**Biomechanics and Modeling of Skeletal Soft Tissues**

Rami K. Korhonen1 and Simo Saarakkala2,3 *1Department of Applied Physics, University of Eastern Finland, Kuopio 2Department of Medical Technology, University of Oulu, Oulu 3Department of Diagnostic Radiology University of Oulu and Oulu University Hospital, Oulu Finland* 

### **1. Introduction**

112 Theoretical Biomechanics

Wangerin, M., Schmitt, S., Stapelfeldt, B. & Gollhofer, A. (2007). Inverse Dynamics in

Weatherill N.P., Soni, B.K. & Thompson, J.F. (1999). *Handbook of grid generation*, CRC press,

Wojtyra, M. (2003). Multibody Simulation Model of Human Walking, *Mechanics Based Design* 

Zatsiorsky, V. M. (2002). *Kinetics of Human Motion* . Human Kinetics, USA, 978-0-7360-3778-5

Springer-Verlag: Berlin and New York, Remagen, September 2007

*of Structures and Machines*, 31(2), pp. 357-379,

978-0-8493-2687-5

Cycling Performance, Proceedings in Physics: Advances in Medical Engineering,

### **1.1 Articular cartilage**

Articular cartilage is a specialized connective tissue that covers the ends of the bones in the diarthrodial joints. The thickness of human articular cartilage is typically between 1-6 mm. The main functions of articular cartilage are to dissipate and distribute contact stresses during joint loading, and to provide almost frictionless articulation in diarthrodial joints. In order to accomplish these demanding tasks, articular cartilage has unique mechanical properties. The tissue is a biphasic material with an anisotropic and nonlinear mechanical behaviour.

Articular cartilage is composed of two distinct phases. Fluid phase of the cartilage tissue consists of interstitial water and mobile ions. The water phase constitutes 68-85 % of the cartilage total weight and is an important determinant of the biomechanical properties of the tissue. Solid phase (or solid matrix) of the cartilage tissue consists mainly of collagen fibrils and negatively charged proteoglycans. The cell density is relatively small – in human adult tissue only ~2% of the total cartilage volume is occupied by the chondrocytes. Collagen molecules constitute 60-80% of the cartilage dry weight or approximately 10-20% of the wet weight. The collagen molecules assemble to form small fibrils and larger fibers that vary in organization and dimensions as a function of cartilage depth. The diameter of collagen fibers is approximately 20 nm in the superficial zone and 70-120 nm in the deep zone, and it varies between different collagen types. The collagen fibrils of the cartilage tissue consist mainly of type II collagen, although small amounts of other collagen types can be also found in cartilage, e.g. collagen type VI is common form in the vicinity of cells (pericellular matrix). In addition to the collagen fibrils, proteoglycan macromolecules constitute 20-40% of the cartilage dry weight or approximately 5-10% of the wet weight. The proteoglycan aggrecan is composed of a protein core and numerous glycosaminoglycan (GAG) chains attached to the core. Many aggrecan molecules are further bound to a single hyaluronan chain to form a proteoglycan aggregate.

The basic structure of the articular cartilage can be divided into four zones based on the arrangement of collagen fibril network (Benninghoff, 1925): 1) *Superficial zone*: here the

Biomechanics and Modeling of Skeletal Soft Tissues 115

more regular and circular-shaped at the lateral side. For a more comprehensive and graphical description of structure and organization of collagen fibril network in the different layers of meniscus, the reader is recommended to consult the study of human meniscus

Similarly as in cartilage, the collagen fibrils are mainly responsible for the tensile properties of meniscus and proteoglycans contribute strongly to the equilibrium response. Fluid has a significant role in carrying impact and dynamic loads. For more information of the general anatomical and functional properties of the meniscus, the reader may consult *e.g.* the review

Ligaments and tendons are soft tissues connecting bones to bones or bones to muscles, respectively. Their primary functions are to stabilize joints and transmit the loads, hold the joints together, guide the trajectory of bones, and control the joint motion area. Ligaments and tendons are also biphasic tissues having fluid and solid phases similarly as in articular cartilage and meniscus. Therefore, they also possess highly viscoelastic mechanical

The fluid phase constitutes 60-70% of the total weight of ligaments and tendons. Solid phase consists of highly organized longitudinal collagen fibril network (over 15 % of the wet weight), elastin network, and proteoglycans. Similarly than in meniscus, the collagen fibrils of ligaments and tendons consist mainly of type I collagen. Since ligaments and tendons have so tightly packed and organized long collagen fibril network they have extremely high

For more information of the anatomical and functional properties of the ligaments and

**Fluid** 

cartilage 60-70% >100 MPa

ligament 60-70% >1000 MPa

**(wet weight) Young's modulus** 

(compression)

(compression)

(tension)

(tension)

**Proteoglycan (wet weight)** 

cartilage 10-20% (type II) 5-10% 68-85% ~0.5 MPa

Meniscus 15-25% (type I) 1-2% 60-70% ~0.1 MPa

less than in

Table 1. Main compositional parameters and elastic properties of articular cartilage,

**2. Experimental mechanical characterization of skeletal soft tissues** 

When skeletal soft tissues are mechanically tested, one can apply either force or deformation to it and then follow the other parameter. For example, constant or changing force may be

tendons, the reader may consult *e.g.* the book chapter by Woo et al. (2005).

structure by Cui&Min (2007).

by Messner&Gao (1998).

properties.

Articular

**2.1 Introduction** 

**1.3 Ligaments and tendons** 

tensile strength and nonlinear stress-strain behavior.

**Collagen (wet weight)** 

Ligament 20-30% (type I) less than in

ligament (type I)

Tendon more than in

meniscus, ligaments and tendons.

chondrocytes are flattened and aligned parallel to the cartilage surface. The collagen fibrils are relatively thin and run parallel to each other. The proteoglycan content is at its lowest and the water content is at its highest. 2) *Middle zone*: here the collagen fibrils have a larger diameter and they are oriented randomly. The cell density and water content is lower and proteoglycan content is higher than in the superficial zone. 3) *Deep zone*: here the diameter of the collagen fibrils is at its largest, and the collagen fibrils are oriented perpendicular to the articular surface. The cell density and water content are at their lowest, the proteoglycan content at its highest but the collagen content is variable. 4) *Calcified cartilage*: this thin layer is located between the deep zone and the subchondral bone and it joins the cartilage tissue to the subchondral bone. Here the chondrocytes usually express a hypertrophic phenotype.

It is nowadays widely accepted that collagen fibrils are primarily responsible for the cartilage tensile stiffness and the dynamic compressive stiffness. In contrast, proteoglycans are primarily responsible for the equilibrium properties during compression, and fluid contributes to the dynamic and time-dependent properties of the tissue. For more comprehensive description of structure-function relationships of cartilage, the reader may consult *e.g.* the book by Mow et al. (2005).

### **1.2 Meniscus**

Meniscus is a wedge-shaped fibrocartilaginous structure between femoral and tibial articular cartilage surfaces inside the knee joint capsule. The function of the meniscus is to bear and dissipate loads, provide stability to the knee joint, and protect articular cartilage from excessive loads by functioning as a shock absorber. Similarly as in articular cartilage, meniscus has complex mechanical properties in order to accomplish these tasks.

Meniscus has also biphasic composition. Such as in cartilage, fluid phase of the meniscus consists of interstitial water and mobile ions. The water phase constitutes 60-70% of the meniscus total weight and is similarly important determinant of the biomechanical properties of the tissue. Solid phase of the meniscus consists of highly organized collagen fibril network, negatively charged proteoglycans and meniscal cells (fibrochondrocytes). Collagen molecules constitute 15-25 % of the meniscus wet weight. In contrast with articular cartilage, the collagen fibrils of meniscus consist mainly of type I collagen, *i.e.* also found in skin and bone tissues, although smaller amounts of types II, III, V, and VI can be also found in meniscus (McDevitt&Webber, 1990). Furthermore, meniscus contains significantly less proteoglycan than articular cartilage, only 1-2% of the wet weight.

