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?