LePing Li

Nonlinear analysis of cartilage in unconfined ramp compression using a fibril reinforced poroelastic model

OBJECTIVE To develop a biomechanical model for cartilage which is capable of capturing experimentally observed nonlinear behaviours of cartilage and to investigate effects of collagen fibril reinforcement in cartilage. DESIGN A sequence of 10 or 20 steps of ramp compression/relaxation applied to cartilage disks in uniaxial unconfined geometry is simulated for comparison with experimental data. BACKGROUND Mechanical behaviours of cartilage, such as the compression-offset dependent stiffening of the transient response and the strong relaxation component, have been previously difficult to describe using the biphasic model in unconfined compression. METHODS Cartilage is modelled as a fluid-saturated solid reinforced by an elastic fibrillar network. The latter, mainly representing collagen fibrils, is considered as a distinct constituent embedded in a biphasic component made up mainly of proteoglycan macromolecules and a fluid carrying mobile ions. The Youngs modulus of the fibrillar network is taken to vary linearly with its tensile strain but to be zero for compression. Numerical computations are carried out using a finite element procedure, for which the fibrillar network is discretized into a system of spring elements. RESULTS The nonlinear fibril reinforced poroelastic model is capable of describing the strong relaxation behaviour and compression-offset dependent stiffening of cartilage in unconfined compression. Computational results are also presented to demonstrate unique features of the model, e.g. the matrix stress in the radial direction is changed from tensile to compressive due to presence of distinct fibrils in the model. RELEVANCE Experimentally observed nonlinear behaviours of cartilage are successfully simulated, and the roles of collagen fibrils are distinguished by using the proposed model. Thus this study may lead to a better understanding of physiological responses of individual constituents of cartilage to external loads, and of the roles of mechanical loading in cartilage remodelling and pathology.

A fibril reinforced nonhomogeneous poroelastic model for articular cartilage: inhomogeneous response in unconfined compression

The depth dependence of material properties of articular cartilage, known as the zonal differences, is incorporated into a nonlinear fibril-reinforced poroelastic model developed previously in order to explore the significance of material heterogeneity in the mechanical behavior of cartilage. The material variations proposed are based on extensive observations. The collagen fibrils are modeled as a distinct constituent which reinforces the other two constituents representing proteoglycans and water. The Youngs modulus and Poissons ratio of the drained nonfibrillar matrix are so determined that the aggregate compressive modulus for confined geometry fits the experimental data. Three nonlinear factors are considered, i.e. the effect of finite deformation, the dependence of permeability on dilatation and the fibril stiffening with its tensile strain. Solutions are extracted using a finite element procedure to simulate unconfined compression tests. The features of the model are then demonstrated with an emphasis on the results obtainable only with a nonhomogeneous model, showing reasonable agreement with experiments. The model suggests mechanical behaviors significantly different from those revealed by homogeneous models: not only the depth variations of the strains which are expected by qualitative analyses, but also, for instance, the relaxation-time dependence of the axial strain which is normally not expected in a relaxation test. Therefore, such a nonhomogeneous model is necessary for better understanding of the mechanical behavior of cartilage.

Strain-rate dependent stiffness of articular cartilage in unconfined compression

The stiffness of articular cartilage is a nonlinear function of the strain amplitude and strain rate as well as the loading history, as a consequence of the flow of interstitial water and the stiffening of the collagen fibril network. This paper presents a full investigation of the interplay between the fluid kinetics and fibril stiffening of unconfined cartilage disks by analyzing over 200 cases with diverse material properties. The lower and upper elastic limits of the stress (under a given strain) are uniquely established by the instantaneous and equilibrium stiffness (obtained numerically for finite deformations and analytically for small deformations). These limits could be used to determine safe loading protocols in order that the stress in each solid constituent remains within its own elastic limit. For a given compressive strain applied at a low rate, the loading is close to the lower limit and is mostly borne directly by the solid constituents (with little contribution from the fluid). In contrast, however in case of faster compression, the extra loading is predominantly transported to the fibrillar matrix via rising fluid pressure with little increase of stress in the nonfibrillar matrix. The fibrillar matrix absorbs the loading increment by self-stiffening: the quicker the loading the faster the fibril stiffening until the upper elastic loading limit is reached. This self-protective mechanism prevents cartilage from damage since the fibrils are strong in tension. The present work demonstrates the ability of the fibril reinfored poroelastic models to describe the strain rate dependent behavior of articular cartilage in unconfined compression using a mechanism of fibril stiffening mainly induced by the fluid flow.

