f
Interstitial fluid flow in tendons or
ligaments: a porous medium finite
element simulation
S. L. Butler 1
S.S.
Kohles =
R.J.
Thieike 3
C. Chen 4
R. V a n d e r b y J r s
I University of Wisconsin-Milwaukee, Milwaukee, Wisconsin
2Department of Biomedical Engineering, Worcester Polytechnic Institute, Worcester,
MA 01609-2280, USA
3Carnegie Mellon University, Pittsburgh, PA, USA
~Cornell University, Ithaca, New York
SDepartment of Mechanical Engineering & Division of Orthopedic Surgery, University of Wisconsin,
Madison WI 53792-3228 USA
Abstract--The purpose of this study is to describe interstitial fluid flow in axisymmetric
soft connective tissue (ligaments or tendons) when they are loaded in tension. Soft
hydrated tissue was modelled as a porous medium (using Darcy's Law), and the finite
element method was used to solve the resulting equations governing fluid flow. A
commercially available computer program (FiDAP) was used to create an axisymmetric
model of a biomechanically tested rat ligament. The unknown variables at element nodes
were pressure and velocity of the inters~ial fluid (Newtonian and incompressible). The
effect of variations in fluid viscosity and permeability of the solid matrix was parametrically
explored. A transient loading state mimicking a rat ligament mechanical experiment was
used in all simulations. The magnitude and distribution of pressure, stream lines, shear
(stress) rate, vorticity and velocity showed regular patterns consistent with extension flow.
Parametric changes of permeability and viscosity strongly affected fluid flow behaviour.
When the radial permeability was 1000 times less than the axial permeability, shear rate
and vorticity increased (approximately 5-fold). These effects (especially shear stress and
pressure) suggested a strong interaction with the solid matrix. Computed levels of fluid
flow suggested a possible load transduction mechanism for cells in the tissue.
Keywords--Fluid dynamics, Ligament, Tendon, Porous medium, Finite element analysis
Med. Biol. Eng. Comput., 1997, 35, 742-746
List of symbols
b
C
f
F
k
K
Kz
Kr
L
AL
M
p
P
= inertial coefficient
= coupling matrix
= body force
= force/force vector
= diffusion matrix related to permeability
= permeability matrix
= p e r m e a b i l i t y in the longitudinal direction
= p e r m e a b i l i t y in the radial direction
= length o f analytical model
= length change o f analytical model
= mass matrix
= nodal pressure or pressttre vector
= fluid pra~sure
= radial axis
r
R = radius of analytical model
AR = radial change of analytic model
t
--- time
T
= matrix transformation
u
= displacement
v
V
= fluid velocity
= velocity/volume vector
V, = velocityin radialdirection
Correspondence should be addressed to Dr. Ray Vanderbw
emaih [email protected]
First received 13 May 1996 and in final form 18 April 1997
r IFMBE:1997
742
I
V~ = velocity in longitudinal direction
V/ = fluid volume
z
= longitudinal axis
= medium porosity
= fluid viscosity
p
= fluid density
V = N a b l a operator (gradient/divergence)
a
= partial derivative
Superscripts
-
9
m
= overlinc; average symbol o f variable
= time rate o f change
= p o w e r index
1 Introduction
6 0 - 7 0 % o f t h e total w e t w e i g h t o f l i g a m e n t s a n d t e n d o n s is
w a t e r (WOO a n d BUCKWALTER, 1988). M o s t o f this w a t e r is
either loosely (translational)or freely bound thus allowing for
flow throughout the solid matr/x during loading. The solid
matrix is primarily composed of collagen fibres with some
ground substance proteins, most of which are proteoglycans.
The collagen fibres are generally arranged parallel to each
other in the direction of the longitudinal structural axis. The
proteoglycans interact with the collagen fibres to form a
physical network throughout the extracellu/armatrix (ECM),
Medical & Biological Engineering & Computing
November 1997
which is strong, cohesive, fibre-reinforced, porous-permeable
and capable of supporting high tensile loadings. This network
is renewed with protein synthesis.
Finite element models of porous materials, in which fluid
dynamics, can be modelled by Darey's law, have been applied to
soft hydrated tissues. Some of the earlier continuum models are
based on classical consolidation theory (GP,EENKORN, 1983;
THIELKEet al., 1995), in which the nodal displacement (u) of the
solid matrix and the nodal pressure (p) of the fluid are expressed
as unknown variables in finite element analysis.
