Mathematical Modeling of Ligaments and Tendons

Transcription

1 S. L-Y. Woo Fellow ASME G. A. Johnson B. A. Smith Musculoskeletal Research Center, Department of Orthopaedic Surgery, University of Pittsburgh, Pittsburgh, PA 523 Mathematical Modeling of Ligaments and Tendons Ligaments and tendons serve a variety of important functions in maintaining the structure of the human body. Although abundant literature exists describing experimental investigations of these tissues, mathematical modeling of ligaments and tendons also contributes significantly to understanding their behavior. This paper presents a survey of developments in mathematical modeling of ligaments and tendons over the past 20 years. Mathematical descriptions of ligaments and tendons are identified as either elastic or viscoelastic, and are discussed in chronological order. Elastic models assume that ligaments and tendons do not display time dependent behavior and thus, they focus on describing the nonlinear aspects of their mechanical response. On the other hand, viscoelastic models incorporate time dependent effects into their mathematical description. In particular, two viscoelastic models are discussed in detail; quasi-linear viscoelasticity (QLV), which has been widely used in the past 20 years, and the recently proposed single integral finite strain (SIFS) model. Introduction Scientific investigation has revealed that major functions performed by ligaments and tendons include transmitting load, maintaining the proper anatomic alignment of the skeleton, and guiding joint motions. Therefore, in the past twenty years, much attention has been given to characterizing the mechanical behavior of ligaments and tendons through both experimental and analytical approaches. Experimental studies of ligament and tendon properties have benefitted from advances in technology, such as the significant improvements made in measuring tissue strain and cross-sectional area (Woo, 982; Woo et al., 990). Ligaments and tendons associated with the knee joint have received the most attention because the knee is often injured. In addition, work related to modeling of the knee joint requires properties of the ligaments and tendons. This manuscript will be devoted to modeling of ligament and tendon tissue, rather than models of entire diarthrodial joints. The focus will be on the evolution of mathematical models used to describe the mechanical behavior of ligaments and tendons. Readers with a special interest in joint modeling should refer to an excellent article by Blankevoort and Huiskes (99). Ligaments and tendons are composed of closely packed collagen fiber bundles organized in a more or less parallel fashion along the length of the tissue so as to resist tensile loads. They are anatomically positioned to guide normal motion, and their mechanical properties are designed to restrain abnormal motion by resisting excessive elongation. The microstructural organization of ligaments and tendons imparts upon these tissues characteristics essential to their physiologic functions. This organization is distinct, consisting of several levels beginning Contributed by the Bioengineering Division of THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS and presented at the 993 ASME/AIChE/ASCE Summer Bioengineering Conference, Forum on the 20th Anniversary of ASME Biomechanics Symposium, Breckenridge, CO, June 25-29, 993. Revised manuscript received July 2, 993. with procollagen molecules, which self assemble into microfibrils. These then aggregate to form subfibrils, which organize into the structural unit referred to as the fibril, the elemental component of fibers (Fig. ). Histologically, fibrils appear in microstructural form in a wave pattern that is referred to as crimp. Viidik and Ekholm (968) have demonstrated this phenomenon using polarized light microscopy. Crimping is thought to have a significant influence upon the biomechanical behavior of ligaments and tendons. In addition to collagen fibrils, ligaments and tendons also contain elastin, proteoglycans, glycolipids, water and cells (Woo and Young, 99). Although the ground substance constituents make up only a small percentage of the total weight x rav x ray Ai5<lTlMiC~R! TROPO- COLLAGEN I5A 35A x ray 35 A staining sites A Evid x ray S ence : S OM A SI OM mm ^^v FASCICLE tm 640 A periodic ty Xf?«P foqg. reticular fibroblasts crimp structure membrane waveform or \ fascicular membrane A >l > SIZE SCALE TENDON Fig. Tendon architectural hierarchy (scale indicated at bottom), from Kastelic et al. (978) 468 / Vol. 5, NOVBER 993 Transactions of the ASME Copyright 993 by ASME