The basic structure of meniscus can be divided into different layers based on the arrangement of the collagen fibril network. Since the meniscus is located between femoral and tibial articular surfaces, it has two surface layers both in top and bottom. Below surface layers are intermediate layers and in the center of the meniscus is the central layer. At the femoral surface layer the collagen fibrils are relatively thick and run parallel to each other and the femoral surface. In contrast, at the tibial surface layer the collagen fibrils are oriented randomly. At inner layers, the arrangement of collagen fibrils is more variable. The central layer can be further divided into four zones in the axial plane: anterior and posterior parts of the central layer exhibit relatively parallelly organized collagen fibrils, middle part of the central layer exhibits irregular organization medially, wheras organization changes more regular and circular-shaped at the lateral side. For a more comprehensive and graphical description of structure and organization of collagen fibril network in the different layers of meniscus, the reader is recommended to consult the study of human meniscus structure by Cui&Min (2007).

Similarly as in cartilage, the collagen fibrils are mainly responsible for the tensile properties of meniscus and proteoglycans contribute strongly to the equilibrium response. Fluid has a significant role in carrying impact and dynamic loads. For more information of the general anatomical and functional properties of the meniscus, the reader may consult *e.g.* the review by Messner&Gao (1998).

### **1.3 Ligaments and tendons**

114 Theoretical Biomechanics

chondrocytes are flattened and aligned parallel to the cartilage surface. The collagen fibrils are relatively thin and run parallel to each other. The proteoglycan content is at its lowest and the water content is at its highest. 2) *Middle zone*: here the collagen fibrils have a larger diameter and they are oriented randomly. The cell density and water content is lower and proteoglycan content is higher than in the superficial zone. 3) *Deep zone*: here the diameter of the collagen fibrils is at its largest, and the collagen fibrils are oriented perpendicular to the articular surface. The cell density and water content are at their lowest, the proteoglycan content at its highest but the collagen content is variable. 4) *Calcified cartilage*: this thin layer is located between the deep zone and the subchondral bone and it joins the cartilage tissue to the subchondral bone. Here the chondrocytes

It is nowadays widely accepted that collagen fibrils are primarily responsible for the cartilage tensile stiffness and the dynamic compressive stiffness. In contrast, proteoglycans are primarily responsible for the equilibrium properties during compression, and fluid contributes to the dynamic and time-dependent properties of the tissue. For more comprehensive description of structure-function relationships of cartilage, the reader may

Meniscus is a wedge-shaped fibrocartilaginous structure between femoral and tibial articular cartilage surfaces inside the knee joint capsule. The function of the meniscus is to bear and dissipate loads, provide stability to the knee joint, and protect articular cartilage from excessive loads by functioning as a shock absorber. Similarly as in articular cartilage, meniscus has complex mechanical properties in order to accomplish these

Meniscus has also biphasic composition. Such as in cartilage, fluid phase of the meniscus consists of interstitial water and mobile ions. The water phase constitutes 60-70% of the meniscus total weight and is similarly important determinant of the biomechanical properties of the tissue. Solid phase of the meniscus consists of highly organized collagen fibril network, negatively charged proteoglycans and meniscal cells (fibrochondrocytes). Collagen molecules constitute 15-25 % of the meniscus wet weight. In contrast with articular cartilage, the collagen fibrils of meniscus consist mainly of type I collagen, *i.e.* also found in skin and bone tissues, although smaller amounts of types II, III, V, and VI can be also found in meniscus (McDevitt&Webber, 1990). Furthermore, meniscus contains significantly less proteoglycan than articular cartilage, only 1-2% of the wet

The basic structure of meniscus can be divided into different layers based on the arrangement of the collagen fibril network. Since the meniscus is located between femoral and tibial articular surfaces, it has two surface layers both in top and bottom. Below surface layers are intermediate layers and in the center of the meniscus is the central layer. At the femoral surface layer the collagen fibrils are relatively thick and run parallel to each other and the femoral surface. In contrast, at the tibial surface layer the collagen fibrils are oriented randomly. At inner layers, the arrangement of collagen fibrils is more variable. The central layer can be further divided into four zones in the axial plane: anterior and posterior parts of the central layer exhibit relatively parallelly organized collagen fibrils, middle part of the central layer exhibits irregular organization medially, wheras organization changes

usually express a hypertrophic phenotype.

consult *e.g.* the book by Mow et al. (2005).

**1.2 Meniscus** 

tasks.

weight.

Ligaments and tendons are soft tissues connecting bones to bones or bones to muscles, respectively. Their primary functions are to stabilize joints and transmit the loads, hold the joints together, guide the trajectory of bones, and control the joint motion area. Ligaments and tendons are also biphasic tissues having fluid and solid phases similarly as in articular cartilage and meniscus. Therefore, they also possess highly viscoelastic mechanical properties.

The fluid phase constitutes 60-70% of the total weight of ligaments and tendons. Solid phase consists of highly organized longitudinal collagen fibril network (over 15 % of the wet weight), elastin network, and proteoglycans. Similarly than in meniscus, the collagen fibrils of ligaments and tendons consist mainly of type I collagen. Since ligaments and tendons have so tightly packed and organized long collagen fibril network they have extremely high tensile strength and nonlinear stress-strain behavior.

For more information of the anatomical and functional properties of the ligaments and tendons, the reader may consult *e.g.* the book chapter by Woo et al. (2005).


Table 1. Main compositional parameters and elastic properties of articular cartilage, meniscus, ligaments and tendons.

### **2. Experimental mechanical characterization of skeletal soft tissues**

### **2.1 Introduction**

When skeletal soft tissues are mechanically tested, one can apply either force or deformation to it and then follow the other parameter. For example, constant or changing force may be

Biomechanics and Modeling of Skeletal Soft Tissues 117

compression (Jurvelin et al., 1997; Korhonen et al., 2002a; Sweigart et al., 2004). On the other hand, Poisson's ratios in tension, shown for anisotropic materials, can be even more

It is also possible to apply load or deformation to a soft tissue sample parallel to the surface. This requires fixed contact between the tester and the surface of the sample. Then, so called

> ȟ݈௦ ݈

where ȟ݈௦ is the deformation of a surface point parallel to the surface, and ݈ is the sample

௦ܨ ܣ

where ܨ௦ is the force applied parallel to the surface, and ܣ is the cross-sectional area of

Mechanical testing geometries for soft tissues can be divided into compression, tension, bending and torsion. We will now consider only compression and tension since they are the

Compression testing is widely used especially for determination of mechanical properties of articular cartilage and meniscus. This is a relevant choice since also in vivo, e.g. during normal walking cycle, articular cartilage and meniscus experiences external compressive forces. When the tissue is mechanically tested in compression, three different measurement configurations can be used: unconfined compression, confined compression and indentation. In unconfined compression, a soft tissue sample is compressed between two smooth metallic plates to a predefined stress or strain. This geometry allows interstitial fluid flow out of the tissue only in the lateral direction (Fig. 1). In confined compression, a soft tissue sample is placed in a sealed chamber and, subsequently, compressed with a porous filter (Fig. 1). In this geometry the interstitial fluid can only flow axially through the tissue surface into the filter. In indentation geometry, a soft tissue is compressed with a cylindrical, typically plane-ended or spherical-ended indenter (Fig. 1). In this geometry, fluid flow outside the indenter-tissue contact point is possible in both the lateral and axial directions. It should be emphasized that the indentation is the only compressive geometry which is not limited into the laboratory use. Since indentation testing does not require a preparation of separate tissue samples it can be also performed in vivo. For example, stiffness of femoral articular cartilage has been measured during

Tensile testing is widely used especially for determination of mechanical properties of ligaments and tendons, while it is less used for the characterization of cartilage and meniscus properties. Again, this is a relevant choice for these tissues since they exhibit mainly tensile stresses in vivo. In tensile testing, a soft tissue sample is fixed with two ends, e.g. by using metallic clamps, and the sample is then streched to a predefined stress

(2.4)

(2.5)

ൌ ߛ

thickness (perpendicular to the surface). Similarly, shear stress (߬) is defined as follows:

߬ ൌ

the contact between the tester and the surface of the sample.

than 1 (Hewitt et al., 2001; Elliott et al., 2002).

shear strain (ߛ (is defined as follows:

**2.2 Mechanical testing geometries** 

most relevant geometries for skeletal soft tissues.

arthroscopy in vivo (Vasara et al., 2005).

or strain.

applied to a tissue and consequent change in the deformation is followed. Similarly, the change in force can be followed when constant or changing deformation is applied. Important parameter to describe the behavior of tissues under loading is strain (ߝ(, defined as follows:

$$
\epsilon = \frac{\Delta l}{l\_0} \tag{2.1}
$$

where ȟ݈ is the change in thickness/length of a tissue sample, and ݈ is the original thickness/length. The normalization with the original thickness/length ensures that the deformation is comparable between tissue samples with different thickness or length. It is important to note that, according to the definition, the strain is a unitless quantity.

