Elsevier

Journal of Biomechanics

Volume 33, Issue 9, September 2000, Pages 1069-1077
Journal of Biomechanics

Three-dimensional deformation and stress distribution in an analytical/computational model of the anterior cruciate ligament

https://doi.org/10.1016/S0021-9290(00)00073-7Get rights and content

Abstract

In this study, a constitutive equation for the ACL composite was formulated, and 3-D finite deformations and stress distributions of the ACL were calculated using a finite element method to simulate knee flexion. The mathematical model of the ACL was created by a structurally motivated phenomenological approach. It was assumed that the ACL can be ideally represented as a homogeneous hyperelastic matrix (Mooney–Rivin material) in which two families of densely distributed extensible fibers are embedded; the fibers in one family have a parallel orientation, the other fibers extend radially in eight equally spaced directions. The non-linear stress–strain characteristic exhibited by collagen fibers was represented by a tri-linear curve. Simulation was performed and the results provided some original data; the stress distribution within the ACL body as well as that over the surface, the 3-D defeormations and stress distributions of the ACL viewed from other sides in addition to those from the medial side, and the variations of the stress distribution pattern in the ACL which occured when the tibia was displaced anteriorly or posteriorly.

Introduction

Quantitative in vivo data on the biomechanics of knee ligaments and their behavior under stress are essential for understanding knee ligament injuries, for successful repair or reconstruction of ligaments, and for the design of prosthetic ligaments. To study the detailed mechanics of the ligaments, especially of the anterior cruciate ligament (ACL), accurate measurement of strains on and in the ligament is necessary.

Although studies measuring the material properties of the ACL are extensive (Woo et al., 1983, Woo et al., 1991), measuring strains on and in the ligament presents difficulties. Even under uniaxial tensile testing, there may be no conditions in which all fiber bundles of the ACL are uniformly loaded and/or stretched. The data on ligament properties vary depending upon which ligament geometry is being measured (Butler et al., 1990). Mommersteeg et al. (1995) have shown that the ligament's structural properties vary considerably as a consequence of changes in the configuration of the ligament; they conclude that there is no such thing, functionally speaking, as “ligament stiffness.” Deductions have not yet been made about ligament strains associated with three-dimensional deformations. The non-uniform distribution of strain on the fibers within the ligaments and the effects of the strain on the ligament are not yet fully understood.

One promising approach to overcome these difficulties may be to measure bi (or tri)-axial strains which vary with the ligament's position, even along the direction of fiber run. Until now the transverse and shear effects within the ACL have been considered almost negligible since the ACL is so highly anisotropic. Yet the transverse and shear effects can never be ignored when we take the following into consideration: the complex fiber anatomy and non-uniform structure of the actual ACL. Compared to studies of other tissues, such as skin (Lanir and Fung, 1974; Manschot and Brakkee, 1986), vessels (Vonesh et al., 1997), the myocardium (Demer and Yin, 1983) and muscle cell (Taber, 1991), studies measuring biaxial deformations in the ligaments under specific conditions have been limited in number. As the ACL is an intra-articular ligament possessing a complex three-dimensional shape, direct measurement of the strain distribution over the entire surface has been extremely difficult. Previous studies have been performed on extra-articular ligaments such as the MCL (Weiss et al., 1992) on which the measurement of bi-axial strain is more feasible. Only recently, have studies measured the strain distribution of the entire surface of the ACL using photoelastic method (Yamamoto et al., 1998).

In addition to experiments on actual ligaments, analysis by mathematical model may be effective because a wide variety of conditions can be simulated by means of a model, thus overcoming some of the difficulties in making actual stress/strain measurements. There are extensive studies which have developed constitutive equations derived from microstructural models of the ligaments and tendons (e.g. Woo et al., 1993). Most have been directed toward determining the effects which various components of the tissues have on the tissues mechanical properties or determining the material properties themselves. However, the microstructural approach tends to introduce highly sophisticated formulations which may make it difficult to implement a finite element analysis to obtain the ligament's three-dimensional deformations and the stress/strain distributions associated with knee motion.

Therefore, to obtain specific insight into the geometric deformations and changes in stress distribution of the ACL caused by knee motion, a phenomenological approach (Hirokawa and Tsuruno, 1997) may be more practical than a microstructural one.

We have sought a compromise between mathematical simplicity and structural accuracy (e.g. Humphrey and Yin, 1987). We based our approach on the assumption, supported by histological studies, that the ACL can be idealized as a homogeneous matrix in which two non-interacting families of densely distributed extensible fibers are embedded. A constitutive equation for this ACL composite was formulated. The material properties of the ACL necessary to formulate the constitutive equation were adopted from the literature. Boundary conditions were applied by prescribing the displacements at the boundary nodes corresponding to the insertions. The finite element method was used to simulate the three-dimensional finite deformation and stress distribution wh ich occur in the ACL during knee flexion. The data were converted into images of the ACL and compared with those in previous studies.

Section snippets

Constitutive equation of the ACL

We begin with two major assumptions: first, the ACL body is homogeneous, with no differences in material properties between the insertions and the middle portion. This assumption is based on data from Nordin and Frankel (1980). Second, characteristics dependent on time such as viscosity, creep and relaxation were neglected because they become significant only if the ligament undergoes accelerations greater than those in a quasi-static state.

Determination of input data for the simulation

The material properties of the ACL necessary to solve its constitutive equation were selected from the literature as shown in Table 1. The MPa value of 1000 we assigned to Ke is the fiber modulus for collagen given by Fung (1993). A value of 1 MPa for the matrix modulus was reported by Ault and Hoffman (1992); thus for the ground substance we gave the Mooney–Rivlin constants C1 and C2 values of 1 and 0.1 MPa, respectively. The volume ratio vm:va:vb was made 0.3 : 0.5 : 0.2, based on studies by Wren

Results

All the data were converted into computer-generated images of the ACL. The data are given in terms of stress values instead of elongation or strain values. Hereafter — unless otherwise specified — the values of the longitu dinal (a0 direction in Fig. 1) components of the Kirchhoff stress will be referred to as longitudinal stresses.

Discussion

A number of investigators, when measuring elongation or strain on separate and distinct fibers of the ligaments, have ignored the changes in the ligament's shape. However, the ligament's shape affects the stress it undergoes. Recognizing this, authors of recent reports (Pioletti et al., 1995; Mommersteeg et al., 1996), have measured the lengths of fiber arrays by considering the ligaments mechanically as multi-bundle structures. Mommersteeg et al. (1997) found that at each flexion angle,

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