ABSTRAcT The stress-strain relations in mammalian tendon are analyzed in terms of the structure and mechanics of its constituents. The model considers the tensile and bending strength of the collagen fibers, the tensile strength of the elastin fibers, and the interaction between the matrix and the collagen fibers. The stress-strain relations are solved through variational considerations by assuming that the fibermaxtrix interactions can be modeled as beam on elastic foundation. The tissue thus modeled is a hyperelastic material. It is further shown that on the basis of the model, the dominant parameters to the tendon's behavior can be evaluated from simple tensile tests. INTRODUCTION

The role of the tendon is to transmit mechanical force. The load-bearing elements in the tendon are collagen fibers. Its efficiency as a force transmission element is exemplified by the very low extensions (1-2%) it undergoes under physiological conditions. Skin on the other hand, undergoes very large physiological deformations (stretch ratio of up to 2.0), whereas its collagen consistency is similar to that of a tendon (Crisp, 1972). The most important reason for this difference is the structure of the collagen fibers in these two tissues: in the tendon they are parallel, nearly straight, and aligned in the direction of applied loads; in the skin they are structured in a three-dimensional wavy array. Hence, the effect of collagen structure on the tissue's function is most important. A most comprehensive review of the structure and function of tendon is given by Elliot (1965) and Crisp (1972). The present work is confined to specific aspects of structure and structure-function relations, namely those which affect the stressstrain relations. The geometry of the collagen fibers in the tendon was described as helical by Lerch (1950), Verzar (1964), Cruise (1958), and Barbenel et al. (1971). Rigby et al. (1959), and more recently, Yannas and Huang (1972) and Diamant et al. (1972) in a most detailed study, determined that the fibers in rat tail tendon are planar and sinusoidally shaped. Evans and Barbenel (1974) suggest that the collagen geometry may be sinusoidal in tail tendon and helical in others. Stress-strain relations in tendons are reversible if strain does not exceed levels of 2-4% (Rigby et al., 1959; Abrahams, 1967; Partington and Wood, 1963). In the reversible range, tendons show marked nonlinear behavior. It is customary to divide the curve into three ranges: the primary range of low strain in which the curve has a

BIOPHYS. J.© Biophysical Society . 0006-3495/78/1101-541 $1.00 Volume 24 November 1978 541-554

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small slope; the secondary range of gradually increasing slope; and the tertiary range of constant, high slope. Gibson and Kenedi (1968) show that the final slope at the tertiary range corresponds to the slope of stress-strain curve of a straight collagen fiber (Morgan, 1960). This is the case with skin and ligamentum flavum as well. From this work and the works of Rigby et al. (1959), Vidiik and Ekholm (1968), Diamant et al. (1972), and Millington et al. (1971), it can be concluded that, at the secondary range, the behavior corresponds to the gradual straightening of the collagen fibers until they are fully straight at the tertiary range. Several models have been proposed for the stress-strain relation of tendons. Two classes seem to be predominant. One is the class of macroscopic, phenomenological models that attempt to approximate experimental results. Thus, Frisen et al. (1969) presented a model consisting of springs, dashpots, and dry friction elements. Hartung (1972) adopted the linear theory of locking materials to nonlinear cases. Haut and Little (1972) used Fung's (1972) quasi-linear viscoelastic concept for rat tail collagen fibers and proposed a simple power law for stress-strain relations under constant rate of stretch. Another class of models are those based on structural considerations. Lerch (1950) proposed a ropelike configuration of the collagen fibers in tendons but did not develop a workable model for stress-strain relations. Diamant et al. (1972) propose a model for stress-strain relations based on their investigations of the structure of the collagen fibers in the tendon. The tendon strength is associated with the bending rigidity of the collagen fibers. The contribution of elastic fibers and matrix-fiber interaction are ignored. The fibers are modeled as zigzag-shaped inflexible hinges. The stress-strain relation is obtained by using the theory of elastica. Their theory agrees with experimental stress-strain data only if the diameter is thick (- 500 nm) compared to the observed amplitude and wavelength (- 200 nm). Under these conditions the theory of elastica used in the model does not apply. Dale et al. (1972) observed the changes of the collagen fiber geometry during strain. By comparing results to theoretical prediction, they concluded that the fiber's shape is a planar sine wave. In a recent report Comninou and Yannas (1976) developed a stress-strain equation for a single, sinusoidally shaped collagen fiber as well as for bundles of fibers embedded in a matrix. The single fiber model results from linearization of Reissner's (1972) one-dimensional finite-strain beam theory. The bundles of fibers in the tendon are modeled as alternating, parallel fibers and matrix layers glued together. It is assumed that initial waviness is small and that the elastic modulus of the matrix is much smaller than that of the fibers. Under these assumptions Bolotin's (1966) analysis of composite material of similar geometry is used to derive the stress-strain law for the tendon. The results are not compared quantitatively to data, but show similarity to experimental curves. Their model of the single fiber corresponds well with morphological evidence. The layered model of fibers and matrix, however, does not agree with observations. Another model in this class is the one proposed by Beskos and Jenkins (1975). 542

