L Mechanics: – a branch of physics that is concerned with the motion and deformation of bodies in response to forces l Applied (or Engineering) Mechanics:

Mechanics:– a branch of physics that is concerned with the motion and deformation of bodies in response to

forces

Applied (or Engineering) Mechanics:– the science of applying the principles of mechanics

Ap

pli

ed M

ech

anic

s

Rigid Body Mech.

Deformable Body Mech

Fluid Mechanics

Statics

Dynamics

Elasticity

Plasticity

Viscoelasticity

Gases and Liquids

Kinematics and Kinetics

Poroelasticity

Physical Properties of Spinal Elements

Spinal Elements:– Bone

• Vertebral body: Cortical bone, cancellous bone, and bony endplate

• Posterior bony elements

– Soft Tissues• Ligaments and intervertebral disc

Physical Properties:– Mass:

• Bone density

• Mass moment of inertia for dynamic analysis

– Morphology:• Structure, shape, size, location, area and polar moment of inertia

– Mechanical (Elastic, elastic, and poroelastic) Properties:• Stiffness: compression, tension, share, bending, torsion

• Moduli and Poisson’s ratio

• Permeability

Measurement of Physical Properties Mass Measurement:

– Direct measurement– Dual X-ray Absorption metry for BMD measurement– QCT

Morphology Measurement:– Direct measurement from cadaveric specimens– Video system– Plain X-rays– CT and MRI

Mechanical Properties– Elastic Property Measurement:

• Static mechanical tests in compression, tension, shear, bending and torsion tests

– Viscoelastic Property measurement:• Creep, relaxation, cyclic loading tests

– Poroelastic Property Measurement:• Elastic property measurement techniques combined with diffusion tests

Measurement of Mechanical Properties Mechanical Tests:

– Based on Loading Direction• Compression/Tension Test

• Bending/Torsion Tests

• Creep and Relaxation Tests

– Based on Loading Speed• Static or dynamic Tests

Input and Output:– Input

• loading by Displacement control vs. Load Control

– Output• measuring resultant force or displacement

• stress-strain curve

Factors for Consideration:– Boundary condition– Loading direction and loading speed– Parameter measurement:– Prevention of dehydration of specimens

Spinal Ligaments

Ligaments:– Uniaxial structures– Effective in carrying tensile loads along the direction in which the fiber runs– Spinal ligaments provide tensile resistance to external loads by developing tension when the spinal

segment is subjected to complex loads

Functions of Ligaments:– Allow adequate physiologic motion and fixed postural attitudes between vertebrae with a minimum

expenditure of muscle energy– Protect the spinal cord by restricting the motions within well-defined limits– Share with the muscles the role of providing stability to the spine within its physiologic ranges of

motion– Protect the spinal cord in traumatic situations (energy absorption)

Quantitative Anatomy

Quantities for describing the functional role of ligaments:– Length, X-sectional Area, 3-D coordinates of the attachment position– Precise data are not available

Region Level Ligament X-area (mm2) Length(mm)

Cervical C1-C2 Transverse 18 20

Alar 22 11

Lumbar ALL 53 13

PLL 16 11

LF 67 19

CL - -

ISL 26 -

SSL 23 11

Tensile Tests Loading Methods

– Load Control– Displacement Control

F-d and Stress-Strain Curves – Stiffness and/or elastic modulus

Displacement or Force Measurements– LVDT installed in MTS machine– Extensometer– Image Analysis of markers on the tissue

Important Factors of Consideration– Loading speed or loading rate– Cross-sectional area measurement for stress calculation– Prevention of dehydration

Typical Load-displacement Curve of LigamentsL

oad

or

Str

ess

Deformation of Strain

NZ EZ PZ

Physiologic Range

TraumaticRange

NZ = Neutral ZoneEX = Elastic ZonePZ = Plastic Zone

Failure

Physical Properties of Ligaments

Ligament Failure Load Deformation Stress Strain(N) (mm) (MPa) (%)

Ant. Atlantooccip. Memb. 233 18.9Post. Atlantooccip. Memb. 83 18.1

C1-C2ALL 281 12.3LF 113 8.7CL 157 11.4Transverse Lig. 354 (170-700)

C2-C7ALL 111.5 (47-176) 8.95 (4.2-13.7)PLL 74.5 (47-102) 6.4 (3.4-9.4)LF 138.5 (56-221) 8.3 (3.7-12.9)CL 204 (144-264) 8.4 (6.8-10)ISL 35.5 (26-45) 7.4 (5.5-9.2)SSL

*Range of values are listed in parentheses.

