bone functional adaptation by remodeling -- …...case study 1: optimization of porous scaffold...

1

　　　　海綿骨の骨梁リモデリングシミュレーションと骨再生用ポーラスScaffold設計への応用

安達　泰治

京都大学大学院工学研究科機械理工学専攻

Department of Mechanical Engineering and ScienceKyoto University

[email protected]

理研シンポジウム RIKEN Symposiumものつくり情報技術統合化研究（第５回）

ＶＣＡＤシステム研究　「ものつくり」ツールの構築と「研究」の基盤ツールへの展開

9-10 June, 2005,　理化学研究所　鈴木梅太郎記念ホール

構造の観察と力学的解釈 → 骨の変形法則 (Law of Bone Transformation, Wolff 1869)力学的機能 → 適応：機能的適応 (Functional Adaptation，Roux 1881)

Wolff’s Law（骨の機能的適応）

生体，構造生体，構造

機能的適応機能的適応

WolffWolff バイオメカニクスバイオメカニクス

Hierarchy in Bone Mechanical System: Tissue to Molecules

Spatial & Temporal Hierarchy in

• Structure - Function• Interaction between

mechanical & biological factors

Tanaka

Macro

Cellular networksCellular networks

Trabecular systemTrabecular system

Molecular systemsMolecular systems

Cytoskeletal systemCytoskeletal system

Osteoblast OsteoclastFormation

marrow

Resorption Trabecular bone

Bone Functional Adaptation by RemodelingBone Functional Adaptation by Remodeling-- Trabecular surface remodelingTrabecular surface remodeling --

Cellular Activities in Bone Remodeling

Microscopic mechanical stimuli activate cellular activities

Parfitt (1994)

1.1. 骨梁レベルの力学刺激（ストレス）の一様化骨梁レベルの力学刺激（ストレス）の一様化から得られるから得られる骨梁構造骨梁構造　　現象論的仮説 → 再構築則

→ 観察： ““WolffWolff’’s Law?s Law?””

2.2. 骨梁レベルの応力一様化を骨梁レベルの応力一様化をもたらすもたらすマイクロマイクロレベルのメカニクスレベルのメカニクス

　　骨細胞ネットワークとの関連付け

Contents

<骨梁レベル>ストレス一様化


<マクロ>Wolff’s Law?


①<骨細胞ネットワークレベル>メカニカルストレス応答の力学

<<骨細胞ネットワークレベル骨細胞ネットワークレベル>>メカニカルストレス応答の力学メカニカルストレス応答の力学②

2

１．骨梁応力一様化（仮説）から得られる骨梁構造


marrow


Bone remodeling

Mechanical stimulus

Structure change





①clnΓ σ= ( )dσ

( )

( )S

d

S

w l ds

w l ds

σσ = ∫

∫(l =|x-xc|)

Local stress nonuniformity on trabecular surface(Adachi98)

σc : Stress at point xc

σd : Representative stress in the neighboring point xd

Driving ForceDriving Force

Rate Equation of Trabecular Surface Remodeling

Uniform stress on trabecular surface Uniform stress on trabecular surface

Trabecular bone

( 0)m <

Marrow

cx

lL

( 0)m >&

l

w(l)

lL0

1

Weight function

Γ > 0 0m>&

0m<&(Resorption)

(Formation)

Γ < 0

Schematic representation of relation between and m& Γ

ΓuΓlΓ Formation

Resorption

m&

0

( )m F Γ=&

Digital Image-Based FEM Model

- Modeling with μCT image data

Rat L1 vertebra

Image data obtained by μCT(32 μm/pixel)

201 slices 15 mm

6 mm

32 μm/voxel

32 μm

Voxel-based Simulation of Trabecular Remodeling

Bone remodeling


Marrow


Mechanical adaptation

Mechanical stimulus

Structure transformation

Resorption Formation

Trabecular surface movement due to remodeling

3D Simulation for Trabecular Bone Remodeling

Sato N Sato N et al. 2004. 2004

displacement

x

yz

1.0 MPa

2.07mm

Remodeling Process

Initial state 1st step 3rd step 5th step

7th step 10th step 15th step 20th step

3

２．ミクロのしくみからマクロの仮説へ

　　

　　

　　Osteoblast OsteoclastFormation

marrow


Bone remodeling

Mechanical stimulus

Structure change

Mechanics in Osteocyte Network System




<骨梁レベル>ストレス一様化①

<骨細胞ネットワークレベル>メカニカルストレス応答の力学

<<骨細胞ネットワークレベル骨細胞ネットワークレベル>>メカニカルストレス応答の力学メカニカルストレス応答の力学②

Osteocyte Network: Viewpoints!

