BME/ME 456 Biomechanics
I. Overview
In this section we discuss how various aspects of both cortical and trabecular bone structure as described in the previous section affect their function represented by the mechanical properties of stiffness and strength. It is very important to note that we may define material properties like stiffness and strength for each level of structure. For example, a section of cortical bone may have an elastic modulus just as an individual osteon may have an elastic modulus. The other critical note to make is that material properties at a more macroscopic level are a function both of material properties and the spatial arrangement or architecture of materials at the more microscopic level. This is the essence of tissue structure function. Thus, cortical bone stiffness, that is stiffness at the 0th level, is a function of osteon arrangement (spatial arrangement or architecture at the 1st level), the stiffness of individual osteons (material properties at the 1st level), the degree of mineralization of the osteons (spatial arrangement and materials properties at the 3rd level, etc. One goal of biomechanics is to determine how changes of a given structural level, due perhaps to disease or a conscious intervention to alter bone structure, affect the load carrying capacity of bone. This fits into the following feedback scenario:
The load on the whole bone (determined for instance by the optimization analysis we discussed previously) generates stress and strain at each level of structure. The level of stress and strain generated by the load is a function of the hierarchical material properties. The subsequent stress or strain is sensed by the cell to determine if it is within acceptable limits, both lower and upper. If not, the cell can act on the matrix to alter the bone structure and thereby alter the mechanical properties. This changes (assuming the load stays the same) the stress and strain felt by the cell. The cells acting together are believed to regulate the bone structure in response to their mechanical environment. This in turn determines tissue stiffness and strength. Diseases that affect either the cells ability to sense mechanical stress or to alter bone matrix can lead to deficient mechanical properties. We next describe one aspect of the loop, namely how different aspects of bone structure affect mechanical properties, generally known as structure function relationshisps. We begin with a definition of stress, strain and constitutive properties viewed in a hierarchical perspective.
II. Hierarchical Stress, Strain and Constitutive Properties
Both cortical and trabecular bone have hierarchical structures, as do all biological tissues. Just as their are structural entities at different length scales from sub-micron to centimeter, so we can define mechanical behavior including stress, strain and material properties at each level of structure. For example, cortical bone properties are often determined by machining a tensile specimen from the cortical diaphysis. This specimen may have a gauge area of 5x5mm and a length of 5 mm. A schematic is shown below:
Stress in this gauge section is computed simply as the force from the load cell in tension divided by the cross sectional area of the gauge. Deformation is measured as the change in length of the gauge section divided by the original length of the gauge section. The gauge section, however, may contain many osteons. It is obvious that the experimental stress and strain are average over many osteons. This indicates that the stress we measure in the testing set-up above cannot represent the stress in a single osteon. We can relate the average stress (the 0th level structure in the cortical bone organizational chart) to the first level stress at the osteonal level using the following equation:
where denotes effective stress 
which is eqiuvalent to stress at the 0th level 
in
our cortical bone organizational
chart, 
denotes
the volume of the cube of bone we are testing, which
is equivalent to a volume at the first level of scale 
,
denotes
stress at the oseon level, which is equivalent to the 1st level stress 
in
the cortical bone organizational
chart. We can also write the same relationship for the average strain:
where the levels are as in the average stress equation and e is strain.
Even though we can write strain and stress for a given macroscopic level as he average over the next microscopic level, we generally cannot do the same for the constitutive equations. This is because the constitutive properties vary over distinct phases of material at the microscopic level. For example, with cortical bone, the microstructure contains stiffness properties for the osteons and the blood vessel space, in addition to the interstitial bone. If we assume that there are n phases of microstructure, then when can write the constitutive equation for the effective or macroscopic level as:
We can rewrite the above equations condensed in matrix form as:
where M is known as a strain localization matrix or local matrix for short. The form of M can vary depending on the assumed model (if we are doing computational or analytical models) or on the experimental measures used to quantify structure. What the above equations represents is what we know intuitively, namely that the effective properties at a macroscopic level depend on the microscopic properties [C], the the spatial arrangement or architecture of those microscopic properties [M]. It is important to note that the above equation governs the relationship between any structural level defined in the cortical and trabecular bone organization chart. That is, the effective properties of osteons may be related to the distribution of lacunae, the collagen organization, degree of mineralization, etc.
