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  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. Introduction • 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. Model • 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 days200 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?