Second important parameter in biomechanical testing is stress (ߪ(, which is defined as:

$$
\sigma = \frac{F}{A\_0} \tag{2.2}
$$

where ܨ is the force applied to tissue, and ܣ is the original cross-sectional area in which the force is acting. Again here, the normalization with the cross-sectional area ensures that the load is comparable between different cross-sectional areas. The unit of stress is Pa, and the definition of stress is fundamentally the same as for pressure.

When both stress (ߪ (and strain (ߝ (are defined as above, mechanical behavior/properties of different skeletal soft tissues can be compared regardless of the size and shape of the samples. If the relation between stress and strain is assumed linear, one obtains the Hooke's linear model for solids from which the stiffness (elastic modulus) of the tissue can be calculated (see section 3.2).

When the compressive or tensile stress is applied to, say, excised soft tissue sample, consequent strain occurs in the direction of the loading. However, when the strain occurs in one direction in a three-dimensional soft tissue sample, there is always corresponding strain in the perpendicular direction. For example, when a soft tissue sample is stretched in one direction it typically simultaneously compresses in perpendicular direction changing its shape. The change of shape is the third important parameter in biomechanical testing. It is quantified with the parameter called the Poisson's ratio (ߥ(, defined as follows:

$$\nu = -\frac{\varepsilon\_{lat}}{\varepsilon} \tag{2.3}$$

where ߝ is the strain in loading direction and ߝ௧ is the corresponding strain in horizontal direction. The Poisson's ratio is the intrinsic parameter of a tissue, and it is unique for different materials. For example, an isotropic elastic material, e.g. rubber, has the Poisson's ratio of 0.5 in compression which means that the volume of the material does not change during mechanical loading. Since the major component of all human soft tissues is interstitial water, mechanical loading causes water to flow out of the tissue. Finally, after the complete relaxation, i.e. in equilibrium state, no fluid flow or pressure gradients exist in a tissue and, consequently, the entire stress is carried by the solid matrix. Because of this time-dependent viscoelastic nature, all human soft tissues have typically lower Poisson's ratios in compression than elastic materials, being in the range of 0.0 - 0.4 in

applied to a tissue and consequent change in the deformation is followed. Similarly, the change in force can be followed when constant or changing deformation is applied. Important parameter to describe the behavior of tissues under loading is strain (ߝ(, defined

(2.1)

(2.2)

(2.3)

߳ ൌ ȟ݈ ݈

important to note that, according to the definition, the strain is a unitless quantity. Second important parameter in biomechanical testing is stress (ߪ(, which is defined as:

definition of stress is fundamentally the same as for pressure.

ൌ ߪ ܨ ܣ

where ܨ is the force applied to tissue, and ܣ is the original cross-sectional area in which the force is acting. Again here, the normalization with the cross-sectional area ensures that the load is comparable between different cross-sectional areas. The unit of stress is Pa, and the

When both stress (ߪ (and strain (ߝ (are defined as above, mechanical behavior/properties of different skeletal soft tissues can be compared regardless of the size and shape of the samples. If the relation between stress and strain is assumed linear, one obtains the Hooke's linear model for solids from which the stiffness (elastic modulus) of the tissue can be calculated

When the compressive or tensile stress is applied to, say, excised soft tissue sample, consequent strain occurs in the direction of the loading. However, when the strain occurs in one direction in a three-dimensional soft tissue sample, there is always corresponding strain in the perpendicular direction. For example, when a soft tissue sample is stretched in one direction it typically simultaneously compresses in perpendicular direction changing its shape. The change of shape is the third important parameter in biomechanical testing. It is quantified with the parameter called the Poisson's ratio (ߥ(,

ൌെߥ

௧ߝ ߝ

where ߝ is the strain in loading direction and ߝ௧ is the corresponding strain in horizontal direction. The Poisson's ratio is the intrinsic parameter of a tissue, and it is unique for different materials. For example, an isotropic elastic material, e.g. rubber, has the Poisson's ratio of 0.5 in compression which means that the volume of the material does not change during mechanical loading. Since the major component of all human soft tissues is interstitial water, mechanical loading causes water to flow out of the tissue. Finally, after the complete relaxation, i.e. in equilibrium state, no fluid flow or pressure gradients exist in a tissue and, consequently, the entire stress is carried by the solid matrix. Because of this time-dependent viscoelastic nature, all human soft tissues have typically lower Poisson's ratios in compression than elastic materials, being in the range of 0.0 - 0.4 in

where ȟ݈ is the change in thickness/length of a tissue sample, and ݈ is the original thickness/length. The normalization with the original thickness/length ensures that the deformation is comparable between tissue samples with different thickness or length. It is

as follows:

(see section 3.2).

defined as follows:

compression (Jurvelin et al., 1997; Korhonen et al., 2002a; Sweigart et al., 2004). On the other hand, Poisson's ratios in tension, shown for anisotropic materials, can be even more than 1 (Hewitt et al., 2001; Elliott et al., 2002).

It is also possible to apply load or deformation to a soft tissue sample parallel to the surface. This requires fixed contact between the tester and the surface of the sample. Then, so called shear strain (ߛ (is defined as follows:

$$\chi = \frac{\Delta l\_{Shear}}{l\_0} \tag{2.4}$$

where ȟ݈௦ is the deformation of a surface point parallel to the surface, and ݈ is the sample thickness (perpendicular to the surface). Similarly, shear stress (߬) is defined as follows:

$$
\pi = \frac{F\_{shear}}{A\_0} \tag{2.5}
$$

where ܨ௦ is the force applied parallel to the surface, and ܣ is the cross-sectional area of the contact between the tester and the surface of the sample.

### **2.2 Mechanical testing geometries**

Mechanical testing geometries for soft tissues can be divided into compression, tension, bending and torsion. We will now consider only compression and tension since they are the most relevant geometries for skeletal soft tissues.

Compression testing is widely used especially for determination of mechanical properties of articular cartilage and meniscus. This is a relevant choice since also in vivo, e.g. during normal walking cycle, articular cartilage and meniscus experiences external compressive forces. When the tissue is mechanically tested in compression, three different measurement configurations can be used: unconfined compression, confined compression and indentation. In unconfined compression, a soft tissue sample is compressed between two smooth metallic plates to a predefined stress or strain. This geometry allows interstitial fluid flow out of the tissue only in the lateral direction (Fig. 1). In confined compression, a soft tissue sample is placed in a sealed chamber and, subsequently, compressed with a porous filter (Fig. 1). In this geometry the interstitial fluid can only flow axially through the tissue surface into the filter. In indentation geometry, a soft tissue is compressed with a cylindrical, typically plane-ended or spherical-ended indenter (Fig. 1). In this geometry, fluid flow outside the indenter-tissue contact point is possible in both the lateral and axial directions. It should be emphasized that the indentation is the only compressive geometry which is not limited into the laboratory use. Since indentation testing does not require a preparation of separate tissue samples it can be also performed in vivo. For example, stiffness of femoral articular cartilage has been measured during arthroscopy in vivo (Vasara et al., 2005).

Tensile testing is widely used especially for determination of mechanical properties of ligaments and tendons, while it is less used for the characterization of cartilage and meniscus properties. Again, this is a relevant choice for these tissues since they exhibit mainly tensile stresses in vivo. In tensile testing, a soft tissue sample is fixed with two ends, e.g. by using metallic clamps, and the sample is then streched to a predefined stress or strain.

Biomechanics and Modeling of Skeletal Soft Tissues 119

return to the original strain although the stress would be completely removed. The yield

After the plastic region, the sudden failure of the tissue occurs and stress disappears (Fig. 3). The location of the breakdown is called the failure point, which is one typical parameter

Relaxation Creep

Deformation (µm)

Force (N)

Time (s)

Time (s)

point is one typical parameter reported for soft tissues under destructive testing.

reported for soft tissues under destructive tensile testing.

Force (N)

Deformation (µm)

Fig. 2. Stress-relaxation (left) and creep (right) testing protocols.