The biphasic model provided by Mow et al. (1980), as well
as other models based on it (VAN-DERBYet aL, 1985; SPILKER
and SUH, 1990a), use the theory of mixtures. Soft eomaective
tissue is considered as a two-phase immiscible mixture. One
phase is an incompressible solid with porous-permeability,
mainly collagen fibres and proteoglycan. The other is an
interstitial fluid phase, primarily incompressible water. Two
different fluid response formulations, a u - P (displacementpressure) method (WAYNE et aL, 1991) and a u - v (displacement-velocity) method (SPILKER et aL, 1990b), have been
established. These models have been applied to cartilage
(SPILKER and SUH, 1990a; SPILKERet aL, 1990b; WAYNE et
at., 1991) and to intervertebral discs (SIMON et al., 1985;
SNIJDERS et al., 1992) to study the influence of fluid exudation
within the tissue upon the mechanical properties of its solid
matrix. Fewer studies have focused on the response within
ligaments and tendons (SHRIVE et al., 1993).
Investigations with osteoblasts and endothelial cells in cell
cultures have shown a biochemical response to shear stress
and streaming potential generated by fluid flows (REICH et aL,
1990; 1991). These findings suggest that mechanically
induced interstitial fluid flow in ligaments and tendons may
also affect tissue cells (fibroblasts) in the regxflation and
maintenance of their ECM.
This study simulated interstitial fluid flow in a rat ligament
using isotropic and anisotropic tissue structures, the purpose
being to establish a foundation for investigating relationships
between fluid flow and cellular response in tendons or ligaments. The resulting model incorporates experimentally measured deformation parameters and is both limited to and
defined by the continuous assumptions and boundary conditions prescribed by Darcy's Law.
2 Methods
2.1 Ligament mechanical evaluation
Mechanical testing was performed on a rat lateral collateral
ligament (LCL). The LCL was selected because it has a crosssection that is nearly round, therefore simplifying computer
simulation with axisymmetry. The ligament was dissected out
of an 80 day old rat that was killed for other experimental
purposes. Sutures were tied to the ligament ends for gripping.
Optical markers were placed on the mid-substance of the
tissue (away from the sutures) for video displacement measurement. The specimen was placed in grips in a miniature
servo-controlled testing system. The tissue was submersed in
physiological saline at room temperature and allowed to
equilibrate.
After an initial load of 0.15 N was quasi-statically applied to
precondition the specimen and snug the sutures, a tensile load
was applied to the specimen at a deformation rate of
0.75rams -1 for 2s. Load and time data were recorded on a
personal computer with an analogue to digital converter and
data acquisition sofhvare, and the test was videotaped through
a microscope for surface deformations in the longitudinal and
transverse directions. After the test, individual video frames
Medical & Biological Engineering & Computing.
were digitally captures on a personal computer and analysed
with software* for surface deformations.
The croSS-Sectional area of the specimen was measured
using a line scanning camera with baeklighting and rotational
grips (TrUELKE, 1995). The transverse width of the specimen
was detected at angular increments of 3.6 ~ by the recording
camera while the specimen was rotated 180 ~ about its longitudinal axis. The shape of the speeimen's transverse section
was then reconstructed, and its cross-sectional area w a s
calculated. Reliability was confirmed with repeated measurements. The area of the specimen was measured at zero and
2.5% strain. The computer model was formulated to have a
radial deformation that would produce the same reduction in
cross-sectional area as was measured in the mechanical
experiment. In the analytical simulation, the specimen was
assumed to be axisymmetric and the loaded and unloaded radii
were calculated from the experimental areas.
2.2 Ligament finite element model
The formulae used to describe fluid passing through porous
medium are based on the Navier-Stokes equation and a
volume averaging theory (GHABOUSSIand WILSON, 1973).
These formulas are summarised in the Appendix. The tissue
was assumed to be a porous solid matrix saturated with a
viscous incompressible fluid. The solid matrix was assumed to
be homogeneous with an anisotropic permeability. The viscous coefficient was constant and turbulence was not included.
The principal axes of anisotropic permeability coincided with
the structural coordinate axes. A finite element formulation,
corresponding to eqn. A5 (Darcy's Law), was constructed in
commercially available soffwaret using the Galerkin method
OV~ALVERN,1969). In matrix notation,
where [ ] aa2_d( ) indicate matrices and vectors, respectively,
and M, F, K, and C represent the mass matrix, force vector,
diffusion matrix related to permeability, and a coupling
component, respectively. Fluid velocity (V) and pressure
(P), and their rates of change with time will also contribute.