For trabecular bone, mechanical properties at the 0th level, most commonly referred to as "effective" or "continuum" level trabecular bone properties, are determined experimentally most often by testing cubes of trabecular bone between 8mm and 1 cm on a side in compression, underneath a loading platen:
Again, as with cortical bone, It is assumed in these tests that one average stress measure can be computed as the force divided by the area of a face of the cube. Strain is computed by divided the change in length of the cube by the original length of the cube. It is important to note that by computing the stress this way we are computing an average or effective stress. In other words, the stress we compute is the same as if we average the stress over all the individual trabeculae and the available pore space. This can be written mathematically as:
where denotes effective stress
which is eqiuvalent to stress at the 0th level in
our trabecular bone organizational
chart, denotes
the volume of the cube we are testing, which is equivalent to a volume at the
first level of scale ,
denotes stress
at the trabecula level, which is equivalent to the 1st level stress in
the trabecular bone organizational
chart. The average strain is defined in the same manner
as:
where the superscripts and the levels are the same as defined for the average stress. The constitutive matrix at the effective level is related to the microscopic level properties using the same rule for the constitutive equation we used for cortical bone.
III Elastic/Strength Properties of Cortical Bone at the 0th
Level: Influence of 1st Level Struct
Mechanical stiffness values have been measured most often for secondary haversian bone human and cow (bovine) bone as well as bovine plexiform bone. Stiffness has been measured using both standard mechanical testing techniques as well as ultrasonic measurements where the velocity of waves propagating through a material is measured. This wave velocity is related to the stiffness and the density of the material.
Reilly et al. (1974) measured the stiffness of human secondary osteonal
bone, bovine secondary haversian bone, and primary bovine bone. Reilly et al. did not measure Hooke's constants directly, but rather
made some measure of moduli in the inferior-superior direction (denoted the
33 direction in Hooke's Law) and the radial direction within the cross section
of the long bone. Reilly et al. found:
Bone Type Modulus I-S Modulus
Radial Strength IST Strenght Radial
Hum. Haversian 17.9 GPa 10.1 GPa 135 MPa 53 MPa
Bovine Haversian 23.1 GPa 10.4 GPa 150 MPa 49 MPa
Bovine Primary 26.5
GPa 11.0 GPa
167 MPa 55 MPa
Note that bone stiffness is greater in the I-S direction along the osteonal length than in the transverse direction across the osteons. Also note that secondary osteons (haversian bone) with more lamellae tend to reduce both the stiffness and strength of cortical bone.
Katz et al. (1984) measured and compared the orthotropic elastic constants
of bovine plexiform bone and human secondary bone:
Elastic Constant Plexiform
Value Secondary Value
C1111 22.4 GPa 21.2 GPa
C2222 25.0 GPa 21.0 GPa
C3333 35.0 GPa 29.0 GPa
C2323 8.2 GPa 6.3 GPa
C1313 7.1 GPa 6.3 GPa
C1212 6.1 GPa 5.4 GPa
C1122 14.0 GPa 11.7 GPa
C2233 13.6 GPa 11.1 GPa
C1133
15.8 GPa 12.7
GPa
Notice that the plexiform bone with the brick like structure has orthotropic symmetry while the secondary haversian canal is nearly transversely isotropic. The type of material symmetry present results from the different types of 1st level structures. Osteons are tube structures which exhibit a transversely isotropic symmetry while the brick like structures of plexiform bone exhibit an orthotropic symmetry depending on the aspect ratios of the brick.
