Vol. 278, Issue 5, H1589-H1597, May 2000
Interstitial flow through the internal elastic lamina affects
shear stress on arterial smooth muscle cells
Shigeru
Tada1 and
John M.
Tarbell2
1 Energy Phenomena Laboratory, Department of
Mechanical Engineering and Science, Tokyo Institute of Technology,
Tokyo 152-8552, Japan; and 2 Biomolecular
Transport Dynamics Laboratory, Chemical Engineering and
Bioengineering Department, The Pennsylvania State University,
University Park, Pennsylvania 16802-4400
 |
ABSTRACT |
Interstitial flow through the tunica media of an artery
wall in the presence of the internal elastic lamina (IEL), which
separates it from the subendothelial intima, has been studied
numerically. A two-dimensional analysis applying the Brinkman model as
the governing equation for the porous media flow field was performed. In the numerical simulation, the IEL was modeled as an impermeable barrier to water flux, except for the fenestral pores, which were uniformly distributed over the IEL. The tunica media was modeled as a
heterogeneous medium composed of a periodic array of cylindrical smooth
muscle cells (SMCs) embedded in a fiber matrix simulating the
interstitial proteoglycan and collagen fibers. A series of calculations
was conducted by varying the physical parameters describing the
problem: the area fraction of the fenestral pore (0.001-0.036),
the diameter of the fenestral pore (0.4-4.0 µm), and the
distance between the IEL and the nearest SMC (0.2-0.8 µm). The
results indicate that the value of the average shear stress around the
circumference of the SMC in the immediate vicinity of the fenestral
pore could be as much as 100 times greater than that around an SMC in
the fully developed interstitial flow region away from the IEL. These
high shear stresses can affect SMC physiological function.
fenestral pore; numerical analysis
| |
INTRODUCTION |
THE WALL SHEAR STRESS of flowing blood on the
endothelial lining layer of blood vessel walls has been studied
extensively because of its central role in the maintenance of vascular
tone, vascular remodeling, and localization of atherosclerosis (6, 16).
Mean values of wall shear stress on endothelial cells are on the order
of 10 dyn/cm2 in arteries and may be higher in capillaries
and lower in postcapillary venules (13). In a previous study, Wang and
Tarbell (22) revealed that smooth muscle cells (SMCs) could be
subjected to significant levels of shear stress associated with normal
transmural interstitial flow, even though the superficial velocity of
interstitial flow is very low (on the order of 10
6
cm/s). In their analytic model, they assumed uniform superficial velocity at the upstream end of an array of cylindrical SMCs, neglecting the more complex entrance conditions that exist at the
intimal-medial boundary, to estimate the magnitude of the shear stress
imposed on the SMCs suspended in a fiber matrix. Their results show
that, under physiological conditions, wall shear stress on SMCs is on
the order of 1 dyn/cm2, a level that has more recently been
shown to affect SMC biology (1, 14).
However, for the SMCs bordering the subendothelial intima, the presence
of internal elastic lamina (IEL) with leaky fenestral pores can greatly
alter the flow field around the boundary SMCs. Because the area
fraction of fenestral pores is in the range 0.002-0.01 and the
mean diameter of the fenestral pores varies from 0.4 to 2.1 µm (8),
the velocity of fluid issuing from an individual pore could be 100-fold
greater than the superficial flow velocity in the tunica media,
resulting in a significant change in the distribution of flow and
associated shear stress on the most superficial layer of SMCs that lie
just beneath the IEL. The elevated shear stresses on the cells in this
layer may affect SMC biology and could play a role in SMC proliferation
and migration that occur in atherosclerosis and intimal hyperplasia
(17).
In this study, under realistic physiological flow conditions,
two-dimensional numerical simulations of the interstitial flow in the
tunica media were performed to estimate the magnitude and spatial
distribution of wall shear stress on SMCs, with the influence of the
IEL on distribution of the flow to the media taken into account. In
addition, the effects of several physiological parameters, the area
fraction of the fenestral pore, the fenestral pore diameter, and the
distance between the IEL and the first SMC, were studied.
