The cross-links between the fibrils are more complex. And this cross-link structure of collagen fibrils provides the strength of the tissue and thus ensures that it performs the task of the tissue under mechanical loads. In the newly formed collagen, these cross bonds are less in number, soluble in salt or acid solution, and can easily break with heat. As collagen matures, the number of cross bonds that can dissolve and break down decreases and decreases to the minimum level.
As a result, organized collagen molecules form microfibril, sub-fibrils, and fibrils. The fibrils are also clustered to form collagen fibers, collagen clusters or fascicles, and the tendon. Tenocytes are arranged between these fascicles and aligned in the direction of the mechanical load [ 10 ]. In the cellular structures of tendons, as mentioned above, there is much less amount of elastin than collagen, because the mechanical properties of the tendons depend not only on the architecture and properties of collagen fibers but also on the extent to which this structure contains elastin.
Because the bond has a special function and the nerve roots of the spine, mechanical stresses, stresses, etc. Blood circulation in tendons is very important, because the current circulation of blood directly affects metabolic activity especially during healing. Therefore, they have a white color when compared to the muscles with a much higher blood vessel density.
However, there are a few factors such as the anatomical location, structure, previously damaged condition, and physical activity level of tendons that contribute to blood supply besides the small amount of vascular structure. There are studies that show that blood flow increases in tendons in the case of increasing physical activity in the literature. There are more vascular tendons due to their anatomical position or shape and function.
The flushing of tendons is primarily derived from the synovium at the point of attachment to the bone or paratenon. However, some tendons feed on the tendon like the Achilles tendon and the paratenon structure, and some tendons are fed by a true synovial sheath they are surrounded. Bone and tendon adhesion is a layer of cartilage where blood flow cannot pass directly from the bone-tendon compound. Instead, they make anastomosis with the veins on the periosteum and make indirect connections [ 16 ].
In contrast, tendons have a very rich neural network and are often innervated from the muscles in which they are associated or from the local cuticle nerves. However, experimental studies on humans and animals have shown that tendons have different characteristics of nerve endings and mechanoreceptors. They play an important role especially for proprioception position perception and nociception pain perception in joints. In fact, studies have shown that there is internal growth in the nervous and vascular systems during the healing of tendon, which causes chronic pain.
Internal growth of the vein is an indicator of the tendon trying to heal, but because of this growth, nerves may feel pain in areas without pain before.
This means that the nerves play an important role not only in the proprioception but also in the nociception. Nerve endings are located below the muscle-tendon junction and typically in the bone-tendon junction in the form of Golgi organs, Pacini bodies, and Ruffini endings.
Of these, the Golgi organs are only mechanically stimulated by pressure and compression, so that they receive information from the power produced by the muscle. Pacinian bodies are rapidly adaptive mechanoreceptors due to nerve endings with a highly sensitive capsular end to deformation, thus dynamically responding to deformation, but are insensitive to constant or stable changes. Ruffin termination results from multiple, thin capsule-tipped, and single axons and has slowly adapting mechanoreceptors and thus continues to receive information until a constant warning level is stimulated during deformation [ 17 ].
The tendons are surrounded by loose, porous connective tissue, which is called paratenon. A complex structure, paratenon, protects the tendon and allows shifting tendon cover format. Tendon sheaths consist of two continuous layers: parietal on the outside and visceral on the inside. The visceral layer is surrounded by synovial cells and produces synovial fluid. In some tendons, the tendon sheath extends along the tendon, while in others it is found only in the binding parts of the bone.
The parietal synovial layer is found only under the paratenon in the body regions where tendons are exposed to high friction. This is called the epitenon and surrounds the fascicles.
In regions where friction is less, tendon is surrounded by paratenon only. At the tendon-bone junction, the collagen fibers of endotenon continue into the bone and become a peritendon. The regions of the tendon bonding to the bone consist of a dense connective tissue, which is able to adhere to the hard bone from the dense connective tissue and is resistant to movement and damage.
Although they occupy a small area in size, the areas of adhesion to the bone have a complex structure that is much different from that of the tendon itself. We next detail these mechanical behaviors and discuss the how the tissue structure contributes to this behavior. 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:. 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 :. 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:. Surface 7. 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:.
If we differentiate with respect to the strain we obtain the stress as:. 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. 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:. 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:.
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:. Solid matrix may be linearly elastic or hyperelastic with isotropic or anisotropic behavior. 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.
Frictional drag of the solid vs. This is diffusive drag. 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. 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:. 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 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. The dermis also contains a ground substance, containing mainly hyaluronic acid, chondroitin sulfate and glycoproteins.
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:.
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:. 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. 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:. 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. 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. Biomech , recently presented both experimental and theoretical models for the nonlinear mechanical behavior of pig kidneys. Farshad et al. 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:.
They found that the kidney tissue was anisotropic with different experimental constants for different testing directions. Specifically, they found:. 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. 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 , 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 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:. 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:.
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:.
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:.
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.
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:.
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. 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 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.
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:. 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. Innermost layer Contains endothelial cells Basal lamina 80 nm thick Subendothelia layer with collagenous bundles, some elastin. 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.
An example of the percentage of all the components is given below in a table from Fung :. 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 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:. 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:.
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:. In addition, tendons can store and release energy during movement contributing to locomotor economy.
Our musculoskeletal system contains several Our musculoskeletal system contains several muscle groups in which different muscles share a tendon. One of the most well-known are the quadriceps muscles, sharing the patellar tendon and the triceps surae muscles, sharing the Achilles tendon.
The structure and mechanical properties of these tendons have been studied extensively, considering them as one unit. However, these shared tendons consist of multiple subtendons with a non-collagenous matrix in between them. In addition, tendon is flexible so that it can bend at joints, as well as acting as a damping tissue to absorb shock and limit potential damage to muscle 1. Tendon also shows a degree of extensibility.
0コメント