Time (s)

Time (s)

Fig. 3. Typical stress-strain curve for destructive tensile testing of skeletal soft tissues. Collagen fibril straightening and failure, related to different regions of the stress-strain

curve, are also schematically shown.

Fig. 1. Unconfined, confined and indentation loading geometries for testing of mechanical properties of articular cartilage.

### **2.3 Destructive and nondestructive testing protocols**

In all experimental mechanical testing geometries it is possible to conduct both destructive and non-destructive testing. In non-destructive protocol tissue is tested with small strains or loads and all the changes induced to the tissue are reversible. In contrast, destructive protocol involves larger strains or loads inducing non-reversible changes to a tissue.

Most common non-destructive testing protocols are called creep and stress-relaxation. These tests can be conducted both in compression and tension geometries. In creep test, constant compressive or tensile stress is applied to a tissue and corresponding strain is followed as a function of time (Fig. 2). In stress-relaxation test, predefined compressive or tensile strain is applied and corresponding stress is followed as a function of time (Fig. 2). All biphasic and viscoelastic soft tissues exhibit first the relaxation phase in both testing protocols, and finally when the tissue reaches its equilibrium state, no fluid flow or pressure gradients exist. Consequently, after the relaxation phase, strain (in creep test) or stress (in stress-relaxation test) stabilizes at the constant level, and then the entire load is carried by the solid matrix of a tissue. Destructive testing is typically conducted for skeletal soft tissues only in tension geometry. Then it is common to follow the tissue mechanical behaviour from the stress-strain curve. At the beginning phase of tension test of skeletal soft tissue, one can observe so called toe region (Fig. 3). In this region, the relation between stress and strain is nonlinear and the slope is increasing with increased loading. The reason for the increasing slope is the straightening of the wavy-like collagen fibrils. After the collagen fibrils are completely straightened begins the elastic region (Fig. 3). In this region, the stress and strain are linearly related and the slope of the curve is called the Young's modulus of tissue. In the elastic range, all changes of a tissue are still reversible, i.e. if the stress is removed tissue returns to the original strain. All non-destructive tests, such as creep and stress-relaxation tests mentioned above, should be conducted in this elastic region. It should be also noted that in human skeletal soft tissues the loading rate affects the slope of the elastic range, i.e. higher loading rate results to steeper slope and higher Young's modulus value.

When the stress is further increased from the elastic region, the slope of the curve changes and the plastic region begins. This is called the yield point (Fig. 3). After the yield point tissue begins to experience destructive changes, e.g. microfractures in the collagen fibril network. In the plastic region irreversible changes have occurred in a tissue and it does not

Unconfined Confined Indentation

Fig. 1. Unconfined, confined and indentation loading geometries for testing of mechanical

Confining chamber

Subchondral bone

Impermeable or permeable indenter

Permeable filter

In all experimental mechanical testing geometries it is possible to conduct both destructive and non-destructive testing. In non-destructive protocol tissue is tested with small strains or loads and all the changes induced to the tissue are reversible. In contrast, destructive

Most common non-destructive testing protocols are called creep and stress-relaxation. These tests can be conducted both in compression and tension geometries. In creep test, constant compressive or tensile stress is applied to a tissue and corresponding strain is followed as a function of time (Fig. 2). In stress-relaxation test, predefined compressive or tensile strain is applied and corresponding stress is followed as a function of time (Fig. 2). All biphasic and viscoelastic soft tissues exhibit first the relaxation phase in both testing protocols, and finally when the tissue reaches its equilibrium state, no fluid flow or pressure gradients exist. Consequently, after the relaxation phase, strain (in creep test) or stress (in stress-relaxation test) stabilizes at the constant level, and then the entire load is carried by the solid matrix of a tissue. Destructive testing is typically conducted for skeletal soft tissues only in tension geometry. Then it is common to follow the tissue mechanical behaviour from the stress-strain curve. At the beginning phase of tension test of skeletal soft tissue, one can observe so called toe region (Fig. 3). In this region, the relation between stress and strain is nonlinear and the slope is increasing with increased loading. The reason for the increasing slope is the straightening of the wavy-like collagen fibrils. After the collagen fibrils are completely straightened begins the elastic region (Fig. 3). In this region, the stress and strain are linearly related and the slope of the curve is called the Young's modulus of tissue. In the elastic range, all changes of a tissue are still reversible, i.e. if the stress is removed tissue returns to the original strain. All non-destructive tests, such as creep and stress-relaxation tests mentioned above, should be conducted in this elastic region. It should be also noted that in human skeletal soft tissues the loading rate affects the slope of the elastic range, i.e. higher

protocol involves larger strains or loads inducing non-reversible changes to a tissue.

loading rate results to steeper slope and higher Young's modulus value.

When the stress is further increased from the elastic region, the slope of the curve changes and the plastic region begins. This is called the yield point (Fig. 3). After the yield point tissue begins to experience destructive changes, e.g. microfractures in the collagen fibril network. In the plastic region irreversible changes have occurred in a tissue and it does not

properties of articular cartilage.

Tissue

**2.3 Destructive and nondestructive testing protocols** 

Impermeable metallic plate

Impermeable metallic plate return to the original strain although the stress would be completely removed. The yield point is one typical parameter reported for soft tissues under destructive testing.

After the plastic region, the sudden failure of the tissue occurs and stress disappears (Fig. 3). The location of the breakdown is called the failure point, which is one typical parameter reported for soft tissues under destructive tensile testing.

Fig. 2. Stress-relaxation (left) and creep (right) testing protocols.

Fig. 3. Typical stress-strain curve for destructive tensile testing of skeletal soft tissues. Collagen fibril straightening and failure, related to different regions of the stress-strain curve, are also schematically shown.

Biomechanics and Modeling of Skeletal Soft Tissues 121

Even though one could determine all required stiffness components for an anisotropic elastic material, the mechanical behaviour of skeletal soft tissues still cannot be described by this linear model. In general, the linear elastic model can be applied for skeletal soft tissues when strains are small and the stress-strain relationship can be assumed linear. However, many soft tissues experience large strains in vivo. Furthermore, time-dependent behaviour (due to viscoelasticity) and different mechanical responses in compression and tension, both typical to skeletal soft tissues, cannot be described with this simple model. Therefore, more sophisticated models are needed for the mechanical characterization of skeletal soft tissues.

Many biological tissues experience large deformations and then the stress-strain relationship becomes nonlinear. These materials are called hyperelastic materials. There are several hyperelastic material models developed, e.g. Neo-Hookean, Arruda-Boyce, Mooney-Rivlin, Ogden models. We will present here one of these models (Neo-Hookean model) that has

The Neo-Hookean material model uses a general strain energy potential for finite strains:

� − 3) <sup>+</sup> �

� = ������ + ������ + ������

<sup>2</sup> � �� <sup>=</sup> 3(1 − 2�)

where �� is the initial shear modulus and ν is the Poisson's ratio. For linear elastic materials,

There are three typical viscoelastic solid materials that have been applied for biological soft tissues; Maxwell, Voigt and Kelvin (Standard linear solid) (Fig. 4). In contrast to the elastic or hyperelastic materials, these models have a time-dependent component that enables the

The solid voscoelastic models are composed of elastic and viscous components. The elastic component is that shown in eq. 3.1, while the viscous component (dashpot) is velocity

> �� ��

where � is the damping coefficient, *F* is force and *x* is deformation/elongation. *F* and *x* can

dashpot experience the same force, while their deformation and velocity are different. The

ε

�=�

) and strain (

the shear modulus can be expressed with the Young's modulus (� = 2��(1+�)).

��

� �� are the deviatoric stretches, � is is the total volume ratio, and �� are the

(��� − 1)�, (3.3)

, (3.4)

��(1 + �) (3.5)

). In the Maxwell model, both the spring and

� is the first

(3.6)

� = ��(��

��

�� <sup>=</sup> ��

where *C*1 and *D*1 are material parameters, ��� is the elastic volume ratio and ��

**3.3 Hyperelastic model** 

where �̅

� = ���

**3.4 Viscoelastic models** 

also be replaced with stress (

total velocity becomes:

dependent as:

been typically applied for many biological soft tissues.

principal stretches. The material parameters are given by:

modelling of creep, stress-relaxation and hysteresis.

