bone ingrowth in a shoulder prosthesis e.m.van aken, applied mathematics
TRANSCRIPT
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
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
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:
1 2[ ] [1 ]mm m m m m m m m
cD c Cc m A c c Fc F c
t
2 3[1 ]cc c c c m c
cA c c F c F c
t
1 3[1 ]bb b b b m c b b
cA c c Fc F c d c
t
Model I
• Matrix densities:
• Growth factors:
(1 )( )ccs c c m c cd c b
mP m c c Q m c
t
(1 )bbs b b b
mP m c
t
[ ]cgc c gc c gc c
gD g E c d g
t
[ ]bgb b gb b gb b
gD g E c d g
t
Model I
• Boundary and initial conditions:
•
maxperiosteummc c
( ,0) 0ic x
( ,0) 0bm x
( ,0) 0.1cm x
fracture gap along boneother boundaries
20 20 0ic b
gg g
x
all boundaries
0ig
x
:Kt t
:Kt tall boundaries
0mc
x
Finite Element Method
• Divide domain in elements
• Multiply equation by test function
• Define basis function and set
• Integrate over domain
i
1
nn
j jj
c c
i
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:2 (1 ) (1 )
(1 ) (1 )
mm m m m c f b m f f m
c c m b b m
cD c P c c c c c F c c
tF c c F c c
2 (1 ) (1 )
(1 ) (1 )
ff f f m c f b f f f m
c c f b b f
cD c P c c c c c F c c
tF c c F c c
(1 ) (1 )( )bb f m c b b b b m f c
cP c c c c c F c c c c
t
(1 ) (1 )( ) (1 )cc f m c b c c c m f b b c
cP c c c c c F c c c F c c
t
Model II
• Matrix densities:
(1 )bb b b
mQ m c
t
(1 )cc b c c b b c tot
mQ m m c D c m m
t
(1 ) ( )ff tot f b b c c f tot
mQ m c D c D c m m
t
Model II
• Boundary and initial conditions:
0 0( , ) ( , ) 0 ( , , , )i ic x t m x t i m f c b
0 at all boundaries ( , , )ic i f c bx
0 max( , ) constant at bone-implant interfacemc x t c
0 at the other boundariesmc
x
Model II
Proliferation and differentiation rates depend on stimulus S, which follows from the mechanical part of the model.
1 bone
1 3 cartilage
3 fibrous tissue
S
S
S
Results
Bone density after 80 days, stimulus=1
Results
Model II
Poro-elastic model
• Equilibrium eqn:
• Constitutive eqn:
• Compatibility cond:
• Darcy’s law:
• Continuity eqn:
div 0p
12 ( )ij j i i ju u
2ij ij ll ij
q u f
q p
Model II
• Incompressible, viscous fluid:
• Slightly compressible, viscous fluid:
( ) ( ) 0
0
u u p
u p
0n p u p
Model II
Incompressible: Problem if
Solution approximates
Finite Element Method leads to inconsistent or singular matrix
0
( ) ( ) 0
0
u u p
u
Model IISolution:
1. Quadratic elements to approximate displacements
2. Stabilization term
0su p p
Model II
• u and v determine the shear strain γ
• p and Darcy’s law determine relative fluid velocity
1 bone
1 3 cartilage
3 fibrous tissue
S
S
S
:Sa b
Model II
Boundary conditions1 2
2
3
: 0
: 0
: 0
p
np
np
1 0
2
3
:
: 0
: 0u
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 days 200 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)
Questions?