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Theories of Bone Modeling and Remodeling

Stresses and Strains at the Organ Level (Example #1):

Numerical Modeling of the Femoral Head

University of Paderborn, Germany 1

Stresses and Strains at the Organ Level (Example #2):

Numerical Modeling of trabecular core of the L1 Vertebra

2Rutgers University, Dept. of Mechanical and Aerospace Engineering

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The above animations display nonlinear finite element simulation of compressive loading of bovine tibial trabecular bone. The volume on the left represents the full specimen (5mm cube). The animation on the right represents a 1.5mm cubed subvolume of the specimen. Red indicates tensile strain failure, while blue represents compressive strain failure.

Stresses and Strains at the Trabecular Level:

Compression Simulation using Finite Element Analysis

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4a

Osteogenesis Modeling

Remodeling

Three major methods of bone adaptation:

4b

3

5

6

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oWolff was a German Anatomist who is credited with general theory of bone adaptation known as Wolff's Law: 1870-1894

oSuggested that bone obtained maximum mechanical efficiency with minimum mass -> optimal configuration

oBone structure could adapt in response to changing Mechanical Environment

Summary of Wolff's Observations:

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Wolff's law:

"Every change in the form and the function of a bone or of their function alone is followed by certain definite changes in their internal architecture, and equally definite secondary alterations in their external confirmation, in accordance with mathematical laws"

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o Roux, a German surgeon, suggests in 1881 that cell activity is modulated by mechanical stress

o Apposition and resorption by cells determines change in bone structure

o Cell based apposition and resorption regulated by value of local stress

Summary of Roux's Observations:

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5

o Koch, American Anatomist Johns Hopkins 1917, performed strength of materials analysis for proximal femur

o Confirms trajectorial theory, trabeculae along principal stress

o Suggests that bone density should be highest in areas of highest shear stress

o Again suggests bone attains maximum strength with minimum material

Summary of Koch's Observations:

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Ø Trabecular orientation aligned with principal stresses

Ø Density highest in areas of highest shear (bending)

stress

Ø Bone structure can be adapted to change in load

Ø Bone cells may be regulated by local stress

Ø Results show no direct relation of stress to cell

activity; Relationship of bone structure to mechanics

derived without regard to physiological mechanisms

Summary of Observations 1865 1920:

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o Glucksman performed experiments on tissue undergoing intramembranous ossification in vitro, Cambridge, 1938-1941

o Arranged ossifying tissues so that as they grew different amounts of tensile stress developed

o Tensile stresses promoted increased ossification of fibrous tissue

o Histological structure of ossifying tissue aligned along principal tensile stresses

A summary of Glucksman's results:

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o Determined in 1966 that bone could be adapted by either modeling or remodeling mechanisms

o Determined that osteoblast and osteoclast activity were coupled during remodeling, no net gain of bone

o Suggested that the relationship between strains and bone mass was different in growing (modeling) and mature (remodeling)

Frost s contributions:

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Chamay & Tschantz (1972)

performed osteotomy of radius in dogs

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o At 9 weeks, significant hypertrophy with 60% to 100% increase in cortical thickness

o Noted several cases of fatigue fracture

o Carter estimated strains were between 5000 and 7000 mstrain; suggested hypertrophy due to damage

Carter & co-workers (1981)

performed osteotomy of ulna in dogs

oMeasured strains in control and experimental limbs cortex with strain gauge

oFound strain increased from 600 mstrain to 1500mstrain after osteotomy

oNo significant change in bone geometry or deposition of mineral after 8 weeks

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Bone Activity Principal Strain (mstrain)

Horse radius Trotting - (-0.28%)

Dog radius Trotting -2400

Goose humerus Flying -2800

Sheep femur Trotting -2200

Pig radius Trotting -2400

Fish hypural Swimming -3000

Monkey mandible Biting -2200

Rubin and Lanyon (1982):

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Rubin and Lanyon (1984) also performed a procedure to isolate the turkey ulna and impose controlled dynamic bending strains on the isolated ulna

o Found that dynamic strains are necessary to maintain bone; Static load led to bone resorption

o 4 cycles of load per day was sufficient to maintain bone; 36 to 1800 cycles per day produced no response

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Frost s description of the different adaptational responses for the adolescent and the adult skeleton.