2 of a ligament or tendon, they are quite significant because of their association with water, which comprises 60 to 70 percent of the total weight of ligaments and tendons. The interactions of these components, as well as the inherent viscoelastic properties of the collagen fibrils themselves, are responsible for the time and history dependent properties of ligaments and tendons. Mathematical models have been developed to complement experimental studies by furthering our understanding of ligament and tendon behavior. Models also have the potential to predict the mechanical behavior of tissue where experiments would be too complex, difficult, and costly. Mathematical models, on the other hand, face the complexities of describing both time dependent and nonlinear effects. Earlier models have neglected the time dependent components of tissue behavior and have concentrated on describing the nonlinear aspects of the stress-strain response. However, since the late 960's viscoelastic models, which incorporate the time and history dependent aspects of the stress-strain relationship, have been developed (e.g., Fung, 968; Viidik, 968; Frisen et al., 969; Decraemer et al., 980b; Lanir, 980). This review will present an overview of the development of mathematical modeling of ligaments and tendons. We will present a chronological summary of models that have been used, separated into elastic and viscoelastic groups, followed by a discussion of microstructural change (i.e., the stress-strain relation has a different mathematical structure for different magnitudes of strain) and its role in formulating mathematical descriptions of tendons and ligaments. Structural and Phenomenological Models For purposes of discussion, we wish to make a further distinction between models on the basis of how they are formulated. By roughly categorizing models as either structural or phenomenological, we can discuss with some generality the common properties of models belonging to each group. Structural models are based on known (or assumed) behavior of the constituents of the tissue. The mechanical responses of the individual components are then combined or generalized to produce a description of gross mechanical behavior. These models are particularly suited to elucidate the connection between structure and mechanical properties, as they include parameters which are directly related to the structure of the tissue. We refer to phenomenological models as those that do not have explicit parameters related to the microstructure of the tissue. This category includes a number of models, from those that are derived simply by curve fitting experimental data, to rigorously formulated continuum models. Because of the manner in which they are formulated, structural models are well suited to relating microstructure with mechanical behavior, whereas phenomenological models are more amenable to generalization, and to predicting behavior in independent tests. It is thus possible to think of structural and phenomenological models as serving complementary roles in describing the mechanical responses of tendons and ligaments. Elastic Models As tensile load is applied to a ligament or tendon, the relationship between load and elongation is initially nonlinear; this area of the stress-strain curve is known as the "toe'' region. Larger applied loads result in a gradual increase in stiffness and an eventual change to a more linear relationship between load and elongation; resulting in the linear region of the stressstrain curve. A simplified representation of this nonlinear behavior was presented by Frisen et al. (969) (Fig. 2). This figure depicts ligaments and tendons as consisting of individual linearly elastic components, each representing a fibril of different initial length in its unloaded and crimped form. Under rela- ^AAAAJ-^o r * A/\A/4<4 F / yf A ' ' Jf Q o *x t, e, a, o Fig. 2 Schematic model of nonlinear elasticity demonstrating progressive recruitment of individual linear components, from Frisen et al. (969) tively small tensile loads, crimped fibrils begin to straighten out. Initially, there is