The influence of geometry on the stress distribution in joints — a finite element analysis

The incongruity of human joints is a phenomenon which has long been recognized, and recent CT-osteoabsorptiometric findings suggest that this incongruity influences the distribution of stress in joints during their normal physiological use. The finite element method (FEM) was therefore applied to five different geometric configurations consistent with the anatomy of articular surfaces, and a program with variable contact areas (Marc) was used to calculate the stress distribution for loads of 100 to 6 900 N. The assumption of congruity between head and socket results in a “bell-shaped” distribution of stress with a maximum value of 61.5 N/mm² in the depths of the socket, decreasing towards zero at its edges. In the model with a flatter socket the von Mises stresses are higher (max. 101.3 N/mm²); with a deeper socket, lower (max. 53.0 N/mm²). If the diameter of the head is greater, the stresses build up from the periphery of the socket and move towards its depths as the load increases. The combination of an oversized head and a deeper socket results in the most satisfactory stress distribution (max. 43.2 N/mm²). These results extend previous photoelastic findings with incongruous joint surfaces. The calculated mechanical conditions show a relationship to the location of osteoarthritic changes, and are reflected by the distribution pattern of subchondral bone density. A more satisfactory stress distribution is found with functionally advantageous, incongruous joint surfaces (oversized head and deepened socket) than in the congruous joint, and a better nutritive situation for the articular cartilage seems likely. The geometry of the joint is therefore a physiologically important and quantifiable factor contributing to an optimized transmission of forces in joints.     

Morphomechanics of the humero-ulnar joint: II. Concave incongruity determines the distribution of load and subchondral mineralization

Background: A deeper joint socket (concave incongruity) is found at most angles of flexion of the humero-ulnar joint and maintained over a wide range of physiological loading. It is, however, unclear how far this incongruity affects the distribution of load and subchondral mineralization of this joint as compared with a congruous configuration. Methods: Two nonlinear, axisymmetrical finite element models with two cartilage layers were constructed, one congruous and one incongruous, with a joint space of realistic magnitude. The distribution of subchondral mineralization was determined by computed tomography osteoabsorptiometry in the same six specimens that were investigated in the first part of the study, and compared with the biomechanical data obtained there and the predictions of the models. Results: In the congruous case, the center of the socket is highly loaded, whereas the periphery does not experience mechanical stimulation. A central bone density maximum is predicted. With concave incongruity the position of the contact areas shifts from the joint margin towards the center as the load increases, and the peak stresses are considerably lower. A bicentric ventro-dorsal distribution pattern of subchondral mineralization is predicted, and this is actually found in the six specimens. Conclusions: Concave incongruity is shown to determine load transmission and subchondral mineralization of the humero-ulnar joint. It is suggested that this shape leads to a more even distribution of stress, provides intermittent stimulation of the cartilaginous tissue, and has beneficial effects on the metabolism, nutrition, and lubrication of the articular cartilage during cyclic loading. © 1995 Wiley-Liss, Inc.     

Physiological incongruity of the humero-ulnar joint: a functional principle of optimized stress distribution acting upon articulating surfaces?

Investigations into the distribution of subchondral bone density in the human elbow have suggested that the geometry of the trochlear notch deviates from a perfect fit with the trochlea, and that the load is transmitted ventrally and dorsally rather than through the centre of the humero-ulnar joint. We therefore decided to make a quantitative assessment of the degree of incongruity between the two components in 15 human specimens (age distribution 60 to 93 years) with different types of joint surface. Polyether casts of the joint cavity were prepared under loads of 10,40,160 and 640 N. The thickness of the casts was then measured at 50 predetermined points, and an area distribution of the width of the joint space represented in a two-dimensional template of the trochlear notch. The reproducibility of this procedure was tested by image analysis. At a load of 10 N, only a narrow space was present ventrally and dorsally in the joint, but in the depths of the trochlear notch a width of 0.5 to 1 mm was recorded in the centre, and up to 3 mm at its medial and lateral edges. Specimens with continuous articular cartilage showed a lower degree of incongruity than those with a divided articular surface. As the load was increased to 640 N, however, the original incongruity between the articular surfaces disappeared almost completely. The joint surfaces became more congruous, probably because of the viscoelastic properties of the articular cartilage and the subchondral bone, and the contact areas merged in the centre of the joint. It is suggested that this physiological incongruity brings about an optimal distribution of stress over the articular surface during the transmission of the load, and it may lead to better nourishment of the articular cartilage by providing intermittent mechanical stimulation and circulation of the synovial fluid.     