Glossary
| a |
Distance between IEL and upstream end of SMC (m)
|
| D |
SMC diameter (m)
|
| d |
Fenestral pore diameter (m)
|
| F |
Volume fraction of SMC
|
| f |
Area fraction of fenestral pore
|
| Kp |
Permeability of media (m2)
|
| Kp eff |
Effective permeability of media (m2)
|
| L |
Distance between the centers of neighboring SMCs (m)
|
| l |
Fenestral pore spacing (m)
|
| P |
Pressure (Pa)
|
| Re |
Reynolds number (=  UD/µ)
|
| r |
Radial coordinate (m)
|
| U |
Mean interstitial flow velocity (m/s)
|
| U0 |
Flow velocity at fenestral pores (m/s)
|
| u |
Velocity vector (m/s)
|
| u |
x component of velocity (m/s)
|
| ur |
r component of velocity (m/s)
|
u |
 component of velocity (m/s)
|
| x |
Positional vector (m)
|
| x |
x-Axis of Cartesian coordinate system (m)
|
| y |
y-Axis of Cartesian coordinate system (m)
|
 |
Boundary layer thickness (m)
|
| |
Angular coordinator
|
| µ |
Viscosity (N · s · m2)
|
| |
Fluid density (kg/m3)
|
 |
Shear stress (N/m2)
|
 |
Dimensionless permeability
|
| Wall |
Value at the SMC surface
|
| * |
Dimensionless value
|
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NUMERICAL ANALYSIS |
A schematic illustration of the typical structure of an arterial wall
is shown in Fig. 1. In the intact vessel
there is a pressure gradient between the blood and the tissue
surrounding the vessel that drives transmural flow across the wall and
exposes the SMCs to interstitial flow shear stress. SMCs are embedded in a tissue matrix beneath the subendothelial intima in most blood vessels and, under normal circumstances, are shielded from the direct
shear forces of flowing blood. The IEL separates the subendothelial intimal layer from the medial layer and provides a complex
entrance flow condition through the fenestral pores.
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Fig. 1.
Transverse sectional view of a blood vessel wall. Bulk of blood vessel
wall consists of media and adventitia. There are some capillaries
(e.g., vasa vasorum) in adventitia. Innermost layer of vessel wall
(intima) is a single layer of endothelial cells and a thin layer of
basement matrix. Internal elastic lamina (IEL), an impermeable barrier
to water flow except for fenestral pores, separates intima from media.
Between smooth muscle cells (SMCs) in media is interstitial phase
containing proteoglycan and collagen fibers.
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To obtain a basic understanding of the flow through the medial layer,
Wang and Tarbell (22) performed an analytic study of the
two-dimensional, steady-state problem by applying Brinkman's model (4)
to describe flow through an infinite periodic array of cylindrical
SMCs. In the present study we consider the same array of cylindrical
SMCs but take into account the fact that the fluid enters the medial
layer of the artery through the fenestral pores, which are randomly
distributed over the IEL. The interstitial flow field in the region
immediately below the IEL is investigated numerically.
The medial layer of the blood vessel wall is modeled as a heterogeneous
medium composed of a periodic square array of cylindrical SMCs and a
continuous interstitial fluid phase filled with proteoglycan and
collagen fibers modeled as a uniform fiber matrix (22). Transmural flow
is distributed into the medial layer from the intima by passing through
the fenestral pores of the IEL. The IEL is assumed to be an impermeable
rigid wall except for its pore openings. SMCs are treated as obstacles
impermeable to fluid because of the low hydraulic conductance of the
cell membrane relative to that of interstitium (22).
Mathematical formulation of the transmural flow problem.
In the present analysis, flow in the interstitial phase filled with
fiber matrix is modeled as Newtonian fluid flow (viscosity, µ)
through a homogeneous porous medium having a Darcy permeability coefficient, Kp. The geometry of the
two-dimensional flow problem is shown in Fig.