little resistance to tension as the fibrils lengthen, but as elongation progresses an increasing number of fibrils become taut. This recruitment of additional fibrils results in the nonlinear behavior characteristic of the toe region. As elongation continues at higher loads all the fibrils become taut, and ligaments display a more linear response. Many early models incorporated the assumption that ligaments and tendons are elastic, that is, they neglect the timeand history-dependent (viscoelastic) aspects of tensile behavior. Although experimental work can be interpreted within the context of the infinitesimal theory of elasticity, Fung (967) recognized that biological tissue normally undergoes finite deformation and proposed an exponential stress-strain relation that accounted for nonlinearity and finite deformation in uniaxial tension. This work was later extended, utilizing the theory of non-linear elasticity, to two- and three-dimensional problems by postulating a stored energy function (Hildebrandt et al., 969). Elastic models have continued to progress, becoming more complex and more suitable mathematical representations of ligament and tendon behavior. Haut and Little (969) fit stressstrain curves of canine anterior cruciate ligaments to a modified "power law" version of Fung's exponential stress-strain relation. Prager (969) proposed a model of locking materials to describe the stress-strain curves of biological tissues. In order to adequately represent the toe region, this theory utilizes the assumption that small strains in a material are associated with zero or negligible stress. Diamant et al. (972) proposed a structural description of tendons and ligaments based on the "elastica" problem in mechanics. The crimped collagen fibers were modeled as having elastic segments joined by rigid hinges, producing a stress-strain curve with the appropriate shape. Tendon has also been modeled as a fiber-reinforced composite. A continuum based model was proposed in which mammalian tendon was described as inextensible fibers arranged in a helical pattern within an incompressible, hollow right circular cylinder (Beskos and Jenkins, 975). Comninou and Yannas (976), taking a more structural approach, modeled collagen fibers as sinusoidal and obtained a stress-strain relation using a linearization of finite strain beam theory. The resulting formulation incorporates an assumption of constant crimp configuration, and, because of the linearization, is restricted to small strains. Lanir (979) proposed a structural elastic model of skin (later extended to model viscoelastic behavior of tendons) in which the crimp of collagen fibrils is induced and then sustained by the contraction of an elastic filament attached to an isolated fibril, at random intervals. The distribution of straightening strains for the numerous buckled loops governs the low stress deformation behavior of the structure. The actual distribution could not be determined experimentally and was chosen based on modeling considerations. I Journal of Biomechanical Engineering NOVBER 993, Vol. 5 / 469

3 The sequential straightening and loading model (Kastelic et al., 980) includes the assumption that resistance arises only from the elasticity of already straightened fibrils. Crimped fibrils are assumed to have negligible resistance to extension. A morphologically based range of crimp angles is assumed for the fibrils within the fascicle. In contrast to previous models in which crimp configuration was constant (Diamant et al., 972; Comninou and Yannas, 976; Lanir, 979), crimp angles in the undeformed tendon were assumed to vary among fibrils throughout the width of the tendon. The resulting equation related crimp angle and crimp angle distribution to the stress generated by elongation. A similar model was proposed by Decraemer et al. (980a), who assumed that soft tissue could be modeled as a large number of purely elastic fibers embedded in a gelatin like liquid. All fibers were assumed to have the same modulus and cross-sectional area but different lengths, being normally distributed about some known mean. Fibers were more or less folded based on their initial length. Stouffer et al. (985) approached the problem of modeling human patellar tendons by replacing the actual configuration of the structural elements with a kinematic chain, composed of a number of short