1. Bone Ingrowth in a shoulder prosthesis E.M.van Aken, Applied Mathematics

2. Outline Introduction to the problem Models First model: Numerical methods First model: Results –1D simulation –Sensitivity analysis Second model Future

3. Problem Osteoarthritis, osteoporosis -> dysfunctional shoulder Possible solution: glenoid replacement

4. Problem

5. Model

6. Cell differentiation:

7. Model I Three models: –Andreykiv: Biophysical stimuli Tissue differentiation –Bailon-Plaza: Tissue differentiation: incl. growth factors –Adam: Growth factors

8. Model I Equations cellular densities:

9. Model I Matrix densities:

10. Model I Boundary and initial conditions:

11. Numerical methods Discretisation in space: –Standard 3-point method: –Central scheme:

12. Numerical methods So leads to Discretisation in time: semi-implicit

13. Results cell densities at t=9: matrix densities at x=0:

14. Results cell density bone at t=15:

15. Results Full ingrowth at time

16. Results Assume bone grown in at time Approximate Determine most influential parameters

17. Results

18. Model II Cellular densities:

19. Model II Matrix densities: Growth factors:

20. Numerical Methods Boundary and initial conditions:

21. Future Finish simulation of the Bailon-Plaza model, using Finite Volume Method Compare and combine the models of Andreykiv and Bailon-Plaza Extend to 3D (FEM)