Morphomechanics of the humero-ulnar joint: I. Joint space width and contact areas as a function of load and flexion angle

Background: Previous studies have shown that the trochlear notch is deeper than necessary for an exact fit with the humerus. However, humero-ulnar joint space width and contact areas have so far not been quantified for variations in the load and angle of flexion. Methods: Six fresh cadaveric specimens were investigated at 30°, 60°, 90°, and 120° of flexion and at loads of 25 and 500 N, simulating resisted elbow extension. The joint space width and contact were determined, using polyether casting material. Results: At 25 N all joints made contact in the ventral and dorsal aspects of the articular surfaces, whereas in the depth of the trochlear notch the joint space was on average between 0.3 and 2.8 mm wide, with some variation between individuals. At 500 N the joint space width was considerably reduced and the contact areas expanded towards the depth of the notch. The size of the dorsal contact areas was significantly smaller at 30° and that of the ventral ones at 120°, their ventro-dorsal ratio decreasing considerably from 30° to 120° (P< 0.01). Conclusion: These results indicate that the size of the contact areas depends to a slight extent on the joint position, but that at all loads and flexion angles a bicentric contact and an important central joint space width emerge because of the concave incongruity of the joint, These data may be used for numerical calculations, analysing the effects of incongruity on the joint stress and on the functional adaptation of the subarticular tissues. © 1995 Wiley-Liss, Inc.     

Energy absorbing ability of articular cartilage during impact

Forty-eight samples of 9 mm diameter articular cartilage and associated subchondral bone from, the tibial plateau of human knee joints were mounted in methylmethacrylate cement and subjected to controlled impact velocities. The controlled velocities provided for the testing of 21 samples at a strain rate of 500 s^?1 and 27 samples at a strain rate of 1000 s^?1. Data are presented on the energy absorption per unit volume of cartilage in relation to the associated stresses and strains. The data in each of the resultant nine graphs are subjected to a least squares fit to the polynomial $$Y = a_1 X + a_2 X^2 $$ The relevant statistical data for each of the strain rates and for the combined data are presented. While the total energy absorption of the articular cartilage of the knee joint at ambulatory stresses of 0·8 to 6·3 N/mm² is calculated to be no more than 0·13 to 3·65 J, it is suggested that the compliance of the cartilage of the knee joint together with the incongruity of the tibia and femur are essential features that should be considered in the design of total knee-replacement prostheses.     

A comparison of subchondral bone mineralization between the glenoid cavity and the humeral head on 57 cadaverous shoulder joints

Purpose

Mineralization distribution of the subchondral bone plate can be used as a marker for long-term stress distribution in diarthrodial joints. Severe injuries or pathological changes of the glenohumeral joint often end in osteoarthritis, where shoulder arthroplasty has become the treatment of choice. The computed tomography osteoabsorptiometry (CT-OAM) is a non-invasive method to determine the distribution of the mineralization of the subchondral bone plate in vivo, which is an important factor concerning the implantation of orthopedic endoprostheses. The aim of this study was to investigate the mineralization of both joint partners of the glenohumeral joint and to compare them with each other.

Methods

The distribution of the mineralization of the subchondral bone plate of 57 shoulder specimens was determined by means of CT-OAM. To evaluate a correlation between age and localization of subchondral mineralization maxima, the Chi-square test correlation test was applied.

Results

Forty-nine glenoid cavities (86 %) showed a bicentric mineralization distribution pattern with anterior and posterior maxima, only 8 glenoid cavities (14 %) revealed a monocentric mineralization pattern with anterior maxima. Forty-five humeral heads (79 %) showed a bicentric distribution pattern with anterior and posterior maxima, 12 humeral heads (21 %) could be classified as monocentric with a centro-posterior pronounced maximum.

