Structure and Function of Ligaments and Tendons

I.          Overview

            Ligaments and tendons are soft collagenous tissues. Ligaments connect bone to bone and tendons connect muscles to bone. Ligaments and tendons play a significant role in musculoskeletal biomechanics. They represent an important area of orthopaedic treatment for which many challenges for repair remain. A good deal of these challenges have to do with restoring the normal mechanical function of these tissues. Again, as with all biological tissues, ligaments and tendons have a hierarchical structure that affects their mechanical behavior. In addition, ligaments and tendons can adapt to changes in their mechanical environment due to injury, disease or excerise. Thus, ligaments and tendons are another example of the structure-function concept and the mechanically mediated adaptation concept that permeate this biomechanics course. In this section, we will review aspects of ligament and tendon structure, function and adaptation. These notes follow very closely Chapter 6 on Structure and Function of Tendons and Ligaments from your text.

II.         Hierarchical Ligament and Tendon Structure

            We start out again emphasizing that ligaments and tendons have a hierarchical structure. One schematic of this hierarchical structure is taken from your text, and is a very famous schematic from Kasterlic:

                            

The largest structure in the above schematic is the tendon (shown) or the ligament itselt. The ligament or tendon then is split into smaller entities called fascicles. The fascicle contains the basic fibril of the ligament or tendon, and the fibroblasts, which are the biological cells that produce the ligament or tendon. There is a structural characteristic at this level that plays a significant role in the mechanics of ligaments and tendons: the crimp of the fibril. The crimp is the waviness of the fibril; we will see that this contributes significantly to the nonlinear stress strain relationship for ligaments and tendons and indeed for bascially all soft collagenous tissues.

The above schematic shows the basic common structures for ligaments and tendons. In terms of specific attributes for tendons:

TENDONS

Anatomy:

            1. Tendons contain collagen fibrils (Type I)
            2. Tendons contain a proteoglycan matrix
            3. Tendons contain fibroblasts (biological cells) that are arranged in parallel rows

Basic Functions

            1. Tendons carry tensile forces from muscle to bone
            2. They carry compressive forces when wrapped around bone like a pulley

Type I Collagen:

            1. ~86% of tendon dry weight
            2. Glycine (~33%)
            3.
Proline (~15%)
            4.
Hydroxyproline (~15%, almost unique to collagen, often used to identify)

Blood Supply:

            1. Vessels in perimysium (covering of tendon)
            2. Periosteal insertion
            3.
surrounding tissues

LIGAMENTS

Anatomy

            1. Similar to tendon in hierarchical structure
            2. Collagen fibrils are slightly less in volume fraction and organization than tendon
            3. Higher percentage of proteoglycan matrix than tendon
            4.
Fibroblasts

Blood Supply

            1. Microvascularity from insetion sites
            2. Nutrition for cell population; necessary for matrix synthesis and repair

III.       General overview of ligament and tendon mechanics

            As with all biological tissues, the hierarchical structure of ligaments and tendons has a signficant influence on their mechanical behavior. Unlike bone, however, not nearly as much quantiative structure function relationships, either experiment/statistical or analytical, have been derived for ligaments and tendons. This is for two reasons: 1) the hierarchical structure of ligaments and tendons is much more difficult to quantify than bone, and 2) ligaments and tendons exhibit both nonlinear and viscoelastic behavior even under physiologic loading, which is more difficult to analyze than the linear behavior of bone.

Nonlinear Elasticity 

           If one neglects viscoelastic behaviour, a typical stress strain curve for ligaments and tendons can be drawn as:

                          

There are three major regions of the stress strain curve: 1) the toe or toe-in region, 2) the linear region and 3) the yield and failure region. In physiologic activity, most ligaments and tendons exist in the toe and somewhat in the linear region. These constitute a nonlinear stress strain curve, since the slope of the toe-in region is different from that of the linear region.

In terms of structure function relationships, the toe-in region represents "un-crimping" of the crimp in the collagen fibrils. Since it is easier to stretch out the crimp of the collagen fibrils, this part of the stress strain curve shows a relatively low stiffness. As the collagen fibrils become uncrimped, then we see that the collagen fibril backbone itself is being stretched, which gives rise to a stiffer material. As individual fibrils within the ligament or tendon begin to fail damage accumulates, stiffness is reduced and the ligament/tendons begins to fail. Thus a key concept is that the overall behavior of ligaments and tendons depends on the individual crimp structure and failure of the collagen fibrils.

A simple model illustrating the dependence of ligament/tendon nonlinear stress/strain relationships is shown below:

                         

In this case, as a spring is stretched to its limit its stiffness increases. This can easily be seen if the effective ligament stiffness is modeled using the Voight model, with each fibril contributing a small part to the overall stiffness. As a fibril becomes uncrimped, its stiffness increases, increasing the overall ligament/tendon stiffness.

Viscoelasticity

            Another important aspect of ligament/tendon behavior is viscoelasticity. Viscoelasticity indicates time dependent mechanical behavior. Thus, the relationship between stress and strain is not constant but depends on the time of displacement or load. There are two major types of behavior characteristic of viscoelasticity. The first is creep. Creep is increasing deformation under constant load. This contrasts with an elastic material which does not exhibit increase deformation no matter how long the load is applied. Creep is illustrated schematically below:

                         

The second significant behavior is stress relaxation. This means that the stress will be reduced or will relax under a constant deformation. This behavior is illustrated below:

                                     

The other major characteristic of a viscoelastic material is hysteresis or energy dissipation. This means that if a viscoelastic material is loaded and unloaded, the unloading curve will not follow the loading curve. The difference between the two curves represents the amount of energy that is dissipated or lost during loading. An example of hysteresis is shown below:

                            

The two figures above show that the amount of hysteresis under cyclic loading is reduced and eventually the stress-strain curve becomes reproducible. This gives rise to the use of pseudo-elasticity to represent the nonlinearity of ligament/tendon stress strain behavior.

Finally, in this overview of ligament/tendon mechanics we discuss briefly experimental considerations in testing ligaments and tendons. Directly measuring ligament or tendon behavior by directly griping the specimen can lead to erroneous measurements. This is because ligaments and tendons are extremely difficulty grip firmly and often slide in the grips leading to errors in displacement measurements. A way around this difficulty is to leave the ligament attached to the bone and use optical methods and markers to measure the strain. A schematic of such a test setup from the text is shown below:

                         

The grips are placed around the bones of the joint to give a much more secure fit. In the case of tendons, there is still a need to grip the muscle end.

IV        Representing Ligament/Tendon Mechanics through hyperelasticity

            To represent the nonlinear elastic behavior of ligaments, Weiss et al. has developed a hyperelastic model including a strain energy function. Weiss sought to represent the contributions of collagen fibers, the ground substance and the interaction between the two. The strain energy function is given below:

            

where Ii are the invariants of the right Cauchy deformation tensor. The 2nd Piola-Kirchoff stress tensor is calculated from the above strain energy function by the relationship:

            

Two recent papers have used the more common isotropic form of the strain energy function for analyzing stress and strain in anteior cruciate ligamens. The first was by Hirokawa and Tsuruno in Medical Engineering and Physics (1997), 19:637-651. They used a strain energy function commonly used for rubber like materials, also called a Mooney-Rivlin strain energy function. It is given below:

           

where I1 and I2 are invariants of the right Cauchy Deformation tensor. There are defined as:

            

           

 Thus, we can rewrite W using the explicit form of the invariants I1 and I2 as:

                  

where a1 and a2 are constants to be determined experimentally.   Let us now assume that the 11 direction lies along the length of the ligament and we would like to determine the stress/strain curve along that direction. We must first take the derivative of W with respect to E11.  First we know the relatonship between E11 and C11 is:

          

The derivative can be taken using the chain rule as:

             

We can rewrite the above equation in terms of E using the general equation relating E to C (the right Cauchy Deformation Tensor):

            

This gives S11 in terms of E as:

           

To plot S11 vs. E11, we need to know E22 and E33.   We will assume for our case that they are 0. In addition, we need to know the material constants a1 and a2. Hirokawa and Tsuruno report these to be 1.687 and 0.106 (kg/cm2), respectively, which, given that 1 kg/cm2 is equivalent ot 98 KPa, gives constants of 165.3 KPa and 5.88KPa. We can make the stress strain plot in MATLAB for s11 vs. e11 (assuming all other strains are zero) using the following commands:

e11 = (0:99)/99;
for i = 1:100
s11(i) = 4.*165.3*e11(i)+6.;
end
plot(e11,s11)
xlabel('strain E11')
ylabel('stress S11 (KPa)')

This yields the following linear relation between the 2nd Piola-Kirchoff stress and Green-Lagrange strain:

A second paper utilizing an isotropic strain energy function to model human anterior cruciate ligament behavior was published by Song et al. in the Journal of Biomechanics, 37:383-390, 2004. Song et al. used a strain energy function that was originally proposed by Veronda and Westmann for skin in 1970 (Veronda and Westmann, J. Biomechanics, 3:111-124, 1970). This strain energy function is given below:

          

where I1 and I2 are the 1st and 2nd invariants of the Green-Lagrange strain tensor, e is the exponential function, and a1,a2,a3 are constants to be experimentally determined.   Based on our previous expansion of the invariants, we write the strain energy function directly in terms of the Green-Lagrange strain tensor components as:

         

To calculate the S11 component of the 2nd Piola-Kirchoff stress, we again take the derivative of W with respect to E11:

         

