
Mechanics:
–
Applied (or Engineering) Mechanics:
–
the science of applying the principles of mechanics
Rigid Body Mech.
Applied Mechanics

a branch of physics that is concerned with the motion and deformation of bodies in response to
forces
Statics
Dynamics
Deformable Body Mech
Elasticity
Plasticity
Viscoelasticity
Fluid Mechanics
Kinematics and Kinetics
Gases and Liquids
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:
–
–
–

Morphology Measurement:
–
–
–
–

Direct measurement
Dual X-ray Absorption metry for BMD measurement
QCT
Direct measurement from cadaveric specimens
Video system
Plain X-rays
CT and MRI
Mechanical Properties
–
Elastic Property Measurement:
•
–
Viscoelastic Property measurement:
•
–
Static mechanical tests in compression, tension, shear, bending and torsion tests
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
•
•
•
–
Based on Loading Speed
•

Static or dynamic Tests
Input and Output:
–
Input
•
–
loading by Displacement control vs. Load Control
Output
•
•

Compression/Tension Test
Bending/Torsion Tests
Creep and Relaxation Tests
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
Cervical
Lumbar
Level
C1-C2
Ligament
Transverse
Alar
X-area (mm2)
18
22
Length(mm)
20
11
ALL
PLL
LF
CL
ISL
SSL
53
16
67
26
23
13
11
19
11
Tensile Tests

Loading Methods
–
–

F-d and Stress-Strain Curves
–

Stiffness and/or elastic modulus
Displacement or Force Measurements
–
–
–

Load Control
Displacement Control
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 Ligaments
Load or Stress
Failure
Physiologic Range
NZ
EZ
PZ
Traumatic
Range
NZ = Neutral Zone
EX = Elastic Zone
PZ = Plastic Zone
Deformation of Strain
Physical Properties of Ligaments
Ligament
Ant. Atlantooccip. Memb.
Post. Atlantooccip. Memb.
C1-C2
ALL
LF
CL
Transverse Lig.
C2-C7
ALL
PLL
LF
CL
ISL
SSL
Failure Load
(N)
Deformation
(mm)
233
83
18.9
18.1
281
113
157
354 (170-700)
12.3
8.7
11.4
111.5 (47-176)
74.5 (47-102)
138.5 (56-221)
204 (144-264)
35.5 (26-45)
8.95 (4.2-13.7)
6.4 (3.4-9.4)
8.3 (3.7-12.9)
8.4 (6.8-10)
7.4 (5.5-9.2)
*Range of values are listed in parentheses.
Stress
(MPa)
Strain
(%)
Elastic Properties of Ligaments
Ligament
Modulus
(MPa)
Thoracic
ALL
PLL
LF
CL
ISL
SSL
Lumbar
ALL
PLL
LF
CL
ISL
SSL
TL
7.8 (<12%) 20 (>12%)
10 (<11%) 20 (>11%)
15 (<6.2%) 19.5 (>6.2%)
7.5 (<25%) 32.9 (>25%)
10 (<14%) 11.6 (>14%)
8.0 (<20%) 15 (>20%)
10 (<18%) 58.7 (>18%)
Failure Load
(N)
Deformation
(mm)
296 (123-468)
106 (74-138)
200 (135-265)
168 (63-273)
75.5 (31-120)
320 (101-538)
10.3 (6.3-14.2)
5.25 (3.2-7.3)
8.65 (6.3-11)
6.75 (3.9-9.6)
5.25 (3.8-6.7)
14.1 (7.2-21)
450 (390-510)
324 (264-384)
285 (230-340)
222 (160-284)
125 (120-130)
150 (100-200)
15.2 (7-20)
5.1 (4.2-7)
12.7 (12-14.5)
11.3 (9.8-12.8)
13.0 (7.4-17.8)
25.9 (22.1-28.1)
*Range of values are listed in parentheses.
Stress
(MPa)
Strain
(%)
11.6 (2.4-21)
11.5 (2.9-20)
8.7 (2.4-15)
7.6 (7.6)
3.4 (1.8-4.6)
5.4 (2.0-8.7)
36.5 (16-57)
26.0 (8-44)
26.0 (10-46)
12.0 (12.0)
13.0 (13.0)
32.5 (26-39)
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
C3
C5
C7
T1
T5
T9
T12
L1
L2
L3
L4
L5
PDW (mm)
6 (4 - 8)
6 (4 - 8)
7 (5 - 9)
8 (5 -10)
5 (3 - 7)
6 (4 - 9)
7 (3 - 11)
9 (5 - 13)
9 (4 -13)
10 (5 - 16)
13 (9 - 17)
18 (9 - 29)
PDH (mm)
8 (6 -10)
7 (5 - 9)
8 (6 -10)
10 (7 - 15)
12 (7 - 14)
14 (11 - 16)
16 (12 - 20)
15 (11 - 21)
15 (10 - 18)
15 (8 - 18)
15 (9 - 19)
14 (10 -19)
PDIs (deg)
41 (20 - 55)
39 (24 - 54)
30 (15 - 45)
27 (16 - 34)
9 (2 - 19)
8 (0 - 11)
-4 (-17 - 15)
11 (7 - 15)
12 (5 - 18)
14 (8 - 24)
18 (6 - 28)
30 (19 -44)
PDIt (deg)
-6 (-16 - 4)
0 (-10 - 10)
6 (4 -16)
13 (4 - 25)
15 (7 - 20)
16 (9 -14)
12 (7 - 16)
2 (-13 - 15)
2 (-10 - 13)
0 (-10 - 12)
0 (-6 - 7)
-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
F

