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10 Biomechanical properties of ligament, tendon, and cartilage

10 Biomechanical properties of ligament, tendon, and cartilage

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420



Skeletal biomechanics



between the three tissues. Because the biomechanical behavior of a tissue is determined by its composition and structure, the mechanical properties of ligaments,

tendons, and cartilage are also considerably different. In this section, we will

explore the structure–function relationships for these three tissues. Before doing

so, we should emphasize the important distinction between structural and material

properties.

Structural properties refer to the mechanical properties of an object as a whole

and so will depend not only on the material of which the object is composed,

but also its shape and size. This should be obvious: it is much easier to break

a thin wooden pencil than a wooden hockey stick, although both are made of

the same material. Structural properties are described by the force–deformation

relationship, the stiffness, the ultimate or failure load, and the ultimate elongation.

For tendons and ligaments, structural properties are determined by measuring the

mechanical response of the entire bone–tendon–muscle or bone–ligament–bone

structure to loading. In this case, the structural properties are influenced not only

by the properties and geometry of the tissue but also by the mechanical properties

of the bone–tissue and muscle–tissue junctions.

Material properties, in contrast, describe the intrinsic mechanical behavior of the tissue constituents only, irrespective of the tissue sample’s overall

geometry. In the pencil and hockey stick example, the material properties would be

those of the wood. Tissue material properties are determined by the biomechanical

properties of the tissue’s constituents (i.e., collagen, elastin, proteoglycans) and

their microstructural organization, orientation, and interaction. Material properties

are described by parameters such as the stress–strain relationship, the modulus,

and the ultimate stress and strain. Since stress and strain are normalized by the

cross-sectional area and length of the sample, respectively, material properties are

independent of the geometry of the specimen.



9.10.1 Structural properties

Experimental studies on the structural properties of soft connective tissues in the

skeletal system have focussed on ligament and tendon. Because ligaments and tendons are primarily responsible for transmitting tensile loads, experimental studies

are performed in tension by clamping the bone–tendon–muscle or bone–ligament–

bone complex in a test machine and measuring the force that is generated with

increasing displacement of the two ends of the complex. The resulting load–

deformation curve can be used to define several structural properties, including

the stiffness (the rate of change of force with deformation), the energy absorbed



421



9.10 Biomechanical properties of ligament, tendon, and cartilage



Figure 9.27

A typical force–displacement (deformation) curve for a rabbit ligament. The schematics at the top of the figure indicate

the state of the collagen at each stage of the force–displacement curve, as explained in the text. From Nigg and

Herzog [28]. Copyright John Wiley & Sons Limited. Reproduced with permission.



(the area under the force–deformation curve), and the ultimate load and ultimate

deformation (at failure).

The load–deformation curves of all collagen-based tissues have a characteristic upwards concave shape in which the stiffness varies non-linearly with force

(Fig. 9.27). In the initial portion of the curve, called the toe region, the tissue

deforms easily without much tensile force. This is followed by a region in which

the load and deformation are approximately linearly related; the slope of this region

is often used to represent the elastic stiffness of the tissue. The linear region is followed by a region in which sharp falls in tensile force are seen, presumably as a

result of sequential microfailure of collagen fibers. Finally, with further loading,

the force reduces to zero as the tissue fails completely. For ligaments and tendons,

the tensile behavior in the toe region permits initial stretching without much resistance. As the force or displacement increases, however, the ligament or tendon

stiffens, providing more resistance to deformation. For ligament, this protects the

joint from excessive relative movement between bones, while for tendon, it ensures

efficient load transfer from muscles to bones.



422



Skeletal biomechanics



Figure 9.28

Simple model to explain how collagen crimp and alignment result in non-linear structural behavior of collagenous

tissues. (A) Collagen fibers, represented by parallel linear springs with stiffness c i , are recruited to resist the applied

force at different displacements, simulating different degrees of crimp and off-axis orientation. (B) As a result of

progressive recruitment, the effective stiffness of the tissue, , increases in a step-wise fashion. (C) The step-wise

increase in stiffness results in a non-linear relationship between force, F, and displacement, X. From Frisen

et al. [36], with permission from Elsevier.



