Abstract-The fundamental objectives of patello-femoral joint biomechanics include the determination of its kinematics and of its dynamics, as a function of given control parameters like knee flexion or applied muscle forces. On the one hand, patellar tracking provides quantitative information about the joint's stability under given loading conditions, whereas patellar force analyses can typically indicate pathological stress distributions asso ciated for instance with abnormal tracking, The determination of this information becomes especially relevant when facing the problem of evaluating surgical procedures in terms of standard (i.e. non-pathological) knee functionality. Classical examples of such procedures include total knee replacement (TKR) and elevation of the tibial tubercle (Maquet's procedure).Following this perspecti ve, the current study was oriented to ward an accurate and reliable determination of the human patella biomechanics during passive knee flexion. To this end, a comprehensive three-dimensional computer model, based on the finite element method, was developed for analyzing articular biomechanics. Unlike previously published studies on patello-femoral biomechanics, this model simultaneously computed the joint's kinematics, associated tendinous and ligamentous forces, articular contact pressures and stresses occurring in the joint during its motion. The components constituting the joint (i.e. bone, cartilage, tendons) were modeled using objective forms of non-linear elastic materials laws. A unilateral contact law allowing for large slip between the patella and the femur was implemented using an augmented Lagrangian formulation, Patellar kinematics computed for two knee specimens were close to equivalent experimental ones (average deviations below 0.5° for the rotations and below 0.5 mm for the translations) and provided validation of the model on a specimen by specimen basis. The ratio between the quadriceps pulling force and the patellar tendon force was less than unity throughout the considered knee flexion range (30-150°), with a minimum near 90° of flexion for both specimens. The contact patterns evolved from the distal part of the retropatellar articular surface to the proximal pole during progressive flexion. The lateral facet bore more pressure than the medial one, with corresponding higher stresses (hydrostatic) in the lateral compartment of the patella. The forces acting on the patella were part of the problem unknowns, thus leading to more realistic loadings for the stress analysis, which was especially important when considering the wide range of variations of the contact pressure acting on the patella during knee flexion.
The development of normal joints depends on mechanical function in utero. Experimental studies have shown that the normal surface topography of diarthrodial joints fails to form in paralyzed embryos. We implemented a mathematical model for joint morphogenesis that explores the hypothesis that the stress distribution created in a functional joint may modulate the growth of the cartilage anlagen and lead to the development of congruent articular surfaces. We simulated the morphogenesis of a human finger joint (proximal interphalangeal joint) between days 55 and 70 of fetal life. A baseline biological growth rate was defined to account for the intrinsic biological influences on the growth of the articulating ends of the anlagen. We assumed this rate to be proportional to the chondrocyte density in the growing tissue. Cyclic hydrostatic stress caused by joint motion was assumed to modulate the baseline biological growth, with compression slowing it and tension accelerating it. Changes in the overall shape of the joint resulted from spatial differences in growth rates throughout the developing chondroepiphyses. When only baseline biological growth was included, the two epiphyses increased in size but retained convex incongruent joint surfaces. The inclusion of mechanobiological-based growth modulation in the chondroepiphyses led to one convex joint surface, which articulated with a locally concave surface. The articular surfaces became more congruent, and the anlagen exhibited an asymmetric sagittal profile similar to that observed in adult phalangeal bones. These results are consistent with the hypothesis that mechanobiological influences associated with normal function play an important role in the regulation of joint morphogenesis.
An augmented Lagrangian formulation is proposed for large-slip frictionless contact problems between deformable discretized bodies in two dimensions. Starting from a finite element discretization of the two bodies, a node-on-facet element is defined. A non-linear gap vector and its first variation are derived in terms of the nodal displacements. The relevant action and reaction principle is stated. The gap distance is then related to the conjugate pressure by a (multivalued non-differentiable) unilateral contact law. The resulting inequality constrained minimization problem is transformed into an unconstrained saddle point problem using an augmented Lagrangian function. Large slip over several facets is possible and the effects of target convexity or concavity are investigated. A generalized Newton method is used to solve the resulting piecewise differentiable equations necessary for equilibrium and contact. The proper tangent (Jacobian) matrices are calculated. The primal (displacements) and dual (contact forces) unknowns are simultaneously updated at each iteration.
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