Cancellous
Bone
Endplate
Fcan


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)
–
Fcor
Fcor
–

McBroom et al, (JBJS 67A:1206, 1985)
–
Cortical
Bone
Endplate
F
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.
The cortical shell provides only 10% (in average) of the
total compressive load even in specimens came from an old
population (63 - 99 yrs)
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 loaddisplacement curve, the mechanical properties represented in the early
part of the curve were quite consistent.
Compressive properties of cancellous bone of vertebrae
Physical Property
Proportional-limit stress
Compression at proportional limit
Modulus of elasticity
Failure stress
Compression at failure
Magnitude
1.37 - 4.0 MPa
6.0 - 6.7 %
22.8 - 55.6 MPa
1.55 - 4.6 MPa
7.4 - 9.5%
Functional Biomechanics of Vertebral Trabecular Bone

Presence of Bone marrow in Cancellous Core:
–
–

Effect of aging on the vertebral trabecular bone structure:
–
–
–

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.
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:
–
–
–
–

Boundary and Loading Conditions:
–
–

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
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 vivo
 Effects of aging, disease, and
instrumentation, etc.
STUDIES OF TRABECULAR
BONE MECHANICS
 Age-related
bone fracture
 Total joint loosening
 Bone 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
–
friction artifacts at specimen-platen interface, damage artifact during
specimen preparation, and specimen geometry, etc.
 Limitations
–
–
Errors
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
Stiffness
–
of Loading Rate on Strength and
Carter and Hayes, 1977; Ducheyne, et al., 1977; Galante et al.;
Linde et al., 1991
TIME-DEPENDENT BEHAVIORS OF
TRABECULAR BONE
 Trabecular
–
–
Kafka et al.
Deligianni et al.
 Limitations
–
–
Bone as a Viscoelastic Material
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
–
mineralized bone tissue with pores
Fluid
–
Phase
Phase
blood vessels, blood, red and yellow marrow, nerve
tissue, miscellaneous cells, and interstitial
ROLE OF FLUID PHASE
 Physiological
–
Transporting nutrients and waste products
 Mechanical
–
–
Role:
Role:
Postulated to cause time-dependent behaviors of trabecular
bone;
Not fully understood yet.
HYPOTHESES
 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
 The
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.
PURPOSE
To 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
–
–
Terzaghi: 1-D model
Rendulic: 3-D model
 Theory
–
–
of Poroelasticity:
Biot; Verrjuit
Rice and Cleary
 Mixture
–
Model:
Theories:
Atkin; Bowen; Morland
COMPARISON OF
POROELASTIC THEORIES
 Biot’s
–
–
Use of model parameters that are not identifiable and difficult measure.
For simplification, Incompressibility of both the solid and fluid phases was
assumed.
 Rice
–
–
–
Formulation:
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:
3(u - )
2G
p = 2Gij +
ij +
kk ij
(1 - 2 )
B(1 - 2 )(1 + u)
ij = Total Stress Tensor (MPa)
ij = Strain Tensor
p = Pore Pressure (MPa)
Poroelastic Equations
(Rice and Cleary, 1976)
Diffusion Equation:
p
t
-
2GB 2(1 - 2)(1 + u)2
9(u - )(1 - 2 u)
2GB (1 + u) kk
 p=3 (1 - 2 u) t
2
- 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
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 - 2 u
3
POROELASTIC PROPERTIES
 G:
drained shear modulus (MPa)
 : drained Poisson’s ratio
 u: 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
Strain Input
Rigid Porous
Loading Platen
•Loading Condition
d3/dt = Constant
2
•Boundary Conditions:
p(0,t) = 0
p(l,t)/ x3 = 0
Bone
Specimen
Impermeable
3 Rigid Boundary
Carters and Hayes, 1977
•Initial Condition:
p(x3,0) = 0
Constitutive Euqations in Uniaxial Strain Condition:
3 =
2G(1 - )
(1 - 2)