elements connected by pins and torsion springs. These mechanical analogs are defined so as to permit extension, but not bending, and the variation in crimp pattern is incorporated by making the link parameters functions of position. Use of this model requires that the material and geometric parameters first be established as a function of position. These investigators utilized a microscope system that allowed measurement of crimp pattern at different positions and at different loads to establish model parameters. Although the governing equations that they ultimately derived were entirely algebraic, the mathematical structure is similar to that used by Diamant et al. (972). Another structural model was proposed by Kwan and Woo (989). This model differed from previous work in that fibrils did not elongate (uncrimp) with zero stress. Instead, each fibril was assumed to have a bilinear stress-strain curve, with different slopes for the toe and linear regions (Fig. 3). Collagen fibrils making up a ligament were assumed to have m different lengths and n different failure groups. The constitutive equations governing the behavior of each fibril were then superposed to produce an equation for the ligament as a whole. A ' 'three-by-three'' (three lengths and three failure groups) model was shown to fit the stress-strain curves of the anteromedial portion of a rabbit anterior cruciate ligament and a canine medial collateral ligament. Belkoff and Haut (99) utilized a structural model based on the models of Lanir (979, 983) and Decraemer et al. (980a) to model skin undergoing uniaxial tension. The model assumes that all fibers are aligned in the direction of the applied <r,, tension, and was later adapted to model human patellar tendon (Belkoff and Haut, 992). The model assumes that the crimp of collagen fibers in the stress-free tendon gradually disappears as fibers become straight and begin to resist deformation and generate load. Fibers are not straightened all at once but sequentially, with the distribution of slack-lengths described by a normal distribution function. Viscoelastic Models As work on elastic ligament and tendon models continued, the first models incorporating viscoelastic behavior were proposed. Viidik (968) formulated a rheological model of parallel fibered viscoelastic tissues consisting of spring and dashpot combinations. In a subsequent investigation, the model was modified to account for nonlinearity of the elastic response (Frisen et al., 969). These types of structural viscoelastic models of ligaments and tendons have been based on the generalization of corresponding structural elastic models. Decraemer et al. (980b) extended their structural elastic model to the viscoelastic case by incorporating internal friction between fibers in soft biological tissues, and between fibers and the material in which they are embedded. Damping was introduced into the model by assuming that all fibers have identical linear viscoelastic properties. By assuming that the individual collagen fibers were linearly viscoelastic, Lanir (980) also extended his elastic model to include viscoelastic behavior. His model was formulated for two special cases; one incorporated a high density of crosslinks between collagen and elastin fibers and the other assumed a low density of crosslinks. Subsequently, the model was generalized to incorporate threedimensional viscoelastic theory (Lanir, 983). Fung (968) first formulated Quasi-Linear Viscoelasticity (QLV), which combines elastic and time dependent components of a tissue's mechanical response using a hereditary integral formulation. The exponential form of the stress-strain relation for uniaxial tension was incorporated into this general viscoelastic model. QLV theory has been used to model a wide variety of tissues. It remains the most commonly used viscoelastic model of ligaments and tendons (e.g., Haut and Little, 972; Woo et al., 98; Woo, 982; Lin et al., 987; Lyon et al., 988). In the QLV theory, Fung incorporates the assumption that stress can be expressed as: a(x, t) = G{t)a e {\), () where G(f) is a function of time, t, called the reduced relaxation function, X is the stretch ratio, and o%\) is the time-independent elastic response. The stress at time t is then given by: o(\, t)= \ v, G(,_ T)^rfT. (2) ax dr Based on association with linear viscoelasticity, but with a e assuming the traditional role of strain within the linear theory, Fung chose the reduced relaxation function, G(t), as: G(t)=- + S(T)e~' I /T dt + S(T)CIT (3) Individual Fiber where: Summation S(T)=- TI<T<T 2, (4) Fig. 3 Schematic of a bilinear stress-strain curve for several collagen fibers. A constitutive equation is obtained for a tendon or ligament by superposing the responses of individual fibers. 470/Vol. 5, NOVBER 993 S(T) = 0, T<Ti,T>T 2. This can be rewritten in terms of exponential integrals as: l + C[E(t/T 2 )-E(t/ Tl )] G(0=- + C lnfom) (5) (6) Transactions of the ASME