Three-dimensional fibril-reinforced finite element model of articular cartilage.

Collagen fiber orientations in articular cartilage are tissue depth-dependent and joint site-specific. A realistic three-dimensional (3D) fiber orientation has not been implemented in modeling fluid flow-dependent response of articular cartilage; thus the detailed mechanical role of the collagen network may have not been fully understood. In the present study, a previously developed fibril-reinforced model of articular cartilage was extended to account for the 3D fiber orientation. A numerical procedure for the material model was incorporated into the finite element code ABAQUS using the “user material” option. Unconfined compression and indentation testing was evaluated. For indentation testing, we considered a mechanical contact between a solid indenter and a medial femoral condyle, assuming fiber orientations in the surface layer to follow the split-line pattern. The numerical results from the 3D modeling for unconfined compression seemed reasonably to deviate from that of axisymmetric modeling. Significant fiber orientation dependence was observed in the displacement, fluid pressure and velocity for the cases of moderate strain-rates, or during early relaxation. The influence of fiber orientation diminished at static and instantaneous compressions.

Strain-rate dependence of cartilage stiffness in unconfined compression: the role of fibril reinforcement versus tissue volume change in fluid pressurization

The strain and strain-rate-dependent response of articular cartilage in unconfined compression was studied theoretically. The transient stress and stiffness of cartilage were determined for strain rates ranging from zero to infinity. It is shown, for a given compressive strain, that the axial stress initially increases quickly as a function of strain rate, and then increases progressively more slowly towards the stress corresponding to the instantaneous response. The volume change of the tissue does not give its transient stiffness uniquely, because of the strong strain-rate dependence. The variation of tissue stiffness is primarily determined by the transient stiffness of the radial fibrils. Load sharing between the solid matrix and fluid pressurization also depends on the strain rate. At 15% axial compression, the matrix bears more than 80% of the applied load at a strain rate of 0.005%/s, while the fluid pressurization contributes more than 80% of the load at a strain rate of 0.15%/s. These results show the interplay between fibril reinforcement and fluid pressurization in articular cartilage: the fluid drives fibril stiffening which in turn produces high pore pressure at high strain rates. As a secondary objective of the present work, a fibrillar continuum element was formulated to replace the fibrillar spring element used previously in fibril-reinforced modeling, in order to eliminate the deformation incompatibility between the spring system and the nonfibrillar matrix. The results obtained using the two fibrillar elements were compared with the closed-form solutions for the static and instantaneous responses for the case of large deformation. It was found for unconfined compression that using the spring elements did not generally result in greater numerical errors than using the fibrillar continuum elements.

A human knee joint model considering fluid pressure and fiber orientation in cartilages and menisci

Articular cartilages and menisci are generally considered to be elastic in the published human knee models, and thus the fluid-flow dependent response of the knee has not been explored using finite element analysis. In the present study, the fluid pressure and site-specific collagen fiber orientation in the cartilages and menisci were implemented into a finite element model of the knee using fibril-reinforced modeling previously proposed for articular cartilage. The geometry of the knee was obtained from magnetic resonance imaging of a healthy young male. The bones were considered to be elastic due to their greater stiffness compared to that of the cartilages and menisci. The displacements obtained for fast ramp compression were essentially same as those for instantaneous compression of equal magnitude with the fluid being trapped in the tissues, which was expected. However, a clearly different pattern of displacements was predicted by an elastic model using a greater Youngs modulus and a Poissons ratio for nearly incompressible material. The results indicated the influence of fluid pressure and fiber orientation on the deformation of articular cartilage in the knee. The fluid pressurization in the femoral cartilage was somehow affected by the site-specific fiber directions. The peak fluid pressure in the femoral condyles was reduced by three quarters when no fibril reinforcement was assumed. The present study indicates the necessity of implementing the fluid pressure and anisotropic fibril reinforcement in articular cartilage for a more accurate understanding of the mechanics of the knee.