1 = 2 =
(1 )
3p(u - )
d3
t dt
B(1 - 2)(1 + u)
3 -
3p(u - )
B(1 - 2)(1 + u)
p
p
Diffusion Equation:
p
t
2GB(1 - 2)(1 + u)2 2p
9(u - )(1 - 2 u)
x32
= -
2GB (1 + u) d3
3 (1 - 2 u) dt
1-D Poroelastic Model
in Uniaxial Strain Condition
Pore Pressure:

p(x3, t) =
6(u - )(d3/dt)
-
n = 1 BL
n
,where eignevalues
3(1
- 2)(1 + u)
[1 - exp(-n2t)]sin(n x3)
n = (2n - 1)p/2L, and  =
2GB2 (1 - 2u)(1 + u)2
9(u - ) (1 - 2)
Total Stress:
3 (x3 , t) =
2G(1 - )
(1 - 2)
3(u - )
d3
t dt
B(1 - 2)(1 + u)
p(x3, t)
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
–
–
Specimens with and without marrow in situ;
Loading with different strain rates (0.001, 0.01, 0.1, 1, 10
/second).
 Luo
–
and Hayes, 1977:
et al., 1993:
Effect of specimen size on hydraulic stiffening of cancellous
bone.
ESTIMATION OF MODEL PARAMETERS

 = 0.3
–

G = 16.17 (MPa)
–

assumed based on literature
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 (MPa)
Compressive Modulus vs. Strain Rate
250
Carter and Hayes’ Experimental Results
200
150
100
50
0
0.001
Model prediction (= 3.4 x 10-5 m2 / MPa-sec)
0.01
0.1
1.0
10.0
Strain Rate (/sec)
Predicted Compressive Moduli of Trabecular Bones
of Different Lengths (B = 0.91 and  = 3.4 x 10-5)
Strain Rate
(/sec)
0.001
0.01
0.1
1.0
10.0
Compressive Moduli (MPa)
5 mm
10 mm
25 mm
54.03
54.27
54.76
54.30
56.74
61.61
57.04
81.38
123.98
84.40
188.69
209.40
192.32
212.33
214.48
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)
Total
Stress
(MPa)
B
K
(m2 / MPa/sec) , B (No unit)
Total Stress vs.  and B
(Strain Rate = 0.1/s)
Total
Stress
(MPa)
K
(m2 / MPa/sec) , B (No unit)
B
Total Stress vs.  and B
(Strain Rate = 10/s)
Total
Stress
(MPa)
B
K
 (m2 / MPa/sec) , B (No unit)
Poroelastic Equations
(Rice and Cleary, 1976)
Constitutive Equation:
3(u - )
2G
p = 2Gij +
ij +
(1 - 2
B(1 - 2 )(1 + u)
)
ij = Total Stress Tensor (MPa)
ij = Strain Tensor
p = Pore Pressure (MPa)
kk ij
Poroelastic Equations
(Rice and Cleary, 1976)
Diffusion Equation:
p
t
-
2GB 2(1 - 2)(1 + u)2
9(u - )(1 - 2 u)
2GB (1 + u) kk
 p=3 (1 - 2 u) t
2
- 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
Permeability Measurement
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.