Conclusions

We could demonstrate that stress distribution in both joint partners of the glenohumeral joint is inhomogeneous and characteristically bicentric due to the physiological incongruity. Monocentric mineralization patterns can result as a cause of age-related loss of incongruity.     

Mechanobiological adaptation of subchondral bone as a function of joint incongruity and loading

Computed tomography (CT) has been employed to determine non-invasively the distribution of subchondral bone density in joints and to evaluate their dominant loading pattern. The objective of this study was to investigate the relationship between subchondral bone adaptation, joint incongruity and loading, in order to determine to what extent the loading conditions and/or geometric configuration can be inferred from the distribution of subchondral density. Finite element models of joints with various degrees of incongruity were designed and a current remodeling theory implemented using the node-based approach. Appropriate combinations of joint incongruity and loading yielded subchondral bone density patterns consistent with experimental findings, specifically a bicentric distribution in the humero-ulnar joint and a monocentric distribution in the humero-radial joint. However, other combinations of incongruity and loading produced similar subchondral density patterns. Both the geometric joint configuration and the loading conditions influence the distribution of subchondral density in such a way that one of these factors must be known a priori to estimate the other. Since subchondral density can be assessed by CT and joint geometry by magnetic resonance imaging, the dominant loading pattern of joints may be potentially derived in the living using these non-invasive imaging methods.     

Morphological and functional studies on the lateral joints of the first and second cervical vertebrae in man

Summary The articular sufaces of the lateral atlanto-axial joints in man were analyzed with respect to their biomechanical aspects. The joint surfaces were investigated macroscopically, and the texture of the tangential fiber layer of the articular cartilage was studied with the aid of the split-line method. Furthermone the thickness of cartilage was measured. From the results obtained the authors draw the conclusion that especially the mode of rotational movement between C₁ and C₂ changes within a life time. As a consequence, the distribution of stress in the lateral atlanto-axial joint is modified.     

Quantitative relationships of normal cartilage volumes of the human knee joint – assessment by magnetic resonance imaging

The objective of this study was to assess the normal range of cartilage volumes in the knee joints of healthy adults, the ratio between the patellar, femoral, and tibial cartilages, and the correlation of the volumes with age, body weight, height, body mass index (obesity), patellar bone size, and the diameter of the tibial head. We examined the knee joints of nine healthy volunteers and eleven normal post-mortem specimens with an age range of 24 to 82 years. The cartilage volumes of the patella, femur, medial tibia and the lateral tibia were quantified, using a fat-suppressed FLASH-3D sequence (resolution 2×0.31×0.31 mm³) and digital postprocessing, involving three-dimensional reconstruction. The mean total volume of the knee joint cartilage was 23,245 mm³, the relative standard deviation (CV%) 19%, and the range 16,341 to 33,988 mm³. In the patella, femur and tibia, the CV% amounted to between 22 and 25%. These joint surfaces occupied a relatively variable proportion of the total knee joint volume, the percentage of the patella being 11 to 22%, that of the femur 54 to 69%, that of the medial tibia 7 to 12%, and that of lateral tibia 11 to 16%. The volumes of the lateral tibia were systematically higher than those of the medial tibia (P<0.001). There was no significant correlation of the knee joint cartilage volume with age (r=+0.05), body weight (r=+0.38), height (r=+0.39) or body mass index (r=+0.29), but a relatively high correlation with the diameter of the tibial head (r=+0.78, P<0.001). After normalising the volumes to this diameter, the CV% of the total knee joint cartilge volume was reduced to 13%, its variation being 12 to 21% in the patella, femur and tibia. MRI is available for measuring cartilage volume during growth, functional adaptation, and tissue loss in degenerative joint disease. The study shows that a wide variation of cartilage volumes exists in the knee joints of normal adults. To reduce the variability between individuals, the cartilage volumes may be normalised to the head of the tibial diameter.