The finite element model was simplified by longitudinal
symmetry and transverse axisymmetry (Fig. 1). The radial
direction was defined as the r-axis, where r = 0 at the transverse symmetric centre (cross-sectional centre), and the longitudinal direction was defined as the z-axis, where z = 0 at the
longitudinal symmetric centre. The ligament cross-sectional
area, at its bony insertion, was assumed constant. The radial
deformation from z = 0 to L was assumed to be linear. An
axisymmetrie finite element model was then formtdated.
Nine node quadrilateral elements were used to build a
longitudinally symmetric and axisymmetric model with a
total of 105 elements and 465 nodes. Permeability was set at
7.6 x 10 - i s m 4 N - l s -1 (MOw et al., 1980). Considering
that water occupies approximately 70% of the volume of the
ligament, we assumed the fluid density to be 1.0 • 103 k g m -3
2(R-,,.~"--~ I '
,T-C C:
::::
F~. 1 Schematic of the deformatio~ of a ligament in tension
*NIH Image, Washington, DCI, USA
~'Fluid dynamics Analysis Package, FiDAP, Evanston, IL, USA
November 1997
743
lr
V,~to~R
, . . . .
a
V,~Oatz.O
Vr=O~r
I-
Fig. 2 Axisymmetrie fintte element boundary conditions
[.
and porosity to be 0.7 (70%). The dynamic viscosity of the
fluid was chosen as 0.1 Nsm - 2 based upon viseometric studies
o f interstitial fluid in cartilage (Mow et ai., 1989) but was
parametrically investigated. When a 1.14ram displacement
was applied in the longitudinal direction to the mechanical
specimen (2.5% grip to grip strain), the longitudinal velocity
(V~) of the end was estimated to be 0.6rams -~ at z = L
(assuming negligible mass transfer at the longitudinal boundary). Simultaneously, transverse contraction of 0.026ram
occurred at the longitudinal symmetric centre ( z = 0 ) . The
deformation of the specimen in its radial direction was defined
as AR while the deformation in one half of the total length was
AL (Fig. 1). Again, assuming negligible mass transfer at the
onset of loading, the average surface velocity, V~(t), in the
radial direction (r = R) of fluid at time t can be expressed as
Vr(t) = (r(t) ~
\~(t))
~,z(t)
.
.
.
.
.
b
I
t
(2)
•
1'
t[
....
.
.
.
.
.
.
I
where V~ at an initial time is expressed as
v,=
R-AR
2(z. + ~ )
Fig. 3
x~
(3)
Thus, a radial surface velocity of - 0 . 0 3 2 3 mm s-~ at z = 0
(linearly distributed to a radial velocity of zero at z = L) was
calculated and incorporated into the model as a moving
boundary condition. Radial and longitudinal deformation of
the ligament at the onset of tension were then mimicked within
the finite element analysis (Fig. 2). A non-linear solution was
required because of the moving boundaries. Additional conditions of V, = 0 at the symmetric centre of the cross-section
(r = 0) and I~ = 0 at the longitudinal symmetric centre (z = 0)
were included.
The analytical evaluation calculated pressure, stream lines,
shear (stress) rate, vorticity, and velocity. Time increments of
At = 0.001 s were used in the transient analysis over a time
range from 0--0.025 s.
Contour plots of analytical results at t = O.001 seconds within
the boundaries defined by Fig. 1: (a) pressure
( m a x = l . O S x 104Nm -2 at z=O), (b) stream lines
(max=2.88 x lO-I~ m3s -1 at r = R , z = L ) , (c) shear rate
(max = 2.23 x I 04 Nm - 2s - 9, (d) vorticity (max = 31.5s - 1),
(e) velocity (ma.~=5.99 • lO-4 m s - t a t z = L )
If the permeability in the transverse direction (Kz) is varied
relative to the permeability in the radial direction (K,), fluid
flow behaviour is substantially altered (Fig. 5). Shear rate,
vorticity and velocity decrease asymptotically as the ratio of
KJK~. increases. When this ratio is >1, the flow parameters
remain relatively constant. Conversely, when the ratio is <1,
substantial increase can occur in shear rate, vorticity and
pressure.
Comparing behaviours immediately after the onset of loading (t=0.001 s) reveals that changes in the viscosity coefficient only affect pressure (Fig. 6). During an initial loading
time from 0.0 to 0.02s at increments of At = 0.001 s, model
3 Results
Immediately after the onset of loading ( t = 0.0Ol s), the
distribution of fluid pressure (Fig. 3a) is nearly uniform at
each transverse section with a maximum pressure of
1.08 x 104Nm -2 appearing at the longitudinal symmetric
centre ( z = 0 ) that gradually reduces to 0 at z = L + AL.