Diffusion Apparatus for
Permeability Measurement
Constant Force for
producing a hydraulic pressure
of 7 kPa
LVDT to measure the piston
displacement h (m)
O-Ring
Trabecular
Specimen
Piston
Spacer
Reservoir filled with
saline solution

Since the rate of flow (Q/t; m3/sec) is proportional to
the pressure difference (p; Pa) according to Darcy’s
law;
=
L
Asp p
Q
t
,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
Flow Volume (m3)
2.0 E-5
1.5 E-5
1.0 E-5
r2 > 0.99
5.0 E-6
0.0
0.0
0.5
1.0
1.5
Time (sec)
2.0
Uniaxial Strain Tests
Loading Piston
MTS Load Cell
Rigid Stainless
Steel Annulus
* u - u and u - p
curves were obtained.
O-Ring
Marrow
In Situ
Trabecular
Specimen
* The specimen was
subjected to a
0.7% strain using
a displacement control
(0.002 mm/sec).
Ram
Pressure
Transducer
Uniaxial Stress Tests
MTS Load Cell
LVDTs
* The specimen was
subjected to a
1.0% strain using
a displacement control
(0.002 mm/sec).
*  -  and L - 
curves were obtained.
MTS
Ram
Data Analyses
(Uniaxial Stress Tests)

From the Elasticity Theory:
Young’s Modulus:
Drained Poisson’s ratio:
Shear Modulus:
E=


L
= - 
E
G = 2(1 + )
* 5th order polynomial curves were used to determine
the  -  and L -  relationships. (Slope at  = 0.0065)
Lateral Strain vs.
Axial Strain
1.
0
0.
8
0.
6
0.4
0.18
Lateral Strain L (%)
Stress (MPa)
Stress vs. Strain
0.2
0.0
0.3
0.6
0.9
Axial Strain  (%)
0.15
0.12
0.09
0.06
0.03
0.0
0.3
0.6
0.9
Axial Strain  (%)
Data Analyses
(Uniaxial Strain Tests)

From the stress-strain relationship in an undrained
uniaxial condition and the definition B:
u
2G(1 - u)
Undrained Poisson’s ratio:
=
u
(1 - 2u)
Skempton’s Coefficient:
3(1 - u) u
B=
(1 + u) u
* 5th order polynomial curves were used to determine
the u - u and p - u relationships. (Slope at u = 0.0065)
Stress (or Pore Pressure)
vs. Strain
Pore Pressure) vs. Stress
4.0
4.0
Undrained Condition
3.0
2.0
Pore Pressure
1.0
0.0
Pore Pressure (MPa)
u or Pressure (MPa)
5.0
3.0
2.0
1.0
0.0
0.1
0.2
0.3
0.4
0.5
u (%)
0.6
1.0
2.0
3.0
4.0
u (MPa)
5.0
Permeability of Human Vertebral Trabecular Bone
Level
Mean Permeability (SD)
L1
L2
L3
L4
L5
Total
(x10-8 m2/Pa/sec)
61.3 (10.4)
53.6 (7.30)
50.4 (6.30)
38.6 (10.7)
45.7 (8.30)
52.2 (10.8)
Permeability of Bovine Vertebral Trabecular Bone
Level
L1
L2
L3
L4
L5
Total
Mean Permeability (SD)
(x10-8 m2/Pa/sec)
14.91 (9.83)
12.39 (7.85)
19.03 (8.55)
18.50 (6.74)
16.85 (6.23)
16.31 (8.02)
RESULTS
All
curves were well represented by a 5th order
polynomial curve (r2  0.97).
Mean
(SD) values of the poroelastic parameters
G

u
B
(x10-8 m2 /Pa/sec)
(MPa)
90.85
(59.59)

0.242
(0.099)
0.399
(0.083)
0.851
(0.144)
16.31
(8.02)
DISCUSSION
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
Measurement of G, , u, and B

Uniaxial Strain Tests in an Undrained Condition:
–

Uniaxial Stress Tests in a Drained Condition:
–

B and u measurements
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.