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o Cowin proposed idea of adaptive elasticity 1976;

Hart performed computational implementation in 1983

o Separates surface and external remodeling

o Rate of Change of bone volume fraction related to strain

o Tensors A,B must be determined by experiment

o Constants difficult to determine:

Qualitative results good,

but accurate validation was not done

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Hart (1983) reached the following conclusions:

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Fyhrie s predictions of bone density distributions in the femoral head

Rem

odel

ing

Vec

tor

gain

loss

2s

Un U

Huiskes et als description of the density change rate in respect to the mechanical stimulus of the Strain Energy Density (U)

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Dynamic Simulations of

Cancellous Bone Resorption Around

Orthopaedic Fixative Implants

Illustrative Example of Using the Algorithm by Huiskes:

Dr. Amit Gefen

The Musculoskeletal Biomechanics Laboratory

Deptartment of Biomedical EngineeringFaculty of Engineering, Tel Aviv University

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Bone screws are well-known and clinically accepted means for fracture fixation or for stabilizing bone transplants

Background

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Background (cont.) Since the screws remain attached to the bone after healing, they may also diminish its strength

The stiffer metallic screws (E>100 GPa) carry most of the shared load, causing the adjacent bone (E=1-20 GPa) to be atrophied in response to the diminished load it is carrying

cancellous bone resorption around the threads due to stress shielding

micromotion and possible loosening

Internal loads will be mainly supported by the screws, which are shielding the bone from carrying the normal mechanical stresses

This stress shielding effect alters the normal stress stimuli for bone growth, and, according to Wolff s law, bone will adapt itself by reducing its mass around the implant

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Objective

Characterize screw designs that provide optimal stress transfer to the surrounding bone, and, thereby, alleviate commonly observed conditions of loosening and failure of fixations due to stress shielding

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Methods Two finite element (FE) model types of the screw-bone mechanical interaction were developed:

cortical bone layer

cancellous bone bulk

80 N

axis

ymm

etri

c co

nstr

aint

s

axis of symmetry

scre

w

fixed constraints between the screw s head and cortical surface

typical meshing around the screw s threads

screw cortical bone layer

gluteus maximus, 900 N

gluteus medius, 1165 N

fixed constraints

joint reaction, 3150 N

cancellous bone

idealized axisymmetrical femoral head30

Methods (cont.)

dead zone

ψref + ωψref − ω

gain

loss ψψ

dρρdt

ψref

calculate loading history

control finite element model

solve von Mises stresses for reference signal ψref

bone-screw finite element model

solve von Mises stresses for transient stress state ψ

+- Σ

actual mechanical stimulus for modeling/ remodeling element stress

within dead zone ?

no

compute local bone density change using remodeling rule

steady state

reached ?

no

yes

yes

construct bone/screw geometries

STOP

An iterative algorithm for simulating bone adaptation around the screw was coupled with the FE models

remodeling rule

remodeling algorithm31

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Methods (cont.)

3αρ=Eα = 2875 MPa cm3/gr

(Carter and Hayes, 1977)

ρψ

U= σε ⋅=

2

1U

( )

( )

−−⋅

+−⋅

=

ωψψ

ωψψρ

cc

cc

dtd

ref

ref

0( )

( )ωψψ

ωψψω

ωψψ

+>−

+≥−≤−

−<−

ref

ref

ref

density-stiffness relation:

stress stimulus for bone remodeling:

remodeling rule:(van Reitbergen et al., 1993 )

c=0.02 (µm/day)/(MPa/day)(Beaupr et al. (1990)

Mat

hem

atic

al F

orm

ula

tio

n

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Methods (cont.)