σ

deviatoric strain invariant defined as:

### **3. Biomechanical modeling of skeletal soft tissues**

### **3.1 Introduction**

In this section, we will present the development of computational models applied for the characterization of biomechanical properties of cartilage, meniscus, ligaments and tendons. We will start from traditional linearly elastic models that can be applied for the characterization of static or dynamic properties of tissues by a simple Hookean relation. As the linear elastic model is only applicable for small strains, we will also introduce hyperelastic models that can be applied for nonlinear problems in larger strains.

Second, we will show traditional solid viscoelastic models, i.e. Maxwell, Voigt and Kelvin models. We will show the basic equations of these models. Then, we will take fluid into account in the model and present a biphasic, poroelastic model. We will present biphasic models with isotropic and anisotropic solid matrixes, improving the prediction of experimentally found mechanical behavior of fluid-saturated soft tissues.

Finally, we will present the fibril reinforced biphasic model of cartilage. In this model, the solid matrix is divided into fibrillar and non-fibrillar parts. We will also present different forms of nonlinearities formulated especially for the collagen fibers and the swelling properties due to the fixed charge density of proteoglycans. At the end of the section, we will summarize the application of the presented constitutive models for cartilage, menisci, ligaments and tendons.

### **3.2 Linear elastic model**

The most traditional and simplest mechanical model for skeletal soft tissues is Hooke's linear elastic model for solid materials. This model assumes the linear relation between stress and strain, corresponding to a spring fixed from one end and compressed or strecthed from the other. Hooke's model can be presented as follows:

$$
\sigma = E \epsilon\_\prime \tag{3.1}
$$

where � is stress, � is strain, and *E* is the elastic (Young's) modulus: This model is easy to apply for various testing geometries and protocols, and consequently stiffness of a tested soft tissue can be expressed by the Young's modulus. However, it should be realized that this simple model is limited to one-dimensional geometry and it assumes tissue as elastic and isotropic material. Hooke's law can be generalized to three-dimensional geometry and then also the Poisson's ratio (�) is needed to describe the mechanical behaviour of the tested soft tissue (see section 3.5.1). Obviously, this is still not adequate for viscoelastic and anisotropic skeletal soft tissues.

Hooke's law can be further generalized for an anisotropic elastic material, when it can be expressed as a matrix form:

$$[\sigma] = [\mathcal{C}][\epsilon],\tag{3.2}$$

where [�] is the stress tensor, [�] is the strain tensor, and [�] is the stiffness matrix. In order to completely characterize the mechanical behaviour of anisotropic and elastic tissue, altogether 21 stiffness components are needed in [�]. For the material with mutually perpendicular planes of elastic symmetry, i.e. orthotropic material, nine elastic constants are needed in [�]. Furthermore, if one assumes the same mechanical properties in one plane (e.g. in x–y plane) and different properties in the direction normal to this plane (e.g. z-axis), five independent elastic constants are needed in [�] and the material is referred as transversely isotropic (see section 3.5.2).

Even though one could determine all required stiffness components for an anisotropic elastic material, the mechanical behaviour of skeletal soft tissues still cannot be described by this linear model. In general, the linear elastic model can be applied for skeletal soft tissues when strains are small and the stress-strain relationship can be assumed linear. However, many soft tissues experience large strains in vivo. Furthermore, time-dependent behaviour (due to viscoelasticity) and different mechanical responses in compression and tension, both typical to skeletal soft tissues, cannot be described with this simple model. Therefore, more sophisticated models are needed for the mechanical characterization of skeletal soft tissues.

### **3.3 Hyperelastic model**

120 Theoretical Biomechanics

In this section, we will present the development of computational models applied for the characterization of biomechanical properties of cartilage, meniscus, ligaments and tendons. We will start from traditional linearly elastic models that can be applied for the characterization of static or dynamic properties of tissues by a simple Hookean relation. As the linear elastic model is only applicable for small strains, we will also introduce

Second, we will show traditional solid viscoelastic models, i.e. Maxwell, Voigt and Kelvin models. We will show the basic equations of these models. Then, we will take fluid into account in the model and present a biphasic, poroelastic model. We will present biphasic models with isotropic and anisotropic solid matrixes, improving the prediction of

Finally, we will present the fibril reinforced biphasic model of cartilage. In this model, the solid matrix is divided into fibrillar and non-fibrillar parts. We will also present different forms of nonlinearities formulated especially for the collagen fibers and the swelling properties due to the fixed charge density of proteoglycans. At the end of the section, we will summarize the application of the presented constitutive models for cartilage, menisci,

The most traditional and simplest mechanical model for skeletal soft tissues is Hooke's linear elastic model for solid materials. This model assumes the linear relation between stress and strain, corresponding to a spring fixed from one end and compressed or strecthed

where � is stress, � is strain, and *E* is the elastic (Young's) modulus: This model is easy to apply for various testing geometries and protocols, and consequently stiffness of a tested soft tissue can be expressed by the Young's modulus. However, it should be realized that this simple model is limited to one-dimensional geometry and it assumes tissue as elastic and isotropic material. Hooke's law can be generalized to three-dimensional geometry and then also the Poisson's ratio (�) is needed to describe the mechanical behaviour of the tested soft tissue (see section 3.5.1). Obviously, this is still not adequate for viscoelastic and

Hooke's law can be further generalized for an anisotropic elastic material, when it can be

where [�] is the stress tensor, [�] is the strain tensor, and [�] is the stiffness matrix. In order to completely characterize the mechanical behaviour of anisotropic and elastic tissue, altogether 21 stiffness components are needed in [�]. For the material with mutually perpendicular planes of elastic symmetry, i.e. orthotropic material, nine elastic constants are needed in [�]. Furthermore, if one assumes the same mechanical properties in one plane (e.g. in x–y plane) and different properties in the direction normal to this plane (e.g. z-axis), five independent elastic constants are needed in [�] and the material is referred as

� = ��, (3.1)

[�] = [�][�], (3.2)

hyperelastic models that can be applied for nonlinear problems in larger strains.

experimentally found mechanical behavior of fluid-saturated soft tissues.

from the other. Hooke's model can be presented as follows:

**3. Biomechanical modeling of skeletal soft tissues** 

**3.1 Introduction** 

ligaments and tendons.

**3.2 Linear elastic model** 

anisotropic skeletal soft tissues.

transversely isotropic (see section 3.5.2).

expressed as a matrix form:

Many biological tissues experience large deformations and then the stress-strain relationship becomes nonlinear. These materials are called hyperelastic materials. There are several hyperelastic material models developed, e.g. Neo-Hookean, Arruda-Boyce, Mooney-Rivlin, Ogden models. We will present here one of these models (Neo-Hookean model) that has been typically applied for many biological soft tissues.

The Neo-Hookean material model uses a general strain energy potential for finite strains:

$$U = \mathcal{C}\_1(\overline{l\_1} - 3) + \frac{1}{\mathcal{D}\_1}(\mathcal{J}\_{el} - 1)^2,\tag{3.3}$$

where *C*1 and *D*1 are material parameters, ��� is the elastic volume ratio and �� � is the first deviatoric strain invariant defined as:

$$
\overline{I\_1} = \overline{\lambda\_1}^2 + \overline{\lambda\_2}^2 + \overline{\lambda\_3}^2,\tag{3.4}
$$

where �̅ � = ��� � �� are the deviatoric stretches, � is is the total volume ratio, and �� are the principal stretches. The material parameters are given by:

$$\mathcal{L}\_1 = \frac{G\_0}{2}, \quad D\_1 = \frac{3(1 - 2\nu)}{G\_0(1 + \nu)}\tag{3.5}$$

where �� is the initial shear modulus and ν is the Poisson's ratio. For linear elastic materials, the shear modulus can be expressed with the Young's modulus (� = 2��(1+�)).

### **3.4 Viscoelastic models**

There are three typical viscoelastic solid materials that have been applied for biological soft tissues; Maxwell, Voigt and Kelvin (Standard linear solid) (Fig. 4). In contrast to the elastic or hyperelastic materials, these models have a time-dependent component that enables the modelling of creep, stress-relaxation and hysteresis.