4 CYCLIC STRESS RELAXATION Theory Experimental (n = 8) 3=f=f=t=Rq PEAK STRESSES $==--f=5= : -I = = i <F --». I STRESSES VALLEY Fig. 4 Stress-strain curves from canine medial collateral ligaments and from QLV predictions, from Woo et al. (98) NUMBER OF CYCLES Fig. 5 Peak and valley stresses from canine medial collateral ligaments subjected to cyclic elongation experimental and QLV predictions, from Woo et al. (98) where: E(y t ) -" y dy, y x = y= (7) The above expression can be simplified further if i\ and TI differ sufficiently so that T X «t «T 2. Equation (6) can then be written as: G(t)- l-cy-cln(t/t 2 ) + C ln(t2/tj) where 7 is the Euler constant (7 = ). Woo et al. (98) employed this formulation to model the canine medial collateral ligament (MCL). The method used to fit data to this model was to initially define the reduced relaxation function, G(0, using stress relaxation data, as G(f) = a{t)/a(q). The parameters C, T\, and r 2 were determined using experimental values for dg(t)/d ln(?), G(co), and G(i). The elastic (time-independent) response was assumed to have the form a e = A{e Be - ), where e is the one-dimensional strain due to simple elongation. The parameters A and B were constants to be determined. This form for <f and the known G(t) were substituted into the original integral constitutive equation and curve fit using constant strain rate data to obtain A and B. The curve fit is shown in Fig. 4. The model predictions using these parameters were then confirmed by comparison with data from a cyclic stress relaxation test between two fixed strain levels (Fig. 5). The QLV theory has also been used to describe the behavior of porcine anterior cruciate ligament (ACL) (Lin et al., 987) and human ACL and patellar tendon (Lyon et al., 988). These studies differed from previous work in that the applied elongation for the stress relaxation test was not assumed to increase instantly at time zero, but was modeled more realistically as a finite ramp. Several other phenomenologically based viscoelastic models have also been used to model ligaments and tendons. Barbenel et al. (973) utilized a generalization of the discrete element (i.e., spring and dashpot) models which incorporated a logarithmic relaxation spectrum. DeHoff (978) and Bingham and DeHoff (979) adapted a continuum-based approximate constitutive equation, which had been used to characterize non- - linear viscoelastic behavior of polymers, to the description of soft biological tissues. Their constitutive equation assumed that ligaments could be modeled as isotropic viscoelastic media with fading memory. Pradas and Calleja (990) developed a onedimensional, nonlinear viscoelastic model to describe the creep behavior of a human flexor tendon. They employed an assumption of quasi-linear behavior, similar to QLV, in formulating a constitutive equation. Recently, we have used a general single integral finite strain (8) (SIFS) viscoelastic theory to model the human patellar tendon (Johnson et al., 992). This structurally motivated continuum model offers the advantages of incorporating a general nonlinear, three-dimensional description of a tissue's mechanical behavior. It is also flexible in that the concept of fading memory is used to model viscoelastic response, allowing a number of physically motivated relaxation functions to be used. The specific form for the three-dimensional equation is obtained by truncating an integral series representation of viscoelastic behavior, and assuming finite deformation and fading memory. This equation may be written as: T =-/>! +C 0 {[+M0B(0 -/xb 2 (0) r -(Co-C.) \ G(Oi[l+Ms)]B(s)-/*B z (s)]cfe. (9) where T is the Cauchy stress, p is the indeterminate part of the stress arising due to the constraint of incompressibility, I is the identity tensor, B