Elastic Properties of Ligaments

Ligament Modulus Failure Load Deformation Stress Strain(MPa) (N) (mm) (MPa) (%)

ThoracicALL 296 (123-468) 10.3 (6.3-14.2)PLL 106 (74-138) 5.25 (3.2-7.3)LF 200 (135-265) 8.65 (6.3-11)CL 168 (63-273) 6.75 (3.9-9.6)ISL 75.5 (31-120) 5.25 (3.8-6.7)SSL 320 (101-538) 14.1 (7.2-21)

LumbarALL 7.8 (<12%) 20 (>12%) 450 (390-510) 15.2 (7-20) 11.6 (2.4-21) 36.5 (16-57)PLL 10 (<11%) 20 (>11%) 324 (264-384) 5.1 (4.2-7) 11.5 (2.9-20) 26.0 (8-44)LF 15 (<6.2%) 19.5 (>6.2%) 285 (230-340) 12.7 (12-14.5) 8.7 (2.4-15) 26.0 (10-46)CL 7.5 (<25%) 32.9 (>25%) 222 (160-284) 11.3 (9.8-12.8) 7.6 (7.6) 12.0 (12.0)ISL 10 (<14%) 11.6 (>14%) 125 (120-130) 13.0 (7.4-17.8) 3.4 (1.8-4.6) 13.0 (13.0)SSL 8.0 (<20%) 15 (>20%) 150 (100-200) 25.9 (22.1-28.1) 5.4 (2.0-8.7) 32.5 (26-39)TL 10 (<18%) 58.7 (>18%)

*Range of values are listed in parentheses.

Functional Biomechanics

Functional properties of a ligament are described as a combination of physical properties and orientation and location with respect to the moving vertebra

Physiological Strain in Ligaments

Future Studies

Physical Properties of Spinal Ligaments:– Tkaczauk et al. (Acta Scand Orthop, 115, 1968)

• Decreases in maximum deformation, the residual (or permanent) deformation, and the energy loss of hysteresis of anterior and posterior longitudinal ligaments with age

• Decrease in maximum deformation and residual deformation with the disc degeneration

– Property changes with age and disc degeneration may be related with segmental instability and low back pain.

Further studies on the changes in the physical properties of spinal ligament with respect to the pathology

THE VERTEBRA

Vertebra:– Vertebral body– Posterior bony ring (neural arch)

• two pedicles and laminae from which arise seven processes of articular, transverse, and spinous processes

– Basic design of the vertebrae from C3-L5 is almost same, but the size and mass increase from the first cervical to the last lumbar vertebra (Mechanical adaptation to the progressively increasing loads).

Functions of the Vertebra:– Protect the spinal cord– Maintain the posture– Provide major load bearing

Pedicles

– Pedicle Height and Pedicle Width (PDH and PDW)– Inclination angles to the sagittal and transverse planes (PDIs and PDIt)

Region PDW (mm) PDH (mm) PDIs (deg) PDIt (deg)

C3 6 (4 - 8) 8 (6 -10) 41 (20 - 55) -6 (-16 - 4)

C5 6 (4 - 8) 7 (5 - 9) 39 (24 - 54) 0 (-10 - 10)

C7 7 (5 - 9) 8 (6 -10) 30 (15 - 45) 6 (4 -16)

T1 8 (5 -10) 10 (7 - 15) 27 (16 - 34) 13 (4 - 25)

T5 5 (3 - 7) 12 (7 - 14) 9 (2 - 19) 15 (7 - 20)

T9 6 (4 - 9) 14 (11 - 16) 8 (0 - 11) 16 (9 -14)