The non-linear force–deformation relationship can be explained by considering the microstructure of the collagen fibers and their organization in connective

tissues. As shown in Fig. 9.22, collagen (the principal component of connective

tissues that provides tensile strength) is crimped. The crimp allows the tissue to be

extended with minimal force; this phenomenon is responsible for the toe region of

the load–deformation curve. As the deformation increases, the crimp flattens out

and the apparent stiffness increases as the force required to deform the tissue is

resisted by the inherent material stiffness of the collagen fibers. A second source

of non-linearity in the force–deformation curves of skeletal connective tissues is

the alignment of the collagen fibers. During tensile loading of tissues like tendon,

little realignment of the fibers occurs because they are already oriented parallel to

the direction of the applied load (Fig. 9.25). As a result, the toe region of the load–

deformation curve is relatively short. In ligament, many of the collagen fibers are

parallel to the long axis, but there are also non-axial fibers that run at a variety of

angles with respect to the long axis (including running perpendicular). As a result,

the toe region for ligament is longer than that of tendon, since some realignment

of collagen fibers must occur during tensile loading to resist deformation.

A simple, but illustrative, model of how the progressive recruitment of individual fibers can result in a non-linear force–displacement curve is shown in

Fig. 9.28. In this model, individual collagen fibers are represented by linear elastic

springs arranged in parallel. The fibrils have different degrees of crimp and different

orientations relative to the loading direction; they are therefore recruited to resist

the applied force at different displacements ( 1 , 2 , . . . , P ). The stiffness of the

tissue, , at any deformation, X, is given by the slope of the force–deformation



423



9.10 Biomechanical properties of ligament, tendon, and cartilage



(F–X) curve and is expressed as:

(X ) = dF/dX.



(9.25)



Because of the progressive recruitment of collagen fibers, the stiffness is a summation of step functions (Fig. 9.28B):

p



(X ) =



ci u(X −



i)



(9.26)



i=0



where ci are the stiffnesses of the individual springs, and u(ξ ) is the Heaviside

function, u(ξ ) = 0 for ξ < 0 and u(ξ ) = 1 for ξ > 0. As a result, with increasing

deformation, more parallel springs are recruited, resulting in increased stiffness

and a non-linear force–displacement response (Fig. 9.28C).

From an engineering design perspective, the collagen arrangements in tendons

and ligaments make sense. Tendon must provide high tensile strength and stiffness to transmit muscle forces to bone without substantial deformation. The highly

oriented parallel fiber bundles provide the high tensile strength and minimal extensibility but allow flexibility in bending (similar to braided wire ropes). Ligaments

must also provide high tensile strength but, depending on the function of the ligament, may experience off-axis loads and therefore require some strength in off-axis

directions. Ligaments also need to be slightly extensible to allow for proper joint

movement. These features are provided by the nearly parallel arrangement of the

collagen fibers in ligaments of the extremities. Some ligaments such as the ligamentum flavum in the spine contain more elastin than collagen; therefore, they

deform substantially before the stiffness increases (long toe region). However, the

ligamentum flavum is “pre-tensed” when the spine is in its neutral position and so

stabilizes the spine to prevent excessive movement between vertebral bodies.



9.10.2 Material properties

The material properties of biological tissues are based on stress–strain relationships

of the tissue substance itself. These properties are more difficult to measure than

structural properties for several reasons: it is difficult to grip the tissue without

damaging it; accurate measurement of tissue cross-sectional area is challenging;

strain is best measured without contacting the tissue; and material properties are

sensitive to external factors such as the source of the tissue and how the tissue

is handled, stored, or prepared prior to testing. Nonetheless, several advances in

test methods and a better understanding of the effects of various external factors

on the measurements have improved characterization of the tensile, compressive,

and time-dependent (viscoelastic) material properties of ligaments, tendons, and

cartilage [37].



Skeletal biomechanics

64



PT



48

X

ACL

PCL

Stress (MPa)



424



32

X



X



X

LCL

16



0

0



5



15



10



20



25



Strain (%)

Figure 9.29

Tensile material (stress–strain) curves for the patellar tendon (PT), anterior cruciate ligament (ACL), posterior cruciate

ligament (PCL), and lateral collateral ligament (LCL) of the knee of a young human donor. Note that the tendon has a

shorter toe region and larger failure stress than the ligament. Modified from Butler et al. [38], with permission from

Elsevier.



425



9.10 Biomechanical properties of ligament, tendon, and cartilage



Stress (MPa)



20



10



0



0



1



2



3



4



5



6



Strain (%)

Figure 9.30

Stress-strain curves for tropocollagen (•), collagen fibrils (◦), and a tendon ( ). Strain in the tropocollagen was

estimated from measurements of the elongation of the helical pitch of the collagen triple helix. Strain in the collagen

fibril was determined using an X-ray diffraction technique that measured the change in D-period spacing. The moduli of

the collagen fibril (430 MPa) and the tendon (400 MPa in the linear portion) are remarkably similar. The modulus of

tropocollagen is about three times greater than that of the fibril. From Sasaki and Odajima [39], with permission from

Elsevier.