An overview (or representative average) of cortical bone properties for human and bovine (cow) were presented by Martin et al. (1998):
Property Human
Value Bovine
Value
Elastic Modulus Transverse 17.4
GPa 20.4
GPa
Elastic Modulus Long 9.6
GPa 11.7
GPa
Shear Modulus 3.5
GPa 4.1
GPa
Tensile Yield Stress Long 115
M Pa 141
MPa
Tensile Ult Stress Long 133
M Pa 156
MPa
Tensile Ult Stress Trans 51
M Pa 50
MPa
Comp Yield Stress Long 182
M Pa 196
MPa
Comp Yield Stress Trans 121
M Pa 150
MPa
Comp Ult Stress Long 195
M Pa 237
MPa
Comp Ult Stress Trans 133
M Pa 178
MPa
Tensile Ultimate Strain 2.9
- 3.2% .67
- .72%
Compressive Ult. Strain 2.2 - 4.6% 2.5
- 5.2%
It is important to note what the quantities of yield and ultimate stress represent. Although bone is not an elastic-plastic material in the classic sense like metals, bone will yield. This means that under high enough loads, permanent deformation will occur in bone. The ultimate strength is the stress at which the bone undergoes catastrophic failure. The elastic modulus is the bone stiffness. We illustrate these concepts on a schematic stress strain curve below:
Porosity
We
saw that cortical bone properties are significantly affect be the microstructural
differences between osteonal and plexiform bone. However, these bone structure
function relationships were qualitative, due to difficulties in quantifying
these types of microstructures. One microstructural variable that has a significant
influence on bone mechanical properties is porosity. Porosity is difficult to
classify however, because it occurs across multiple scales. In cortical bone,
porosity may result from haversian canals and resorption cavities on the 1st
level (~ 100 to 200 microns) down to lacunae and canaliculi at the 2nd level
(~5 to 20 microns). In cortical bone, although porosity measurements take all
these voids into account, it is believed that the larger scale voids, the haversian
canals, have a large affect on mechanical properties.
Both Schaffler and Burr (1988) and Currey (1988) measured tensile elastic modulus
of cortical bone and compared the results to measures of porosity. Both found
statistically significant relationships and derived empirical relationships
between porosity p and Young's modulus E:
Note that Young's modulus is given in GPa and that the exponent on porosity is very large and nonlinear. This means that the moment one introduces porosity into bone, there is a large decrease in stiffness. The effect of ultimate stress on porosity is similar. Martin et al. (1998) present a composite graph showing multiple experiments relating ultimate stress to porosity:
Mineralization
A significant component of bone that differentiates it from soft tissues is the presence of the HA like ceramic mineral. It is postulated that the mineral component gives bone its high stiffness compared to soft tissues while the type I collagen contributes to the post-yield behavior of bone. Currey (1986) found that specific mineralization and porosity could explain 84% of the variance in cortical bone stiffness. Specific mineralization is defined as the volume of mineral per volume of bone matrix exclusive of voids. Schaffler and Burr (1988) found that mineral did explain some variance in cortical bone stiffness, but not a significant amount due to the fact that there was little variability in specific mineralization between specimens. They derived an equation relating mineral to elastic modulus:
Note that the exponent for mineralization is not as high for porosity, indicating that porosity has a large affect on bone stiffness than mineralization. This of course is only true if the variation in mineralization is small.
IV Effect
of 2nd Order Structure on 1st Order
Very little mechanical data is available on 2nd level cortical bone mechanics. Ascenzi and colleagues (see Martin and Burr, 1998) have performed the most mechanical testing of individual osteons. The procedure by which Ascenzi and co-workers have tested osteons is illustrated below:
. The results from Ascenzi's testing indicate
that secondary osteon stiffness is less than that of large cortical bone specimens.
This would indicate that some other structures in cortical bone, perhaps
interstitial bone, contribute more to the overall stiffness of cortical bone.
