a feasibility study on the acceleration and upscaling of bone ingrowth simulation an investigation...

A feasibility study on the acceleration and upscaling of bone ingrowth simulationAn investigation into the feasibility of applying a homogenization scheme to bone ingrowth model as well other options of accelerating the bone ingrowth model developed by A. Andreykiv.

MSc colloquium by Yoeng Sin KhoeDelft Technical University

Faculty of Biomedical Engineering – Tissue Biomechanics & ImplantsDate: June 30th

, 2009

Introduction• Uncemented Shoulder Implant

• Total Shoulder Arthroplasty• Osteoarthritis, rheumatoid arthritis• Porous surface in which bone can

grow to ensure fixation of the implant

• Cemented implant– Immediate fixation– Tissue necrosis– Cement fracture

Introduction• Finite Element Modelling of uncemented

implants – 2 approaches

• How to transfer knowledge from the detailed model to the larger (macroscopic) model?

Andreykiv, 2006 Folgado, 2009

Recommendations

Original Bone Ingrowth model

Contents

Model Optimizations

Options

Results

Code Optimization

Possibilities

Results

Computational Homogenization

Theory

Implementation

Results

Introduction

Summary & Conclusions

The original bone ingrowth model

The original bone ingrowth modelGeneral Overview

• Coupled Simulation• Prendergast tissue differentiation model for the

production of bone, cartilage and fibrous tissueMechanical

Model

Biophysical

Stimulus

Biological

Model

Material Properti

es

General Overview Mechanical Model Biophysical Stimulus Biological Model Numerical Implementation

The Original Bone Ingrowth ModelMechanical Model

• Biological Tissue response– Non-linear– History dependent– Viscoelastic

• Biphasic Model– 80% is made up of fluid– Solid & Fluid component


The Original Bone Ingrowth ModelMechanical Model

• Solid (3 displacements)– Neo-Hookean Hyperelastic Material

model• Fluid (pressure)

– Mass balance

0

0

0

pvdt

dp

K

n

fv

fv

sf

f

ffffff

ssssss

σ

σ

0

0

pvdt

dp

K

n

p

sf

f

s

Iσ


The Original Bone Ingrowth ModelBiophysical Stimulus

• Input for the biological model• Determines the preference of the

formation of bone, cartilage of fibrous tissue

• Based on maximal shear strain and fluid velocity

baS


The Original Bone Ingrowth Model Biological Model

• Diffusion – Mesenchymal stem cells– Fibroblast

• Cell Proliferation• Cell Differentiation• Tissue Production• Tissue Degradation• As a function of the biophysical stimulus


The Original Bone Ingrowth Model Biological Model

Mesenchymal Cells

Osteoblast Cells

Chondrocyte Cells

Fibroblast Cells

Bone

Fibrous Tissue

Cartilage

Differentiation

Proliferation

Tissue production


The Original Bone Ingrowth Model Numerical Implementation

• Implemented in subroutines of MSC Marc• Non Linear equations requires an iterative

solver


force

displacement

force

displacement

The Original Bone Ingrowth Model Numerical Implementation & Results

• Loading based on the results of animal experiments– applied displacement in the first week, – average force over the remaining days

• Result: Tissue Evolution & Micromotions• Total simulation length 28 days• Computation time: ~200.000 sec

(2 days and 7 hours)


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 290

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0

10

20

30

40

50

60

Results of the Bone Ingrowth Simulation

BoneCartilageFibrous TissueMicromotions

Time [Days]

Tiss

ue fr

actio

ns

Micr

omoti

ons [

µm]

Model Optimization

Model OptimizationPossibilities for model simplification (1)

• Investigate the necessity of the biphasic model

<Weighted Fluid Velocity Weighted Shear Strain

baS

Potential Simplifications(1) Results: Removing the fluid phase Results: Linearized Biological Model Results: 1D Diffusion


• Linearization of the biological model

• Acceptable approximation?



• 1-D Diffusion approximation


Model OptimizationResults – Removal of Fluid phase

• Fluid phase is essential for correct calculations• Increased fibrous tissue production• Reduced bone & cartilage production

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 290

0.01

0.02

0.03

0.04

0.05

0.06

0.07

Effect of removing the fluid phase

Bone - OriginalBone - NoFluidCartilage - OriginalCartilage - NoFluidFibrous Tissue - OriginalFibrous Tissue - NoFluid

Day nr.