2. The fenestral pore center is aligned
with the SMC center to investigate the extreme case in which the shear stress around SMCs attains the maximum possible value. Although in real
arteries there can be fenestral pores that are not aligned with the SMC
array, this case has not been taken into account in the present
calculation, because it is expected to produce minor changes in shear
stress relative to the fully developed flow values computed by Wang and
Tarbell (22). SMCs are arranged in a square-array configuration. This
is an obvious idealization, but it is not far from real features,
because the flow field is not very sensitive to array structure unless
the volume fraction of SMCs is high (22).
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Fig. 2.
Geometrical model for numerical simulation of interstitial flow in
media. Continuous fiber matrix between SMCs is assumed to be a
homogeneous porous medium with hydraulic permeability
(Kp). SMCs are rigid circular cylinders, arranged
in a square array. Fluid is distributed to media through fenestral
pores in IEL. See Glossary for definitions.
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Once the SMC diameter (D) and volume fraction for SMC (F) are
given, the distance between centers of neighboring SMCs (L) can
be calculated. In the same manner, the pore spacing (l) is also
calculated when the pore diameter (d) and area fraction of pores (f) are given. In the present calculation, four different values
of l (calculated from 4 different values of f) are selected (Fig. 3). In Fig. 3, A-D, when
d is kept constant, f is changed according to the following
definition: f = 
d2/4l2. On
the other hand, when we examine the effect of d under the condition of a constant f, values of l and d are
changed simultaneously. Moreover, the distance between the IEL and
upstream end of the SMC (a) is varied in accordance with recent
experimental data.
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Fig. 3.
Schematic illustration showing various geometric configurations of
fenestral pores used in calculation for different fenestral pore area
fractions: f = 0.0160 (A), f = 0.0040 (B), f = 0.0017 (C), f = 0.0010 (D). See Glossary for
definitions.
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|
To simplify the problem to be solved, the following additional
assumptions are invoked: 1) The flow is steady and
perpendicular to the longitudinal axis of the cylindrical SMCs.
2) The IEL is a rigid impermeable wall with zero thickness
except for its fenestral openings. 3) The circular
fenestral pores are distributed with a periodic square-array
configuration over the IEL surface. 4) The no-slip condition is
applied on all surfaces of the SMCs and IEL. 5) A
uniform velocity profile is specified at the entrance to the fenestral pore.
The governing equations are Brinkman's equation (4), a generalization
of Darcy's law for flow in a saturated porous medium that allows for
satisfaction of no-slip boundary conditions
|
(1)
|
and
the equation of continuity
|
(2)
|
where
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, which drives the flow, the first term on the
right-hand side represents the viscous term, which allows satisfaction
of the no-slip condition, and the last term on the right-hand side represents the Darcy-Forchheimer term, which characterizes flow in the
porous medium away from the solid boundaries.
After definition of 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)
|
where
Re is Reynolds number defined as
|
(6)
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Physical parameters and constants.
To determine physiological values of constants and parameters for the
numerical analysis, data for transmural flow in the rabbit thoracic
aorta have been employed. Typical values of parameters defining the
flow problem are taken from a range of experimental data available from
Huang and Tarbell (10). For the fiber matrix in the interstitial phase,
the value of hydraulic permeability is Kp = 1.432 × 1018 m2, which is consistent
with the value used by Wang and Tarbell (22) based on the data of
Tedgui and Lever (19) for the rabbit aorta. The mean U is
obtained from Darcy's law
|
(7)
|
where
Kp eff is the effective permeability defined by
Wang and Tarbell and 
P/x is pressure gradient along the
flow direction. The effective permeability is given by Wang and Tarbell
|
(8)
|
where
F is the volume fraction of the SMC defined in terms of D and
L (Fig. 2) and O represents order of magnitude
|
(9)
|
Taking
F = 0.4, µ = 6.86 × 103
N · s · m2
(viscosity of water at 36.9°C), and P/x = 0.5 mmHg/µm (10), we obtain U = 5.8 × 108 m/s from Eq. 7.
Besides 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 SMCs (a)
are considered to be important parameters defining the flow field.