Kamioka Kamioka et al.et al.Osteocyte network in chick calvaria

Osteocyte　　　　courtesy Kamioka

Osteocyte Mechanosensor Distribution Mechanosensor Distribution Cellular CommunicationCellular Communication

in Differential or Integral Mannersin Differential or Integral Manners

Mechanical loading

Bone matrix deformation

Pressure gradient in interstitial fluid

Interstitial fluid flow

Shear stress on cell process

Mechanics

Mechanobiology in Osteocyte Network

Flow Osteocyte Cell process

Canaliculus

Gap junction

Lacuna Interstitial fluid

Mechanosensing

Communication

Bone formation and resorption

Change in bone structure

Biology

Framework

x y

Mechanical loadingEH( H, Vf )

KH( H, Vf ) V(x)P(x)

σ (x)ε (x)

dc, dp, μN(x), A

T(x)S(x)N(x), dp, lp w(L)

( i ) Macro-scale (Homogenized)

( ii ) Micro-scale

( iii ) Interstitial fluid scale

( )&M x Change in structure

v τ

Macro-scale Mechanical load

Micro-scale Shear stress

Macro-scale Surface remodeling

Modeling of Mechanosensing by Osteocyte

( ) ( ) ( )i b S rS w l P dΩ

Ω= ∇∫x x

r bl = −x x

(0 )1( )

( )0S

SS

l lw l

l l≤ ≤⎧

= ⎨ <⎩

1( ) tr ( )3

P = −x σ xPressure :

Mechanical quantity sensed by osteocyte: Pressure gradient (→ Fluid shear stress)

Mechanical stimulusMechanical stimulus:

Trabecula Cell process

(a) Osteocytes embedded in bone matrix. (b) Isotropic cell processes.

Osteocyte cell body

Mechanosensing region

(c) Modeling of mechanosensing region.

(d) Evaluation of mechanical Stimulus, Si(xb).

Cell body

lSxb

xr,l

( )rP∇ x

Mechanosensing radius: lS

Intercellular Communication

( ) ( ) ( )ocN

c T i bi

T w l S= ∑x x

(0 )1 /( )

( )0TT

TT

l ll lw l

l l≤ ≤−⎧

= ⎨ <⎩

Mechanical signalMechanical signal which surface cell receives:

(on surface)

Communication radius, lT

Surface cell, xc

Osteocyte i, xbTrabecula

l

c bl = −x x

Intercellular communication radius: lT

4

2D Simulation Model

Marrow

Trabecula

1x

2x1.36mm

1.44mm

σ2=1MPa (Uniform displacement)

Trabecular bone

Marrow Et = 20GPa , νt = 0.3

Em = 20MPa , νm = 0.49

Mechanical properties

Finite element divisionVoxel size : δ = 8μm x1: 180 voxel x2 : 170 voxel Plain strain condition

Osteocyte density : 1 cell / 3000μm2

Mechanosensing radius : lS=35μm

Model parameters

Intercellular communication radius : lT=150μm Remodeling thresholds : Tl=2000, T0=2500, Tu=3000 (×δ 3 N)

Change in Stress Distribution

Initial 20th step 50th step 100th step 0

12

Equiv. stress(MPa)

Resorption

Resorption

Formation

Formation

Resorption

Quantitative Evaluation of Functional Adaptation

0

1

2

3

4

5

6

0 20 40 60 80 100

Model XModel YModel Z

Step Step

(a) Change in bone volume (b) Change in apparent stiffness in loading direction

X Y Z

BV Stiffness Less material with higher stiffness

Functional adaptation by remodelingFunctional adaptation by remodeling

0.2

0.3

0.4

0.5

0.6

0 20 40 60 80 100

Model XModel YModel Z

App

aren

t stif

fnes

s, σ 2/ε

2(G

Pa)

Bon

e vo

lum

e fr

actio

n, B

V/TV

Stress Uniformity at a Single Trabecular Level

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0 2 4 6 8

Initial100step

0

0.02

0.04

0.06

0.08

0.1

0.12

0 2 4 6 8

Initial100step

0

0.02

0.04

0.06

0.08

0.1

0 2 4 6 8

Initial100step

Equivalent stress, σ (MPa) Equivalent stress, σ (MPa)

Equivalent stress, σ (MPa)