2low-stiffness titanium core

polymeric external layer and threads

following insertion of the screw, the steel sphere is pushed downward

the resulted opening of the screw s tip exerts controlled compression on adjacent bone

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hollow canal

Newly Proposed ScrewsCommercial Designs

Graded-Stiffness Active-Compression

constant dimensions- shaft diameter- screw-head diameter- thread pitch- screw length

modified properties- material stiffness- thread profile shape- thread diameter- no. of threads

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Methods (cont.)

cortical bone layer

cancellous bone bulk

σσ t

σσ fbσσ ft

j

σσb i

tf

bf

σ

σα =

∑

∑

∑

∑=

=

=

=

== N

jjt

N

iib

N

jjt

N

iib

N

N

2

2

2

2

1

1

σ

σ

σ

σβ

for the first thread:

for all other threads:

In order to quantify the screw-bone load sharing (and its evolution with bone adaptation) for the purpose of rating the different designs, two stress transfer parameters (STP) were defined,

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Results

135100

65

35

25

2017

14

11

8

42

0

MPa

rectangular

wide profile

9 threads

* each time step represents 10 weeks of bone adaptation

triangular

wide profile

9 threads

trapezoidal

standard profile

9 threads

trapezoidal

wide profile

9 threads

Evolution of stresses around different screws idealized model

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* each time step represents 10 weeks of bone adaptation

Results (cont.)

135100

65

35

25

2017

14

11

8

42

0

MPa

displacements(magnified ×× 30)

evolution of stresses

stresses around the threads

stresses around a graded-stiffness triangular screw

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Results (cont.)Evolution of the STP for bone adaptation behavior around commercial screw designs

αα STP ββ STP

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Results (cont.)Evolution of the STP for bone adaptation behavior around the newly proposed graded-stiffness and active-compression screws

0.5

0 .55

0.6

0 .65

0.7

0 .75

0.8

0 .85

0.9

0 .95

1

0 50 100 150 200Time [Weeks ]

αα

" A c t i v e - C o m p r e s s i o n " W i d e R e c t a n g u l a r

" A c t i v e - C o m p r e s s i o n " T r i a n g u l a r

" G r a d e d- S t i f f n e s s" W i d e R e c t a n g u l a r G r a d e d - S t i f f n e s s T r i a n g u l a r

0

0 .1

0 .2

0 .3

0 .4

0 .5

0 .6

0 5 0 1 0 0 1 5 0 2 0 0

T i m e [W e e k s ]

" A c t i v e - C o m p r e s s i o n " W i d e R e c t a n g u l a r

" A c t i v e - C o m p r e s s i o n " T r i a n g u l a r

" G r a d e d - S t i f f n e s s " W i d e R e c t a n g u l a r

" G r a d e d - S t i f f n e s s " T r i a n g u l a r

αα STP ββ STP

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Conclusions

A method of evaluating and rating designs and expected biomechanical performances of orthopaedic fixative screws was presented.

Wide (6-mm thread diameter) rectangular and trapezoidal screw profiles have superior biomechanical compatibility with bone compared with other commercial designs

The promising computational STP results obtained for the graded-stiffness and active-compression screw designs will have to be validated with data from animal experiments

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Conclusions (cont.)

Finally, it was demonstrated that bone remodeling computer simulations can be used as a highly effective tool for evaluation of several design parameters of the screws, such as geometry, material characteristics and even coating

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üState of bone (growth, healing, mature) determines its ability to respond to mechanical strain

üMature bone may seek to exist within limited strain range

üResulting structure (from growth) shows attributes of efficient or optimized structure -> fully stressed / strained?

üComputational models have the ability to predict qualitatively correct bone structure distribution

üBone cells can respond directly to mechanical strain

Summary of State-of-the-Art Knowledge in Bone Adaptation Theory:

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Future Work:

Ø Strain measured on cortical surface or computed by single level solid continuum models is not the same as that experienced by cells due to hierarchical bone structure

Ø Specific relationships between strain and adaptation for different states of bone response have not been clearly delineated

Ø Nature of loads on bone are difficult to determine, but are necessary to fully understand its mechanics and adaptation