Primary cilia act as mechanosensors during bone healing around an implant.

The primary cilium is an organelle that senses cues in a cell's local environment. Some of these cues constitute molecular signals; here, we investigate the extent to which primary cilia can also sense mechanical stimuli. We used a conditional approach to delete Kif3a in pre-osteoblasts and then employed a motion device that generated a spatial distribution of strain around an intra-osseous implant positioned in the mouse tibia. We correlated interfacial strain fields with cell behaviors ranging from proliferation through all stages of osteogenic differentiation. We found that peri-implant cells in the Col1Cre;Kif3a(fl/fl) mice were unable to proliferate in response to a mechanical stimulus, failed to deposit and then orient collagen fibers to the strain fields caused by implant displacement, and failed to differentiate into bone-forming osteoblasts. Collectively, these data demonstrate that the lack of a functioning primary cilium blunts the normal response of a cell to a defined mechanical stimulus. The ability to manipulate the genetic background of peri-implant cells within the context of a whole, living tissue provides a rare opportunity to explore mechanotransduction from a multi-scale perspective.

A biphasic finite element model of in vitro plowing tests of the temporomandibular joint disc.

Disorders of the temporomandibular joint (TMJ) afflict 3-29% of people aged 19-40 years. Degenerative joint disease (DJD) of the TMJ generally occurs 15 years earlier than in other human joints and 1.5-2 times more often in women than men. The TMJ disc is the primary stress distribution mechanism within the joint. Mechanical failure of the TMJ disc precedes clinical signs of DJD. Unlike postcranial synovial joints, biomimetic replacements of the disc have not been successful, probably due to the paucity of knowledge about TMJ biomechanics. Translation of stress-fields mediolaterally across the TMJ disc may lead to fatigue failure because of the effect of traction forces on the tissue surface and because the disc is relatively weak in this aspect. Traction forces are composed of friction forces, which are known to be low in the TMJ, and plowing forces which are relatively much higher and result from movement and pressurization of fluids within the tissues due to translating surface loads. In the in vitro plowing experiment, a rigid curve-ended indenter is lowered into a TMJ disc that has been mounted on a stage with pressure gauges, and the indenter is then translated in a prescribed mediolateral motion that is intended to simulate the motion of the mandibular condyle on the TMJ disc in vivo. As a first step, these plowing experiments have quantified the variables thought to be important in tissue failure. A next step is to define the full role of these variables in the pathomechanics of TMJ disc tissue through a validated model. Therefore, the aim of this study was to develop and test a finite element model of the plowing experiments based on an orthotropic biphasic description of the soft tissue behavior of the TMJ disc. For this plowing model, the arbitrary Lagrange Eulerian method was used to approximate the moving load problem, where in vitro the indenter slid along the tissue's superior surface. Approximate validation of the plowing model was based on comparisons of model-predicted temporal and spatial distribution of indenter displacement and total normal stresses (+/-15%) and laboratory measurements during one complete cycle of plowing motion. Other useful predictions from the plowing model include spatial and temporal distributions of biomechanical variables of interest that cannot be measured experimentally, such as total stress, pressure, strain, and the relative significance of the orthotropic solid phase properties.

An evaluation of three-dimensional diarthrodial joint contact using penetration data and the finite element method.

We have developed an approximate method for simulating the three-dimensional contact of soft biphasic tissues in diarthrodial joints under physiological loading. Input to the method includes: (i) kinematic information describing an in vitro joint articulation, measured while the cartilage is deformed under physiological loads, (ii) geometric properties for the relaxed (undeformed) cartilage layers, obtained for the analyses in this study via stereophotogrammetry, and (iii) material parameters for the biphasic constitutive relations used to represent cartilage. Solid models of the relaxed tissue layers are assembled in physiological positions, resulting in a mathematical overlap of the cartilage layers. The overlap distribution is quantified and converted via the biphasic governing equations into applied traction boundary conditions for both the solid and fluid phases for each of the contacting layers. Linear, biphasic, three-dimensional, finite element analysis is performed using the contact boundary conditions derived for each of the contacting layers. The method is found to produce results consistent with the continuity requirements of biphasic contact. Comparison with results from independent, biphasic contact analyses of axisymmetric problems shows that the method slightly underestimates the contact area, leading to an overestimation of the total traction, but yields a good approximation to elastic stress and solid phase displacement.

