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Bone Ingrowth. in a shoulder prosthesis E.M.van Aken, Applied Mathematics. Outline. Introduction to the problem Models: Model due to Bailon-Plaza: Fracture healing Model due to Prendergast: Prosthesis Numerical method: Finite Element Method Results

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bone ingrowth

Bone Ingrowth

in a shoulder prosthesis

E.M.van Aken, Applied Mathematics

  • 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
  • Osteoarthritis, osteoporosis

 dysfunctional shoulder

  • Possible solution:
    • Humeral head replacement (HHR)
    • Total shoulder arthroplasty(TSA): HHR + glenoid replacement
  • 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
  • Cell differentiation:
  • Two models:
    • Model I: Bailon-Plaza:
      • Tissue differentiation: incl. growth factors
    • Model II: Prendergast:
      • Tissue differentiation
      • Mechanical stimulus
model i
Model I
  • Geometry of the fracture
model i1
Model I
  • Cell concentrations:
model i2
Model I
  • Matrix densities:
  • Growth factors:
model i3
Model I
  • Boundary and initial conditions:
finite element method
Finite Element Method
  • Divide domain in elements
  • Multiply equation by test function
  • Define basis function and set
  • Integrate over domain
numerical methods
Numerical methods
  • Finite Element Method:
      • Triangular elements
      • Linear basis functions
results model i
Results model I

After 2.4 days: After 4 days:

results model i1
Results model I

After 8 days: After 20 days:

model ii
Model II
  • Geometry of the bone-implant interface
model ii1
Model II
  • Equations cell concentrations:
model ii2
Model II
  • Matrix densities:
model ii3
Model II
  • Boundary and initial conditions:
model ii4
Model II

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


Bone density after 80 days, stimulus=1

model ii5
Model II

Poro-elastic model

  • Equilibrium eqn:
  • Constitutive eqn:
  • Compatibility cond:
  • Darcy’s law:
  • Continuity eqn:
model ii6
Model II
  • Incompressible, viscous fluid:
  • Slightly compressible, viscous fluid:
model ii7
Model II

Incompressible: Problem if

Solution approximates

Finite Element Method leads to inconsistent or singular matrix

model ii8
Model II


1. Quadratic elements to approximate displacements

2. Stabilization term

model ii9
Model II
  • u and v determine the shear strain γ
  • p and Darcy’s law determine relative fluid velocity
model ii10
Model II

Boundary conditions

results model ii
Results Model II

Arm abduction 30 ° Arm abduction 90 °

results model ii1
Results Model II

30 ° arm abduction, during 200 days

results model ii2
Results Model II

Simulation of 200 days: first 100 days: every 3rd day arm abd. 90°,

rest of the time 30 °.

100 days200 days

  • 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)