The solid voscoelastic models are composed of elastic and viscous components. The elastic component is that shown in eq. 3.1, while the viscous component (dashpot) is velocity dependent as:

$$F = \eta \frac{d\mathbf{x}}{dt} \tag{3.6}$$

where � is the damping coefficient, *F* is force and *x* is deformation/elongation. *F* and *x* can also be replaced with stress (σ) and strain (ε). In the Maxwell model, both the spring and dashpot experience the same force, while their deformation and velocity are different. The total velocity becomes:

$$\frac{d\mathcal{X}}{dt} = \frac{1}{\mu} \frac{dF}{dt} + \frac{F}{\eta} \tag{3.7}$$

$$F = \mu \mathfrak{x} + \eta \frac{d\mathfrak{x}}{dt} \tag{3.8}$$

$$F + \tau\_{\epsilon} \frac{dF}{dt} = E\_R \left( \chi + \tau\_{\sigma} \frac{d\chi}{dt} \right) \tag{3.9}$$

$$
\tau\_{\epsilon} = \frac{\eta\_1}{\mu\_1}, \tau\_{\sigma} = \frac{\eta\_1}{\mu\_0} \left( 1 + \frac{\mu\_0}{\mu\_1} \right), E\_R = \mu\_0. \tag{3.10}
$$

$$
\sigma\_s = -\phi\_s pI + \sigma\_E,\tag{3.11}
$$

$$
\sigma\_f = -\phi\_f pI,\tag{3.12}
$$

$$
\sigma\_t = \sigma\_s + \sigma\_f = -pI + \sigma\_E,\tag{3.13}
$$

$$
\sigma\_E = \mathbb{C}\epsilon,\tag{3.14}
$$

$$\nabla \cdot (\phi\_s \boldsymbol{\nu}\_s + \phi\_f \boldsymbol{\nu}\_f) = \mathbf{0},\tag{3.15}$$

$$
\nabla \cdot \sigma\_a + \pi\_a = \mathbf{0},
\tag{3.16}
$$

$$
\pi\_s = -\pi\_f = \frac{\phi\_f^2}{k} (\upsilon\_f - \upsilon\_s) \tag{3.17}
$$

$$
\nabla \cdot \boldsymbol{\sigma}\_t = \mathbf{0},\tag{3.18}
$$

$$k = \frac{\phi\_f^2}{K}.\tag{3.19}$$

$$k = \left. k\_0 \left( \frac{1+e}{1+e\_0} \right)^{\mathcal{M}} \right|, \tag{3.20}$$

$$
\phi\_t = 0.80 - 0.10\text{z},\tag{3.21}
$$

$$
\sigma\_E = \frac{\,\_E}{(1+\nu)(1-2\nu)} \begin{bmatrix}
1-\nu & \nu & \nu & 0 & 0 & 0 \\
\nu & 1-\nu & \nu & 0 & 0 & 0 \\
\nu & \nu & 1-\nu & 0 & 0 & 0 \\
0 & 0 & 0 & 1-2\nu & 0 & 0 \\
0 & 0 & 0 & 0 & 1-2\nu & 0 \\
0 & 0 & 0 & 0 & 0 & 1-2\nu
\end{bmatrix} \epsilon. \tag{3.22}
$$

$$
\sigma\_{E} = \begin{bmatrix}
\frac{1 - \nu\_{p2}\nu\_{xp}}{E\_p E\_x \Delta} & \frac{\nu\_p + \nu\_{xp}\nu\_{p2}}{E\_p E\_x \Delta} & \frac{\nu\_{xp} + \nu\_p \nu\_{xp}}{E\_p E\_x \Delta} & 0 & 0 & 0 \\
\frac{\nu\_p + \nu\_{p2}\nu\_{xp}}{E\_x E\_p \Delta} & \frac{1 - \nu\_{xp}\nu\_{p2}}{E\_x E\_p \Delta} & \frac{\nu\_{xp} + \nu\_{xp}\nu\_p}{E\_x E\_p \Delta} & 0 & 0 & 0 \\
\frac{\nu\_{p2} + \nu\_p \nu\_{p2}}{E\_p^2 \Delta} & \frac{\nu\_{p2}(1 + \nu\_p)}{E\_p^2 \Delta} & \frac{1 - \nu\_p^2}{E\_p^2 \Delta} & 0 & 0 & 0 \\
0 & 0 & 0 & 2G\_{xp} & 0 & 0 \\
0 & 0 & 0 & 0 & 2G\_{xp} & 0 \\
0 & 0 & 0 & 0 & 0 & \frac{E\_p}{1 + \nu\_p}
\end{bmatrix} \epsilon,\tag{3.23}
$$

$$
\Delta = \frac{(1+\nu\_p)(1-\nu\_p-2\nu\_{pz}\nu\_{xp})}{E\_p^2 E\_z}.\tag{3.24}
$$

$$E\_{pz}, E\_p, \nu\_{pz} = 0.5, \nu\_p = 1 - 0.5 \frac{\nu\_p}{E\_{px}}, G\_{xp}. \tag{3.25}$$

$$
\sigma\_t = \sigma\_{nf} + \sigma\_{flbrill} - pI,\tag{3.26}
$$

$$E\_f = E\_f^0 + E\_f^\xi \epsilon\_f,\text{for }\epsilon\_f > 0,\tag{3.27}$$

$$E\_f = 0, \text{for } \epsilon\_f \le 0,\tag{3.28}$$

$$\sigma\_f = -\frac{\eta}{2\sqrt{(\sigma\_f - E\_f^0 \varepsilon\_f)E\_f^\varepsilon}} \dot{\sigma}\_f + E\_f^0 \varepsilon\_f + \left(\frac{\eta E\_f^0}{2\sqrt{(\sigma\_f - E\_f^0 \varepsilon\_f)E\_f^\varepsilon}} + \eta\right) \dot{\varepsilon}\_f, \text{for } \varepsilon\_f > 0,\tag{3.29}$$

Biomechanics and Modeling of Skeletal Soft Tissues 127

implementation of swelling properties, the fixed charge density can be taken from

Other anisotropic and nonlinear representation have also been presented for biological soft tissues. Specifically the collagen fibrils and their nonlinear stress-strain tensile behavior has

where *Pf* is the first Piola-Kirchhoff fibril stress, *εf* is the total fibril strain, *εe* is the strain of the

2008). Tensile stress-stretch relationship for collagen fibrils has also been presented in the

 � �0, < 1, ������(���) − 1� 1 < � < �∗, ������ � � �∗,

In these equations, *F2* is the strain energy function for the collagen fibers, usually in conjunction with the hyperelastic model, such as Neo-Hookean (eq. 3.3), λ is fiber stretch, λ\* is the stretch where collagen fibers are straightened, and ��, ��, �� and �� are material

Articular cartilage has been modelled using almost all the above mentioned models (Mow et al., 1980; Lai et al., 1991; Li et al., 1999; Garcia et al., 2000; Guilak&Mow, 2000; Soltz&Ateshian, 2000; DiSilvestro&Suh, 2001; Korhonen et al., 2003; Laasanen et al., 2003; Wilson et al., 2004; Julkunen et al., 2007). The choice of the material model has been mainly based on the study purpose and loading protocol. Recently, however, the fibril reinforced material description has been applied by many researchers and it is probably the most realistic approach for cartilage (Li et al., 1999; Li et al., 2000; Korhonen et al., 2003; Wilson et al., 2004; Wilson et al., 2005b; Julkunen et al., 2007; Korhonen et al., 2008; Julkunen et al., 2009). It should also be noted that in articular cartilage negative fixed charges create tissue swelling pressure and is very important for the mechanical behaviour of the tissue. Thus, tissue swelling model or triphasic approaches are important phenomena. Meniscus, ligaments and tendons have only a small amount of fixed charges and swelling mechanisms

Meniscus has been typically modelled as isotropic or transversely isotropic material (Spilker et al., 1992; Meakin et al., 2003; Sweigart et al., 2004; Guess et al., 2010). Poroelastic properties have also been included in meniscus models. Typical models for ligaments and tendons have been transversely isotropic nonlinear with hyperelastic behaviour (Pena et al.,

*<sup>1</sup>* (Fig. 3c), and *E1*, *E2*, *k1* and *k2* are constants (Wilson et al., 2006; Julkunen et al.,

�� � ��(���� − 1), (3.34)

�� � ��(���� − 1), (3.35)

�� � ������(�∗��) − 1� − ���∗. (3.38)

�� � �� � ��, (3.36)

(3.37)

experimental measurements (Maroudas, 1968; Chen et al., 2001).