We can then plot a stress-strain curve using the values for the experimental coefficients a1,a2 and a3. If we assume that all strains except E11 are zero due to the test set up, we can write the following MATLAB commands to plot the stress versus strain curve:

e11 = (0:99)/99;
for i = 1:100
s11(i)=0.26*11.35*exp(11.35*e11(i)-11.35*3.)
end
plot(e11,s11)

V.        Structure-Function in Ligaments/Tendons: Aging Effects

            Much of our knowledge of structure-function relationships for ligaments and tendons is empirical and not strictly quantiative as with bone, for reasons we touched upon earlier. Most studies on structure-function relationships in ligaments and tendons come from comparisons of mechanical properties and histology/biochemistry of ligaments and tendons from young versus old animals. For example, in rat tail tendons, the diameter of collagen fibrils increase during age from skeletally immature to mature animals. There is a corresponding increase in the ultimate tensile strength of the tendons. In a study of New Zealand white rabbits at 1.5 months, 4-5 months, 6-7 months, and 12-15 months, it was determined that the stiffness and strength of both the mid-ligament substance and the ligament bone attachment increased with increasing maturity in the femur-mcl-tibia complex (FMTC). The change in ligament attachment strength was due to rapid remodeling of bone near the insertion site in younger animals. A schematic drawing showing this change in material properties with age is shown below:

                            

You can see from this graph that as the strength of the ligament insertion increases, failures change from being avulsion failures to being mid-substance failures. A quantitative example of the increase in mechanical properties with maturity is shown in the graph from your text below:

                            

            While increasing age from child to adult also increases the mechanical properties of ligaments and tendons, further increasing age from young adulthood decreases the properties of ligaments and tendons. One study found that structural properties of the FATC (femur-acl-tibia complex) decreased by 2-3 times for older versus younger adult knees. Structural properties refer to the stiffness of the ligament in the bone-ligament-bone testing mode when it is not calculated using normalized stress and strain values, ie force vs displacement.
Woo and colleagues tested FATC from young cadaver knees with an average age of 35 and older cadaver knees with an age of 76. They found that the linear structural stiffness of the ACL decreased both when tested at 30 degrees of knee flexion and when tested along the axis of the ligament complex. The structural stiffness in the ligament axis averaged 183 N/mm for the younger group but only 158 N/mm for the older group. Tests under flexion were 150 N/mm and 129 N/mm for the younger and older group respectively. In addition, more specimens in the older group suffered avulsion failures than the younger group.

VI.       Mechanically Mediated Ligament and Tendon Adaptation: Immobilization versus Exercise

            Ligaments and tendons are adapted in response to changes in mechanical stiffness. The changes in ligaments and tendons generally occur more slowly than adaptation in bone, because ligaments and tendons have less vascular supply. Again, our knowledge of how mechanical stimulus mediates ligament and tendon structure is more empirical and less rigorous than that for bone. Most of our knowledge comes from two extremes in mechanical stimulus: immobilization and exercise.

            Immoblization of a joint for a long period of time leads to significant changes in joint structure and function, including decreased range of motion for the joint. The affects of both ligaments and tendons can be severe. Woo et al. studied rabbit knees in the following experimental groups:

1. 9 weeks immobilization
2.
12 weeks immobilization
3.
9 weeks immobilization, then 9 weeks active
4.
12 weeks immoblization, then 9 weeks active

9 weeks of immoblization led to a 69% decrease in ultimate load and an 82% decrease in energy to failure. After 12 weeks of immoblization led to a 71% decrease in ultimate load. Affects on the stress strain curve from Woo are shown below:

                        

If the rabbits became active, there was an increase in stiffness and strength almost back to the level of controls.

Corresponding to the reduction in mechanical properties, there is a reduction in the ligament structure. During immobilization, the cross sectional area of the ACL is reduced. This implies a loss of collagen fibrils as well as a loss of glycosaminoglycans that form the ground substance of the ligament. In addition, the may be alterations in collagen fibril orientation that reduce properties. Upon remoblization, it appears that the mechanical properties of the ligament are gained back first, followed by the structural properties. This indicates that structural loss at the ligament insertion site may take longer to be removed that changes in ligament substance.

Excercise and increased load on tendons and ligaments is believed to alter their structural makeup and lead to increased mechanical properties, although experimental data is far from conclusive. Woo et al studied the affect of exercise on swine digital tendons and the FMTC. Animals were run on a track at speeds of 6 to 8 km/hr for an average of 40km/week for 3 months and 12 months.. A sedentary group was used as a control. The short term group showed no significant changes in mechanical properties for either the tendons or the FMTC. There was an increase in cross-sectional area of the tendon as well as a 22% increase in tensile strength. For the FMTC, however, there was little change in most mechanical properties, although there was a significant increase in maximum load to failure when normalized by animal weight. Another study in dogs also found higher ultimate load to body strength ratios for the FMTC. Woo put the findings on immobilization and exercise together in a graph showing how changes in mechanical load may alter ligament/tendon structure, in a statement he characterized as Wolff's law for ligaments/tendons. This hypothesis from the text is shown below:

                         

As you can see, immobilization has a more rapid and substantial affect on mechanical properties than does increased load from exercise. This is in some ways similar to Frosts theory on adult bone adaptation, where he believed it was difficult to achieve substantial increase in bone structure through mechanical loading unless damaged was caused, but that losses in bone mass were realized quite readily when loads were significantly reduced as in immobilization. The same can be seen in the above graph for ligaments and tendons.

VII       Mechanics of and Mechanical Affects on Healing Tendons/Ligaments

            One of the most important topics in mechanically mediated ligament and tendon adaptation is the effect of mechanical stimulus on ligament and tendon repair. Ligament and tendon repair are a critical area of orthopaedic surgery, especially in sports medicine. Important issues in ligament and tendon repair are how the ligament should be repaired, if applying mechanical load hinders or helps repair, and when repair has progressed to the point where complete load bearing is possible.

            In the case of tendons, which glide within a sheath, the introduction of passive motion for healing and repaired tendons is believed to be important because it prevents adhesion between the sheath and tendons that restricts motion. In one study, the flexor tendons of skeletally mature dogs were lacerated and then repaired. There were three experimental groups: 1) compete immobilization, 2) delayed mobilization and 3) early passive motion. The results indicated that tendon strength and motion increased more quickly for the early passive motion group than for the other two groups.

The relationship between mobilization following repair and alterations of ligament structure and function is complex, depending on how long the ligament is immobilized following repair. In a study of Medial Collateral Ligament (MCL) repair, MCL were lacerated and then repaired. In one group, the MCL was not repaired, but it was immobilized. In the second group, the MCL was repaired and immobilized for three weeks while in the third group immoblization lasted for 6 weeks. Suprisingly, the group without repair that was immobilized early showed the best gain in strength over time. This reflected changes in the structural makeup of the ligament. The amount of type I vs. type III collagen (type III collagen is a type of collagen associated with wound healing) was closer to normal for the early mobilized ligament without repair. The change in strength over time is shown in the figure below (group 1 is the early mobilization without repair, group 2 is repair with 3 week immobilization and group 3 is repair with 6 week immobilization):

                      

These results demonstrate two basic concepts: 1) in a confirmation of tissue structure function relationships, the stiffness and strength of healing ligaments correlates with the type and amount of collagen fibrils present, and 2) that mechanical stimulus has a significant affect on ligament structure.

                         

Cartilage Structure and Function

I.          Overview

            There are three major types of cartilage in the body: 1) hyaline cartilage, 2) fibrocartilage, and 3) elastic cartilage. Elastic cartilage exists in the epiglottis and the eustachian tube. Fibrocartilage, as we saw in the section on fracture fixation, often exists temporarily at fracture sites. However, fibrocartilage is permanently present in three major locations in the body: 1) the intervertebral disks of the spine, 2) as a covering of the mandibular condyle in the temporomandibular joint, and 3) in the meniscus of the knee. The third type of cartilage, hyaline cartilage, is most prominently found in diarthroidal joints covering long bones. In addition, hyaline cartilage forms the growth plate by which long bones grow during childhood. In this section, we review the structure and mechanical behavior of hyaline cartilage in diathroidal joints, typically called articular cartilage, and the meniscus of the knee. The accompany information in the text is Chapter 4 "Structure and Function of Articular Cartilage and Meniscus" by Van Mow and Anthony Ratcliffe.

II.         Diarthroidal Joint Anatomy and Hierarchical Cartilage Structure

            Again, although it begins to sound redundant at this point, articular cartilage itself has a hierarhical structure and is also part of a diarthroidal joint which is a composite structure. The nature of the hierarchical structure of both diarthroidal joints and articular cartilage is illustrated in the figure from your text shown below:

                  

The top row of the figure illustrates the composite strcuture of diarthroidal joints which consist of bone, articular cartilage, ligaments, tendons, muscle and the joint capsule. This whole organ level exists at a scale of > .5 cm. The next level in this schematic indicates in more detail the actual bearing surface of the joint, in this case the knee joint as evidenced by the meniscus. The scale is between 100 microns (.1 mm) and 1 cm. At this point the articular cartilage may be viewed as a solid homogeneous material. In the next level of structure, called the microstructure between .0001 mm (.1 microns) and .1 mm (100 microns), we see the existence of the structural features of articular cartilage including the chondrocytes (cells that make cartilage matrix) and the organization of the type II collagen fibrils. The organization at this level can actually be divided into four zones: 1) the superficial tangential zone (10-20% of the cartilage thickness, 2) the middle zone, 60% of the cartilage thickness, 3) the deep zone, 30% of the cartilage thickness, and 4) the calcified cartilage zone where the cartilage interfaces with the bone. The zones contain different collagen organization as well as different amounts of proteoglycans. A schematic of these zones from your text is shown below:

             

The superficial or tangential zone contains the highest collagen content, about 85% by dry weight. In addition, the collagen fibrils are oriented parallel to the joint surface, indicating that the purpose of this zone may be primarily to resist shear stresses. The amount of collagen decreases in each zone moving closer to the tidemark, dropping to 68% in the middle zone.