A mixed-penalty biphasic finite element formulation incorporating viscous fluids and material interfaces.

The fluid viscosity term of the fluid phase constitutive equation and the interface boundary conditions between biphasic, solid and fluid domains have been incorporated into a mixed-penalty finite element formulation of the linear biphasic theory for hydrated soft tissue. The finite element code can now model a single-phase viscous incompressible fluid, or a single-phase elastic solid, as limiting cases of a biphasic material. Interface boundary conditions allow the solution of problems involving combinations of biphasic, fluid and solid regions. To incorporate these conditions, the volume-weighted mixture velocity is introduced as a degree of freedom at interface nodes so that the kinematic continuity conditions are satisfied by conventional finite element assembly techniques. Results comparing our numerical method with an independent, analytic solution for the problem of Couette flow over rigid and deformable porous biphasic layers show that the finite element code accurately predicts the viscous fluid flows and deformation in the porous biphasic region. Thus, the analysis can be used to model the interface between synovial fluid and articular cartilage in diarthrodial joints. This is an important step toward modeling and understanding the mechanisms of joint lubrication and another step toward fully modeling the in vivo behavior of a diarthrodial joint.

Contact analysis of biphasic transversely isotropic cartilage layers and correlations with tissue failure.

Failure of articular cartilage has been investigated experimentally and theoretically, but there is only partial agreement between observed failure and predicted regions of peak stresses. Since trauma and repetitive stress are implicated in the etiopathogenesis of osteoarthritis, it is important to develop cartilage models which correctly predict sites of high stresses. Cartilage is anisotropic and inhomogeneous, though it has been difficult to incorporate these complexities into engineering analyses. The objectives of this study are to demonstrate that a transversely isotropic, biphasic model of cartilage can provide agreement between predicted regions of high stresses and observed regions of cartilage failure and that with transverse isotropy cartilage stresses are more sensitive to convexity and concavity of the surfaces than with isotropy. These objectives are achieved by solving problems of diarthrodial joint contact by the finite-element method. Results demonstrate that transversely isotropic models predict peak stresses at the cartilage surface and the cartilage-bone interface, in agreement with sites of fissures following impact loading; isotropic models predict peak stresses only at the cartilage-bone interface. Also, when convex cartilage layers contacted concave layers in this study, the highest tensile stresses occur in the convex layer for transversely isotropic models; no such differences are found with isotropic models. The significance of this study is that it establishes a threshold of modeling complexity for articular cartilage that provides good agreement with experimental observations under impact loading and that surface curvatures significantly affect stress and strain within cartilage when using a biphasic transversely isotropic model.

Indentation analysis of biphasic articular cartilage: nonlinear phenomena under finite deformation.

The nonlinear indentation response of hydrated articular cartilage at physiologically relevant rates of mechanical loading is studied using a two-phase continuum model of the tissue based on the theory of mixtures under finite deformation. The matrix equations corresponding to the governing mixture equations for this nonlinear problem are derived using a total Lagrangian penalty finite element method, and solved using a predictor-corrector iteration within a modified Newton-Raphson scheme. The stress relaxation indentation problem is examined using either a porous (free draining) indenter or solid (impermeable) indenter under fast and slow compression rates. The creep indentation problem is studied using a porous indenter. We examine the finite deformation response and compare with the response obtained using the linear infinitesimal response. Differences between the finite deformation response and the linear response are shown to be significant when the compression rate is fast or when the indenter is impermeable. The finite deformation model has a larger ratio of peak-to-equilibrium reaction force, and higher relaxation rate than the linear model during the early relaxation period, but a similar relaxation time. The finite deformation model predicts a slower creep rate than the linear model, as well as a smaller equilibrium creep displacement. The pressure distribution below the indenter, particularly near the loaded surface is also larger with the finite deformation model.

Biomechanics of diarthrodial joints: a review of twenty years of progress.

