cell-scale variations in shear stress
may activate mechanotransduction signaling path ways (2,
6) suggest a potential role for shear stress gradients on SMC.
Recent in vitro studies have shown that vascular smooth muscle cells
are responsive to shear stress in the range 1-25
dyn/cm2 and increase their synthesis of transforming growth
factor-
, tissue plasminogen activator (25), heme
oxygenase-1 (26), nitric oxide (7, 15),
prostaglandins (1, 28), and basic fibroblast growth factor
(16). These results and our simulations suggest that in a
normal artery with intact IEL, the innermost layer of SMC in the
intimal-medial border may be the most active biochemically because of
elevated shear stresses. In addition, several in vitro studies have
shown that shear stress is an inhibitor of vascular SMC proliferation
(21, 25) and migration (18). Other in vivo
studies in balloon-injured rabbit carotid arteries (3)
have demonstrated that increasing blood flow on the injured artery
inhibits matrix metalloproteinase-2 mRNA and intimal hyperplasia. Taken
together, these studies suggest that high levels of shear stress on SMC
are beneficial because they suppress SMC proliferation and migration,
which might otherwise contribute to intimal hyperplasia.
| |
ACKNOWLEDGEMENTS |
This work was supported by National Heart, Lung, and Blood
Institute Grant HL-35549.
| |
FOOTNOTES |
Address for reprint requests and other correspondence: J. M. Tarbell, 155 Fenske Laboratory, The Pennsylvania State Univ., Univ.
Park, PA 16802-4400 (E-mail: jmt{at}psu.edu).
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.
10.1152/ajpheart.00751.2001
Received 21 August 2001; accepted in final form 31 October 2001.
| |
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Am J Physiol Heart Circ Physiol 282(2):H576-H584
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