According to experimental data (rat thoracic aorta) at 0 mmHg pressure
reported by Huang et al. (8, 9) and Roach and Song (15), f is
0.002-0.01 and d is 0.4-2.1 µm. The parameter a was estimated to be 0.36 µm from electron micrographs of
rabbit aortic intima (Y. Huang, personal communication). To investigate the influence of these parameters on the flow field and associated shear stress on SMCs, we consider the values of f, d, and
a within the ranges 0.001-0.036, 0.4-4.0 µm, and
0.2-0.8 µm, respectively. All the constants and parameters
described above are listed in Table 1.
Computational method.
The FIDAP software package (version 7.62, FLUENT) was used for the
numerical simulations. The set of governing equations was solved by
direct Gaussian elimination. The global matrix arising from the finite
element method discretization was decomposed into smaller submatrices
to save memory space. Numerical simulations were carried out on the IBM
RS/6000 SP system at the Penn State Center for Academic Computing. One
of the finite element mesh realizations used in our calculations is
shown in Fig. 4. The curvilinear grid
system was generated by using FI-GEN, which is a submodule of the FIDAP
solver. Velocity grid points are arranged on each nodal point of the
grid system, while the pressure is solved at the centroid of each cell.
In the vicinity of solid boundaries, the flow is dominated by
viscosity. Therefore, a finer mesh of grid points taken perpendicular
to each SMC surface was employed to ensure sufficiently high resolution
within the boundary layer.
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Fig. 4.
One of computational grid configurations used in present calculation.
This grid system consists of 12 subdomains of half-space SMC and 1 rectangular subdomain with finer-mesh resolution at fenestral pores.
Number of SMC subdomain elements is changed according to fenestral pore
spacing. When pore spacing is larger, number of SMCs between
neighboring pores is increased (see Fig. 3).
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In the present problem, the Brinkman boundary layer thickness () is
on the order of
or, in
dimensionless form, * = O(
). * is estimated to
be 1 × 103; hence, a minimum mesh size of 1 × 105 was taken next to the SMC and IEL
surfaces. After it was established that the numerical results were
independent of mesh density, a computational mesh consisting of 2,400 (40 × 60, radial × circumferential) four-node quadrilateral
elements was applied for the half-space of each SMC. The total mesh
size was changed according to the combination of f and d
values. The maximum mesh size was ~75,000 (containing 6 × 5 half-space SMC units); on the other hand, the minimum mesh size was
~16,000 (containing 6 × 1 half-space SMC units). As the
convergence criteria for implicit Gaussian elimination iterations, a
relative error of velocity
|
(10)
|
was
applied on the value of the nth iteration. In each element,
biquadratic interpolation functions were used to approximate the
velocities. For the pressure variable, a piecewise constant approximation was applied. A symmetry boundary condition was applied on
the top and bottom end lines of the computational domain. At the
downstream end, a gradient-free (u*/x* = 0) outlet boundary condition was applied. The effect of the location of
the outlet boundary condition on the resulting flow field was examined
by increasing the number of columns of SMCs parallel to the IEL
surface. The vertical (y) component of velocity at the
downstream end of the last column of SMCs was used as an indicator of
the fully developed outlet flow condition. Results were independent of
exit length when six or more layers of SMCs were employed.
Numerical validation.
To validate the present interstitial flow simulation method, the wall
shear stress on an SMC () was computed for several values of the SMC
volume fraction (F) for an infinite periodic square array of SMC (with
no IEL). The computed results were compared with the analytic solution
of Wang and Tarbell (22).
The local shear stress is defined as
|
(11)
|
where
Wall represents the value taken on the SMC surface (r* = 1),
u is the angular component of the velocity, 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; local is calculated to first-order
discretization accuracy by use of the numerically obtained velocity
variables ux and
uy.