At the next level, denoted as the ultrastructural level, between .00001 mm (.01 microns) and .001 mm (1 micron), we see the existence of the major biochemical constituents of articular cartilage including the individual collagen fibrils and the proteoglycan matrix. Finally, at the nanostructural level (.0000001 mm (.1 nanometer) to .000001 mm (1 nanometer) we see the interior structure of the collagen and proteoglycan molecules. As you can see, definition of these structural levels is not cut and dried, but exists as a conceptual tool to help us understand cartilage structure and how that structure influences cartilage function. The other very important point to note is that the micro and ultra structures contain water and electrolytes that are bound to the molecules (mainly proteoglycans and collagen) that constitute the solid matrix of articular cartilage. As we will see later, the fluid-solid interaction is a major determinant of articular cartilage mechanical behavior.

III.        Articular Cartilage and Meniscus Composition

            Section II outlined the hierarchical structure of articular cartilage. In this section, we outline the chemical composition that is common to articular cartilage and meniscus. There are two major phases of articular cartilage and meniscus: 1) a fluid phase containing water and electrolytes, and 2) a solid phase containing collagen (type I in meniscus and type II in articular cartilage), protoeglycans, glycoproteins and chondrocytes. Chondrocytes are the cells that produce cartilage matrix.    The specific percentages of the major constituents for articular cartilage and meniscus are given in table I of the text chapter, reproduced below:

Tissue                           Water               Collagen                     Proteoglycans

Articular Cartilage          68-85%           10-20%  (type I)        5-10%
Meniscus                       60-70%           15-25% (type II)        1-2%

            As you can imagine, these three major components act together to determine the mechanical behavior of cartilage. Changes in the relative amounts of these components due to disease will change the time dependent mechanical properties of cartilage.

Of the three major components, the most prevalent is water. About 30% of the total water exists within the intrafibrillar space of collagen. The collagen fibril diameter and the amount of water within the collagen is determined by the swelling pressure due to the fixed charge density (FCD) of the proteogylcans. In other words, the proteogylcans have strong negative electric charges. The proteoglycans are constrained within the collagen matrix. Because the proteogylcans are bound closely, the closeness of the negative charges creates a replusion force that must be neutralized by positive ions in the surrounding fluid. The higher concentration of ions in the tissue compared to outside the tissue leads to swelling pressures. The exclusion of water raises the density of fixed charge, which in turn raises the swelling pressure and charge-charge repulsion. The amount of water present in cartilage depends on 1) the concentration of proteoglycans which determines FCD and swelling pressure, 2) the organization of the collagen network, and 3) the stiffness and strength of the collagen network. The collagen network resists the swelling of the articular cartilage. If the collagen network is degraded, as in the case of OA, the amount of water in the cartilage increases, because more negative ions are exposed to draw in fluid. The increase in fluid can significantly alter the mechanical behavior of the cartilage.

In addition, with a pressure gradient or compression, fluid is squeezed out of the cartilage. When the fluid is being squeezed out, there are drag forces between the fluid and the solid matrix that increase with increasing compression and make it more difficult to exude water. This behavior increases the stiffness of the cartilage as the rate of loading is increased.

Collagen is the component of cartilage that is believed to contribute most to tensile behavior of the tissue. The predominant collagen in articular cartilage is type II while the predominant collagen in meniscus is type I.

The third major component of cartilage are the proteoglycans. Proteoglycans are large biomolecules that consist of a protein core with glycosaminoglycan side chains. These molecules normally occupy must large space when not compacted by a collagen network. The compaction of the proteoglycans affects swelling pressure as well as fluid motion under compression.

IV.       Structure-Function Relationships in Articular Cartilage and Meniscus

            As perhaps can be gleaned from the previous sections, there are three major factors that contribute to articular cartilage mechanical behavior. First, there is the swelling pressure due to the ionic affects in the tissue. Second, there are the elastic behavior of the solid matrix itself. Third, there is the the fluid-solid interaction in the cartilage under compressive load. We next detail these mechanical behaviors and discuss the how the tissue structure contributes to this behavior.

Solid Matrix Properties

First, let us consider the tensile properties and behavior of the cartilage solid matrix. As with the other soft collagenous tissues that we have studied, the mechanical behavior of the solid matrix is determined by the amount and crimp of collagen in the matrix. Thus, this matrix follows the classic nonlinear stress strain curve for soft tissues as shown below:

                         

where we see a toe region, a linear region, and a failure region. These regions correspond the unfolding of the crimp. A typical dumbell specimen is used to test the matrix tensile properties as shown below:

                            

In terms of structure function relationships, we can see the effect of increasing collagen content on tensile properties by looking at the tensile moduli from the linear portion of the above stress strain curved measured in the different cartilage zones. Some experimental data is shown below in  MPa:

                                   Bovine                                            Canine                                     Human
                      Glenoid           Humerus          Femoral Groove    Femoral Condyle     Groove        Condyle
Superficial        5.9                    13.4                    27.4                     23.3                    13.9             7.8
Middle             0.9                     2.7                                                                             3.4              4.0
Deep                0.2                    1.7                                                                              1.0

This result can also be confirmed looking at the plot in your text that relates tensile modulus to the ratio of collagen to proteoglycan in the cartilage matrix:

                     

Osteoarthritis, or OA, a major disease that affects cartilage can have signficant effects on the tensile properties of the solid matrix. In OA, we know histologically that there is a disruption in the collagen fibrils in the solid matrix. This is reflected in decreases in the tensile modulus of the solid matrix, as shown below in the table from your text:

                              Normal (MPa)                      Fibrillated (MPa)                      OA (MPa)

Surface                   7.8                                       7.2                                           1.4
Subsurface              4.9                                       7.5                                           0.85
Middle                    4.0                                       4.9                                           2.11

Finally, as with other soft tissues that had a nonlinear stress strain curve and could be considered hyperelastic, we can derive a strain energy function that can be used to calculate the dependence of stress on strain. Let us consider the following strain energy function:

                         

where k and B are constants and E is the Green-Lagrange strain. If we differentiate with respect to the strain we obtain the stress as:

                        

which, if we consider the constant A = Bk, is the same result as your text. In addition to articular cartilage, the tensile modulus of the meniscus is significantly dependent on the amount and orientation of collagen fibrils. In the outer meniscus, collagen fibrils are arranged circumferentially in a more organized manner than in the middle of the meniscus. This gives rise to higher tensile moduli in the circumferential zone.

Compressive Fluid-Solid Properties

          As was mentioned in the section of cartilage composition, the interaction between the fluid and solid phase of the cartilage plays a significant role in the mechanical behavior of cartilage. The flow of water out of the tissue and the drag this creates on the solid phase are major determinants of the compressive behavior of the tissues. Thus, in this sense, the mechanical behavior of the cartilage is very dependent on how easy it is for the fluid to move in and out of the tissue, a property known as permeability. Flow of fluid through solid, permeable matrices is governed by Darcy's law. Darcy's law states that the rate of volume discharge through a porous solid is related to the pressure gradient applied to the solid and the hydraulic permeability coefficient k. Mathematically, Darcy's law is stated as follows:

                     

where Q is the rate of volume discharge in m^3/sec, k is the permeability coefficient in m^4/Ns, A is the area in m^2, delta P is the pressure gradient in N/m^2 and h is the height of the specimen given in m. Thus, the units work out as:

                     

The permeation speed V is related to Q by dividing Q by the A times the volume fraction of the fluid. The diffusive drag coefficient, how much drag the fluid creates on the solid, determined as:

                    

where K is the drag, k is the permeability and the remaining term is the volume fraction of fluid.

Permeability and load sharing between the solid and fluid components form the basis for the biphasic theory of cartilage behavior. The tenets of biphasic theory are the following:

1. Solid matrix may be linearly elastic or hyperelastic with isotropic or anisotropic behavior.

2. The solid matrix and interstitial fluid are incompressible. This means that cartilage as a whole can only be compressed if fluid is exuded from the cartilage.

3. Energy dissipation is result of fluid flow relative to solid matrix.

4. Frictional drag of the solid vs. the fluid is proportional to relative veloctiry. This is diffusive drag.

The standard stress equilibrium equations are modified for biphasic theory as follows:

                  

where is the solid stress, is the fluid stress, K is the drag coefficient, is the solid velocity and is the fluid velocity.

This theory captures the basic behavior of cartilage under compression. As example of the behavior of cartilage under compression from the text is shown below:

              

In this case, cartilage is subjected to a fixed displacement at point B. We see a large rise of stress in the graph at the right at point B. Because the fluid cannot immediately leave, it carries a good portion of the load. As the fluid leaves the cartilage, load is shifted to the solid matrix and stress is reduced.

Two key material properties in biphasic theory are equilibrium modulus and permeability. Equilibrium modulus is the stiffness of the cartilage as all the fluid flows out. In progressive OA, permeability increases and the equilibrium modulus decreases. As the permeability increases, this means that less load is shared by the fluid phase, increasing stress on the solid phase.