is the left Cauchy-Green strain tensor, G(t) is the time-dependent relaxation function, C 0 is the instantaneous modulus, C a is the long time modulus, /x is the shear modulus, and I(s) = trc, where C is the right Cauchy- Green strain tensor. This model reduces to the appropriate case of finite elasticity at very short times and, if linearized, yields classical linear viscoelasticity. Having developed an equation describing finite viscoelasticity, we can incorporate nonlinear effects in the form of microstructural change. The idea of microstructural change, whether due to damage or merely to straightening of crimp, is implicit in the structural theories of both elasticity and viscoelasticity. At different strain levels there is assumed to be a different micromechanical mechanism supporting load. In structural models, this change is modeled by recruitment of additional collagen fibers, or by a change in interaction between tissue components. The idea that deformation imposed on polymeric materials can alter the micromechanism responsible for generating stress in those materials was introduced by Tobolsky and Andrews (945). This idea may also be applied to the mechanical description of tissues. Chu and Blatz (972) noted that simple viscoelasticity was insufficient to describe hysteresis of living tissue, as it predicts relaxation times to be the same for both loading and unloading. These investigators formulated a onedimensional model for hysteresis based on a theory of cumulative microdamage. In their model, the constitutive equation describing the stress-strain response of a tissue changed with deformation to account for differences in loading and unloading behavior. As mentioned previously, microstructural change is included implicitly in structural theories. Several studies, however, have Journal of Biomechanical Engineering NOVBER 993, Vol. 5/47

5 6.0 Experimental Theoretical ra i 4.0 *w in - 0) 9> Strain (%) Fig. 6 Experimental and curve (it stress-strain curves for human patellar tendon Number of Cycles Fig. 7 Peak and valley stresses from a human patellar tendon subjected to cyclic elongation experimental and SIFS predictions made this type of mathematical structure explicit. Kwan and Woo (989), in the development of their structural elastic model, assumed that the modulus of an individual fiber changed from one value to another at a discrete value of strain. Pradas and Calleja (990) also utilized a constitutive equation that represented two separate responses patched together. In the single integral finite strain viscoelastic model this is accomplished by utilizing different constitutive equations for different levels of strain and "patching" them together mathematically. Within the context of a continuum theory the specific change taking place within the ligament substance is not of paramount concern, but the form of the proposed constitutive equation must still reflect the different mechanisms involved. Thus, an expression for stress, o"i(x), is used in the toe region and a second expression, a 2 (K) in the linear region. The model stretch parameter, X, marks the transition from nonlinear to linear response on the stress-strain curve. For X < X the stress is given by a = o\ and for X > X the stress is given by a = /(cr,, o- 2 ):,ffi(x) Wrl+ffiW X<Xj X>X (0) It is important to note that for the second region a new stretch should be defined so that X is the reference stretch. Stress-strain equations for the two regions were curve fit to data obtained from experiments on a human patellar tendon. The value of X is determined from the stress-strain curve and the model parameters are then determined by fitting to stressrelaxation and stress-strain data (Fig. 6). For confirmation, the parameters determined from the curve fits are used to predict stress in a human patellar tendon resulting from a cyclic elongation. Figure 7 shows the predicted stress and the experimentally measured stress resulting from this cyclic test. The model provides a reasonable fit, but with some discrepancies for the first few cycles. The parameters in the relaxation function are extremely sensitive to variations in tissue response at short times, and thus, like many viscoelastic models, the SIFS model may have difficulty describing the transient behavior of the early cycles. The single integral finite strain (SIFS) viscoelastic