��

constants (Pena et al., 2006; Zhang et al., 2008).

**3.6 Models applied for skeletal soft tissues** 

have thus been neglected in the models.

been presented as follows:

spring μ

where

following form

$$
\sigma\_f := 0, \text{for } \,\, \varepsilon\_f \le 0,
$$

where � is the viscoelastic damping coefficient, and �� and �� are the stress- and strain-rates, respectively.

The fibrillar part has also been modeled with primary and secondary fibrils (Wilson et al., 2004). The primary fibrils represent the collagens detected with polarized light microscopy (Arokoski et al., 1996; Korhonen et al., 2002b), which cause a depth-dependent tensile modulus for the tissue. The fibrils are oriented vertically in the deep zone, curve in the middle zone, and reach a parallel orientation with the articular surface in the superficial zone (Benninghoff, 1925). Two parameters are needed to describe the fibril orientation: thickness of the superficial zone (*d*vec) and bending radius of the collagen fibrils in the middle zone (*r*vec). The secondary fibrils mimic the less organized collagen network which are observed in scanning electron microscopy (Kaab et al., 2003). The stresses for primary and secondary fibrils can be formulated as:

$$
\sigma\_{f,p} = \rho\_{\rm z} \mathcal{C} \sigma\_f,\tag{3.30}
$$

$$
\sigma\_{f,\mathbb{S}} = \rho\_{\mathbb{Z}} \sigma\_f,\tag{3.31}
$$

where �� represents the depth dependent fibril density, and *C* is the density ratio of primary and secondary fibrils. The stress of the fibril network is then determined as the sum of the stresses in each individual fibril (��,��� � ),

$$
\sigma\_f = \Sigma\_{l=1}^{totf} \sigma\_{f,all}^l. \tag{3.32}
$$

### **3.5.4 Other models of skeletal soft tissues**

There are also other models of biological soft tissues than those presented above. The conewise linear elastic model is able to characterize compression-tension nonlinearity of the tissues (Soltz&Ateshian, 2000). The poroviscoelastic model includes both fluid flow dependent and fluid flow independent viscoelasticities (DiSilvestro&Suh, 2001). The triphasic model includes ion flow (Lai et al., 1991) and it is equivalent to the biphasic swelling model at equilibrium (Wilson et al., 2005a). In the biphasic fibril reinforced swelling model, after inclusion of osmotic swelling and chemical expansion, the total stress becomes:

$$
\sigma\_t = \sigma\_{nf} + \sigma\_{f1brill} - \Delta\pi I - T\_c I - \mu\_f I,\tag{3.33}
$$

where �� is the osmotic pressure gradient, �� is the chemical expansion stress, and �� = � � �� is the chemical potential of fluid (Huyghe&Janssen, 1997; Wilson et al., 2005a; Wilson et al., 2005b; Korhonen et al., 2008). The osmotic pressure gradient is caused by the difference in ion concentration of the cartilage and that of the surrounding fluid (Huyghe&Janssen, 1997; Wilson et al., 2005a; Wilson et al., 2005b). It is also referred to as the Donnan swelling pressure gradient. The chemical expansion stress comes from the repulsion between negative charges in the solid matrix (Lai et al., 1991; Wilson et al., 2005a; Wilson et al., 2005b). Swelling of the tissue is resisted by the collagen network, inducing pre-stresses in the collagen fibrils. This model has been applied specifically for cartilage since its swelling properties due to the fixed charge density have a significant role for the deformation behavior of the tissue, especially under static loading. For the implementation of swelling properties, the fixed charge density can be taken from experimental measurements (Maroudas, 1968; Chen et al., 2001).

Other anisotropic and nonlinear representation have also been presented for biological soft tissues. Specifically the collagen fibrils and their nonlinear stress-strain tensile behavior has been presented as follows:

$$P\_1 = E\_1(e^{k\_1\varepsilon\_f} - 1),\tag{3.34}$$

$$P\_2 = E\_2(e^{k\_2 \varepsilon\_\theta} - 1),\tag{3.35}$$

$$P\_f = P\_1 + P\_2,\tag{3.36}$$

where *Pf* is the first Piola-Kirchhoff fibril stress, *εf* is the total fibril strain, *εe* is the strain of the spring μ*<sup>1</sup>* (Fig. 3c), and *E1*, *E2*, *k1* and *k2* are constants (Wilson et al., 2006; Julkunen et al., 2008). Tensile stress-stretch relationship for collagen fibrils has also been presented in the following form

$$
\lambda \frac{\partial F\_2}{\partial \lambda} = \begin{cases} 0, & \lambda < 1, \\ \mathcal{C}\_3 \{ e^{\mathcal{C}\_4(\lambda - 1)} - 1 \} & 1 < \lambda < \lambda^\*, \\ \mathcal{C}\_5 \lambda + \mathcal{C}\_6 & \lambda > \lambda^\*, \end{cases} \tag{3.37}
$$

where

126 Theoretical Biomechanics

�� = 0, ��� �� 0, where � is the viscoelastic damping coefficient, and �� and �� are the stress- and strain-rates,

The fibrillar part has also been modeled with primary and secondary fibrils (Wilson et al., 2004). The primary fibrils represent the collagens detected with polarized light microscopy (Arokoski et al., 1996; Korhonen et al., 2002b), which cause a depth-dependent tensile modulus for the tissue. The fibrils are oriented vertically in the deep zone, curve in the middle zone, and reach a parallel orientation with the articular surface in the superficial zone (Benninghoff, 1925). Two parameters are needed to describe the fibril orientation: thickness of the superficial zone (*d*vec) and bending radius of the collagen fibrils in the middle zone (*r*vec). The secondary fibrils mimic the less organized collagen network which are observed in scanning electron microscopy (Kaab et al., 2003). The stresses for primary

where �� represents the depth dependent fibril density, and *C* is the density ratio of primary and secondary fibrils. The stress of the fibril network is then determined as the sum of the

> �� = ∑ ��,��� ���� �

There are also other models of biological soft tissues than those presented above. The conewise linear elastic model is able to characterize compression-tension nonlinearity of the tissues (Soltz&Ateshian, 2000). The poroviscoelastic model includes both fluid flow dependent and fluid flow independent viscoelasticities (DiSilvestro&Suh, 2001). The triphasic model includes ion flow (Lai et al., 1991) and it is equivalent to the biphasic swelling model at equilibrium (Wilson et al., 2005a). In the biphasic fibril reinforced swelling model, after inclusion of osmotic swelling and chemical expansion, the total

where �� is the osmotic pressure gradient, �� is the chemical expansion stress, and �� = � � �� is the chemical potential of fluid (Huyghe&Janssen, 1997; Wilson et al., 2005a; Wilson et al., 2005b; Korhonen et al., 2008). The osmotic pressure gradient is caused by the difference in ion concentration of the cartilage and that of the surrounding fluid (Huyghe&Janssen, 1997; Wilson et al., 2005a; Wilson et al., 2005b). It is also referred to as the Donnan swelling pressure gradient. The chemical expansion stress comes from the repulsion between negative charges in the solid matrix (Lai et al., 1991; Wilson et al., 2005a; Wilson et al., 2005b). Swelling of the tissue is resisted by the collagen network, inducing pre-stresses in the collagen fibrils. This model has been applied specifically for cartilage since its swelling properties due to the fixed charge density have a significant role for the deformation behavior of the tissue, especially under static loading. For the

� ),

��,� = �����, (3.30)

��,� = ����, (3.31)

��� . (3.32)

�� = ��� � ������� � ��� � �������, (3.33)

respectively.

and secondary fibrils can be formulated as:

stresses in each individual fibril (��,���

stress becomes:

**3.5.4 Other models of skeletal soft tissues** 

$$\mathcal{L}\_6 = \mathcal{L}\_3 \{ e^{\mathcal{L}\_4(\lambda^\*-1)} - 1 \} - \mathcal{L}\_5 \lambda^\*. \tag{3.38}$$

In these equations, *F2* is the strain energy function for the collagen fibers, usually in conjunction with the hyperelastic model, such as Neo-Hookean (eq. 3.3), λ is fiber stretch, λ\* is the stretch where collagen fibers are straightened, and ��, ��, �� and �� are material constants (Pena et al., 2006; Zhang et al., 2008).