A survey of some of the advances made over the past twenty years in understanding diarthrodial joint biomechanics is presented. Topics covered in this review include: biotribology (i.e., friction, lubrication and wear of diarthrodial joints); contact area determinations; stereophotogrammetric rendering of articular surfaces; deformational field analysis using canonical problems; and finite element formulations for both infinitesimal and finite deformations of biphasic materials and precise anatomic surfaces. Suggestions are made for future research directions as well.

A transversely isotropic biphasic finite element model of the meniscus.

A finite element model of the meniscus is presented, based on an axisymmetric geometric approximation of the menisci and a biphasic description of the tissue as a mixture of solid and fluid components. The highly fibrous nature of the meniscal tissue is accounted for by using a fiber-reinforced, transversely isotropic description of the solid phase. This model is used to study the response of a meniscus resting on a perfectly lubricated tibial surface and subjected to distributed loads applied to the femoral surface, and to examine the effects of changes in loading conditions at the femoral and tibial interfaces. Quantities of interest include the stress, pressure and strain distributions at discrete times early in the meniscal response, and the flow of the fluid phase relative to the solid phase. Of particular interest are regions of large tensile strains which could lead to meniscal failures such as the bucket-handle tear. We show that all components of strain are positive in regions of the outer third of the meniscus, and that the maximum tensile strain perpendicular to the circumferentially arranged fibers (largest principal strain in the axisymmetric cross section) is positive throughout most of the cross section. Changing the partition of the load on the femoral surface and the permeability at the tibial surface changes the time-dependent response, but has little effect on the strain distributions at times of the order of 5 s considered in this study. The inclusion of a transversely isotropic, fibrous representation of the solid phases is shown to be essential to proper meniscal simulation. The results demonstrate the importance of the biphasic representation since the fluid phase is shown to carry a significant part of the applied load.

A finite element analysis of the indentation stress-relaxation response of linear biphasic articular cartilage.

The indentation problem of a thin layer of hydrated soft tissue such as cartilage or meniscus by a circular plane-ended indenter is investigated. The tissue is represented by a biphasic continuum model consisting of a solid phase (collagen and proteoglycan) and a fluid phase (interstitial water). A finite element formulation of the linear biphasic continuum equations is used to solve an axisymmetric approximation of the indentation problem. We consider stress-relaxation problems for which analytic solution is intractable; where the indenter is impermeable (solid) and/or when the interface between the indenter and tissue is perfectly adhesive. Thicknesses corresponding to a thin and thick specimen are considered to examine the effects of tissue thickness. The different flow, pressure, stress and strain fields which are predicted within the tissue, over time periods typically used in the mechanical testing of soft tissues, will be presented. Results are compared with the case of a porous free-draining indenter with a perfectly lubricated tissue-indenter interface, for which an analytic solution is available, to show the effects of friction at the tissue-indenter interface, and the effects of an impermeable indenter. While these effects are present for both thin and thick tissues, they are shown to be more significant for the thin tissue. We also examine the effects of the stiffness of the subchondral bone on the response of the soft tissue and demonstrate that the subchondral bone substrate can be modeled as a rigid, impermeable boundary. The effects of a curved tissue-subchondral bone interface, and the early time response are also studied. For physiologically reasonable levels of curvature, we will show that the curved tissue-subchrondal bone interface has negligible influence on the tissue response away from the interface. In addition, the short-time stress-relaxation responses of the tissue (e.g., at times less than 1s) demonstrate the essential role of the fluid phase in supporting the load applied to the tissue, and by extrapolation to shorter times characteristics of normal joint motion, suggest the essential role of a biphasic model in representing soft tissue behavior in joint response.

Finite element methods for the biomechanics of soft hydrated tissues: nonlinear analysis and adaptive control of meshes.

