A Mechanically Stimulated Fracture Healing Model Using a Finite Element Framework - Biomedical Technology - page 43

Here D m , C m are diffusive and advective coefficients as above, A m denotes the
proliferation parameter and F 1 , F 2 and F 4 are the differentiation coefficients for
osteogenic, chondrogenic and fibrogenic differentiation. By multiplication of F l with
the respective ˈ l the intensity of the differentiation is scaled.

3 Methods

To solve the presented mathematical model the Finite Element method is employed.
However, when a standard Galerkin approach is chosen, spurious oscillations often
pollute the solutions of advection-diffusion-reaction equations. These oscillations
occur in the vicinity of steep gradients in the solution, which are present from the
beginning of the calculation (e.g. at Dirichlet boundaries) or build up during the cal-
culation. Dominance of the advection or reaction terms often lead to steep gradients
in the solution. This problem is well documented in literature and various techniques
were developed to prevent oscillations.
In this work the Time Discontinuous Galerkin method (TDG) was combined
with the finite calculus scheme (FIC) developed by Oñate [ 13 , 14 ] to achieve a
stable solution of the system of instationary advection-diffusion-reaction equations
described above, for details it is referred to [ 11 ].

3.1 Simulation

As a geometrical model for the simulation a simplified axisymmetric representation
of an osteotomy is chosen, as shown in Fig. 2 . The dimensions of the callus region
are chosen according to [ 4 ] to indicate a sheep metatarsal model. The fracture gap is
set to 3mm throughout the presented simulation.
The boundary conditions for biochemical simulation are shown on the right side
of Fig. 2 . They describe cell and growth factor reserviors in the surrounding tissue.
Stem cells and fibroblasts are therefore recruited from the marrow cavity and soft
tissue outside of the periosteal callus. Endothelial cells originate from the periosteum,
endosteum and marrow. The source for chondrogenic growth factors is the fractured
bone end and osteogenic growth factors invade the callus region from the periosteum
and endosteum. The initial condition of the callus is chosen to be a connective tissue of
low density, indicating the granulation tissue formed during the inflammation phase
of bone healing. And a high concentration of angiogenic growth factor is present
throughout the callus in accordance with observations documented in literature [ 15 ].
The mechanical simulation is executed at the start of each day of healing. At the
beginning of the healing process a maximal displacement acts on the cortical bone.
The displacement driven simulation continues until the reaction forces calculated on
the upper boundary are equal to the maximal load of a sheep leg (i.e. about 600N
taken from [ 12 ]). Then this maximal force is applied to the former displacement