Osteon Properties from
Ascenzi
Longitudinal Compression 6.3 110
Transverse Compression 9.3 164
Alternating Compression 7.4 134
Longitudinal Tension 11.7 114
Alternating Tension 5.5 94
Longitudinal Shear 3.3 46
Transverse Shear 4.2 57
Alternating Shear 4.1
55
The terms longitudinal and alternating refer to how the collagen fiber bundles are oriented with respect to the plane of the osteon section. Notice that collagen fiber bundles oriented with the direction of testing produce a higher normal stiffness while collagen fiber bundles oriented out of the plane of testing produce a lower normal stiffness but a higher shear stiffness. Ascenzi also reported the affect of mineralization on the properties of these different types of osteons. The effect is illustrated in the stress strain diagram below:
Another important aspect of how 2nd level structure affects cortical bone mechanics is crack propagation and fatigue life. Since it seems that both primary and plexiform bone have higher strength and stiffness than secondary osteonal bone, the question arises as to why humans and other active mammals have secondary bone rather than plexiform or primary bone. Although there may be many metabolic reasons, one mechanical reason has to do with crack initiation and arrest. Since humans and other smaller mammals are much more active than cows and sheep, their bones are subject to many cycles of loading. This would make human bone much more subject to fatigue failure. Once mechanical advantage of secondary bone is that it has many compliant interfaces such as cement lines. These weak interfaces provide many opportunities for crack arrest , thus making secondary osteonal bone perhaps more fatigue resistant.
V. Dependence of Trabecular Bone Effective (0th Level) Properties on 1st Level Structure: Part I Quantifying Trabecular Structure
The main features of trabecular bone structure at the 1st level are the high porosity and the intricate architecture and orientation of the complicated rod and plate structure of trabeculae. We can expect therefore that these features along with mineralization to be the major factors contributing to the effective stiffness of trabecular bone. This is similar to the determinants of cortical bone effective stiffness, which were osteons versus plexiform structure, the amount of porosity and the degree of mineralization. However, a significant difference between cortical bone 1st level structure and trabecular bone 1st level structure is the substantial variation in 1st level trabecular bone structure compared to cortical bone structure. Whereas most human adult structure is osteonal we a tight porosity range of 5 to 10%, human adult trabecular bone has a much larger range of porosity (10 to 50%) and a much more varied architecture. Osteons by and large have fairly consistent orientation in cortical bone but rods and plates in trabecular bone have a much greater variation in orientation. Therefore in addition to quantifying porosity in trabecular bone, we must also come up with a way to quantify orientation of rods and plates in trabecular bone.
We quantify trabecular structure using the techniques of stereology. These methods use points and lines laid across a structure and count the number of points that fall in each phase of the structure and the number of intersections between structural boundaries of phases. Initially, this was done using a microscope using 2D sections, but now with 3D imaging these approaches have been automated in computer algorithms. The basic idea of making stereology measurements on trabecular bone (or any tissue structure for that matter) is shown below:
In the above picture, trabeculae are shown in cross hatch and void or marrow space is shown in white. A square grid is laid across this 2D section of bone. The dark circles within the bone are denoted as Pp. This is one measurement made on trabecular architecture. Pp is basically a ratio of how many grids intersections fall within the bone divided by the total number of grid intersections. This is a measure of the volume fraction of bone, or equivalently the inverse of the porosity. A second measurement, Pl, is made by counting the number of intersections between bone and the surrounding marrow space. This measure is actually a function of orientation, since the grids are rotated from 0 to 360 degrees and intersect counts are made on the structure. Using CT voxel data, this anisotropy measurement is made using a sphere. We thus have Pl as a function of orientation, ie Pl(q). This measurement gives us and indication of the distances between bone marrow intersections, thus giving us some indication of the width of a trabecuale. Because of this, Pl is often called a mean intercept length. When one plots the number of intersects versus theta in polar plot for trabecular bone, the result is an ellipse. In 3D, the result is an ellipsoid. The general equation for an ellipsoid is (neglecting terms that reflect orientation of the ellipsoid):
where the quantities Aij are coefficients of the ellipsoid and in this case the xi are the mean intercept length measures in the base coordinate system. Harrigan and Mann (1984) recognized that the above equation is essentially the inner product of a second order tensor with two vectors and that the above equation could be rewritten as:
Since A has nine quantities in 3D space, we recognize that it is a second order tensor. A fundamental property of tensors is that we may calculate principal directions and characteristic values of tensors. The characteristic values are called eigenvalues while the characteristic directions for a second order tensor are vectors called eigenvectors. The coefficients A for the tensor are calculated by performing a statistical fit between the experimentally measured intercept lengths as a function of orientation and the ellipsoid equation. The resulting coefficients are typically termed the anisotropy tensor, since the give us and indication of structural symmetry in bone. Note that this tensor is NOT a measure of the anisotropy in mechanical properties of the bone, although as we will see it is related. If we perform an eigenanalysis on the anisotropy tensor we will find the eigenvalues as well as the eigenvectors. The eigenvectors give us the principal orientation of trabeculae within the cube, while the eigenvalues give us the relative average length of bone trabeculae in each of those directions. If we are given a matrix (2nd order tensor) of mean intercept length coefficients, we can use MATLAB to calculate the eigenvalues and associated eigenvectors. An example is shown below:
Suppose we are given the following coefficients of an anisotropy tensor:
» a=[1. .2 .3; .2 .9 .1; .3 .1 .7] ; This is the MATLAB input for the tensor
a = 1.0000 0.2000 0.3000
0.2000 0.9000 0.1000
0.3000 0.1000 0.7000
To solve for the eigenvalues and associated eigenvectors, we use the MATLAB command eig, as:
» [e,v]=eig(a) ; In this case, e will hold the eigenvectors and v will hold the eigenvalues.