(a) Model X (b) Model Y

(c) Model Z

Freq

uenc

y, V

(σ)/V

All

Freq

uenc

y, V

(σ)/V

All

Freq

uenc

y, V

(σ)/V

All

Initial state: Stress in wide range

After remodeling: Narrow range

Stress uniformity in trabecular levelStress uniformity in trabecular level

Effect of Loading Magnitude

0

2

4

6

0 0.2 0.4 0.6 0.8 1.0 1.2 1.4App

aren

t stif

fnes

s σ 2/ε

2(G

Pa)

Apparent stress σ2 (MPa)

0

0.1

0.2

0.3

0.4

0.5

0 20 40 60 80 100

0.500.671.001.33

Step

Bon

e vo

lum

e fr

actio

n, B

V/TV

2 (MPa)σ

σ2

Width and stiffness of trabecula were regulated

depending on loading magnitude.

σ2 = -1.33MPa σ2 = -1.00MPa

σ2 = -0.67MPa σ2 = -0.50MPa

(100th step)

Relation to “Uniform stress hypothesisUniform stress hypothesis”10mm

ln( / )c dΓ σ σ=

( )

( )S

d

S

w l dS

w l dS

σσ = ∫

∫( ) 0 (0 )Lw l l l> ≤ <

( )M F Γ=&

“Uniform stress hypothesisUniform stress hypothesis”

Functional adaptation (alignment to principal stress direction)Stress uniformity at trabecular levelStress uniformity at trabecular level

The proposed model considering The proposed model considering mechanosensory systemmechanosensory system

in osteocyte networkin osteocyte network

5

海綿骨欠損部再生ための海綿骨欠損部再生ためのScaffoldScaffold内部構造内部構造

設計支援シミュレーション設計支援シミュレーション

Background

Defected

Bone

Scaffold

OsteoblastGrowth factor

Replaced

Scaffoldの内部構造は骨再生に大きな影響を与える

Defected

Bone

Scaffold

OsteoblastGrowth factor

Replaced

海綿骨欠損部におけるScaffoldを用いた

骨再生を計算機シミュレーションにより模擬

Scaffoldの内部構造は骨再生に大きな影響を与える

Scaffold内部構造の設計手法を提案

・内部構造の設計変数・骨再生過程の評価法

定める

Introduction: Bone regeneration using scaffold

DefectBone

Scaffold Osteoblast

BiodegradablePolymer

New bone

Degradation

Regenerated

(1) (2) (3) (4)

(1)(1) Defect in BoneDefect in Bone(2)(2) Scaffold ReplacementScaffold Replacement(3)(3) Scaffold Degradation & Scaffold Degradation &

New Bone FormationNew Bone Formation(4)(4) RegeneratedRegenerated

Mec

hani

cal F

unct

ion

BoneScaffold

(1)

(2)

(4)

DegradationFormation

Regeneration Time

Bone-scaffold system

Simulation Model of Bone-Scaffold System

Ingrowth into scaffold

Formation on scaffold and bone surface

ex. Surface Remodeling

Mech. Integrity: Eb

Water contents: h- Diffusion eq.

Molecular weight: W- hydrolysis

Mech. Integrity: Es

+

Degradation of ScaffoldDegradation of ScaffoldDegradation of Scaffold

Change in Mechanical Function of Bone-Scaffold System

Change in Mechanical Function of Change in Mechanical Function of BoneBone--Scaffold SystemScaffold System

New Bone FormationNew Bone FormationNew Bone Formation

( )W c cβ= −&

2c cα= ∇&

b mm

( ) tE t ET

=S S00

( ) WE W EW

=

Optimization of Scaffold DesginOptimization of Scaffold Optimization of Scaffold DesginDesgin

Structural Optimization of Porous Scaffold

Eval

uatio

n fu

nctio

n Ev

alua

tion

func

tion

Design parameter Design parameter

Minimum Minimum

InitialScaffold

Design parameter … structure, size,

molecular weight, etc.