T12 7 (3 - 11) 16 (12 - 20) -4 (-17 - 15) 12 (7 - 16)

L1 9 (5 - 13) 15 (11 - 21) 11 (7 - 15) 2 (-13 - 15)

L2 9 (4 -13) 15 (10 - 18) 12 (5 - 18) 2 (-10 - 13)

L3 10 (5 - 16) 15 (8 - 18) 14 (8 - 24) 0 (-10 - 12)

L4 13 (9 - 17) 15 (9 - 19) 18 (6 - 28) 0 (-6 - 7)

L5 18 (9 - 29) 14 (10 -19) 30 (19 -44) -2 (-8 - 6)

Neural Arch- Most failures occurred through the

pedicles.

- In Lamy et al.’s study, 1/3 of the failures were through the pars interarticularis (Spondylolysis). This number increased when the tests were conducted at higher rates of loading.

- No strength difference between males and females as well as between normal and degenerated discs

Facet Joints- Shape and position of the

articulating processes are the important factors for determining the pattern of spinal motion.

- Cervical spine

- Thoracic spine

- Lumbar spine:- Curved mating surfaces not plane

- Facet orientations in the figure are only approximate

Facet Joints

Physical Properties of the Vertebral Body

- The variation in the vertebral strength with the spinal level is most probably due to the size of the vertebrae alone.

- Strength decreases with age. A rapid rate of decrease was observed from age 20 - 40 years, while the strength remained more or less constant after age 40.

Physical Properties of the Vertebral Body

- Strength decrease with relative ash content or osseous tissue of the vertebrae.

- 25% osseous tissue loss results in a more than 50% strength decrease.

- Bone mineral content (BMC) decrease with age.

- Mechanical Strength BMC

Cortical Shell and Cancellous Core

The vertebral body carries most of the compressive loads that are transmitted from the superior to the inferior endplate.

Mechanical properties of the vertebral cortical shell has not been clearly investigated yet.

Rockoff et al. (Calcif. Tissue Res 3:163, 1969)– Trabecular bone contributes 25 - 55% of the strength

depending upon the ash content of the bone.– 55% vs 35% carried under and after 40 yrs of age.

McBroom et al, (JBJS 67A:1206, 1985)– The cortical shell provides only 10% (in average) of the

total compressive load even in specimens came from an old population (63 - 99 yrs)

F

CancellousBone Endplate

Fcor

F

Fcor

Fcan

EndplateCorticalBone

Cancellous Bone

- Failure Type I:decreasing strength after the maximum load reached (13% of the specimens)

- Failure Type II:maintaining strength (about 50% of the specimens)

- Failure Type III:increasing strength (38% of the specimens)

- Type III failure was found most frequently in males under 40 yrs of age and least frequently in women over 40.

Compressive Properties of Vertebral Cancellous Bone

- Despite of much variation after the maximum strength in the load-displacement curve, the mechanical properties represented in the early part of the curve were quite consistent.

Compressive properties of cancellous bone of vertebrae

Physical Property Magnitude

Proportional-limit stress 1.37 - 4.0 MPaCompression at proportional limit 6.0 - 6.7 %Modulus of elasticity 22.8 - 55.6 MPaFailure stress 1.55 - 4.6 MPaCompression at failure 7.4 - 9.5%

Functional Biomechanics of Vertebral Trabecular Bone

Presence of Bone marrow in Cancellous Core:– significantly increase the compressive strength as well as the energy capacity.– This suggests that the function of cancellous core is not only to share the load with the cortical shell but

also to act as the main resistor of the dynamic peak loads.

Effect of aging on the vertebral trabecular bone structure:– Loss of the horizontal trabeculae with simultaneous thickening of the vertebral trabeculae– Loss of the horizontal trabeculae occurs in the central region of the vertebral body while those in the

peripheral regions remained unaltered.– In another study, however, it was found that both trabeculae get thinner and decrease at the same rate,

but the horizontal trabeculae are lesser in number than the vertical trabeculae at all density levels. Thus, the spacing between horizontal trabeculae increases more rapidly than the spacing between vertical trabeculae.