 

Constitutive Properties of Other Soft Tissues

Overview

            As mentioned often in class, many soft tissues have the same general nonlinear stress-strain curve as those we have seen for ligaments, tendons, blood vessels, and the cartilage solid matrix. This nonlinear stress-strain relationship is illustrated schematically below:

            

Where S is the 2nd Piola-Kirchoff stress and E is the Green-Lagrange strain. For the stress-strain relationship above, we assume that the tissue has been cyclically loaded and that the stress-strain curve has a repeatable loading and unloading portion. We then neglect viscoelastic influences and model the tissue as pseudo-elastic, where the loading and unloading curves are treated as separate elastic materials. We can characterize the constitutive or stress strain equations of pseudo-elastic nonlinear soft tissues using a strain energy function. A strain energy function contains a measure of tissue deformation like the Green-Lagrange strain or Right Cauchy deformation tensor, plus constants that must be determined experimentally. The ability to quantify the constitutive equations of soft tissues this way is important for studying structure-function relationships and mechanically mediated tissue adaptation. We have already seen examples of this showing changes in experimental constants of blood vessel strain energy functions due to disease and adaptation. In this section, we present examples of strain energy functions and constitutive behaviors for other soft tissues including skin, kidney and brain tissue. We also present a general strain energy function for soft tissues proposed by Fung.

Skin

            Skin is the largest organ in the body. It is composed of two layers, the epidermis and the dermis. The epidermis is the outermost layer and is between 15 and a hundred cells thick. The cell types are keratinocytes. The epidermis has no blood vessels. It relies on the dermis for nutrients. The dermis itself consists of two layers, the more superficial papillary dermis and the deeper reticular dermis. The papillary dermis is the thinner of the two layers, and contains blood vessels, elastic fibers, collagen and reticular fibers. The deeper reticular dermis contains larger blood vessels, interlaced elastin fibers, and parallel bundels of collagen fibers. It contains fibroblasts and mast cells. The fibroblasts are the major cell type and produce the elastin and collagen within the papillary dermis. Collagen makes up 70% of the papillary dermis by weight. Of this total, type I collagen is 85% and type III collagen is 15%. Elastin makes up about 1%. The dermis also contains a ground substance, containing mainly hyaluronic acid, chondroitin sulfate and glycoproteins. An example showing the structure of skin from the site http://www.medic.mie-u.ac.jp/derma/anatomy.html is shown below:

This shows the major layers, epidermis and dermis, cell types, and matrix components of skin.

. It has the same general nonlinear stress-strain curve as other soft tissues.

 

You will notice, however, that in comparison with other soft tissues, skin has a very long toe region. Tong and Fung characterized soft tissue mechanics using a strain energy function of the form:

where

As with any other strain energy function, to determine the 2nd Piola-Kirchoff stress as a function of the Green-Lagrange strain for skin, we differentiate the strain energy function with respect to the appropriate Green-Lagrange strain component. Thus, to determine stress components for the skin we have:

Lets look at an an example of calculating S11 using the above strain energy function. We can actually calculate the derivative using symbolic manipulation in MATLAB, which is useful since the derivative, although straight-forward requires a fair amount of bookkeeping. This is done as:

First, we define the symbols or variables in the strain energy function:

>>syms al1 al2 al3 al4 a1 a2 a3 a4 gam1 gam2 gam3 gam4 C e11 e22 e12

We then input the strain energy function as:

>> w = 0.5*(al1*e11^2+al2*e22^2+2.*al3*e12^2+2.*al4*e11*e22)+... 0.5*C*exp(a1*e11^2+a2*e22^2+2.*a3*e12^2+2.*a4*e11*e22+... gam1*e11^3+gam2*e22^3+gam3*(e11^2)*e22+gam4*e11*e22^2)

and MATLAB gives:

w = 1/2*al1*e11^2+1/2*al2*e22^2+al3*e12^2+al4*e11*e22+1/2*C*exp(a1*e11^2+a2*e22^2+2*a3 *e12^2+2*a4*e11*e22+gam1*e11^3+gam2*e22^3+gam3*e11^2*e22+gam4*e11*e22^2)

we then ask the symbolic manipulator in MATLAB to differentiate the strain energy function W with respect to E11 to obtain the 2nd Piola-Kirchoff stress S11:

>> diff(w,'e11')

MATLAB gives the following answer:

ans = al1*e11+al4*e22+1/2*(2*a1*e11+2*a4*e22+3*gam1*e11^2+2*gam3*e11*e22+gam4*e22^2)*C* exp(a1*e11^2+a2*e22^2+2*a3*e12^2+2*a4*e11*e22+gam1*e11^3+gam2*e22^3+gam3*e11^2*e2 2+gam4*e11*e22^2)

Writing this as a formula gives:

This result shows that the stress depends nonlinearly on strain.

doing the same for S22 gives:

>> diff(w,'e22')

ans = al2*e22+al4*e11+1/2*(2*a2*e22+2*a4*e11+3*gam2*e22^2+gam3*e11^2+2*gam4*e11*e22)*C* exp(a1*e11^2+a2*e22^2+2*a3*e12^2+2*a4*e11*e22+gam1*e11^3+gam2*e22^3+gam3*e11^2*e2 2+gam4*e11*e22^2)

writing in terms of a formula:

Finally, for the shear stress S12 we have:

>> diff(w,'e12')

ans = 2*al3*e12+2*a3*e12*C*exp(a1*e11^2+a2*e22^2+2*a3*e12^2+2*a4*e11*e22+gam1*e11^3+gam2 *e22^3+gam3*e11^2*e22+gam4*e11*e22^2)

Again, writing as a formula:

Brain

            Although brain tissue does not spring right to mind when thinking about tissue mechanics, the mechanical properties of brain tissue are of interest for at least two significant applications: understanding head injury and in simulating neurosurgery. In the first case, we need to know the response of brain tissue under very high loading rates. Under these circumstances, the ability to model viscoelastic effects in brain tissue loading would be necessary. For neurosurgical simulation, the loading would be expected to be much slower, approaching in the limit a quasi-static loading. Miller and Chinzei (J. Biomechanics, 11/12:1115-1121, 1997) recently presented a combined nonlinear elastic-viscoelastic constitutive of brain tissue. We will focus on the applications of the model for very slow loading, where the brain tissue can be modeled as nonlinear elastic. Miller and Chinzei used a platen loading device to test samples of brain tissue as shown below:

Due to the delicacy of the brain tissue, only one load cycle was applied for specimen.

A typical stress-strain curve for brain tissue at the slowest loading rate along with the model fit from Miller and Chinzei is shown below:


For finite deformation of brain tissue, Miller and Chinzei proposed the following strain energy function:

It is important to note, that in constrast to strain energy functions we have studied so far, this one is a function of the Left Cauchy Deformation tensor not the Right Cauchy Deformation tensor. In the case of slow speed test results that are best modeled without viscoelastic influences, Miller and Chinzei found that the experimental data could best be fit with only two constants. Using these constants, which is the case where N = 1 in the summation, the above strain energy function becomes:

Kidney

            Kidney is another tissue that is not thought of in terms of its mechanical properties. However, even though kidney is not a load bearing tissue, its mechanical properties are important in at least three instances: trauma, surgical simulation, and simulation for radiation treatment, where deformation of the kidney may affect the envelope to which radiation is delivered. Farhad et al. (1999, J. Biomech, 417-425) recently presented both experimental and theoretical models for the nonlinear mechanical behavior of pig kidneys. Farshad et al. performed extensive mechanical tests to determine the multiaxial behavior of kidney tissue including uniaxial compression, triaxial compression, uniaxial tension, and triaxial compression. They found that only 5 Newtons of load was sufficient to cause significant deformation of kidney tissue. They found that 20 N of force would be sufficient to rupture the tissue.

To model the nonlinear stress stain behavior of kidney, Farhad utilized a mechanical model known as the Blatz-Ko model. This model relates stress S to the principal stretch ratios l as:

where g and a are constants that are fit to experimental data. They found that the kidney tissue was anisotropic with different experimental constants for different testing directions. Specifically, they found:

a = 6.8 and g = .005 for a tangential direction

a = 3.9 and g = .0025 for a radial testing direction

By integrating the above expression for stress, we can derive a strain energy function for the kidney tissue:

Thus, as with other soft tissues, we can derive a strain energy function to describe the constitutive behavior of kidney tissue. For structure-function purposes, it would now be possible to relate measures of tissue structure to the experimental constants in the strain energy function.