### **3.6 Models applied for skeletal soft tissues**

Articular cartilage has been modelled using almost all the above mentioned models (Mow et al., 1980; Lai et al., 1991; Li et al., 1999; Garcia et al., 2000; Guilak&Mow, 2000; Soltz&Ateshian, 2000; DiSilvestro&Suh, 2001; Korhonen et al., 2003; Laasanen et al., 2003; Wilson et al., 2004; Julkunen et al., 2007). The choice of the material model has been mainly based on the study purpose and loading protocol. Recently, however, the fibril reinforced material description has been applied by many researchers and it is probably the most realistic approach for cartilage (Li et al., 1999; Li et al., 2000; Korhonen et al., 2003; Wilson et al., 2004; Wilson et al., 2005b; Julkunen et al., 2007; Korhonen et al., 2008; Julkunen et al., 2009). It should also be noted that in articular cartilage negative fixed charges create tissue swelling pressure and is very important for the mechanical behaviour of the tissue. Thus, tissue swelling model or triphasic approaches are important phenomena. Meniscus, ligaments and tendons have only a small amount of fixed charges and swelling mechanisms have thus been neglected in the models.

Meniscus has been typically modelled as isotropic or transversely isotropic material (Spilker et al., 1992; Meakin et al., 2003; Sweigart et al., 2004; Guess et al., 2010). Poroelastic properties have also been included in meniscus models. Typical models for ligaments and tendons have been transversely isotropic nonlinear with hyperelastic behaviour (Pena et al.,

Biomechanics and Modeling of Skeletal Soft Tissues 129

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Springer, New York.

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405.

2006; Zhang et al., 2008). Also viscoelastic solid models (Thornton et al., 1997) and poroelastic models have been applied for ligaments (Atkinson et al., 1997). However, the fluid-flow dependent viscoeasticity may not be that important in ligaments and tendons because they experience mainly tensile forces under physiological loading and it has been suggested that fluid has only a minor role in contributing to soft tissue response in tension (Li et al., 2005). Furthermore, viscoelastic models with anisotropic nonlinear stress-strain behaviour have been developed to capture the strain rate dependent nonlinearity of ligaments and tendons (Pioletti et al., 1998; Limbert&Middleton, 2006).

### **3.7 Optimization of material parameters**

The optimization of material parameters of the model can be done by typically minimizing the mean squared error (MSE), root mean squared error (RMSE) or mean absolute error (MAE) between the simulated and experimental force curves (Fig. 5). This can be done for instance using a multidimensional unconstrained nonlinear minimization routine (fminsearch) available in Matlab (Mathworks Inc., Natick, MA, USA). The optimization should be first tested with different initial values of the material parameters, and the optimized parameter values should be always the same, independent on the initial guess. Then one of the equations for MSE, RMSE and MAE,

$$MSE = \frac{1}{n} \sum\_{j=1}^{n} \left( F\_{model,j} - F\_{exp,j} \right)^2,\tag{3.39}$$

$$RMSE = \frac{1}{n} \Sigma\_{f=1}^{n} \sqrt{(F\_{model,f} - F\_{exp,f})^2},\tag{3.40}$$

$$MAE = \frac{1}{n} \sum\_{j=1}^{n} \left| F\_{model,j} - F\_{exp,j} \right|,\tag{3.41}$$

where �������� is the model output and ������ is the experimental result at any time point (*j*), can be applied. The optimizations have also been conducted using normalized MSE, RMSE and MAE, i.e. by dividing equations 3.39-3.41 with ������ at each time point.

Fig. 5. A typical stress-relaxation measurement of articular cartilage and corresponding optimized model fit using a fibril reinforced poroviscoelastic model.

### **4. References**

128 Theoretical Biomechanics

2006; Zhang et al., 2008). Also viscoelastic solid models (Thornton et al., 1997) and poroelastic models have been applied for ligaments (Atkinson et al., 1997). However, the fluid-flow dependent viscoeasticity may not be that important in ligaments and tendons because they experience mainly tensile forces under physiological loading and it has been suggested that fluid has only a minor role in contributing to soft tissue response in tension (Li et al., 2005). Furthermore, viscoelastic models with anisotropic nonlinear stress-strain behaviour have been developed to capture the strain rate dependent nonlinearity of

The optimization of material parameters of the model can be done by typically minimizing the mean squared error (MSE), root mean squared error (RMSE) or mean absolute error (MAE) between the simulated and experimental force curves (Fig. 5). This can be done for instance using a multidimensional unconstrained nonlinear minimization routine (fminsearch) available in Matlab (Mathworks Inc., Natick, MA, USA). The optimization should be first tested with different initial values of the material parameters, and the optimized parameter values should be always the same, independent on the initial guess.

> � <sup>∑</sup> ��������� ��������� � �

� <sup>∑</sup> ���������� ��������) � �

� <sup>∑</sup> ��������� ��������� �

where �������� is the model output and ������ is the experimental result at any time point (*j*), can be applied. The optimizations have also been conducted using normalized MSE, RMSE

Fig. 5. A typical stress-relaxation measurement of articular cartilage and corresponding

optimized model fit using a fibril reinforced poroviscoelastic model.

��� , (3.39)

��� , (3.40)

��� , (3.41)

ligaments and tendons (Pioletti et al., 1998; Limbert&Middleton, 2006).

��� � �

���� � �

��� � �

and MAE, i.e. by dividing equations 3.39-3.41 with ������ at each time point.

**3.7 Optimization of material parameters** 

Then one of the equations for MSE, RMSE and MAE,


Biomechanics and Modeling of Skeletal Soft Tissues 131

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**7** 

*Universitat Jaume I* 

*Spain* 

**Biomechanical Models** 

**of Endodontic Restorations** 

Pablo J. Rodríguez-Cervantes and José L. Iserte-Vilar

Antonio Pérez-González, Carmen González-Lluch, Joaquín L. Sancho-Bru,

Endodontic treatment is one of the most widely used techniques in present-day odontology owing to the tendency to save teeth whenever possible. In endodontic therapy, the injured pulp of a tooth (located in the interior of the tooth and containing nerves and other vital tissues) is cleaned out and then the space is disinfected and subsequently filled with restorative material. This process is commonly known as root canal treatment. The devitalised tooth resulting from endodontics, has a different stiffness and resistance as compared to the original tooth and is less resistant as a consequence of the loss of tooth structure (Walton & Torabinejad, 2002). The use of intraradicular posts has extended as a technique to restore teeth that have lost a considerable amount of coronal tooth structure. After removing the pulp, the intraradicular post is introduced into the devitalised root. The post helps to support the final restoration and join it to the root (Christensen, 1998). Fig. 1a shows the typical structure of a tooth endodontically restored with a post. The post is inserted into the devitalised root canal, which has previously been obturated at its apical end with a biocompatible polymer called gutta-percha. Cement is used to bond the post to the root canal and a core is placed over the remaining dentine and the post. Finally, an artificial crown is used to achieve an external appearance that is similar to that of the original tooth. Nowadays most of the posts are prefabricated in a range of different materials and designs (Scotti & Ferrari, 2004). However, before prefabricated post became generalised, cast post and cores were used as a single metal alloy unit (Fig. 1b). Cast postcore systems take longer to make and involve an intermediate laboratory stage in which the retention system is created, which makes the whole process more costly. In comparison, prefabricated posts do not need this intermediate stage, which means that the whole restoration process can be performed in a single visit and is obviously easier and cheaper for the patient (Christensen, 1998). Nonetheless, the adaptation of the prefabricated posts to the

As the endodontically restored tooth is composed of materials that are different to those of natural teeth, it is expected to have a different biomechanical response under oral loads. The deformation of the system under flexural, compressive or tension forces could be different and so its mechanical strength under static or fatigue loads. Ideally, it seems interesting that the biomechanical behaviour of the restored system should preferably resemble that of the original tooth as much as possible in order to avoid failure of the repaired tooth or its

**1. Introduction**

root canal may be less accurate (Chan et al., 1993).

Zhang, X., Jiang, G., Wu, C., Woo, S. L., (2008). A subject-specific finite element model of the anterior cruciate ligament. In *30th Annual International Conference of the IEEE Engineering in Medicine & Biology Society*. Vancouver, British Columbia, Canada, August 20-24, pp. 891-894.