Biomechanical adaptation:– Changes in the trabecular bone was found with the disc degeneration.– With less disc degeneration, the trabecular bone is stronger in the center.– In case of degenerated discs, the trabecular bone strength has uniform distribution.

Biomechanical Factors for Bone Tests

Experimental Artifacts:– Specimen preparation methods: Damages on the specimen surface– Orientation and anatomical site of the specimen– Specimen condition: wet or dry; repeated use of specimens– Specimen shape and size: A cylindrical specimen with at least 1:2 aspect ratio

Boundary and Loading Conditions:– End-artifact: Friction between the specimen and the platen and also the deformation

measurement points– Loading rates

Effect of bone marrow on the mechanical properties of the trabecular bone (poroelastic effect)

BIOMECHANICAL ANALYSIS OF TRABECULAR BONE

AS A POROELASTIC MATERIAL

STUDIES OF TRABECULAR BONE MECHANICS

Bone behavior in vivoEffects of aging, disease, and

instrumentation, etc.

STUDIES OF TRABECULAR BONE MECHANICS

Age-related bone fractureTotal joint looseningBone remodeling, etc.

TRABECULAR BONE(ELASTIC MATERIAL)

Elastic Properties– Young’s Modulus (E), Shear Modulus (G), and Poisson’s ratio ()

Stiffness and Strength– Relationship with Bone Density

Anisotropy– Transversely isotropic or orthotropic

Effect of the Architectural Features of Trabecular Bone

TRABECULAR BONE(ELASTIC MATERIAL)

Micromechanics– Mathematical models– Material properties of individual trabecular tissue

Experimental Errors– friction artifacts at specimen-platen interface, damage artifact during

specimen preparation, and specimen geometry, etc.

Limitations in using the Theory of Elasticity– Considering trabecular bone as a single-phase solid material;– unable to describe the time-dependent behaviors.

TIME-DEPENDENT BEHAVIORS OF TRABECULAR BONE

Creep and Stress Relaxation– Zilich et al., 1980; Schoerfeld et al., 1974; Deligianni et al.,

1994; Bowman et al., 1994

Influence of Loading Rate on Strength and Stiffness

– Carter and Hayes, 1977; Ducheyne, et al., 1977; Galante et al.; Linde et al., 1991

TIME-DEPENDENT BEHAVIORS OF TRABECULAR BONE

Trabecular Bone as a Viscoelastic Material– Kafka et al.– Deligianni et al.

Limitations of a Viscoelastic Theory– Unable to experimentally determine the time-dependent

viscoelastic properties of trabecular bone;– Difficult to describe the mechanical role of fluid-phase.

BONE STRUCTURE

Solid Phase– mineralized bone tissue with pores

Fluid Phase– blood vessels, blood, red and yellow marrow, nerve

tissue, miscellaneous cells, and interstitial

ROLE OF FLUID PHASE

Physiological Role:– Transporting nutrients and waste products

Mechanical Role:– Postulated to cause time-dependent behaviors of trabecular

bone;– Not fully understood yet.

Fluid phase may change the mechanical behaviors of trabecular bone .

Two Coupled Interaction Mechanisms between the Interstitial Fluid and the Porous Trabecular Tissue

– Compression of trabecular bone causes a rise of pore pressure;

– An increase in pore pressure induces dilation of trabecular bone.

HYPOTHESES

HYPOTHESESThe apparent elastic and time-dependent

behaviors of trabecular bone can be well described by using the theory of poroelasticity.

Trabecular bone can be characterized using poroelastic properties.

THREE STUDIES

Poroelastic Model of Trabecular Bone;

Effect of the Fluid Flow in Trabecular Bone on the Relaxation Behavior;

Measurement of Poroelastic Properties of Trabecular Bone.

PURPOSETo investigate:if the apparent mechanical behavior of

trabecular bone can be well described with poroelasticity theory.

what affects the poroelastic behavior.