Streamlines are characteristic of extensional flow (Fig. 3b)
with no flow occurring at the longitudinal and transverse
symmetrical centres (r = z = 0). Stream line magnitude
varies from 0 to 2.88 x 10-~~
-~ at r = R
and
z = L + AL. There is some similarity for the distribution of
shear rate (Fig. 3c) and vorticity (Fig. 3</). The maximum
shear rate, 2.23 x 10~Nm-~s '-l, and vorticity, 31.5 s -~, take
place near the longitudinal boundary (z "~ L) of the anal tytical
model. The minimum shear rate is 7.62 x 10 -2 N m - 2 s - and
minimum vorticity is 0. The velocity vector (Fig. 4) shows a
linear increase from zero at z = 0 to its maximum of
5.99 x 1 0 - 4 m s -~ at z = L + A L ,
which is consistent with
the boundary velocity 0.6 mm s-~ in the mechanical evaluation of the ligament.
744
z
Fig. 4 Surface plot of longitudinal velocity (Vz) throughout the
length of the model
Medical & Biological Engineering & Computing
November 1997
t40
T
k
\
\
~
~e~sum, P
- - Q - v ~ . v.
\
. ,
shearr.
J
E
~5
40"
-10
0,001
/
0.01
0.I
1
10
100
~rmea~ay, k,/k,
Fig. 5 Effect of changes in radial permeability on fluid pressure,
velocity, shear rate, and vortici~, while longitudinal permeability remains constant. These results represent the effects
of the possible range o f a transverse isotropic structure
within ligamentous tissue
nodes at z = 0, 89 and L (Fig. 2) were chosen as evaluation
points for the temporal observation of pressure, shear rate,
vorticity and velocity (z and r directions). These variables
remained nearly constant with time. The temporal response of
the other variables was similar.
4 Discussion
The purpose of this study was to examine interstitial fluid
flow in tendons or ligaments under tensile loading. A tensile
test with a rat lateral collateral ligament provided boundary
conditions for the computer model. A porous medium finite
element model based upon Darcy's Law was then used to
describe the resulting fluid flow under parametric constitutive
assumptions. Important findings from this study are that under
a known or specified viscosity, fluid characteristics of vorticity, shear rate and pressure are all affected by tissue anisotropy. Also, computed fluid velocities and pressure are
sufficiently high that they might serve as mechanical transduction mechanisms for cellular autoregulation in these tissues.
Limitations in this study arise from a number of possible
sources. The finite deformation of the solid matrix and the
boundary conditions are not well defined by Darcy's Law.
Furthermore, the tissue is microstmcturally anisotropic, but
the constitutive properties are not well defined so a parametric
10000
&,
1000'
p~ssure, P
- - C ~ veso~ty, v,
z,
shearrate
100"
10
1!
0,1
0.01"
0.01
0,1
1
10
100
viscosity rat~
Fig. 6 Effect of fluid viscosity on pressure, velocity, and shear rate
Medical & Biological Engineering & Computing
study was required. The fluid is mostly water. However there
is the potential for non-Newtonian behaviour if there a r e
sufficient unbound proteins or other large molecules in the
ground substance. Finite deformations in the solid matrix may
also alter the actual permeability and anisotropy which were
assumed to be constant in the Darcy's Law coefficients.
In this study, a small displacement assumption for the
porous solid matrix with constant permeability neglects
these effects. In our mechanical experiment, we measured
boundary deformation and not fluid velocity. We then
assumed a steady average deformation velocity for our
model and applied moving boundaries in the radial and longitudina! directions. Another potential source of error is the
assumption of no fluid flux across the ligament surface.
Because of the above assumptions and uncertainties, the
model should not be interpreted in a strict quantitative
sense. To improve the adopted finite element model, the
above factors should be explored.
Some insight can be obtained by comparing Frangos'
research (REICH et al., 1990; 1991) with our analytical results.
In their investigations, osteoblasts are found to be quite
sensitive to flow, as noted by stimulation of the production
of prostaglandin (PGE2) and increases in the intracellular
concentration of inositol triphosphate (IP3).
Fluid flow induced by mechanical stress may thus be an
important mediator of bone remodelling. In our analysis,
results show pressure, shear rates, vorticity and velocityfields in the ligament that could have a similar physiological
effect. Changes of permeability affect these field-variables
especially at levels of K~/K~ < 1, which have been shown to
exist in ligamentous tissue (TANG et aL, 1993), with a
concurrent pressure dependence upon viscosity. Significant
shear stresses may exist in the walls of the solid matrix pores.
The fluid within the collagen matrix may also be stagnant at
some locations because of the complicated geometry of the
pores.