9.10.3 Material properties: tension

The material (stress–strain) behavior of connective tissues under tension is nonlinear, in large part because of the crimping and alignment of collagen fibers

(Fig. 9.29). The linear region after the toe region is believed to represent the

resistance to deformation of the collagen fibers themselves. Consistent with this

theory, stress–strain curves of collagen fibrils are linear and have a slope similar

to that of the linear region for tendon (Fig. 9.30). It is possible, however, that the

non-linear tissue behavior is also partly a result of complex molecular interactions

between the various components of connective tissues, which is a phenomenon

that is not well understood at present.



426



Skeletal biomechanics



Table 9.8. Equilibrium tensile moduli of skeletal connective tissues. Cartilage specimens were

harvested from high-weight-bearing areas (HWA) or low-weight-bearing areas (LWA). The higher tensile

moduli in the LWA correlated with a higher collagen to proteoglycan ratio in these regions.

Tissue



Species



Location



Tensile

modulus (MPa)



Source



Tendon



Human



Achilles tendon



819



Wren [40]



Human



Patellar tendon



643



Butler [38]



Ligament



Articular

cartilage



Human



Cruciate ligaments



345



Butler [38]



Human, young



Ligamentum flavum



98



Nachemson [29]



Human, aged



Ligamentum flavum



20



Nachemson [29]



Human



HWA of femoral condyle,

superficial zone



5.0



Akizuki [41]



Human



HWA of femoral condyle,

middle zone



3.1



Akizuki [41]



Human



LWA of femoral condyle,

superficial zone



10.1



Akizuki [41]



Human



LWA of femoral condyle,

middle zone



5.4



Akizuki [41]



The tensile material properties of ligament, tendon, and cartilage can be

described by the tensile modulus, E, which is equal to the slope of the linear

portion of the stress–strain curve. When a soft connective tissue is subjected to

an applied force, its length will change with time until it reaches an equilibrium

length. This process is called “creep” and is the result of the viscoelastic

properties of the tissue. Ideally, the tensile modulus should be measured once the

tissue stops deforming and has reached equilibrium. The equilibrium modulus

therefore describes the quasi-static behavior of the tissue and is dependent on the

intrinsic tensile properties of the tissue matrix (mostly collagen). The transient,

time-dependent properties of the tissue must be described with other (viscoelastic)

parameters, which we will consider below. Typical equilibrium tensile moduli

for skeletal connective tissues are listed in Table 9.8. By comparison with the

composition of the tissues shown in Table 9.7, you can see there is a correlation

between collagen content and tensile stiffness.



9.10.4 Material properties: compression

Because articular cartilage is loaded in compression, it makes sense to study its

compressive properties. The compressive properties of cartilage can be measured

using several test configurations, the most popular of which are shown in Fig. 9.31.

A typical compression test for cartilage is a confined compression test, in which



427



9.10 Biomechanical properties of ligament, tendon, and cartilage



Figure 9.31

Schematics of three test configurations used to measure the compressive material properties of cartilage: confined

compression (top), unconfined compression (middle), and indentation (bottom). From Mow and Guo [42] with kind

permission of Annual Reviews.



428



Skeletal biomechanics

Ant



0.75

1.86



0.79

2.72



0.91

2.75



0.87

2.32

Med



0.42

4.67



0.36

3.50



1.11

2.29



0.77

2.19



0.69

2.16



0.80

4.30



Lat



0.40

4.08



0.74

2.20



0.47

3.07



0.81

2.03



0.65

4.27



Pos

HA (MPa)

Thickness (mm)



Figure 9.32

A map of aggregate compressive equilibrium moduli (H A , the number above the line) and cartilage thickness (the

number below the line) measured in the tibial plateau of a normal 21-year-old male. Significant regional differences in

both parameters are evident. The tibial plateau is the surface of the tibia (shin bone) that forms the base of the knee joint.

Ant, anterior; Pos, posterior; Med, medial; Lat, lateral; see text for definition of H A . From Mow et al. [43] with permission

of Lippincott Williams & Wilkins and Dr. Shaw Akizuki.



a small cylindrical plug of cartilage tissue is placed in a cylindrical chamber

(Fig. 9.31, top). The walls of the chamber, which contact the cartilage plug, prevent

the cartilage from expanding laterally, thereby ensuring that deformation occurs

only in the direction of loading. The cartilage is loaded under a constant compressive load applied with a porous filter. The pores in the filter allow fluid from the

cartilage to flow through the filter as the cartilage is compressed. Once equilibrium

is reached (i.e., the fluid stops flowing out of the cartilage), the stress and strain

are recorded and the ratio is defined as the aggregate equilibrium modulus (HA ).