The wall shear stress averaged around an SMC is defined as
|
(12)
|
Furthermore,
is nondimensionalized by using the reference shear stress defined
by Wang and Tarbell (22) as
|
(13)
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Figure 5 shows a comparison of the
numerical results with the analytic result of Wang and Tarbell (22)
calculated from the following equation
|
(14)
|
For
smaller values of F, the agreement is very good. However, for the
largest value of F simulated, the numerical shear stress is slightly
greater than the analytic result. We attribute the deviation to
truncation error in the analytic infinite series expansion.
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Fig. 5.
Relationship between averaged shear stress over SMCs (*) and volume
fraction of SMCs (F) for fully developed interstitial flow.  ,
Present numerical results; solid line, analytic result of Wang and
Tarbell (22).
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RESULTS AND DISCUSSION |
The incoming flow velocity at each fenestral pore
(U0) is derived from the relation
|
(15)
|
which
ensures that the volumetric flow rate far away from the IEL is equal to
that supplied by the fenestral pores. A schematic view of the
relationship between the superficial velocity (U) and
U0 is shown in Fig. 6.
For all results to be presented, U is taken to be 5.8 × 108 m/s.
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Fig. 6.
Relationship between mean velocity downstream and inlet flow velocity
at fenestral pore. See Glossary for definitions.
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To reveal basic characteristics of the interstitial flow, intermediate
values of the model parameters were used to compute the results
displayed in Fig. 7: d = 0.4 µm,
f = 0.004, and a = 0.36 µm. F was maintained at 0.4 for all calculations, because it was expected to have a minor influence
on the shear stress distribution over SMC near the IEL (Fig. 5).
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Fig. 7.
Streamline contour plot of interstitial flow in media. Flow direction
is from left to right. Fluid is distributed from 2 fenestral pores at
top and bottom end of IEL at left end of flow domain.
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Characteristics of the interstitial flow.
Figure 7 shows a streamline contour plot of the interstitial flow in
the media. The fluid coming from the fenestral pores (located at the
top and bottom of the IEL at the left end of the computational domain)
is distributed into the whole region of the media. Near the IEL, fluid
spreads laterally into the upstream region of the media as soon as it
enters from the fenestral pore, in contrast to a conventional Stokes
fluid flow with the same Re (i.e., 1.657 × 107). The Darcy resistance tends to make the jet
broader, as discussed by Friedrich and Rudraiah (7). The flow is
symmetrical about the center line in the flow direction, because the
volumetric flow rate at the entrance of each fenestral pore is the
same. At the narrowest point in the path between neighboring SMCs, the intervals between streamlines are almost equal, suggesting that a
uniform velocity distribution is established, except in a thin boundary
layer near the SMC surfaces. This feature is also quite different from
conventional Stokes flow, in which the fluid displays a parabolic
velocity distribution. Furthermore, streamlines penetrate more deeply
into the wake region behind each SMC, in contrast to those of Stokes flow.
Changes of shear stress for different pore diameter.
As suggested by Fig. 7, the shear stress on the first SMC is elevated
as a result of the concentrated flow entering from the fenestral pore.
On the other hand, the second to the sixth SMCs have an almost
identical distribution. Clearly, the existence of the fenestral pore
significantly affects shear stress on the SMC located directly beneath
the pore but has no effect on the shear stress distribution around the
second and more distal SMCs. Therefore, we will focus on the first SMC,
which has elevated shear stress.
Figure 8 shows the local shear stress
distribution around the circumference of the SMC directly beneath a
fenestral pore for four different values of d. F and f are kept
constant at 0.4 and 0.004, respectively;
*local is the
nondimensional shear stress defined in Eq. 13. To obtain a
sense of the dimensional magnitude of these shear stresses, the
normalization factor,
µU, takes on a value of 0.52 dyn/cm2 on the basis of
interstitial flow data in the rabbit thoracic aorta, as discussed
earlier (19). The value of * averaged over an SMC in the fully
developed flow regimen is ~1.3 (Fig. 5). So we are observing a
significant elevation of shear stress on the first SMC. A constant
value of f implies that the incoming flow velocity at the pore entrance
is constant, independent of d. It is interesting that the
maximum value of the local shear stress appearing on the upstream side
of the SMC increases with d. This reflects the fact that the
inflow jet impinges on a larger fraction of the SMC surface as
d increases. In addition, a qualitative change in the
distribution of shear stress is also observed. The shear stress
distribution changes from a complex one having two maxima and a minimum
to a triangular shape having a steep gradient at the upstream side as
d becomes larger. The position on the SMC surface where the
maximum shear stress appears also changes. It moves downstream as
d becomes larger. Clearly, the spatial gradient of shear stress
(*local) becomes
large at the leading edge of the SMC. The largest value of
*local is 750 for
d = 1.6 µm;
*local = 1 corresponds to 0.26 dyn · cm2 · µm1.