This chapter addresses computationally demanding numerical formulations in the biomechanics of soft tissues. The theory of mixtures can be used to represent soft hydrated tissues in the human musculoskeletal system as a two-phase continuum consisting of an incompressible solid phase (collagen and proteoglycan) and an incompressible fluid phase (interstitial water). We first consider the finite deformation of soft hydrated tissues in which the solid phase is represented as hyperelastic. A finite element formulation of the governing nonlinear biphasic equations is presented based on a mixed-penalty approach and derived using the weighted residual method. Fluid and solid phase deformation, velocity, and pressure are interpolated within each element, and the pressure variables within each element are eliminated at the element level. A system of nonlinear, first-order differential equations in the fluid and solid phase deformation and velocity is obtained. In order to solve these equations, the contributions of the hyperelastic solid phase are incrementally linearized, a finite difference rule is introduced for temporal discretization, and an iterative scheme is adopted to achieve equilibrium at the end of each time increment. We demonstrate the accuracy and adequacy of the procedure using a six-node, isoparametric axisymmetric element, and we present an example problem for which independent numerical solution is available. Next, we present an automated, adaptive environment for the simulation of soft tissue continua in which the finite element analysis is coupled with automatic mesh generation, error indicators, and projection methods. Mesh generation and updating, including both refinement and coarsening, for the two-dimensional examples examined in this study are performed using the finite quadtree approach. The adaptive analysis is based on an error indicator which is the L2 norm of the difference between the finite element solution and a projected finite element solution. Total stress, calculated as the sum of the solid and fluid phase stresses, is used in the error indicator. To allow the finite difference algorithm to proceed in time using an updated mesh, solution values must be transferred to the new nodal locations. This rezoning is accomplished using a projected field for the primary variables. The accuracy and effectiveness of this adaptive finite element analysis is demonstrated using a linear, two-dimensional, axisymmetric problem corresponding to the indentation of a thin sheet of soft tissue. The method is shown to effectively capture the steep gradients and to produce solutions in good agreement with independent, converged, numerical solutions.

Effects of friction on the unconfined compressive response of articular cartilage: a finite element analysis.

A finite element analysis is used to study a previously unresolved issue of the effects of platen-specimen friction on the response of the unconfined compression test; effects of platen permeability are also determined. The finite element formulation is based on the linear KLM biphasic model for articular cartilage and other hydrated soft tissues. A Galerkin weighted residual method is applied to both the solid phase and the fluid phase, and the continuity equation for the intrinsically incompressible binary mixture is introduced via a penalty method. The solid phase displacements and fluid phase velocities are interpolated for each element in terms of unknown nodal values, producing a system of first order differential equations which are solved using a standard numerical finite difference technique. An axisymmetric element of quadrilateral cross-section is developed and applied to the mechanical test problem of a cylindrical specimen of soft tissue in unconfined compression. These studies show that interfacial friction plays a major role in the unconfined compression response of articular cartilage specimens with small thickness to diameter ratios.

Material constants for a finite element model of the intervertebral disk with a fiber composite annulus.

A simple axisymmetric finite element model of a human spine segment containing two adjacent vertebrae and the intervening intervertebral disk was constructed. The model incorporated four substructures: one to represent each of the vertebral bodies, the annulus fibrosus, and the nucleus pulposus. A semi-analytic technique was used to maintain the computational economies of a two-dimensional analysis when nonaxisymmetric loads were imposed on the model. The annulus material was represented as a layered fiber-reinforced composite. This paper describes the selection of material constants to represent the anisotropic layers of the annulus. It shows that a single set of material constants can be chosen so that model predictions of gross disk behavior under compression, torsion, shear, and moment loading are in reasonable agreement with the mean and range of experimentally measured disk behaviors. It also examines the effects of varying annular material properties.

Mechanical response of a simple finite element model of the intervertebral disc under complex loading.

A simple axisymmetric finite element model of a human spine segment containing two adjacent vertebrae and the intervening intervertebral disc was constructed. The bodies and disc were modeled by three substructures; one to represent each of the vertebral bodies, the annulus fibrosus, and the nucleus pulposus. A semi-analytic technique was used to maintain the computational economies of a two-dimensional analysis when non- axisymmetric loads were imposed on the model. The response of the model to compression, shear, torsion and bending loads applied to the superior vertebral body was examined to determine the effects of disc geometry and material properties on response. Comparisons of model responses with experimentally measured responses were made to estimate material property values for which model behaviors are in agreement with measured behaviors.