Trabeculated Myocardium

            A classification of soft tissues for which material models have only recently been developed is that of muscle tissue. There are three types of muscle tissue: 1) skeletal or striated, 2) smooth and 3) cardiac. A major challenge in modeling muscle tissue is that in addition to passive nonlinear properties, muscle tissue can generate force, termed activation force. In addition, as we have seen in blood vessels with adaptation of the media layer, muscle tissue adapts in reponse to mechanical stimulus. The ability of cardiac muscle to adapt to mechanical stimulus is believed to play a major role in cardiac development and normal heart function depends significantly on cardiac development (Xie and Perucchio, 2001). It is believed that development of the trabeculated myocardium in the heart is modulated by stress and strain fields. To test hypotheses concerning mechanical influence on myocardium development, one must first be able to calculate stress and strain fields. This requires development of a material model for the myocardium. In addition to having nonlinear material behavior, the myocardium has a hierarchical structure (as do all biological soft tissues). An example of the trabeculated microstructure from Xie and Perucchio (2001) is shown below:

Thus, to determine the overall or effective behavior of the trabecular myocardium, Xie and Perucchio assumed a strain energy function for the myocardial microstructure, based on earlier work by Taber:

where A, B and Cf are experimentally determined constants, I1 is the first invariant of the Green Lagrange strain tensor, and eff is the Green-Lagrange strain in the direction of the muscle fiber. The first term is similar to other soft tissues and represents the passive nonlinear properties of the trabeculated myocardium microstructure muscle. The second term is new and accounts for the fact that muscle fibers can generate active stress. When modeling muscle as a nonlinear material with active stress generation potential, a common approach is write the strain energy function in two parts: 1) a part representing passive tissue properties and 2) a part representing active force generation. This general approach can be written as:

where Wp is the strain energy function written for the passive properties and Wa is the strain energy function written for the active for generation capability. It is imporant to note that the strain energy function above is written for the myocardial trabeculated microstructure. To determine the overall mechanics of the heart muscle, we must also have a material model for the overall effective heart mechanics. The effective heart mechanics will be a function of the microstructural material properties, as well as the architecture of the trabeculated myocardium. To determine effective behavior, we must also propose a material model for the effective level. Xie and Perucchio proposed the following passive and active strain energy functions:

In the active strain energy function Wa, the normal strains are used as a scaling factor to represent alignment and stiffening of muscle fibers with increasing strain. To compute the experimental constants a1 - a7, Xie and Perucchio simulated the response of the trabeculated myocardium assuming the microstructural material properties being subject to a biaxial state of strain, and the third direction of strain fixed to zero:

where eij are components of the Green-Lagrange strain tensor. The corresponds to the boundary conditions illustrated below:

An additional set of boundary conditions was then used to test the fitting of the first boundary condition representing uniaxial stretch. An example of the numerical model from Xie and Perucchio is shown below:

After running the numerical simulation, computing the average 2nd Piola-Kirchoff stress and Green-Lagrange strain, an optimization procedure was used to compute the coefficients for the proposed material model. The optimization model computes the model coefficients such that the stress computed from the material model matches that from the finite element calculation:

where the components of the 2nd Piola-Kirchoff stress tensor are computed as usual by differentiating the strain energy function with respect to the Green-Lagrange strain components:

The results showing stress strain behavior for both passive and active tissue under biaxial deformation is shown below:

The results showing the data from the numerical simulation and the optimal fit for the uniaxial case is shown below:

In this work, the finite element calculation plays the role of the mechanical test. This manuscript demonstrates the hierarchical nonlinear behavior of soft tissues and the use of optimization technique to compute the constants for the material model.

A General Proposal for a Strain Energy Function

            Due to the consistent nature of soft tissue nonlinear mechanical behavior, Fung proposed a general form of a strain energy function that could be adapted to any soft tissue. Since it is a general function, it contains many experimental constants that may be neglected depending on the tissue. This general strain energy function contains the two major features of any strain energy function we have examined so far: a measure of deformation and constants to be fit to experimental data:

where a, b, g and k are all constants to be experimentally and E is the Green-Lagrange strain tensor. This general form would a framework for consistent modeling of soft tissues and development of structure-function relationships.

 

Structure, Function and Adaptation of Blood Vessels

I.          Overview

            Although we don't often view them in this context, blood vessels are subject to mechanical stress during the pumping of blood. Thus, blood vessels must have mechanical properties that can withstand these stresses. Again, the mechanical properties of blood vessels are a function of the underlying tissue structure. Since blood vessels are soft collagenous tissues (with a good deal of elastin, another biomolecule), their stress-strain behavior resembles that of other soft collagenous tissues like ligaments and tendons. Thus, we can approximate their behavior under cyclic stress as pseudoelastic, nonlinear material, which implies hyperelasticity modeling. In addition, it seems that blood vessels like other biological tissues like to live in a homeostatic stress/strain range. Values of stress/strain outside of this range will lead to adaptation and changes in the tissue structure. In this section, we will give a brief overview of blood vessel structure, followed by an overview of modeling blood vessels as hyperelastic materials and the relationship of blood vessel properties to their structure, and finally, a description of mechanically mediated adaptation of blood vessels.

II.         Blood Vessel Structure

            In general the circulatory system of blood vessels may be broken down into those vessels that deliver oxygenated blood to tissues: the arteries, arterioles, and capillaries, and those vessels that return blood with carbon dioxide for gas exchange: the veins and venules. The basic structure of all these vessels can be broken down into three layers:

1. The Intima
2. The Media
3.
The Adventia

It is the materials that make up these layers and the size of these three layers themselves that differentiates arteries from veins and indeed even one artery from another artery or one vein from another vein. A schematic from Fung's "Mechanical Properties of Living Tissues" shown below gives an overview of the different structures in the different types of blood vessels:

                            

Although a little bit difficult on the reproduced schematic, arteries have a large media layer than veins. Since smooth muscle is generally found in the media layer, this means that arteries have more smooth muscle to contract than do veins. Arteries have a higher amount of elastin than veins. Thus, veins have a higher ratio of collagen to elastin than do arteries. In addition, veins have a thicker adventia layer in proportion to the media layer than do arteries.

Here is the composition of each layer of a blood vessel:

Intima:

            Innermost layer
            Contains endothelial cells
            Basal lamina (80 nm thick)
            Subendothelia layer with collagenous bundles, some elastin

Media:

            Middle Layer
            Contains mainly smooth muscle cells
            Collagenous fibrils (type III collagen)
            Divided from adventia by elastin layer (elastin is a protein which is very elastic, can undergo a
            stretch ratio of 1.6, about 80% strain)

Adventia:

            Outermost layer
            Collagen fibers (mainly type III, differ in amino acid sequence from I and II)
            Ground substance
            Fibroblasts

An example of the percentage of all the components is given below in a table from Fung:

          

III.        Blood Vessel Mechanical Characterization and Structure-Function

            Once we know something about tissue structure, the next natural question is: How does this structure contribute to mechanical function? If we view the mechanical behavior of blood vessels using the typical non-linear soft tissue stress strain curve, we can make qualitative statements about how tissue constituents affect mechanical behavior. Roach and Burton in 1957 digested collagen out of blood vessels using trypsin, and digested elastin from blood vessels using formic acid. They found that collagen contributed mainly to the linear region of the nonlinear stress-strain curve while elastin contributed mainly to the toe part of the stress-strain curve. We can see this in stress strain curve from a human vena cava below:

                      

Another critical aspect of blood vessel behavior is residual stress. This means that even in the unloaded state, there is still stress in the artery. This state of residual stress is dependent on the thickness and the composition of the artery. In fact, as arteries are remodeled in response to mechanical stress, the amount of residual stress changes, as we will see in the section on mechanically mediated vessel adaptation. A mark of the amount of residual stress is how much the blood vessel will open when cut. Since the blood vessel is under stress, when we cut the vessel, the stress holding the vessel together is removed and the blood vessel springs open. A figure from Fung below shows that different amounts of residual stress are present in different arteries:

                      

             If we desire a more quantitative description of blood vessel mechanics than toe versus linear region, than we can model the blood vessel as a pseudoelastic material using hyperelastic strain energy functions. In that case, the blood vessel is often described as a cylinder, with stress and strain represented using cylindrical coordinates. We use the 2nd Piola-Kirchoff stress tensor and Green-Lagrange strain tensor to represent the stress and strain in the blood vessel, respectively. These are denoted below:

                         

The tests used to impose these stress/strain states are torsion (), internal pressue (), and longitudinal stretch ().

An example of a test set-up to test blood vessels from Fung's laboratory is shown below:

                             

The test set-up allows for torsional, tensile and pressure testing. The blood vessel itself must be kept in a saline bath during testing.

Of course, when performing these tests we need to have a constitutive model in mind to describe the tissue mechanical behavior. For a hyperelastic model, we need to use a strain energy function. For blood vessel mechanics, there are two types of strain energy functions often used. The first form often used is the polynomial form, given below in terms of cylindrical Green-Lagrange strain components:

      

where A1 through A7 are material constants and the strains are the same as those described above.  The second form uses an exponential function:

            

The above forms neglect shear stress, assuming a very thin vessel. To calculate the stress components, we differentiate the strain energy function with respect to the strain components:

             

As can be expected from differences in tissue structures, there are differences in the constants for the strain energy functions for different arteries.