By comparison, the maximum value of
*local for the
fully developed flow is only ~3.2. Clearly, the elevation of shear
stress gradient on the first cell is very significant.
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Fig. 8.
Local shear stress distribution around circumference of SMC directly
beneath fenestral pore for 4 different values of diameter of fenestral
pore. Volume fraction of SMC and area fraction of fenestral pore are
kept constant at 0.4 and 0.004, respectively. is taken
counterclockwise from downstream end of SMC.
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* varies almost linearly with the variation of d (Fig.
9). This suggests that * around the
first SMC is proportional to the area that the fenestral pore projects
onto the surface of the SMC in the streamwise direction.
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Fig. 9.
Relationship between averaged sheer stress (*) and pore diameter for
4 different values of fenestral pore diameter.
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To obtain a more detailed understanding of the shear stress behavior,
which depends strongly on d, another set of calculations for a
large f (i.e., 0.016) was performed. d was varied from 0.8 to
4.0 µm, which corresponds to the diameter of the SMC. The local shear
stress distributions (not shown) are similar in shape to those in Fig.
8. The largest value of
*local is 450 for
d = 3.2 µm.
Figure 10 shows the relationship between
* and d for the case F = 0.4 and f = 0.016. The value of
* increases linearly with d for d < 2.4 µm;
however, it decreases at d = 4 µm after displaying a maximum
at d = 3.2 µm. From these results, we can see that
* increases with increasing d until d approaches the
SMC diameter. Beyond that, * will decrease with increasing
d. Ultimately, for d 
D (as in a damaged
IEL), * should asymptotically approach the fully developed flow
value (without the IEL) given in Fig. 5.
Changes of shear stress for different pore area fraction.
Figure 11 shows the relationship between
* on the SMC directly beneath a fenestral pore and f, for constant
values of F and d. The local shear stress distributions (not
shown) are similar in shape to those shown in Fig. 8. The largest value
of *local is 2,600 for f = 0.001. The value of * varies with
1/f2, as expected. This means that the magnitude of * is
simply proportional to the velocity of incoming flow approaching the
SMC. For f = 0.001, * is significantly elevated, up to 135. This
corresponds to a value ~100 (135/1.3) times greater than experienced
by SMC in the fully developed flow field away from the IEL. In other words, a very strong shear force can be imposed on the SMC directly beneath the pore, particularly when f is sufficiently small.
Changes of shear stress for different spacing between IEL and SMC.
The distance between the IEL and upstream end of the nearest SMC
(a) is expected to be one of the most important parameters that
affect the shear stress distribution over an SMC. The value of
a = 0.36 µm was chosen as a reference value on the basis of the scanning electron microscope data (rat thoracic aorta) provided by
Dr. Yaqi Huang (personal communication). However, inasmuch as a
may vary in different blood vessels and in different species, we
simulated three different values (0.2, 0.36, and 0.8 µm) to examine
the dependence of the shear stress behavior on a.
Figure 12 displays the shear stress
distribution around the circumference of the SMC directly beneath the
fenestral pore for three different values of a. A significant
change in the shear stress distribution appears at the upstream side of
the SMC surface when a varies. The magnitude of
*local changes
drastically in the region > 3/4. The location where
*local takes on its
maximum value moves slightly toward the upstream end as a
becomes smaller. *local takes on a
maximum value of 1,200 for a = 0.2 µm.
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Fig. 12.