HISTORY OF POROELASTICITY THEORY

Consolidation Model:– Terzaghi: 1-D model– Rendulic: 3-D model

Theory of Poroelasticity:– Biot; Verrjuit– Rice and Cleary

Mixture Theories:– Atkin; Bowen; Morland

COMPARISON OF POROELASTIC THEORIES

Biot’s Formulation:– Use of model parameters that are not identifiable and difficult measure.– For simplification, Incompressibility of both the solid and fluid phases was

assumed.

Rice and Cleary’s Formulation:– Use of model parameters that are elastically identifiable and measurable.– Full incorporation of the compressibility. – Simplified the interpretation of asymptotic poroelastic phenomena

Poroelastic Equations(Rice and Cleary, 1976)

Constitutive Equation:

2G(1 - 2 )

= 2Gij +ij +3(u - )

B(1 - 2 )(1 + u) p kk ij

ij = Total Stress Tensor (MPa)

ij = Strain Tensor

p = Pore Pressure (MPa)

Poroelastic Equations(Rice and Cleary, 1976)

Diffusion Equation:

kk

t2p = -

- Governing pore pressure generation with volumetric deformation of the control element

- Rate of flow through the pores is proportional to the gradient

of pore pressure (p): Darcy’s Law

p

t2GB (1 + u) 2GB 2(1 - 2)(1 + u)2

9(u - )(1 - 2 u) 3 (1 - 2 u) -

Asymptotic Poroelastic Phenomena

Drained Deformation:– Quasi-static deformation in a drained condition in which free-fluid flow is

allowed;– No pore pressure generation and thus elastic behavior only

Undrained Deformation:– Deformation in an undrained condition in which the fluid is prevented

from flowing out across the boundary;

3 = 2G(1 - u)

1 - 2u

3

POROELASTIC PROPERTIES

G: drained shear modulus (MPa): drained Poisson’s ratiou: undrained Poisson’s ratio

– Poisson’s ratio for an undrained deformation;

– Theoretical range 0.5 < u <

B: Skempton’s coefficient– describes the undrained pore pressure change with a change in mean

stress (0.0 < B < 1.0).

: permeability (m2/MPa/sec)

UNIAXIAL STRAIN CONDITION

3

2

Impermeable Rigid Boundary

Rigid PorousLoading Platen

Carters and Hayes, 1977

•Boundary Conditions:

•Initial Condition:

•Loading Condition

p(0,t) = 0p(l,t)/ x3 = 0

p(x3,0) = 0

d3/dt = Constant

Bone Specimen

Strain Input

1 = 2 =3p(u - )

B(1 - 2)(1 + u) p

(1 - )

3 -

p

t

2GB (1 + u) -2GB(1 - 2)(1 + u)2

9(u - )(1 - 2 u)

2p

x32

= -3 (1 - 2 u)

d3

dt

3p(u - )

B(1 - 2)(1 + u) pt -

2G(1 - )(1 - 2)3 =

d3

dt

Constitutive Euqations in Uniaxial Strain Condition:

Diffusion Equation:

1-D Poroelastic Model in Uniaxial Strain Condition

Pore Pressure:

Total Stress:

6(u - )(d3/dt)

BLn3(1 - 2)(1 + u)

3(u - )

B(1 - 2)(1 + u) p(x3, t) t -

2G(1 - )

(1 - 2)3 (x3 , t) =

d3

dt

p(x3, t) = - n = 1

[1 - exp(-n

2t)]sin(n x3)

2GB2 (1 - 2u)(1 + u)2

,where eignevalues n = (2n - 1)p/2L, and = 9(u - ) (1 - 2)

Assumptions for Poroelastic Modeling of Trabecular Bone

Interconnective Pores (Proven) Rate of flow through the pores proportional to

the gradient of pore pressure (Proven) Solid trabecular tissue is assumed to be isotropic

and elastic. Pores are assumed to be uniformly distributed.

Compression Tests of Trabecular Bone in Uniaxial Strain Condition

(Carter and Hayes, 1977)

Carter and Hayes, 1977:– Specimens with and without marrow in situ;– Loading with different strain rates (0.001, 0.01, 0.1, 1, 10

/second).

Luo et al., 1993:– Effect of specimen size on hydraulic stiffening of cancellous

bone.