Plotting Nonlinear Stress Strain Curves using MATLAB:

To gain some insight into how the coefficients in the strain energy function affect the shape of the stress strain curve, we will use MATLAB to plot the stress strain curve for the Carotid and Aorta arteries modeled using a polynomial strain energy function. The strain energy function is shown below:

            

To obtain the 2nd Piola-Kirchoff stress component Sqq, we differentiate the strain energy function with respect to Eqq (we can also get Szz by differentiating W with respect to Ezz):

            

To plot Sqq in MATLAB, we write a program in a .m file called ves1plot, as shown below:

eww = (0:99)/99;
ezz = .1*ones(1,100);
for i = 1:100
sww(i) = 2.93*(2.5*eww(i)+.176*ezz(i))*exp(2.5*eww(i)*eww(i)+...
.46*ezz(i)*ezz(i)+2.*.176*eww(i)*ezz(i));
end
plot(eww,sww)
xlabel('strain Eww, Ezz = .1')
ylabel('stress S11 (KPa)')
hold on
for i = 1:100
sww(i) = 3.39*(2.8*eww(i)+.58*ezz(i))*exp(2.8*eww(i)*eww(i)+... .52*ezz(i)*ezz(i)+2.*.58*eww(i)*ezz(i));
end
plot(eww,sww)

where eww and sww are eqq and sqq respectively. We first create an array of strains with eww = (0:99)/99. We fix the ezz strain at 10%. This illustrates a very important aspect of nonlinear stress strain relationships. The amount of strain in one direction can influence uniaxial strain in the other direction. Let us use the following constants in the above stress relation for the plot:

Artery                 C (KPa)                   a1                     a2                      a4

Carotid                2.9                          2.5                     .46                    .176
Upper Aorta        3.38                        2.8                     .52                    .58

We we run the above code, we obtain the following plots, where the upper curve is the aorta and the lower curve is the carotid artery:

                                 

To see the senstivity of stress derived from the strain energy function to the parameters in the strain energy function, we first vary the constant C, changing from 2.9 to 3.9. We get the plot shown below:

                                     

We see that C increasing C slightly shifts the curve to become stiffer, along almost the whole graft. If instead we increase a1 from 2.5 to 4.5, we get the following graph:

                                          

Here we see a dramatic stiffening of the material, especially in the linear zone. Although quantitative statistical results are not reported in the text, you can see that relating specific tissue attributes like the amount of collagen vs. elastin to constants in the strain energy function like C, a1, etc. are one way to characterize structure function in soft collagenous tissues, as long as we use consistent, the same strain energy functions.

In addition to derving strain energy functions for the whole blood vessel, Fung performed bending experiments on arteries and used composite beam theory to back out some constants for each layer. He found significant differences in the linear portion of the stress strain curve for the intima-media layer vs. the adventia layer. In the thoracic arteries of pigs, he found a modulus of 43.25 KPa for the intima-media layer but a modulus of only 4.7 KPa for the adventia layer. These results indicate that the difference in structure between the layers affects the mechanical properties..

III.        Mechanically and Disease mediated Blood Vessel Adaptation

            There primary ways that blood vessel tissue structure changes in through aging, disease, and change in mechanical load. Sometimes it is a combination of all three factors. For example, hypertension or high blood pressure is a disease that raises the mechanical load on the blood vessel. Due to higher stresses, the structure of the blood vessel is altered. One example of a disease that alters blood vessel structure and consequently mechanical properties is diabetes. An example of changes in mechanical properties due to diabetes is seen in rats after a single injection of stretozocin. Fung presents the changes in material properties based on the strain energy function shown below:

                        

where a1, a2, a4, and C are material constants, and E11 and E22 are components of the Green-Lagrange strain tensor. Again, we obtain the second Piola-Kirchoff stress tensor if we differentiate the strain energy function with respect to the strain:

                        

For the above strain energy function, we obtain the stress component S11 for example as:

                        

were we see that stress is definitely a nonlinear function of strain with the higher order terms and the exponential. In the rats we diabetes, Fung and colleagues measured the material constants for the above strain energy function of the thoracic aorta artery in normal rats and those 20 days after the onset of diabetes. Although he did not report changes in tissue structure in the text, he noted profound changes in the nonlinear stress strain curve and the material constants in the strain energy function for the diabetic rats, with their aorta becoming stiffer, as shown below:

                          

You will also note that the constants in the strain energy function change significantly. This indicates that we can use the material constants in proposed strain energy functions to quantify changes in blood vessel function due to changes in structure. Thus, the strain energy function becomes a conduit to quantify structure-function of soft collagenous tissues just as the anisotropic Hooke's law is a way to quantify bone structure function relationships.

            As we mentioned, increase in vessel mechanical stress due to increased blood pressure can cause changes in tissue structure and mechanical properties. Fung and Liu performed an experiment where they puts rats in a low oxygen chamber, similar to changes due to elevation. Nitrogen was added so that the total pressure was the same as that at sea level. They found that the systolic blood pressure increased from 2.0 KPa to 2.93 KPa within minutes after the rat was in the chamber. After one month, the pressure rose to 4.0 KPa. Histologically, they note significant changes in tissue structure of the pulmonary artery even after a few days. Even after a few hours, there are histological staining changes that indicate a change in the total amount of elastin in the vessel. After 12 hours, there is a significant thickening of the media layer in the pulmonary artery. After 96 hours the adventia has also experienced a significant increase in thickness. The histological changes that Fung saw are shown below:

                            

In terms of mechanical properties, Fung reported the change in opening angle of the artery, a measure of the change in residual stress. They note that early after exposure to higher pressure, the residual stress in the artery was greater than that of the controls. However, after prolonger exposure, the residual stress, as measured by the opening angle decreased, indicating that the adaptation changes had reduced the residual stress.

 

Bone Structure

I.          Overview

            We start our section on tissue structure function and mechanically mediated tissue adaptation with bone tissue. This is for two reasons: 1) from a mechanical standpoint, bone is historically the most studied tissue, and 2) due to 1) and the simpler behavior of bone compared to soft tissues, more is known about bone mechanics in relation to its structure. Bone is also a good starting point because it illustrates the principle of hierarchical structure function that is common to all biological tissues. In this section, we illustrate the anatomy and structure of bone tissue as the basis for studying tissue structure function and mechanically mediated tissue adaptation. We first begin by describing the hierarchical levels of bone structure (anatomy) and then describe how these levels are constructed by bone cells removing and adding matrix (physiology).

II.         Cortical Bone versus Trabecular Bone Structure

            Bone in human and other mammal bodies is generally classified into two types 1: Cortical bone, also known as compact bone and 2) Trabecular bone, also known as cancellous or spongy bone. These two types are classified as on the basis of porosity and the unit microstructure. Cortical bone is much denser with a porosity ranging between 5% and 10%.  Cortical bone is found primary is found in the shaft of long bones and forms the outer shell around cancellous bone at the end of joints and the vertebrae. A schematic showing a cortical shell around a generic long bone joint is shown below:                             

                       

 

The basic first level structure of cortical bone are osteons. Trabecular bone is much more porous with porosity ranging anywhere from 50% to 90%.   It is found in the end of long bones (see picture above), in vertebrae and in flat bones like the pelvis. Its basic first level structure is the trabeculae.

III.       Hierarchical Structure of Cortical Bone

            As with all biological tissues, cortical bone has a hierarchical structure. This means that cortical bone contains many different structures that exist on many levels of scale. The hierarchical organization of cortical bone is defined in the table below:

Cortical Bone Structural Organization 

Level Cortical Structure Size Range h                          ____________________________________________________

0 Solid Material              > 3000 mm

____________________________________________________

1 Secondary Osteons (A)  100 to 300 mm         < 0.1
             Primary Osteons
(B)
                                                Plexiform (C)
                                                Interstitial Bone

____________________________________________________

2 Lamellae (A,B*,C*) 3 to 20 mm                 < 0.1
      Lacunae (A,B,C,D)
  Cement Lines (A)

_____________________________________________________

3 Collagen- 0.06 to 0.6 mm         <0.1                                                 Mineral
   Composite  (A,B,C,D)

            A - denotes structures found in secondary cortical bone
                             B - denotes structures found in primary lamellar cortical bone
C - denotes structures found in plexiform bone
                               D - denotes structures found in woven bone
                       * - indicates that structures are present in b and c, but much less than in a

                Table 1.      Cortical bone structural organization along with approximate physical scales.
                                   The parameter
h is a ratio between the level i and the next most macroscopic level i - 1.
                                   This parameter is used in RVE analysis.

 

There are two reasons for numbering different levels of microstructural organization. First, it provides a consistent way to compare different tissues. Second, it provides a consistent scheme for defining analysis levels for computational analysis of tissue micromechanics. This numbering scheme will later be used to define analysis levels for RVE based analysis of cortical bone microstructure. The 1st and 2nd organization levels reflect the fact that different types of cortical bone exist for both different species and different ages of different species. Note that at the most basic or third level, all bone, to our current understanding, is composed of a type I collagen fiber-mineral composite. Conversely, all bone tissue for the purpose of classic continuum analyses is considered to be a solid material with effective stiffness at the 0th structure. In other words, a finite element analysis at the whole bone level would consider all cortical bone to be a solid material.

Different types of cortical bone can first be differentiated at the first level structure. However, different types of first level structures may still contain common second level entities such as lacunae and lamellae. We next describe the different types of 1st level structure based on the text by Martin and Burr (1989). As you will see, the different structural organizations at this level are usually associated with either a specific age, species, or both.

III.1 First Level Cortical Bone Structure

As discussed by Martin and Burr (1989), there are four types of different organizations at what we have described as the 1st structural level. These four types of structure are called woven bone, primary bone, plexiform bone, and secondary bone. A general view of cortical bone structure showing some of the 1st and 2nd level structures is shown below:

III.1.1 Woven-fibered cortical bone

Woven cortical bone is better defined at the 1st structural level by what it lacks rather than by what it contains. For instance, woven bone does not contain osteons as does primary and secondary bone, nor does it contain the brick-like structure of plexiform bone (Fig. 1). Woven bone is thus the most disorganized of bone tissue owing to the circumstances in which it is formed. Woven bone tissue is the only type of bone tissue which can be formed de novo, in other words it does not need to form on existing bone or cartilage tissue. Woven bone tissue is often found in very young growing skeletons under the age of 5. It is only found in the adult skeleton in cases of trauma or disease, most frequently occurring around bone fracture sites. Woven bone is essentially an SOS response by the body to place a mechanically stiff structure within a needy area in a short period of time. As such, woven bone is laid down very rapidly which explains its disorganized structure. It generally contains more osteocytes (bone cells) than other types of bone tissue. Woven bone is believed to be less dense because of the loose and disorganized packing of the type I collagen fibers (Martin and Burr, 1989). It can become highly mineralized however, which may make it somewhat more brittle than other cortical bone tissue with different level one organization. Very little is known, however, about the mechanical properties of woven bone tissue. Christel et al., (1981) suggested that woven bone is less stiff than other types of bone tissue based on the premise that fracture callus is composed mainly of woven bone and is much less stiff than normal bone tissue. Direct measurements of woven bone tissue stiffness have not been made.