We obtain the following results:
e = 0.3733 -0.5426 0.7525
-0.8748 0.0640 0.4802
0.3087 0.8376 0.4508
v = 0.7794 0 0
0 0.5133 0
0 0 1.3073
The first column of e is the eigenvector
associated with the first column of v and so forth. This tells us that the highest
amount of bone is 1.3 mm located along the unit vector
.75i + .48j + .45k.
To summarize, when we use the grid measurements of trabecular bone, we obtain the following measures of structure:
Pp - # of grids in bone/total # of grids => Volume fraction of bone
Pl - # of bone/marrow intercepts as a function of orientation => fit to an ellipsoid with a subsequent eigenanalysis gives the principal direction of bone and the amount of bone along that direction.
V. Dependence of Trabecular Bone Effective (0th Level) Properties on 1st Level Structure: Part I Relationship between Trabecular Structure Measures and Stiffness
A great deal of work has been done
to measure the 0th level mechanical properties of trabecular bone which result
from its 1st level structure.
Many investigators would characterize trabecular bone on the 0th level
as an orthotropic material. Many measurements of trabecular bone stiffness
at the 0th level have only
characterized the axial stiffness of trabecular bone. This is because of the difficulties involved in measuring orthotropic
properties. Trabecular bone 0th
level stiffness and strength are a function of its 1st level structure. Many studies have therefore related measures
of first level structure including density or volume fraction of trabeculae
and orientation of trabeculae to 0th
level trabecular stiffness. The stiffness
of trabecular bone has been found to range anywhere from 1 to 1000 MPa.
The ultimate strength of trabecular bone has been found to range between
0.12 MPa to 310 MPa (Goldstein, 1987). Thus,
trabecular bone tends to exhibit a much wider range of stiffness and strength
than cortical bone, perhaps due to the large variations in both the density
and organization of 1st level structure inherent to trabecular
bone. Many investigators have tried
to find a correlation between density and trabecular bone stiffness and density
and trabecular bone strength. In general,
equations of the form:
where A and B are constants in both
cases, E is the Young modulus, and nf is the volume
fraction. The power relationship has
found many uses in the cellular foam literature. Since 1st
level trabecular bone structure in many ways resembles a cellular foam, it was
thought that a power relationship would provide a better fit. The B coefficient in the power relationship
has generally been found to range between 2 and 3 for trabecular bone. However, it appears as though the fit provided
by the linear and the power relationship provides comparable predictive capability
for trabecular bone stiffness. Ciarelli et al. (1991) found that linear relationships worked better
for some bone while power relationships provided better fits for other regions.