Evaluation functionEvaluation function

Tiss

ue fu

nctio

n Ti

ssue

func

tion

Time

Time

Bone regeneration process

Optimal Structure Optimal Structure

Evaluation Function

Minimize: Difference in mechanical function between bonebone--scaffoldscaffold and desired bonedesired bone

w.r.t Design Valuables

(iii)

(i)

(ii)

Regeneration Time

Mec

hani

cal F

unct

ion

(Str

ain

Ener

gy)

Desired boneDesired bone

BoneBone--scaffoldscaffold

(i) Maximum difference

,peak bonemax ( )U U t UΦ = −

(ii) Final diff. after regeneration

final final boneU UΦ = −

(iii) Average difference

boneaverage 0

( )T U t Udt

TΦ

−=∫

6

Case Study 1:

Optimization of Porous Scaffold Optimization of Porous Scaffold MicrostructureMicrostructure

(3D Simulation)(3D Simulation)

Scaffold Model

・24×24×24 voxel elements・2.0MPa in Z direction

・Time interval 1daytΔ =

S0 20GPaE =

S 0.3ν =

0 70000W =4000 / dayβ =

ScaffoldMarrow

m 20GPaE =

b 0.3ν =

m 2daysT =

Bone

Regeneration Process

Initial 5 days 10 days 20 days 30 days

40 days 70 days 90 days 130 days 150 days

Scaffold New bone Matured bone

Bone formation

Scaffold degradation

結果2 剛性の変化

0

0.2

0.4

0.6

0.8

1.0

1.2

0 20 40 60 80 100 120 140 160

BoneScaffoldTotal

Stra

in e

nerg

y [1

0-5 J

]

Days

Strain Energy of Bone-Scaffold System

Initial

10 days

20 days

40 days

70 days

(b) Spherical cavity(a) Lattice

0.4 mm < l < 2.2 mm0.4 mm < l < 2.2 mm 2.5 mm < d < 3.2 mm2.5 mm < d < 3.2 mm

ld

Design Valuables

(iii) boneaverage 0

( )T U t Udt

TΦ

−= ∫

Lattice : l = 1.4 mm Sphere : d = 2.5 mm

Lattice : l = 1.4 mm Sphere : d = 2.5 mm

S( 0.62)ρ =S( 0.59)ρ =

0

0.2

0.4

0.6

0.8

1.0

0 0.2 0.4 0.6 0.8 1

LatticeSpherical cavity

Cavity volume fraction

Eval

uatio

n fu

nctio

n,

[10

-5J]

Φav

erag

e

, Sρ Optimal design variables were determined for both lattice and spherical pore microstructure of scaffold

Optimal design variables were determined for both lattice and spherical pore microstructure of scaffold

Results

( i )

(ii)

(iii)

Time

Stra

in e

nerg

y

Desired bone

7

Case Study 2:

Effect of Scaffold Design Parameters Effect of Scaffold Design Parameters on Regenerated Trabecular Boneon Regenerated Trabecular Bone

(2D simulation)(2D simulation)

Simulation Model

μ

μ

Design Variables

D : 空孔径　L : 中心間距離

設計変数

D，Lの2つの設計変数の値を

定めることにより初期構造を決定

(Vf : 体積分率)

強度を保つため周囲に0.36mmの厚みを付加

Bone Regeneration Process (Initial)

Bone Regeneration Process (150 days) Optimal Internal Structure

最適な内部構造寸法は,(D , L) = ( 480 μm, 960 μm)

　　　　　　　　不十分な骨再生

8

骨再生用ポーラス骨再生用ポーラスScaffoldScaffoldのの設計・造形プロセスの検討設計・造形プロセスの検討（医用デジタルイメージの活用）（医用デジタルイメージの活用）

V-CADへの期待

Background & Purpose

二段階の鋳造工程を経たScaffold造形プロトコル(1)の確立

三次元CAD 三次元造形型A HA懸濁液注入

型A除去型B (HA製)熱分解焼結

PLA注入 PLA Scaffold

(1) S.J.Hollister et al. (2003)

造形プロトコル

近年，Scaffoldを用いた実験的な検討例は報告されているが，現状では，Scaffoldの確固たる造形法は確立されていない．

Rapid Prototyping

溶融金属など液化させた材料を必要な個所のみに

噴射・堆積させて立体モデルを作製する造形法

インクジェット積層法

繰り返し球形粒子

造形ベース造形ベース

カッター積層ピッチ

プリンタヘッド球形粒子：直径約76μm最小積層ピッチ：13μm

型Aの評価

(a) D = 1.8 mm , L = 0.6 mm

造形モデル

(b) D = 0.75 mm , L = 0.25 mm

CT画像

X線CT画像を用いたScaffold形状構築

欠損部位別Scaffold

設計支援システム

Thank You!

Taiji AdachiMechanics of Adaptive Structures and Materials Laboratory

Department of Mechanical Engineering and ScienceKyoto University

[email protected]

bone functional adaptation by remodeling -- …...case study 1: optimization of porous scaffold...

Documents