Tiss

ue fr

actio

n

Potential Simplifications Results: Removing the fluid phase Results: Linearized Biological Model Results: 1D Diffusion

Model OptimizationResults – Linear Biological Model

• Results of linear biological model do not match non linear model

• No more fibrous tissue production• Overestimate bone production• Reasonable cartilage estimate

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 290

0.010.020.030.040.050.060.07

Effect of a linear Biological model

Bone - OriginalBone - LinearBIOCartilage - OriginalCartilage - LinearBIOFibrous Tissue - OriginalFibrous Tissue - LinearBIO

Day nr.

Tiss

ue fr

actio

n

Mesenchymal Cells

Osteoblast Cells

Chondrocyte Cells

Fibroblast Cells

Bone

Fibrous Tissue Cartilage

Differentiation

Proliferation

Production

Tissue production


Model OptimizationResults – 1D diffusion

• Simulations failed– Non-Positive definite stiffness matrix– Snap back behaviour? Causing the Newton-Raphson

method to fail?– Perhaps an arc length method can improve the

results• Potentially gained simulation time is marginal

– A estimated decrease of 200 seconds over the complete simulations.


Code Optimizations

Code OptimizationsPotential for increasing speed

• Culprit: Disk operations• Example: Reading the stimuli.dat file • Reduce waiting times• Reduce number

of disk operationsSend command to open

stimulus.dat file. Is file in free for use?

Read element number and stimulus value.

Is file operation complete?

Yes

NoWait 1 second

START

Wait 1 secondNo

Yes

Is this the last record in stimuli.dat

Yes

Close file and go to next record

Close file and continue subroutine

No

Potential Optimizations Results:MINISLEEP Results:Batchwrites

Code OptimizationsMinisleep

• Reduce waiting time• FORTRAN limits the waiting period to 1 sec• Write the MINISLEEP

0 50 100 150 200 2500

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2x 10

5

time

(s)

Comparison of timings

Iteration nr

Original

MINISLEEP(1)MINISLEEP(10)

MINISLEEP(250)

20%


Code OptimizationsBatchwrites

• Reduce the number of disk writes.• Store all variables in memory and write at the

end of an iteration• Keep in mind data sharing

• Reduction of 65% in computation time


Computational Homogenization

Computational HomogenizationTheory (1)

• Exploit periodicity to bridging the gap• Microscopic level Macroscopic level

Global Idea Deformation Tensor Localisation Upscale Stresses Macroscpoic Tangent Implementation Results

• Macroscopic deformation• Microscopic deformations / microfluctuations

Computational HomogenizationTheory (2)

deformedundeformed

wXxxdXd M

F


Computational HomogenizationTheory (3) - Localisation

• Translation between microscopic and macroscopic deformation tensor

00

00 00

11

dNwdVV V

mM

FF

00 0

1dV

V V

mM FF

01

00 0

dNw

wXx M

F

wXx M

F


Computational HomogenizationTheory (4) – Stresses

• Hill-Mandel condition• Work conjugated couple:

– Deformation tensor & 1st Piola-Kirchhoff stress

mM WW

00 0

1

dXpM

P


Computational HomogenizationTheory (4) – Macroscopic tangent

• Tangent describes how small variations affect the stresses in the system

• Numerical differentiation• Very cumbersome method, but Miehe (1996)

developed a more efficient method.


Computational HomogenizationImplementation in MSC Marc

• Macroscopic Model– Loading– Deformation tensor– Macroscopic tangent

• Microscopic Model– Periodic Boundary

Conditions– Upscale stresses


Computational HomogenizationResults / Issues

• CH implementation requires a lot a additional computing time– Computational overhead in the numerical

differentiation scheme• Application of loading

1 2 3 4 5 6 7 80

0.05

0.1

0.15

0.2

0.25

Effect of different loading

Bone - OriginalBone - TopLoadedCartilage - OriginalCartilage - TopLoadedFibrous Tissue - OriginalFibrous Tissue - TopLoaded

Day nr.

Tiss

ue fr

actio

n


Summary & Conclusions• Sections of the constitutive equations that are

responsible for long calculations cannot be neglected– Fluid phase, non-linear biological model, diffusion

• Acceleration of the simulation was obtained by efficiently directing disk activity.

• Computational Homogenization increases simulation time and cannot account for specific loading

Summary & Conclusions Recommendations Questions

Recommendationsalternatives for upscaling results

• How to bridge the gap?– Use the model to investigate time to fixation under

different loadings– Use a larger model to asses post-surgery

micromotions– Develop an element that adapts the stiffness in

order to ensure that at the time to fixation the micromotions are reduced to zero.


Questions?

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Questions?

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a feasibility study on the acceleration and upscaling of bone ingrowth simulation an investigation...

Documents

detailed model

larger macroscopic model

production of bone

formation of bone

cartilage of fibrous

msc colloquium

homogenization scheme

feasibility study