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

2. Outline Introduction to the problem Models: –Model due to Bailon-Plaza: Fracture healing –Model due to Prendergast: Prosthesis Numerical method: Finite Element Method Results –Model I: model due to Bailon-Plaza -> tissue differentiation, fracture healing –Model II: model due to Prendergast -> tissue differentiation, glenoid –Model II: tissue differentiation + poro elastic, glenoid Recommendations

3. Introduction Osteoarthritis, osteoporosis  dysfunctional shoulder Possible solution: –Humeral head replacement (HHR) –Total shoulder arthroplasty(TSA): HHR + glenoid replacement

4. Introduction

5. Need for glenoid revision after TSA is less common than the need for glenoid resurfacing after an unsuccesful HHR TSA: 6% failure glenoid component, 2% failure on humeral side

6. Model

7. Cell differentiation:

8. Models Two models: –Model I: Bailon-Plaza: Tissue differentiation: incl. growth factors –Model II: Prendergast: Tissue differentiation Mechanical stimulus

9. Model I Geometry of the fracture

10. Model I Cell concentrations:

11. Model I Matrix densities: Growth factors:

12. Model I Boundary and initial conditions:

13. Finite Element Method Divide domain in elements Multiply equation by test function Define basis function and set Integrate over domain

14. Numerical methods Finite Element Method: Triangular elements Linear basis functions

15. Results model I After 2.4 days:After 4 days:

16. Results model I After 8 days:After 20 days:

17. Model II Geometry of the bone-implant interface

18. Model II Equations cell concentrations:

19. Model II Matrix densities:

20. Model II Boundary and initial conditions:

21. Model II Proliferation and differentiation rates depend on stimulus S, which follows from the mechanical part of the model.

22. Results Bone density after 80 days, stimulus=1

23. Results

24. Model II Poro-elastic model Equilibrium eqn: Constitutive eqn: Compatibility cond: Darcy’s law: Continuity eqn:

25. Model II Incompressible, viscous fluid: Slightly compressible, viscous fluid:

26. Model II Incompressible: Problem if Solution approximates Finite Element Method leads to inconsistent or singular matrix

27. Model II Solution: 1. Quadratic elements to approximate displacements 2.Stabilization term

28. Model II u and v determine the shear strain γ p and Darcy’s law determine relative fluid velocity

29. Model II Boundary conditions

30. Results Model II Arm abduction 30 ° Arm abduction 90 °

31. Results Model II 30 ° arm abduction, during 200 days

32. Results Model II Simulation of 200 days: first 100 days: every 3rd day arm abd. 90°, rest of the time 30 °. 100 days 200 days

33. Recommendations Add growth factors to model Prendergast More accurate simulation mech. part: –Timescale difference between bio/mech parts Use the eqn for incompressibility (and stabilization term) Extend to 3D (FEM)

34. Questions?