ESTIMATION OF MODEL PARAMETERS

= 0.3 – assumed based on literature

G = 16.17 (MPa)– comp modulus of 56.6 of the specimen with marrow responding to the slowest strain rate,

0.001/sec (Carter and Hayes, 1977)

u = 0.459– comp modulus of 211.1 of the specimen with marrow responding to the fastest strain rate,

10.0/sec (Carter and Hayes, 1977)

B = 0.91 (0.82 ~ 1.0) = 3.54 x 10-5 (m2/MPa/sec)

– Permeability of bovine proximal tibia measured by Ochoa and Hilbery, 1992.

Compressive Modulus vs. Strain Rate

Model prediction (= 3.4 x 10-5 m2 / MPa-sec)

Carter and Hayes’ Experimental Results

0.001 0.01 0.1 1.0 10.0

Strain Rate (/sec)

Co

mp

ress

ive

Mo

du

lus

(MP

a)

0

50

100

150

200

250

Predicted Compressive Moduli of Trabecular Bones of Different Lengths (B = 0.91 and = 3.4 x 10-5)

0.001 54.03 54.27 54.76

0.01 54.30 56.74 61.61

0.1 57.04 81.38 123.98

1.0 84.40 188.69 209.40

10.0 192.32 212.33 214.48

Compressive Moduli (MPa)5 mm 10 mm 25 mm

Strain Rate(/sec)

Effect of Trabecular Bone Length

At a strain of 0.001/sec, near zero change in the compressive modulus was predicted for the longer trabecular bones.

Greater strain rate effect on the compressive modulus was in the longer specimens.

Similar effects were observed in the study of bovine tibial cancellous bone (Luo et al., 1993)

Total Stress vs. and B (Strain Rate = 0.001/s)

BK

TotalStress(MPa)

(m2 / MPa/sec) , B (No unit)

(m2 / MPa/sec) , B (No unit)

TotalStress(MPa)

KB

Total Stress vs. and B(Strain Rate = 0.1/s)

Total Stress vs. and B

(Strain Rate = 10/s)

TotalStress(MPa)

(m2 / MPa/sec) , B (No unit)

K

B

Poroelastic Equations(Rice and Cleary, 1976)

Constitutive Equation:

2G(1 - 2 )

= 2Gij +ij +3(u - )

B(1 - 2 )(1 + u) p kk ij

ij = Total Stress Tensor (MPa)

ij = Strain Tensor

p = Pore Pressure (MPa)

Poroelastic Equations(Rice and Cleary, 1976)

Diffusion Equation:

kk

t2p = -

- Governing pore pressure generation with volumetric deformation of the control element

- Rate of flow through the pores is proportional to the gradient

of pore pressure (p): Darcy’s Law

p

t2GB (1 + u) 2GB 2(1 - 2)(1 + u)2

9(u - )(1 - 2 u) 3 (1 - 2 u) -

A total of 40 bovine and 22 human lumbar vertebrae were used.

Cylindrical Trabecular Specimens (9.8 mm in diameter and 15 mm in length) were obtained using a diamond coring tool and a low-speed bone saw.

Bone marrow was removed by a water jet.

Permeability Measurement

Diffusion Apparatus for Permeability Measurement

Constant Force forproducing a hydraulic pressureof 7 kPa

Piston

Reservoir filled with saline solution

Spacer

TrabecularSpecimen

O-Ring

LVDT to measure the pistondisplacement h (m)

Since the rate of flow (Q/t; m3/sec) is proportional to the pressure difference (p; Pa) according to Darcy’s law;

=L

Asp pQt

,where t = time (sec); Asp = specimen X-area (301.7 x 10-6 m2);

p = pressure diff. (7.0 kPa); and L = specimen length(0.015 m).

Flow Volume (Q; m3) = h x piston X-area

Typical Relationship of Flow Volume vs. Time

Flo

w V

olu

me

(m3)

Time (sec)

0.0 0.5 1.0 1.5 2.00.0

5.0 E-6

1.0 E-5

1.5 E-5

2.0 E-5

r2 > 0.99

Uniaxial Strain Tests

Loading Piston

Rigid StainlessSteel Annulus

O-Ring

MarrowIn SituTrabecularSpecimen

MTS Load Cell

Ram PressureTransducer

* The specimen was

subjected to a

0.7% strain using

a displacement control

(0.002 mm/sec).