Local shear stress distribution around circumference of SMC directly
beneath a fenestral pore for 3 different values of spacing parameter
(a). Volume fraction of SMC, diameter of fenestral pore, and
area fraction of fenestral pore are kept constant at 0.4, 0.4 µm, and
0.004, respectively. is taken counterclockwise from downstream end
of SMC.
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Figure 13 shows the relationship between
* and a; * increases nonlinearly as a decreases.
This reflects the fact that when a is large, the flow velocity
just outside the boundary layer on the first SMC is reduced, because
there is a greater distance over which the flow can spread laterally;
when a is small, the fluid can maintain a higher velocity.
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Fig. 13.
Relationship between * and distance between IEL and upstream end of
SMC. Volume fraction of SMC, diameter of fenestral pore, and area
fraction of fenestral pore are kept constant at 0.4, 0.4 µm, and
0.004, respectively.
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Concluding remarks.
The two-dimensional numerical simulations of interstitial flow
immediately below the IEL reveal a fluid mechanical environment around
the most proximal layer of SMCs that is quite distinct from that around
the more distal cells. Depending on f and d, the average shear
stress on this proximal layer could be 10-100 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 (22). Thus the first layer of SMCs may experience
shear stress levels that are even higher than that of endothelial cells
exposed to normal blood flow [on the order of 10 dyn/cm2 (13)].
The shear stress gradients on the leading edge of the first layer of
SMCs are also highly elevated relative to the distal layer. Depending
on f and d, the maximum shear stress gradients on this layer
range from 200 to 700 dyn · cm2 · µm1,
whereas for the distal layers the maximum gradient may only be ~1
dyn · cm2 · µm1.
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, which are hypothesized to transduce mechanical signals through
shear stress gradients (2, 6), suggest a potential role for shear
stress gradients in SMCs.
The magnitudes of shear stresses estimated by two-dimensional
calculations must be interpreted with some caution. They represent upper bounds on the magnitudes that would actually be obtained in a
full three-dimensional calculation. With reference to Fig. 6, it should
be realized that our two-dimensional calculations do not account for
spreading of the jet along the axis of the SMC. Our results estimate
the shear stresses on the first SMC only over an axial length scale on
the order of d. Clearly, the velocity will be reduced along the
axis of the SMC moving away from the fenestral pore. Full
three-dimensional calculations are required to determine the axial
distribution of shear stress along the first SMC.
Recent in vitro studies have shown that vascular SMCs are responsive to
shear stress in the range 1-25 dyn/cm2 and increase
their synthesis of transforming growth factor-
, tissue plasminogen
activator (20), heme oxygenase-1 (21), nitric oxide (14), and PGs (1).
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.
On the other hand, several in vitro studies have shown that shear
stress is an inhibitor of vascular SMC proliferation (18, 20). In
addition, 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. These
studies suggest that high levels of shear stress on SMCs are
beneficial, because they suppress SMC proliferation and migration,
which might otherwise contribute to intimal hyperplasia. Thus, because
an intact IEL promotes high shear stresses on the superficial SMCs
associated with transmural flow (as we have demonstrated in this
study), an intact IEL may contribute to the suppression of intimal
hyperplasia. A damaged IEL would reduce transmural flow shear stress on
the superficial SMCs and upregulate mechanisms supporting intimal
hyperplasia. This scenario must be considered speculative and requires
additional studies to assess its physiological relevance.
| |
ACKNOWLEDGEMENTS |
The authors are grateful to Dr. Yaqi Huang (Dept. of Mechanical
Engineering, Massachusetts Institute of Technology) for providing his
scanning electron micrograph experimental data to us.
| |
FOOTNOTES |
The research is supported by National Heart, Lung, and Blood Institute
Grant HL-35549.
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. §1734 solely to indicate this fact.
Address for reprint requests and other correspondence: J. M. Tarbell,
155 Fenske Laboratory, The Pennsylvania State University, University
Park, PA 16802-4400 (E-mail: jmt{at}psu.edu).
Received 14 May 1999; accepted in final form 10 November 1999.
| |
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