Bone Ingrowth

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 -gt tissue
  differentiation, fracture healing
• Model II model due to Prendergast -gt 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, stimulus1

23
Results

24
Model II

• Poro-elastic model
• Equilibrium eqn
• Constitutive eqn
• Compatibility cond
• Darcys 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 ?
• p and Darcys 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?