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

The strain dependent permeability, k was also implemented using the variable void ratio
as:
!
1 + !
! = ! ! , ! =
1 + ! !
Θ !
(4)
Θ !
where k0 is the initial permeability (equal to 10 -14 m 2 ), e is void ratio (defined as the
ratio of fluid volume fraction Θ ! to solid volume fraction Θ !), e0 is initial void ratio
(equal to 4), and M is the permeability coefficient, with the value 1.3 from [8].
This model quantified synovial fluid flow inside the cartilage when the cartilage was
deformed sinusoidally. In this situation, both cartilages were deformed at 2 Hz
frequency and 15% amplitude to simulate slow running [9]. A free boundary condition
(p = 0) was defined at the free boundary of cartilage to let the fluid flows in and out of
the porous cartilage during deformation. During compression, the fluid exudes out of
the cartilages and during unloading phase, cartilage imbibes the synovial fluid. At each
time step, the pressure and velocity distribution in modeled cartilages were calculated.
Heat transfer interface: calculation of the thermal field. This interface uses the following
version of the heat equation for heat transfer in cartilage [10]:
!"
(!! !) !"
!" + ! (5)
!! !!. ∇! = ∇. ! !"∇! + !
In this equation u is the fluid velocity (introduced in equation 2). At each time step the
fluid velocity magnitude is solved from the poroelastic model and is used in equation 5
for calculating the temperature distribution. The equivalent conductivity of the solid-
fluid system, keq is related to the conductivity of the solid, kp and to the conductivity of
the fluid, kl by keq = ! pk p + ! lkl . The equivalent volumetric heat capacity of the solidfluid
system is related to the heat capacity of the solid, Cp and to the heat capacity of the
fluid, Cl by (!Cp ) eq = ! p! pCp + ! l! lCl . Here Θp denotes the solid material’s volume
fraction and Θl fluid fraction. The heat capacity and conductivity of cartilage and
synovial fluid were measured experimentally.
In equation 5, Q is the heat source. In this model we have 2 heat sources; first the
cylindrical hydrogel, which dissipation properties have to be optimized and then the
cartilage dissipation, which is modeled as a heat source distributed uniformly in
cartilage. The amount of cartilage dissipation was measured experimentally. There was
a temperature boundary condition at the free boundary of the cartilage, which keeps its
temperature in equilibrium with intra-articular temperature. It has been reported that
intra-articular temperature increase during joint activity mainly because of friction in
the joint. The knee temperature is also influenced by the muscle metabolism and the
circulating blood during activity [11]. Some studies measured the temperature increase
by direct intra-articular measurement [11, 12]. We applied temperature profile reported
by Becher at this boundary [11].
3.2. Experimental determination of cartilage thermal parameters
Dissipation of cartilage. Cartilage dissipation was measured by applying cyclic
compression test at 2 Hz and 15% amplitude on cylindrical pieces of bovine cartilage.
Dissipation of cartilage was calculated by integrating the surface of the hysteresis curve.
Heat capacity of cartilage and synovial fluid. Heat capacity of bovine cartilage and
synovial fluid were measured using a differential scanning calorimeter (DSC-Q100).
Measurements were performed at a rate of 10°C/min and under a 4 min isothermal