理研シンポジウム 「生体力学シミュレーション」

1999年7月1日，2日

骨のリモデリングの計算バイオメカニクス

‐階層的モデルとその応用‐

安達 泰治，坪田 健一，冨田 佳宏

神戸大・工，理研

Hierarchical Structure of BoneMacro

Micro

Tanaka

Internal Structure of Trabeculae

Trabecular microstructure of cancellous bonechanging / maintained by remodeling, mechanical factors

Adaptation to mechanical environmentregulated by Oc / Ob activities on trabecular surface

Surface movement by cellular activities lead to macroscopic changes of trabecular architecture

Introduction: Trabecular Surface Remodeling

Macro Micro

Trabecular Surface Remodeling (Parfitt, 1994)

Theoretical models & Computational simulations

Microscopic resorption and formation (Parfitt84)

Local mechanical signals play an important role(e.g. Guldberg97)

Trabecular level mechanical stimulus related to morphological changes of

trabecular architecture

Introduction: Computational simulation

Microscopic MechanismCowin92, Sadegh93,

Mullender94

Macroscopic PhenomenaCowin76, Carter87, Huiskes87,

Beaupre90, Weinans92

Macroscopic Model: Huiskes et al.

Fig. Feedback Mechanism of Remodeling(Huiskes & Hollister, 1993)

ρ : Apparent DensityX : Coordinate of Surface PointS : Mechanical StimulusS0 : Reference Value of SCI,CE : Remodeling Constant

)( 0SSCdtd

I −=ρ

)( 0SSCdtd

E −=X

Internal Remodeling (Huiskes87)

External Remodeling :

Macroscopic Model and Simulation

)]()()[( 0 QEQEQCU ijijij −=

- 2D Beam Theory- Long Bone

-P

-P(a) Simulation Model

(b) Immobilization (c) Ostectomy

(i) Predicted Geometry (i) Predicted Geometry

(ii) Experiment (Uhthoff78) (ii) Experiment (Lanyon82)

Ulna

Radius

1mm1mm

U : Rate of surface movement

- Adaptive Elasticity (Cowin85)

- Self Optimization Model (Carter87)

Macroscopic Model: Cowin, Carter et al.

)()()( 0ijijijeAeadtde εε −+=

)( ASbbcdtd Ψ−Ψ=ρ

0ρρ −=ewhere

where )(1

∑==

Ψ N

i

mbb

iσσm/1

∑Ψ=

=N

iib n

1

(

- Adaptive Elasticity (Cowin76)

(b) Predicted Density(a) FEM Model

Remodeling under Single Load

(Adachi97)

703 N, 28

2317 N, 24

351 N, -8

1158 N, -15

468 N, 35

1548 N, 56

(a) One-legged stance (b) Abduction (c) Adduction

Remodeling under Multi‐Loads

Computational Biomechanics of Bone Remodeling

Purpose: basic understanding & applicationTo understand mechanism of adaptive bone remodelingTo predict remodeling, around bone-implant interfaceTo design implant, screw ...To apply in bone tissue engineering, design scaffold ...

Approaches:Phenomenological modeling and simulation “Macro”down toward mechanism at cellular level “Micro”

Macro- & Microscopic Viewpoints

Macro

Micro

力学負荷 外形状・密度変化：巨視的連続体

骨梁形状・構造変化：微視的連続体

階層的，微視構造

力学刺激

巨視的力学特性特性変化

骨リモデリングの連続体モデル？微視的力学刺激を反映できない

微視レベルでの直接的モデル？巨視的力学負荷を反映できない

•微視的な力学刺激を巨視的力学負荷と関連付けるモデル

•微視構造を考慮した巨視的連続体モデル

Approaches: Model & Simulation

Lattice Continuum Model of BoneMechanics and RemodelingCosserat ContinuumHierarchy of structure in continuum

cf. Homogenization and localization method

Trabecular Surface RemodelingMicrostructural changes --> Macro structural and mechanical properties

Voxel finite element model

Mechanics and Remodeling

Lattice Continuum Model

),,,(, ⋅⋅⋅== SHEEεET ρ

Macroscopic Constitutive Eq.

),,(),( ⋅⋅⋅== eeff εTSSE &&&

- Macro- Remodeling Rate Eq. ),,( ⋅⋅⋅= εTE f&

S: Structure microεT, ee εT ,- Micro- Remodeling Rate Eq.