Like the tensile equilibrium modulus, the aggregate modulus is a measure of the

stiffness of the solid matrix and is independent of fluid flow (since it is measured

at equilibrium once fluid movement has ceased).

Aggregate compressive equilibrium moduli for human articular cartilage range

from 0.1 to 2 MPa and vary significantly with location, even in a single joint

(Fig. 9.32). Experiments have shown that HA is directly related to cartilage



429



9.10 Biomechanical properties of ligament, tendon, and cartilage



proteoglycan content [44], but not collagen content. Hence, proteoglycans, with

their ability to bind water electrochemically, are primarily responsible for providing the compressive stiffness of cartilage. In osteoarthritis, a degenerative disease

of cartilage, there is often loss of proteoglycans, leading to decreased compressive

stiffness of the tissue [34].



9.10.5 Material properties: viscoelasticity

An important characteristic of the material properties of skeletal connective tissues

is that they display properties of both elastic and viscous materials and so are

known as “viscoelastic.” Some of the viscoelastic characteristics displayed by most

biological tissues are time-dependent responses, hysteresis, and history-dependent

responses.

Time-dependent responses. Such responses include creep, an increase in deformation over time under constant load, and stress relaxation, a decline in stress

over time under constant deformation (Fig. 9.33A,B). The material properties of viscoelastic tissues are also time-dependent in that they respond differently to variations in the rate at which the material is loaded or strained

(Fig. 9.33C).

Hysteresis. During loading and unloading of viscoelastic materials, the force–

displacement curves do not follow the same path but instead follow a hysteresis

loop, indicating that internal energy is dissipated during the loading–unloading

cycle (Fig. 9.33D).

History-dependent responses. If a viscoelastic tissue is subjected to several

different stress paths that arrive at the same stress value, σ0 , at time t1 , the corresponding strains, ε, at time t1 will differ (Fig. 9.34). In other words, the strain at

time t1 depends on the entire sequence or “history” of stresses up to that time.

The mechanisms responsible for these viscoelastic behaviors in soft connective

tissues likely include the intrinsic viscoelasticity of the solid phase of the tissue

arising from intermolecular friction between the collagen, elastin, and proteoglycan

polymeric chains; deformation of these molecules; and other complex intermolecular interactions. For biological tissues that contain significant amounts of water,

viscoelasticity also results, in part, from the interactions of the water with the proteins, specifically the frictional drag the water creates when it flows through the

porous ECM.

As we demonstrated in Section 2.6.1 for cells, linear viscoelastic behavior can be

modeled by using lumped parameter approaches in which the viscoelastic material

is represented by linear springs and linear dashpots. Consider the model shown

in Fig. 2.35 (p. 56); as we will see below, this “standard linear model” is the simplest



Skeletal biomechanics



430



A



F



B



Creep



F0



F



t



t

d



Stress relaxation



d

d0

t



C



t



D



F



F



increasing

deformation rate

d



d



Figure 9.33

Typical force–deformation (F–d) characteristics of viscoelastic materials. (A) Creep, which occurs following a change in

applied load, in this case a step change from zero to a constant F 0 . After the instantaneous deformation when the load is

applied, a continued gradual increase in deformation is seen until equilibrium is reached. (B) Stress relaxation, which is

a gradual decrease in stress under constant deformation (d 0 ) until equilibrium is reached. (C) Rate dependency of

force–displacement relationship, in this case showing that increased force is required to obtain a given deformation if the

rate of deformation is increased. (D) Hysteresis, or different loading and unloading paths indicating dissipation of energy.

The arrows show the direction of loading and unloading.



A



B



s



C



e0



e



s0

e2



s2



s

s0



s0 − e0

s2 − e2



e1



s1− e1



s1



t1



t



t1



t



e



Figure 9.34

The response of viscoelastic materials to loading depends on the “stress history.” Although the three different stress

paths in (A) arrive at the same stress level at time t = t 1 , the corresponding strains differ (B), and hence the resulting

stress–strain relationships are each different (C). Thus, the strain at t = t 1 depends on the stress path (or its history) to

that point. From Wineman and Rajagopal [45]. Reproduced with the permission of the authors and publisher.



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