BIOPHYSICAL JOURNAL VOLUME 24 1978

The tendon is modeled as an incompressible fiber-reinforced composite with continuously distributed inextensible fibers of helical shape embedded in a hollow right circular cylinder. They solve the stress-strain relations of such a system by using 'the theory of internal constraints which was developed by Ericksen and Rivlin (1954). Since the collagen fibers extensibility is not taken into consideration, the model predicts infinite strength at high strain, which is not the case. In the present work a model for the tendon structure and structure-function relationship is developed. The fiber's geometry and mechanics as well as the interactions between the fibers and the matrix is taken into consideration. Attention is paid to the following questions: (a) How is the observed structure of collagen fibers maintained under equilibrium? (b) What is the function and interplay between various tissue constituents under stretch? (c) What is the effect of the fiber mechanics and geometry on the tendon's stress-strain relations? (d) What are the key parameters dominating the tissue's behavior? (e) How can these parameters be evaluated from simple mechanical tests? THE PRESENT MODEL

Following the works of Diamant et al. (1972), Dale et al (1972), Vidiik (1968), and Yannas and Huang (1972), we shall assume that the collagen fibers in mammalian tendons are parallel, planar, and sinusoidally shaped. It will be further assumed that this observed configuration represents the state of minimum energetic level of this system, which contains the collagen fibers themselves, the matrix in which they are embedded (ground substance), and the elastin fibers. Upon stretching, the energy of the system changes due to the tensile and bending strength of the collagen fibers, the matrix-fiber interaction, and the tensile strength of the elastin fibers. A simple workable model of one section of this system is shown in Fig. 1. Although the role of elastin in undulating the tendon's collagen fibers has not yet been proved, it has been shown by Daly (1969) that in the skin (which resembles the tendon in many ways), and by Wood (1954, Fig. 2) for ligamentum nuchae, the role of elastin is predominant on the mechanical behavior at low ranges of strain. Similar results were obtained for arteries (Roach and Burton, 1957). Consider an element of the collagen fiber (Fig. 2). In equilibrium its shape is given by:

(1) YO= AOsinf8X, where the x-axis coincides with the length of the fiber and I8 = 2wr/A. AO is the COLLAGEN FF

Fin-

-Z

for

ELASTIN

Fe

F

A/2 f

FIGURE 1 A simple model for the basic functional unit of the collagen-elastin system.

Y. LANIR Structure-Strength Relations in Mammalian Tendon

543

V

xo x

X0+AXO X+AX

x

FIGURE 2 Free body diagram of an infinitesimal section of the collagen fiber. F, axial load; Q, shear load; M, moments; qy, intensity of lateral matrix reaction; m, intensity of matrix reactive moments.

initial amplitude and X is the wavelength. Upon stretching, the length changes from ds to ds (1 + e) where e is the tensile strain of the collagen fibers. The additional internal forces in the fiber (F, M, and Q) and the additional reactions of the matrix on the wire (qy and m) are shown in Fig. 2. Comninou and Yannas (1976) analyzed the behavior of a single fiber in a similar manner, but excluded the reactions of the matrix (qy and m). By assuming that the horizontal force in each element of the fiber is constant throughout its length, they formulated equilibrium conditions in the form of a nonlinear second-order differential equation of th,. angle 4 (the angle between the tangent to the collagen fiber at a point and the horizontal x-axis) in terms of the length s. They solve the load-strain relations of the fiber by several approximations based on the fact that the strain e is very small compared to unity. Herrmann et al. (1967) also used this approach in their analysis of the response of reinforcing wires to compressive state of stress. The reaction of the matrix was introduced by considering the wire as a "beam on elastic foundation" where the reaction of the foundation is characterized by foundation constants. It was assumed that the wires' shape remains sinusoidal upon deformation. Since e