Trabecular Surface Remodeling

Rate Equation: MicroscopicApproach to Cellular level mechanism

Simulation: Microscopic--> Structural changes in Macro--> Change in Apparent Mechanical Properties

Remodeling at trabecular level: microscopic level

stimulusmechanical cmicroscopi:

)(Γ

Γ= FM&

Rate of trabecular surface movement

3D Simulation Model Using Voxel Micro FE

Representative Stress

Driving Force∫∫= SS rd dSlwdSlw )()( σσ

)/( dclnΓ σσ=

・ µCT Image-Based Cancellous Bone Model

・ Uniform Stress Criterion

MechanicalStimulus

MorphologicalChange

Trabecular-LevelMacroscopicPhenomenon

Cellular-LevelMechanism

M : Rate of Surface Movement.

(Adachi et al., 1997; Tsubota et al., 1998)

Application:Computational Biomechanics of Bone Remodeling

Evaluation and prediction of bone remodeling around artificial joint, implant, artificial joint, …Design of implant, screw, joint considering remodelingDesign of scaffold structure in tissue engineeringScaffold & new bone ingrowth

mechanical function and biochemical degradation

Comparison

(Mosekilde 1990)

Vertebral Body with a Rod ScrewF1 F2

F3

50.0mm

25.0

mm

3.0mm3.5mm

25.0mm 25.0mm

4.0m

m

20 elements

Cancellous Bone Cortical Bone

Annulus FibrosusNucleus Pulposus

Cartilaginous End-plate

Rigid Plate

Rod Screw

Caud.

Cran.

VertebraIntervertebral Disc

Artificial Vertebra

Instrument( Ikebuchi K. et al. 1995 )

Screw

10/)(N8.58

21

3

FFF

+==

0.4,5mm1

elements128290

=Γ−=Γ=×

ul

Ll

Applied Load to Rod Screw

Remodeling: With a Rod Screw

Remodeling Rod Screw Interface

~ ~ ~~ ~ ~ ~~12 3 4

5 6 7

θ = 10.7

a = 1.240 mmb = 0.774 mm

2.50 mm

2.50 mmθ = 15.8

a = 1.108 mmb = 0.800 mm

2.50 mm

2.50 mmθ = -13.3

a = 1.322 mmb = 0.914 mm

2.50 mm

2.50 mmθ = -19.6

a = 1.407 mmb = 0.884 mm

2.50 mm

2.50 mmθ = -0.31

a = 1.136 mmb = 0.896 mm

2.50 mm

2.50 mmθ = 4.1

a = 1.129 mmb = 0.904 mm

2.50 mm

2.50 mmθ = -2.6

a = 1.499 mmb = 0.792 mm

2.50 mm

2.50 mm

1 2 3 4 5 6 7

Model around Rod Screw Interface

5.6mm

2.8m

m

20 elements

Rod Screw Cancellous Bone σ = 1.0 MPa

τ = 1.0 MPa

Compression: Case Sc

Shear: Case Ss

θ = -6.7

a = 0.315 mmb = 0.297 mm

0.500 mm

0.500 mm

elems70140×

elems140280×

Remodeling around Rod Screw Interface

θ = -5.9

a = 0.316 mmb = 0.283 mm

0.500 mm

0.500 mm

θ = 19.4

a = 0.319 mmb = 0.286 mm

0.500 mm

0.500 mm

θ = 4.0

a = 0.358 mmb = 0.235 mm

0.500 mm

0.500 mm

θ = 3.1

a = 0.349 mmb = 0.256 mm

0.500 mm

0.500 mm

θ = 15.6

a = 0.310 mmb = 0.284 mm

0.500 mm

0.500 mm

θ = 29.2

a = 0.323 mmb = 0.288 mm

0.500 mm

0.500 mm

Compression: Case Sc Shear: Case Ss

4thstep

12thstep

20thstep

Strain Energy Densityin Tissue

Application: Bone tissue engineering(1) Digital Image

(3) Design of Scaffold

(2) 3D Model of Bone, Screw and Scaffold

(Hollister et al., 1998)

Topology Optimization

Initial ArchitecturePorous Scaffold

Fixation Screw

Original CondyleAnatomyHigh

LowVon Mises Stress

in Scaffold

・Structural analysis of Scaffold・Mechanical Bone Remodeling

Manufactured Condylewith Scaffold

Rendering of CT Image

(4) SFF Manufacture Directly or by Casting

Application: Design structure of scaffold

as a load bearing constructcompatible with cell ingrowth and migration as a transducer of mechanical signal to cellsto consider transition between

degradation of scaffoldnew bone formation and remodeling

Porous Scaffold

Fixation Screw

Original CondyleAnatomy (Hollister et al., 1998)

Conclusions

Computational biomechanics in bone remodeling

Macro- and Micro- Hierarchyin Bone mechanics and remodeling

Basic science --> Application