What do ligaments bind

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. 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 S qq , we differentiate the strain energy function with respect to E qq we can also get Szz by differentiating W with respect to Ezz :.

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. 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.

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. 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. 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. In the thoracic arteries of pigs, he found a modulus of These results indicate that the difference in structure between the layers affects the mechanical properties.. 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:. Again, we obtain the second Piola-Kirchoff stress tensor if we differentiate the strain energy function with respect to the strain:.

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.

After one month, the pressure rose to 4. 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. 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.

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 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. 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. 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. 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 As you will see, the different structural organizations at this level are usually associated with either a specific age, species, or both.

As discussed by Martin and Burr , there are four types of different organizations at what we have described as the 1 st 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:. 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. 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, 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. Direct measurements of woven bone tissue stiffness have not been made. 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, 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 microns m m across Martin and Burr, 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.

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 2 nd 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 hypothesized that primary osteonal cortical bone may be mechanically stronger than secondary osteonal cortical bone. 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 2 nd 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 to microns denoted as m m; equal to 0. In addition to osteons , secondary cortical bone tissue also contains interstitial bone, as shown in the cortical bone schematic.

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 m m. Within the haversian canal is a blood vessel typically 15 m m in diameter Martin and Burr, 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 m m 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. 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 2 nd level structural entities are lamellae, osteocyte lacunae, osteocyte canaliculi , and cement lines. Lamellae are bands or layers of bone generally between 3 and 7 m m 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 m m. 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. One of the most intriguing 2 nd 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. 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.

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. 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. 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. 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 , 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 90 o 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.

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 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. Another schematic of the twisted plywood model from Martin et al. 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 90 o 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 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. 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. 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 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. Within the packing of the collagen fibers are distinct gaps sometimes called hole zones Fig. 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. 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. 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. 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. 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 1 st and 2 nd level scalewise. Table 2. Trabecular bone structural organization along with approximate physical scales.

The major difference between trabecular and cortical bone structure is found on the 1 st and 2 nd structural levels. It should be noted that the 3 rd 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 0 th and 1 st structural level.

Trabecular bone is more compliant than cortical bone and it is believe to distribute and dissipate the energy from articular contact loads. However, trabecular bone has a much greater surface area than cortical bone.

Within the skeleton, trabecular bone has a total surface area of 7. A comparison between the general features of cortical bone and trabecular bone including volume fraction and surface area is given below Jee , :. Volume Fraction 0. Total Bone Volume 1. Total Internal Surface 3. Table 3. Comparison of some structural features of cortical and trabecular bone. One of the biggest differences between trabecular and cortical bone is noticeable at the 1 st level structure.

As seen in the first table, trabecular bone is much more porous than cortical bone. 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. Early finite element models of 1 st level trabecular structure did indeed model trabeculae using plate and beam finite elements. Trabecula are in general no greater than m m in thickness and about m m 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 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 1 st 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 , A typical trabecular packet is about 50 m m 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. The 2 nd level structure of trabecular bone has most of the same entities as the 2 nd 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 m m; Kragstrup et al. Lamellae are not arranged concentrically in trabecular bone as in cortical bone, but are rather arranged longitudinally along the trabeculae within trabecular packets Fig.

The elastic fibers allow the ligaments to stretch to some extent. Ligaments surround joints and bind them together. They help strengthen and stabilize joints, permitting movement only in certain directions. Ligaments also connect one bone to another such as inside the knee. Merck and Co. From developing new therapies that treat and prevent disease to helping people in need, we are committed to improving health and well-being around the world.

The Manual was first published in as a service to the community. As a muscle contracts, the attached tendon pulls the bone into movement. Think of what happens to your bicep when you bend your elbow. Tendons also help absorb some of the impact muscles take as they spring into action.

Many sprains happen suddenly, either from a fall, awkward movement, or blow. Sprains commonly happen in the ankle, knee, or wrist. For example, a misstep can cause you to twist your ankle in an awkward position, snapping a ligament and causing your ankle to be unstable or wobbly.

You might hear a pop or feel a tear when the injury occurs. A wrist is often sprained when you reach out your extended hand to break a fall, only to have the wrist hyperextend back. That hyperextension overstretches the ligament.

Symptoms of a sprained ligament generally include pain, swelling, and bruising in the affected area. The joint may feel loose or weak and may not be able to bear weight. The intensity of your symptoms will vary depending on whether the ligament is overextended or actually torn. Doctors classify sprains by grades, from grade 1 a mild sprain with slight stretching of the ligament to grade 3 a complete tear of the ligament that makes the joint unstable.

Common areas affected by strains are the leg, foot, and back. Strains are often the result of habitual movements and athletics. Athletes who overtrain their bodies without adequate time for rest and muscle repair in between workout sessions are at increased risk.

Much like a sprain, symptoms include pain and swelling. You may also experience muscle cramping and weakness. Tendonitis, another tendon injury, is an inflammation of the tendon. This can occur as a result of the natural aging process. Like other parts of the body, tendons weaken as we age, becoming more prone to stress and injury. Tendonitis can also occur from overuse of a tendon. Golfers and baseball pitchers, for instance, often experience tendonitis in their shoulders.

Symptoms of tendonitis include pain when the muscle is moved and swelling. The affected muscle may feel warm to the touch. Telling the difference between a ligament or tendon injury on your own can be hard.