III.1.2 Plexiform Cortical Bone Tissue

Like woven bone, plexiform bone is formed more rapidly than primary or secondary lamellar bone tissue. However, unlike woven bone, plexiform bone must offer increased mechanical support for longer periods of time. Because of this, plexiform bone is primarily found in large rapidly growing animals such as cows or sheep. Plexiform bone is rarely seen in humans. Plexiform bone obtained its name from the vascular plexuses contained within lamellar bone sandwiched by nonlamellar bone (Martin and Burr, 1989). In the figure below from Martin and Burr lamellar bone is shown on the top while woven bone is shown on the bottom:

Plexiform bone arises from mineral buds which grow first perpendicular and then parallel to the outer bone surface. This growing pattern produces the brick like structure characteristic of plexiform bone. Each "brick" in plexiform bone is about 125 microns (mm) across (Martin and Burr, 1989). Plexiform bone, like primary and secondary bone, must be formed on existing bone or cartilage surfaces and cannot be formed de novo like woven bone. Because of its organization, plexiform bone offers much more surface area compared to primary or secondary bone upon which bone can be formed. This increases the amount of bone which can be formed in a given time frame and provided a way to more rapidly increase bone stiffness and strength in a short period of time. While plexiform may have greater stiffness than primary or secondary cortical bone, it may lack the crack arresting properties which would make it more suitable for more active species like canines (dogs) and humans.

III.1.3 Primary Osteonal Cortical Bone Tissue

When bone tissue contains blood vessels surrounded by concentric rings of bone tissue it is called osteonal bone. The structure including the central blood vessel and surrounding concentric bone tissue is called an osteon. What differentiates primary from secondary osteonal cortical bone is the way in which the osteon is formed and the resulting differences in the 2nd level structure. Primary osteons are likely formed by mineralization of cartilage, thus being formed where bone was not present. As such, they do not contain as many lamellae as secondary osteons. Also, the vascular channels within primary osteons tend to be smaller than secondary osteons. For this reason, Martin and Burr (1989) hypothesized that primary osteonal cortical bone may be mechanically stronger than secondary osteonal cortical bone.

III.1.4 Secondary Osteonal Cortical Bone Tissue

Secondary osteons differ from primary osteons in that secondary osteons are formed by replacement of existing bone. Secondary bone results from a process known as remodeling. In remodeling, bone cells known as osteoclasts first resorb or eat away a section of bone in a tunnel called a cutting cone. Following the osteoclasts are bone cells known as osteoblasts which then form bone to fill up the tunnel. The osteoblasts fill up the tunnel in staggered amounts creating lamellae which exist at the 2nd level of structure. The osteoblasts do not completely fill the cutting cone but leave a center portion open. This central portion is called a haversian canal (see cortical bone schematic). The total diameter of a secondary osteon ranges from 200 to 300 microns (denoted as mm; equal to 0.2 to 0.3 millimeters). In addition to osteons, secondary cortical bone tissue also contains interstitial bone, as shown in the cortical bone schematic. A histologic view of compact bone (from http://www.vms.hr/vms/atl/a_hist/ah062.htm) is seen below:

                                               

Some other pictures of compact bone histology may be viewed at http://www.grad.ttuhsc.edu/courses/histo/cartbone/intro.html, such as the histology section of cortical bone shown below:

                                    

showing another cross section of osteonal cortical bone, and the following longitudinal section shown below:

                                      

Notice the haversian canals (large dark circles) and the rings of lamellae that surround them to form an osteon. The smaller dark circles are lacunar spaces within the bone.

. The haversian canal in the center of the osteon has a diameter ranging between 50 to 90 mm. Within the haversian canal is a blood vessel typically 15 mm in diameter (Martin and Burr, 1989). Since nutrients which are necessary to keep cells and tissues alive can diffuse a limited distance through mineralized tissue, these blood vessels are necessary for bringing nutrients within a reasonable distance (about 150 mm) of osteocytes or bone cells which exist interior to the bone tissue. In addition to blood vessels, haversian canals contain nerve fibers and other bone cells called bone lining cells. Bone lining cells are actually osteoblasts which have taken on a different shape following the period in which they have formed bone.

III.2 Second Level Cortical Bone Structure

The second level cortical bone structure consists of those entities which make up the osteons in primary and secondary bone and the "bricks" in plexiform bone. Woven bone is again distinguished by the fact that no discernible entities exist at the second structural level. Within osteonal (primary and secondary) and plexiform bone the four major matrix 2nd level structural entities are lamellae, osteocyte lacunae, osteocyte canaliculi, and cement lines. Lamellae are bands or layers of bone generally between 3 and 7 mm in thickness. The lamellae are arranged concentrically around the central haversian canal in osteonal bone. In plexiform bone the lamellae are sandwiched in between nonlamellar bone layers. The lamellae in osteonal bone are separated by thin interlamellar layers in which the orientation of bone mineral may be altered. Lamellae contain type I collagen fibers and mineral.

The osteocyte lacunae and canaliculi are actually holes within the bone matrix that contain bone cells called osteocytes and their processes. Osteocytes evolve from osteoblasts which become entrapped in bone matrix during the mineralization process. As such, the size of osteocyte lacunae if related to the original size of the osteoblast from which the osteocyte evolved. Osteocyte lacunae have ellipsoidal shapes. The maximum diameter of the lacunae generally ranges between about 10 to 20 mm. Within the lacunae, the osteocytes sit within extracellular fluid. Canaliculi are small tunnels which connect one lacunae to another lacunae. Canalicular processes starting at osteocytes travel through the osteocytes canaliculi to connect osteocytes. Many people believe that these interconnections provide a pathway through which osteocytes can communicate information about deformation states and thus in some way coordinate bone adaptation. A color view of 2nd level cortical bone structure is shown below (this picture was posted on the website http://medocs.ucdavis.edu/CHA/402/studyset/lab5/lab5.htm, which has a good collection of bone and cartilage histology):

One of the most intriguing 2nd level structural entities from a mechanical point of view is the cement line. Cement lines are only found in secondary bone because they are the result of a remodeling process by which osteoclasts first resorb bone followed by osteoblasts forming bone. The cement line occurs at the point bone resorption ends and bone formation begins. Cement lines are about 1 to 5 microns in thickness. Cement lines are believed to be type I collagen deficient structures. Beyond this, the nature of cement has been widely debated. Schaffler et al. (1987) found that cement lines were less mineralized than the surrounding bone tissue. Many people have suggested that cement lines may serve to arrest crack growth in bone being that they are very compliant and likely to absorb energy.

III.3 Third Level Cortical Bone Structure

The farther down the hierarchy of cortical bone structure we go, the more sketchy and less quantitative the information. This is because it becomes more difficult to measure both bone structure and mechanics at increasingly small levels. Most information about third level cortical bone structure mechanics is based on some quantitative measurements mixed with a great deal more theory.

Third level cortical bone structure may be separated into two basic types, lamellar and woven. Each type contains the basic type I collagen fiber/mineral composite. What differentiates these two structures is how the composite, primarily the collagen fibers are organized. In woven bone, the collagen fibers are randomly organized and very loosely packed. A picture of woven bone forming at a fracture from the website http://www.pathguy.com/lectures/bones.htm is shown below:

             

As noted earlier, this results from the rapid manner in which bone is laid down. Lamellar bone, which is found in plexiform, primary osteonal, and secondary osteonal bone, is laid down in a more organized fashion (as seen in the picture above) and constrasts very clearly to the woven bone above.. Although there is probably some continuum of structure between woven and lamellar bone, both bone structure is most frequently organized into these two categories. The structure of lamellar bone is still widely debated, so we will discuss here the competing theories

III.3.1 Intra and Inter-Lamellar Type I Collagen Orientation

One of the earliest theories to gain acceptance will be denoted here as the parallel collagen fiber orientation theory. This is based largely on the work of Ascenzi and Bonucci (1970, 1976). This theory suggests that collagen fibers within the same lamella are predominantly parallel to one another and have a preferred orientation within the lamellae. The orientation of collagen fibers between lamellae may change up to 90o in adjacent lamellae. Based on this, three types of osteons containing three different type of lamellar sub-structures have been defined as drawn in Martin et al. 1998:

. In the figure above, a is Type T, b is Type A, and c is Type L. Type L osteons are defined because there lamellae contain collagen fibers which are oriented perpendicular to the plane of the section, or parallel to the osteon axis. These type of osteons appear dark under polarized light. Type A osteons contain alternating fiber bundle orientations and thus give an alternating light and dark pattern under polarized light. Finally, type T osteons contain lamellae with fiber bundles that are oriented parallel to the plane of the section. With respect to the osteon axis, these bundles are oriented in a transverse spiral or circumferential hoop perpendicular to the center of the osteon.