model shows promise as a general theoretical framework that includes nonlinear, three-dimensional, finite viscoelasticity. It is our hope that this representation of viscoelastic behavior will be extended to describe the anisotropic properties of ligaments and tendons. It will also be possible to use this model in describing deformations that are more complex than uniaxial extension, thus providing a valuable tool for further understanding and more accurate prediction of the mechanical behavior of ligaments and tendons under complex loading conditions. Ultimately, a combination of theories and experiments will be needed to elucidate the function of these tissues as no single approach is sufficient by itself, making it crucial that we continue to make advances in both areas. Acknowledgments This work was supported by NIH grants AR4820 and AR References Barbenel, J. C, Evans, J. H., and Finlay, J. B., 973, "Stress-Strain-Time Relations for Soft Connective Tissues," Perspectives in Biomedical Engineering, Kenedi, Ed., McMillan, London, pp Belkoff, S. M., and Haut, R. 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6 Uniform Biaxial Deformation of Nearly Isotropic Incompressible Tissues," Biophysical Journal, Vol. 9, pp Johnson, G. A., Rajagopal, K. R and Woo, S.L-Y., 992, "A Single Integral Finite Strain (SIFS) Model of Ligaments and Tendons," Advances in Bioengineering, Vol. 22, pp Kastelic, J., Palley, I., and Baer, E., 978, "The Multicomposite Ultrastructure of Tendon," Connective Tissue Research, Vol. 6, pp Kastelic, J., Palley, I., and Baer, E., 980, "A Structural Mechanical Model for Tendon Crimping," Journal of Biomechanics, Vol. 3, pp Kwan, M. K., and Woo, S.L-Y., 989, "A Structural Model to Describe the Nonlinear Stress-Strain Behavior for Parallel-Fibered Collagenous Tissues," ASME JOURNAL OF BIOMECHANICAL ENGINEERING, Vol. Ill, pp Lanir, Y., 979, "A Structural Theory for the Homogeneous Biaxial Stress- Strain Relationships in Flat Collagenous Tissues," Journal of Biomechanics, Vol. 2, pp Lanir, Y., 980, "A Microstructure Model for the Rheology of Mammalian Tendon," ASME JOURNAL OF BIOMECHANICAL ENGINEERING, Vol. 02, pp Lanir, Y., 983, "Constitutive Equations for Fibrous Connective Tissues," Journal of Biomechanics, Vol. 6, pp. -2. Lin, H. C, Kwan, M. K. W., and Woo, S. L-Y., 987, "On the Stress Relaxation Properties of Anterior Cruciate Ligament (ACL)," Advances in Bioengineering, pp Lyon, R. M., Lin, H. C, Kwan, M. K. W., Mollis, J. M., Akeson, W. H., and Woo, S. L-Y., 988, "Stress Relaxation of the Anterior Cruciate Ligament (ACL) and the Patellar Tendon (PT),'' Transactions of the Orthopaedic Research Society, Vol. 3, p. 8. Pradas, M. M., and Calleja, R. D., 990, "Nonlinear Viscoelastic Behavior of the Flexor Tendon of the Human Hand," Journal of Biomechanics, Vol. 23, pp Prager, W., 969, "On the Formulation of Constitutive Equations for Living Soft Tissues," Quarterly of Applied Mathematics, Vol. 27, pp Stouffer, C. D., Butler, D. L., and Hosny, D.', 985, "The Relationship Between Crimp Pattern and Mechanical Response of Human Patellar Tendon- Bone Units," ASME JOURNAL OF BIOMECHANICAL ENGINEERING, Vol. 07, pp Tobolsky, A. V., and Andrews, R. D., 945, "Systems Manifesting Superposed Elastic and Viscous Behavior," Journal of Chemical Physics, Vol. 3, pp Viidik, A., 968, "A Rheological Model for Uncalcified Parallel-Fibred Collagenous Tissue," Journal of Biomechanics, Vol., pp. 3-. Viidik, A., and Ekholm, R., 968, "Light and Electron Microscopy of Collagen Fibers Under Stress," Z. Anat. EntwGesch., Vol. 27, p. 54. Woo, S. L-Y., Gomez, M. A., and Akeson, W. H., 98, "The Time and History-Dependent Viscoelastic Properties of the Canine Medial Collateral Ligament," ASME JOURNAL OF BIOMECHANICAL ENGINEERING, Vol. 03, pp Woo, S. L-Y., 982, "Mechanical Properties of Tendons and Ligaments I. Quasi-Static and Nonlinear Viscoelastic Properties," Biorheology, Vol. 9, pp Woo, S. L-Y., Young, E. P., and Kwan, M. K., 990, "Fundamental Studies in Knee Ligament Mechanics," Knee Ligaments: Structure, Function, Injury, and Repair, D. Danial et al., eds., Raven Press, New York, pp Woo, S. L-Y., and Young, E. P., 99, "Structure and Function of Tendons and Ligaments," Basic Orthopaedic Biomechanics, V. C. Mow and W. C. Hayes, eds., Raven Press, New York, pp Journal of Biomechanical Engineering NOVBER 993, Vol. 5 / 473