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Interstitial fluid flow in tendons or ligaments: A porous medium finite element simulation

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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 interstitial 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.

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Abbreviations

b :

inertial coefficient

C:

coupling matrix

f :

body force

F:

force/force vector

\(\bar K\) :

diffusion matrix related to permeability

K :

permeability matrix

K z :

permeability in the longitudinal direction

K r :

permeability in the radial direction

L :

length of analytical model

ΔL :

length change of analytical model

M:

mass matrix

p:

nodal pressure or pressure vector

P:

fluid pressure

r:

radial axis

R :

radius of analytical model

ΔR :

radial change of analytic model

t:

time

T :

matrix transformation

u:

displacement

ν:

fluid velocity

V:

velocity/volume vector

V r :

velocity in radial direction

V z :

velocity in longitudinal direction

V f :

fluid volume

z :

longitudinal axis

ϕ:

medium porosity

η:

fluid viscosity

ρ:

fluid density

∇:

Nabla operator (gradient/divergence)

ϖ:

partial derivative

−:

overline; average symbol of variable

⊙:

time rate of change

m :

power index

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Authors and Affiliations

  1. University of Wisconsin-Milwaukee, Milwaukee, Wisconsin

    S. L. Butler

  2. Department of Biomedical Engineering, Worcester Polytechnic Institute, 01609-2280, Worcester, MA, USA

    S. S. Kohles

  3. Carnegie Mellon University, Pittsburgh, PA, USA

    R. J. Thielke

  4. Cornell University, Ithace, New York

    C. Chen

  5. Department of Mechanical Engineering & Division of Orthopedic Surgery, University of Wisconsin, 53792-3228, Madison, WI, USA

    R. Vanderby Jr

Authors
  1. S. L. Butler
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  2. S. S. Kohles
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  3. R. J. Thielke
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  4. C. Chen
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  5. R. Vanderby Jr
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Correspondence to R. Vanderby Jr.

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Butler, S.L., Kohles, S.S., Thielke, R.J. et al. Interstitial fluid flow in tendons or ligaments: A porous medium finite element simulation. Med. Biol. Eng. Comput. 35, 742–746 (1997). https://doi.org/10.1007/BF02510987

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  • DOI: https://doi.org/10.1007/BF02510987

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