Giraud-Guille (1988) presented the twisted and orthogonal plywood model of collagen fibril orientation within cortical bone lamellae. Giraud-Guille noted that the twisted plywood model as shown in Martin et al., 1998:

allows for parallel collagen fibrils which continuously rotate by a constant angle from plane to plane in a helical structure. Another schematic of the twisted plywood model from Martin et al. (1998) is shown below:

The orthogonal plywood model consists of collagen fibrils which are parallel in a given plane but unlike the twisted plywood fibrils do not rotate continuously from plane to plane. Instead, the fibrils can only take on one of two directions which are out of phase 90o with each other. Giraud-Guille believed that the orthogonal plywood model most closely resembles the type L and type T osteons from Ascenzi's model while the twisted plywood model would most likely explain the type A or alternating osteons from Ascenzi's model. However, instead of three distinct structures creating three different polarized light patterns there would now be only two.

Whereas both Ascenzi and colleagues and Giraud-Guille proposed models of collagen orientation assuming parallel fibers, Marotti and Muglia (1988) proposed that collagen fibrils were not parallel to each other, but instead had random orientations. The alternating dark and light patterns seen in polarized light Marotti and Muglia believed were not the product of changes in orientation but were rather the result of different packing densities of collagen fibrils. They defined dense and loose packed lamella (shown in Martin et al., 1998):

. The light bands in polarized light microscopy they attributed to the loosely packed lamellae while the dark bands could be attributed to the densely packed lamellae. Marotti and Muglia that the dense and loose packed lamellar model corresponded better with how bone was formed. They suggested that alternating collagen orientations would require that osteoblasts somehow rotate when they were laying down bone. Their model would require that osteoblasts would lay down an intertwined mesh of collagen fibers, but the density with which osteoblasts would lay down collagen fibers would change.

III.4.2 Mineral Packing within Collagen Fibrils

A very thorough review of bone structure (as thorough as possible) from the angstrom level (mineral crystal) to the micron level (lamellae) was recently presented by Weiner and Traub (1992). In that work, Wiener and Traub reviewed mineral structure, the mineral collagen composite, and how the mineral collagen composite fit into lamellae. Collagen fibers, with a typical length of 0.015 mm, or .000015 mm, and a length of 3 mm, or .003 mm, packed together form collagen fibrils. Within the packing of the collagen fibers are distinct gaps sometimes called hole zones (Fig. 14). The structure of these holes is currently the focus of some debate. In one model, the holes are completely isolated from each other. In another model, the holes are contiguous and together from a groove about 0.015 mm thick and .370 mm long. Within these holes mineral crystals form. The mineral crystals in final form are believed to be made from a carbonate apatite mineral called dahllite which may initially resemble an octacalcium crystal. The octacalcium crystal naturally forms in plates. These mineral plates are typically 0.25 by 0.5 mm in length and width and have a thickness of 0.02 to 0.03 mm. It is these plates which are packed into the type I collagen fibrils. Because of the nature of the packing, the orientation of the collagen fibrils will determine the orientation of the mineral crystals. One such model is provided by Weiner and Traub (shown in Martin et al., 1998):

              

IV. Trabecular Bone Structure

Trabecular bone is the second type of bone tissue in the body. It fills the end of long bones and also makes up the majority of vertebral bodies. As with cortical bone, we will organize trabecular bone structure according to physical scale size.

Trabecular Bone Structural Organization

Level Trabecular Structure       Size Range h

____________________________________________________________

0 Solid Material       > 3000 mm

____________________________________________________________

1 Secondary Trabeculae (A) 75 to 200 mm          < 0.1                                     Primary Trabeculae (B)
Trabecular Packets  (D)

____________________________________________________________

2 Lamellae(A,B*) 1 to 20 mm <0.1 Lacunae (A,B,C*)
Cement Lines
(A)
Canaliculi

____________________________________________________________

3 Collagen- 0.06 to 0.4 mm     <0.1                                      Mineral
                                     Composite(
A,B,C)

A - denotes structures found in secondary trabecular bone

B - denotes structures found in primary trabecular bone

C - denotes structures found in woven bone

D - trabecular packets fall in between the 1st and 2nd level scalewise

but we have classified them as first level structures.

* - indicates that structures are present in b and c, but much less than in a

Table 2. Trabecular bone structural organization along with approximate physical scales. The parameter h is a ratio between the level i and the next most macroscopic level i - 1. This parameter is used in RVE analysis.

The major difference between trabecular and cortical bone structure is found on the 1st and 2nd structural levels. It should be noted that the 3rd level of trabecular bone structure is the same (as far as we know) as cortical bone structure. The major mechanical property differences (as far as we know) between trabecular and cortical bone are the effective stiffness of the 0th and 1st structural level. Trabecular bone is more compliant than cortical bone and it is believe to distribute and dissipate the energy from articular contact loads. Trabecular bone contributes about 20% of the total skeletal mass within the body while cortical bone contributes the remaining 80%. However, trabecular bone has a much greater surface area than cortical bone. Within the skeleton, trabecular bone has a total surface area of 7.0 x 106 mm2 while cortical bone has a total surface area of 3.5 x 106 mm2. A comparison between the general features of cortical bone and trabecular bone including volume fraction and surface area is given below (Jee,1983):

Structural Feature Cortical Bone Trabecular Bone

Volume Fraction 0.90 (0.85 - 0.95) 0.20 (0.05 - 0.60)

(mm3/mm3)

Surface/Bone Volume 2.5 20

(mm2/mm3)

Total Bone Volume 1.4 x 10^6 0.35 x 10^6

(mm3)

Total Internal Surface 3.5 x 10^6 7.0 x 10^6

(mm2)

Table 3. Comparison of some structural features of cortical and trabecular bone.

IV.1 First Level Trabecular Bone Structure

One of the biggest differences between trabecular and cortical bone is noticeable at the 1st level structure. As seen in the first table, trabecular bone is much more porous than cortical bone. Trabecular bone may have bone volume fraction ranging from just over 5% to a maximum of 60%. Bone volume fraction is defined as the volume of bone tissue (including internal pores like lacunae and canaliculi) per total volume. The trabecular bone volume fraction varies between different bones, with age, and between species. The basic structural entity at the first level of trabecular bone is the trabecula. Trabecula are most often characterized as rod or plate like structures (as seen in these renderings from the website http://www.npaci.edu/envision/v15.3/keaveny.html).

                                  

Early finite element models of 1st level trabecular structure did indeed model trabeculae using plate and beam finite elements. Trabecula are in general no greater than 200 mm in thickness and about 1000 mm or 1 mm long. Unlike osteons, the basic structural unit of cortical bone, trabeculae in general do not have a central canal with a blood vessel. (Note: we are characterizing the basic or 1st level structural unit of trabecular bone as the trabecula based on the fact that it has similar size ranges as the osteon. Jee (1983) denotes the trabecular packet as the basic structural unit of trabecular bone based on the fact that it is the basic remodeling unit of trabecular bone just as the osteon is the basic remodeling unit of cortical bone). In rare circumstances it is possible to find unusually thick trabeculae containing a blood vessel and some osteon like structure with concentric lamellae.

Another structure found within the trabecula is the trabecular packet. We have chosen to define the trabecular packet as a 1st level structure because of its size. The trabecular packet is only found in secondary trabecular bone because it is the product of bone remodeling in which bone cells called osteoclasts first remove bone and bone cells called osteoblasts then deposit new bone were the old bone was removed. Trabecular bone can only be remodeled from the outer surface of trabeculae. The typical trabecular packet has a crescent shape (Jee, 1983). A typical trabecular packet is about 50 mm thick and about 1 mm long. Trabecular packets contain lamellae and are attached to adjacent bone by cement lines similar to osteons in cortical bone.

IV.2 Second Level Trabecular Bone Structure

The 2nd level structure of trabecular bone has most of the same entities as the 2nd level structure of cortical bone including lamellae, lacunae, canaliculi, and cement lines. Trabecular bone, as noted before, does not generally contain vascular channels like cortical bone. What differentiates trabecular bone from cortical bone structure is the arrangement and size of these entities. For instance, although lamellae within trabecular bone structure are of approximately the same thickness as cortical bone (about 3 mm; Kragstrup et al., 1983), the arrangement of lamellae is different. Lamellae are not arranged concentrically in trabecular bone as in cortical bone, but are rather arranged longitudinally along the trabeculae within trabecular packets (Fig. 5). Krapstrup et al. noted that the thickness of lamellae tended to increase in age for females. Cannoli et al. (1982) found a higher density and larger lacunae within metaphyseal and epiphyseal trabecular bone than in diaphyseal or metaphyseal cortical bone. They found that the lacunae were ellipsoidal in both areas. The cross-sectional area of lacunae in trabecular bone ranged between 50.6 and 53.8 mm2 while the cross-sectional area of lacunae in cortical bone ranged between 35 and 26 mm2. Thus, the lamellar pattern as well as the lacunae size differ between trabecular and cortical bone.

IV.3 Third Level Trabecular Bone Structure

The third level of trabecular bone structure consists of the same entities as the third level of cortical bone structure, namely the collagen fibril-mineral composite. As no detailed studies have been perfomed on trabecular bone at this level, it is presumed for now that the structure at this level, i.e collagen fibril organization within lamellae and collagen-mineral structure, is the same as for cortical bone.

References:

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