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This study examines bone in-growth in a shoulder prosthesis using finite element method models. It contrasts two models based on tissue differentiation and features results after various time intervals. Recommendations include adding growth factors and enhancing the mechanical simulation for greater accuracy and extending the study to 3D using FEM.
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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 • 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
Introduction • Osteoarthritis, osteoporosis dysfunctional shoulder • Possible solution: • Humeral head replacement (HHR) • Total shoulder arthroplasty(TSA): HHR + glenoid replacement
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
Model • Cell differentiation:
Models • Two models: • Model I: Bailon-Plaza: • Tissue differentiation: incl. growth factors • Model II: Prendergast: • Tissue differentiation • Mechanical stimulus
Model I • Geometry of the fracture
Model I • Cell concentrations:
Model I • Matrix densities: • Growth factors:
Model I • Boundary and initial conditions:
Finite Element Method • Divide domain in elements • Multiply equation by test function • Define basis function and set • Integrate over domain
Numerical methods • Finite Element Method: • Triangular elements • Linear basis functions
Results model I After 2.4 days: After 4 days:
Results model I After 8 days: After 20 days:
Model II • Geometry of the bone-implant interface
Model II • Equations cell concentrations:
Model II • Matrix densities:
Model II • Boundary and initial conditions:
Model II Proliferation and differentiation rates depend on stimulus S, which follows from the mechanical part of the model.
Results Bone density after 80 days, stimulus=1
Model II Poro-elastic model • Equilibrium eqn: • Constitutive eqn: • Compatibility cond: • Darcy’s law: • Continuity eqn:
Model II • Incompressible, viscous fluid: • Slightly compressible, viscous fluid:
Model II Incompressible: Problem if Solution approximates Finite Element Method leads to inconsistent or singular matrix
Model II Solution: 1. Quadratic elements to approximate displacements 2. Stabilization term
Model II • u and v determine the shear strain γ • p and Darcy’s law determine relative fluid velocity
Model II Boundary conditions
Results Model II Arm abduction 30 ° Arm abduction 90 °
Results Model II 30 ° arm abduction, during 200 days
Results Model II Simulation of 200 days: first 100 days: every 3rd day arm abd. 90°, rest of the time 30 °. 100 days200 days
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)