Creep behavior of the intact and meniscectomy knee joints

The mechanical functions of the menisci may be partially performed through the fluid pressurization in articular cartilages and menisci. This creep behavior has not been investigated in whole knee joint modeling. A three-dimensional finite element knee model was employed in the present study to explore the fluid-flow dependent creep behaviors of normal and meniscectomy knees. The model included distal femur, tibia, fibula, articular cartilages, menisci and four major ligaments. Articular cartilage or meniscus was modeled as a fluid-saturated solid matrix reinforced by a nonlinear orthotropic and site-specific collagen network. A 300 N compressive force, equal to half of body weight, was applied to the knee in full extension followed by creep. The results showed that the fluid pressurization played a substantial role in joint contact mechanics. Menisci bore more loading as creep developed, leading to decreased stresses in cartilages. The removal of menisci not only changed the stresses in the cartilages, which was in agreement with published studies, but also altered the distribution and the rate of dissipation of fluid pressure in the cartilages. The high fluid pressures in the femoral cartilage moved from anterior to more central regions of the condyles after total meniscectomy. For both intact and meniscectomy joints, the fluid pressure level remained considerably high for thousands of seconds during creep, which lasted even longer after meniscectomy. For the femoral cartilage, the maximum principal stress was generally in agreement with the fiber direction, which indicated the essential role of fibers in load support of the tissue.

Recent advances in computational mechanics of the human knee joint.

Computational mechanics has been advanced in every area of orthopedic biomechanics. The objective of this paper is to provide a general review of the computational models used in the analysis of the mechanical function of the knee joint in different loading and pathological conditions. Major review articles published in related areas are summarized first. The constitutive models for soft tissues of the knee are briefly discussed to facilitate understanding the joint modeling. A detailed review of the tibiofemoral joint models is presented thereafter. The geometry reconstruction procedures as well as some critical issues in finite element modeling are also discussed. Computational modeling can be a reliable and effective method for the study of mechanical behavior of the knee joint, if the model is constructed correctly. Single-phase material models have been used to predict the instantaneous load response for the healthy knees and repaired joints, such as total and partial meniscectomies, ACL and PCL reconstructions, and joint replacements. Recently, poromechanical models accounting for fluid pressurization in soft tissues have been proposed to study the viscoelastic response of the healthy and impaired knee joints. While the constitutive modeling has been considerably advanced at the tissue level, many challenges still exist in applying a good material model to three-dimensional joint simulations. A complete model validation at the joint level seems impossible presently, because only simple data can be obtained experimentally. Therefore, model validation may be concentrated on the constitutive laws using multiple mechanical tests of the tissues. Extensive model verifications at the joint level are still crucial for the accuracy of the modeling.

Alterations in Mechanical Behaviour of Articular Cartilage due to Changes in Depth Varying Material Properties--a Nonhomogeneous Poroelastic Model Study

The depth dependence of the material properties is present in normal adult cartilage and is believed to have significant implications in its normal mechanical function. Cartilage pathology may alter the depth dependence, e.g. a reduced depth dependence of the fibril stiffness has been observed in osteoarthritic cartilage. The objective of the present study is to investigate the alterations in the mechanical response of articular cartilage when the depth dependence of the material properties is varied to simulate healthy and pathological situations. This study is made possible by a recently developed nonhomogeneous poroelastic model. Depth variations of the strains and stresses for individual material phases (collagen, proteoglycan and fluid) are obtained for cartilage disks in unconfined compression using the finite element method. The mean nominal axial strain considered is up to 15%, while the axial strain at the articular surface can reach 33%. This paper demonstrates how the mechanical behaviours of cartilage are affected by individual depth dependent cartilage properties, while such observations are not fully available in experimental investigations. This study suggests the possibility of diagnosing cartilage health by analysing its mechanical behaviours.

Partial Meniscectomy Changes Fluid Pressurization in Articular Cartilage in Human Knees

Partial meniscectomy is believed to change the biomechanics of the knee joint through alterations in the contact of articular cartilages and menisci. Although fluid pressure plays an important role in the load support mechanism of the knee, the fluid pressurization in the cartilages and menisci has been ignored in the finite element studies of the mechanics of meniscectomy. In the present study, a 3D fibril-reinforced poromechanical model of the knee joint was used to explore the fluid flow dependent changes in articular cartilage following partial medial and lateral meniscectomies. Six partial longitudinal meniscectomies were considered under relaxation, simple creep, and combined creep loading conditions. In comparison to the intact knee, partial meniscectomy not only caused a substantial increase in the maximum fluid pressure but also shifted the location of this pressure in the femoral cartilage. Furthermore, these changes were positively correlated to the size of meniscal resection. While in the intact joint, the location of the maximum fluid pressure was dependent on the loading conditions, in the meniscectomized joint the location was predominantly determined by the site of meniscal resection. The partial meniscectomy also reduced the rate of the pressure dissipation, resulting in even larger difference between creep and relaxation times as compared to the case of the intact knee. The knee joint became stiffer after meniscectomy because of higher fluid pressure at knee compression followed by slower pressure dissipation. The present study indicated the role of fluid pressurization in the altered mechanics of meniscectomized knees.