* u - u and u - p

curves were obtained.

Uniaxial Stress Tests

MTS Load Cell

MTS Ram

LVDTs

* The specimen was

subjected to a

1.0% strain using

a displacement control

(0.002 mm/sec).

* - and L -

curves were obtained.

Data Analyses (Uniaxial Stress Tests)

From the Elasticity Theory:

G =E

2(1 + )Shear Modulus:

Drained Poisson’s ratio:L

= -

E =Young’s Modulus:

* 5th order polynomial curves were used to determine

the - and L - relationships. (Slope at = 0.0065)

Axial Strain (%)

Str

ess

(MP

a)

Lat

eral

Str

ain

L (

%)

0.0 0.3 0.6 0.9

Axial Strain (%)

0.03

0.06

0.09

0.12

0.15

0.18

Stress vs. Strain Lateral Strain vs. Axial Strain

0.0 0.3 0.6 0.9

0.2

0.4

0.6

0.8

1.0

Data Analyses (Uniaxial Strain Tests)

From the stress-strain relationship in an undrained uniaxial condition and the definition B:

B =3(1 - u)

(1 + u)

u

uSkempton’s Coefficient:

u

u=

2G(1 - u)

(1 - 2u)Undrained Poisson’s ratio:

* 5th order polynomial curves were used to determine

the u - u and p - u relationships. (Slope at u = 0.0065)

u o

r P

ress

ure

(M

Pa)

u (%)

5.0

2.0

3.0

4.0

1.0

0.00.1 0.2 0.3 0.4 0.5 0.6

Pore Pressure

Undrained Condition

Stress (or Pore Pressure) vs. Strain

Po

re P

ress

ure

(M

Pa)

u (MPa)

2.0

3.0

4.0

1.0

1.0 2.0 3.0 4.0 5.00.0

Pore Pressure) vs. Stress

Permeability of Human Vertebral Trabecular Bone

Level Mean Permeability (SD)

(x10-8 m2/Pa/sec)

L1 61.3 (10.4)

L2 53.6 (7.30)

L3 50.4 (6.30)

L4 38.6 (10.7)

L5 45.7 (8.30)

Total 52.2 (10.8)

Permeability of Bovine Vertebral Trabecular Bone

Level Mean Permeability (SD)

(x10-8 m2/Pa/sec)

L1 14.91 (9.83)

L2 12.39 (7.85)

L3 19.03 (8.55)

L4 18.50 (6.74)

L5 16.85 (6.23)

Total 16.31 (8.02)

All curves were well represented by a 5th order polynomial curve (r2 0.97).

Mean (SD) values of the poroelastic parameters

G u B (MPa) (x10-8 m2 /Pa/sec)

90.85 0.242 0.399 0.851 16.31

(59.59) (0.099) (0.083) (0.144) (8.02)

RESULTS

First measurement of the poroelastic properties of trabecular bone.

Feasibility for measuring the poroelastic properties of human trabecular bone.

Similar methods can be used for the measurement of cortical bone properties.

Knowledge of these parameters may improve our understanding of the mechanical behavior of trabecular bone in vivo.

DISCUSSION

Linear relationship between the fluid flow (Q) and time (t)– The fluid flow in trabecular bone follows Darcy’s law as observed in

another study (Simkin, 1985).

First measurement of permeability for human trabecular bone

Significantly larger permeability in human trabecular bone than in bovine trabecular bone

In vivo, permeability would be smaller because of 67 times higher viscosity of bone marrow than that of water

DISCUSSION

Uniaxial Strain Tests in an Undrained Condition:– B and u measurements

Uniaxial Stress Tests in a Drained Condition:– E, , and G measurements

Each bovine vertebral trabecular specimen (9.8 mm in diameter and 20 mm in length) was used for both tests.

– Uniaxial strain tests were always performed first to minimize the loss of bone marrow.

Measurement of G, , u, and B