A comparison of the stiffness (compressive modulus) in the inferior-superior
direction for different bones from Ciarelli et al. is shown below:
Region R2 for Linear
Model R2
for Power Model
Proximal Femur 0.50 0.55
Distal Femur 0.65 0.65
Proximal Tibia
0.41 0.40
Proximal Humerus 0.65 0.66
Distal Radius 0.17 0.13
As you may have noted, the density alone does not provide a very good
prediction of 0th level stiffness
in many cases. This is because a scalar
cannot be used to estimate a tensor quantity like stiffness. It is also because the 0th
level stiffness of trabecular bone depends on other aspects of 1st level structure, as you would guess from
its anisotropic nature. Thus, some investigators
have tried to measure the orthotropic constants of trabecular bone while others
have used other measures of trabecular bone structure to relate to stiffness.
Ashman et al. (1989) measured the orthotropic axial and shear moduli
of proximal tibial trabecular bone using ultrasound, finding:
Modulus Mean
Stiffness (MPa)
Axial, E1 346.8 (218)
Axial, E2 457.2 (282)
Axial, E3 1107.1 (634)
Shear, G12 98.3 (66.4)
Shear, G13 132.6 (78.1)
Shear, G23
165.3 (94.4)
where the numbers in parentheses denote standard deviation of experimental measurements. In this case, the 3 direction is the IS (inferior-superior) direction, the 1 direction is the AP (anterior-posterior) direction, and the 2 direction is the ML (medial-lateral) direction. Since the tibia is loaded primarily along the IS direction, you can see that the bone stiffness has been adapted to the predominant loading.
Snyder et al. (1989) measured the complete orthotropic elastic constants
of proximal femoral trabecular bone by mechanical testing.
They also measured trabecular bone 1st level structure using the Mean
Intercept Length (MIL). This essentially
tells how much bone is oriented along any particular direction.
It is measured by laying test lines across a bone face and counting the
number of times the test line intersects a bone surface (Fig. 4).
The lines are rotated through 180o
and the number of intersections is counted at fixed angular increments. These intercepts can then be fit to the form
of an ellipsoid, or equivalently a second order tensor which is a measure of
the anisotropy of trabecular bone. Snyder
et al. related both the bone volume fraction and the MIL to the orthotropic
elastic constants. They found the following
relationships:
Sii = 1/Ei
= .00166/Vf + .00117/MIL
Skk = 1/Gij
= .00552/Vf + .0299/(MILi +
MILj) + .00615/(MIL+ MIL)
Sij = nij/Ei
= -.06/(MILi +
MILj) + .0105/(MILi*MILj)
- .000348/(MIL+ MIL)
where Sii are the axial compliance coefficients,
Skk are the shear compliance
coefficients, and Sij are the
off axis compliance coefficients, Vf is the bone volume fraction and MILi is the mean intercept length in the ith antomical direction. The compliance matrix [S] is the inverse of
the material stiffness matrix [C]. In
other words: 
. The
results verify the significant dependence of 0th
level trabecular bone stiffness on 1st
level trabecular bone structure.
VI Analytical Models for Structure Function Relationships
In addition to statistical experimental approaches to bone structure function relationships, there are also analytical computational models for bone structure function relationships. Two of the simplest yet most often used are the Voight and Reuss models. The Voight model assumes that the strain in all areas of the microstructure are equal. This leads to computational of effective stiffness by volume fraction weighting of the stiffness of each phase. In terms of the general relationship between stiffness and structure, the local matrix is simply the identity matrix. If we perform the integration of the volume of each microstructure, we obtain an averaged matrix, that has the volume fraction of bone on the diagonal:
where the quantity vf is the volume fraction of the nth phase. This is the same as the rule of mixtures. If we assume the stress in each phase is equal, the we obtain the Reuss model. This models is the inverse of the Voight model in that it weighs the complicance matrices by the volume fraction of each phase:
It is important to remember that
the compliance S is the inverse of the stiffness C,
We note here that because of assumptions about strain distributions at the microscopic level, computational and analytical models provide bounds on the actual stiffness derived from structure. In fact, the Voight and Reuss models provide absolute upper and lower bounds respectively. This is useful because the models are simple and we can immediately gain a bound on the stiffness range based on the volume fraction of each structure. There are more advanced computational and analytical models for estimating effective properties based on structure, but these are beyond the scope of this class.
