ARUP; ISBN: 978-0-9562121-5-3 - CMBBE 2012 - Cardiff University

3. METHODS
3.1. Model of heat transfer in cartilage
Analytically, the problem was formulated as a two steps model. In the first step, the
velocity field of synovial fluid was determined in a poroelastic model. In the second
step, the solution of the velocity field was introduced into the thermal diffusion
equation. The thermal field within the porous cartilage and hydrogel sample was then
determined.
Comsol multiphysics 4.2. finite element software (COMSOL Inc., Burlington, MA,
USA) was used to develop the poroelastic, heat transfer model. The axisymmetric
model consisted in a femoral and tibial cartilage layer each with the thickness of 4
millimeters. Both cartilage layers are attached to their corresponding femoral and tibial
bones. A cylindrical hydrogel (self-heating material considered as heat source) is placed
inside the tibial cartilage. The heat source is a viscoelastic material, which dissipates
heat when it deforms (Figure 1). The geometry was meshed with triangular elements.
Figure 1: geometry of the problem, consisted in a femoral and tibial layer of cartilage with their corresponding
femoral and tibial bones and the hydrogel. The whole structure is meshed with triangular elements.
We modeled the cartilage as a poroelastic material. We assumed that 80 percent of the
cartilage is composed of water. The porosity was set accordingly to 0.8 [6]. The
simulation consisted of two coupled multiphysics: poroelasticity and heat transfer
interfaces.
Poroelasticity interface: calculation of the velocity field for the synovial fluid. Darcy’s
law describes the flow field in the poroelastic model [7], which in the pressure
formulation translates to:
ρ ! ! !! !
!" + ∇. ρ ! ! = −ρ ! ! !
where u and S are:
! = − !
! ∇! !
! = ! !! ! + (! ! − ! !)
!
!ε !"#
!"
∂εvol/∂t is the rate of change in volumetric strain, pl is the pressure field and αB is the
Biot-Willis coefficient [7], which is considered equal to 1 in our case. The material
properties needed for the poroelastic model (in equations 1-3) are listed in Table 1.
Bulk modulus of cartilage, K [Mpa] 0.36
Viscosity of synovial fluid, µ [Pa.s] 0.6
Porosity of cartilage, ! ! [-] 0.8
Compressibility of synovial fluid, ! ! [1/Pa] 1e-10
Density of fluid, ρl [kg/m 3 ] 1000
Table 1: material properties used in the model.
(1)
(2)
(3)