Accompanying the moving boundary and matrix deformation, physical elements such as the fluid, solid matrix, pore
space, and interfaces between fluid and solid may relocate
while varying their shapes. As a result, characteristics such as
shear rate, vorticity and velocity, which are a measurement of
shear stress and fluid pressure, may exert strong counteractions upon the solid matrix. The fibroblasts within the
solid matrix may in turn be stimulated by the action of these
fluid stresses.
In other experiments on fluid flow and pressure within
tendonous tissue, the pressure changes measured were of the
same order as those predicted within this study (10.8 kPa). In a
study of eleetrokinetic phenomena (CI-mN et al., 1995), maximum pressure ehanges within a patellar tendon were measured at 1-2 kPa during a 5 MPa tensile test. These kinds of
pressure are above the general range that have been noted to
stimulate a response in endothelial cells (0.25 to 1.5 Pa)
(JAMES et aL, 1995), but are less than those reported for
stimulation of fibroblasts (100 kPa) (WI~GnT et aL, 1992) and
chondrocytes (15 MPa) ( P ~ , ' q
et aL, 1995). The experiments with the fibroblasts and chondrocytes did not, however,
include the added stimulation of fluid flow predicted by this
analysis.
The fluid flow in a ligament under tensile loading was
analysed with a porous medium finite element model. This
model was based on an actual ligament tensile loading
experiment completed in our laboratory. The resulting analysis
is both limited to and defined by the constitutive descriptions
and boundary conditions prescribed by Darcy's Law. Results
suggest that interstitial fluid dynamics in ligaments and
tendons may play an important role in the stimulation of
cells and thereby provide a transduction mechanism for the
November 1997
745
maintenance or adaptation of the extracellular matrix in
response to tensile loading.
References
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746
Appendix
In this analysis it is assumed that the fluid is incompressible
and Newtonian (i.e. the viscosity coefficient o f the fluid does
not change). The Navier-Stokes equation for this fluid is
Ov
P ~ + P(Vv) 9v = - V p + pV. (Vv) + pf
(A1)
in which p is density, v is veloeity, p is pressure, t is time, ~t is
viscosity, f is body force and V is the gradient operator.
According to the local volume-average theorem equations
(St.ArrERY, 1962), the volume-averaged Navier--Stokes equation becomes
0~
P'gi + p(V~). ~ = -V~ + ~,V- (V~) + p? - r
(A2)
where F is a resistance tensor and the overlines represent
average properties of v,p, and f This resistance component
obstructs the motion of a particle at a point inside an
elementary channel and is influenced by the structure of the
solid matrix, fluid velocity, velocity gradient, density and
viscosity. By Forchbeimer's hypothesis (BEAR, 1972), F is
considered to be non-linear. If the permeability o f the solid
matrix is anisotropic (as in ligamentous tissue), F is then
defined by
F =/~[K]-I~ + bp[K1/2] -l I~['~
K=k~/
(A3)
i,j=1,2,3
where KV is the intrinsic permeability of the solid matrix and b
and m are. the inertial coefficient and power index, respectively. If @ is considered as the porosity of the medium, then
by substituting v/& for v and dropping the overlines, Eqn. A2
becomes
p~+p
V
-~=
(A4)
+ p.f - ~ [ K ] - % + bp[K1P'] -~ Ivl"v]
This is a general equation for an incompressible fluid in a
porous medium. If we ignore the influence of nonlinear and
inertial terms in Eqn. A4, the tensor equation used in the Fluid
Dynamics Analysis Software Package (FiDAP) is obtained
p~
~
+/~[K]-Iv = - V p + #V- (Vv) + p f
(AS)
Author's biography
Ms. Butler received an MS in Engineeaing
Mechanics from the Bcijing University of Aeronautics and Astronautics (1982) and a BS in
Mechanical Engineering from Dalian University
of Technology (1970), both in the People's
Republic of China. Presently she is standing for
a Phi) in Mechanical Engineering at the Universit), of Wisconsin-Milwaukee, Milwaukee, Wiscortsin, USA, while concurrently working as an
Engineer in the Computer-Aided Engineering Department within the
Falk Corporation, Milwaukee, Wisconsin, USA. The research undertaken for this paper was completed during Ms. Buffer's previous
position as a Visiting Scholar within the Division of Orthopedic
Surgery, University of Wisconsin-Madison, Madison, Wisconsin,
USA. Her professional interests include the finite element method
and analysis; structural analysis and optimisation, structural stress,
stability, vibration, and dynamics; mechanics of compositie materials; computer modelling; computer-aided design and engineering;
and biomechanics.
Medical & Biological Engineering & Computing
November 1997