Judith R. Meakin

Statistical methods in finite element analysis

Finite element analysis (FEA) is a commonly used tool within many areas of engineering and can provide useful information in structural analysis of mechanical systems. However, most analyses within the field of biomechanics usually take no account either of the wide variation in material properties and geometry that may occur in natural tissues or manufacturing imperfections in synthetic materials. This paper discusses two different methods of incorporating uncertainty in FE models. The first, Taguchis robust parameter design, uses orthogonal matrices to determine how to vary the parameters in a series of FE models, and provides information on the sensitivity of a model to input parameters. The second, probabilistic analysis, enables the distribution of a response variable to be determined from the distributions of the input variables. The methods are demonstrated using a simple example of an FE model of a beam that is assigned material properties and geometry over a range similar to an orthopaedic fixation plate. In addition to showing how each method may be used on its own, we also show how computational effort may be minimised by first identifying the most important input variables before determining the effects of imprecision.

Compression or tension? The stress distribution in the proximal femur.

BackgroundQuestions regarding the distribution of stress in the proximal human femur have never been adequately resolved. Traditionally, by considering the femur in isolation, it has been believed that the effect of body weight on the projecting neck and head places the superior aspect of the neck in tension. A minority view has proposed that this region is in compression because of muscular forces pulling the femur into the pelvis. Little has been done to study stress distributions in the proximal femur. We hypothesise that under physiological loading the majority of the proximal femur is in compression and that the internal trabecular structure functions as an arch, transferring compressive stresses to the femoral shaft.MethodsTo demonstrate the principle, we have developed a 2D finite element model of the femur in which body weight, a representation of the pelvis, and ligamentous forces were included. The regions of higher trabecular bone density in the proximal femur (the principal trabecular systems) were assigned a higher modulus than the surrounding trabecular bone. Two-legged and one-legged stances, the latter including an abductor force, were investigated.ResultsThe inclusion of ligamentous forces in two-legged stance generated compressive stresses in the proximal femur. The increased modulus in areas of greater structural density focuses the stresses through the arch-like internal structure. Including an abductor muscle force in simulated one-legged stance also produced compression, but with a different distribution.ConclusionThis 2D model shows, in principle, that including ligamentous and muscular forces has the effect of generating compressive stresses across most of the proximal femur. The arch-like trabecular structure transmits the compressive loads to the shaft. The greater strength of bone in compression than in tension is then used to advantage. These results support the hypothesis presented. If correct, a better understanding of the stress distribution in the proximal femur may lead to improvements in prosthetic devices and an appreciation of the effects of various surgical procedures affecting load transmission across the hip.

Effect of removing the nucleus pulposus on the deformation of the annulus fibrosus during compression of the intervertebral disc

Eighteen frozen ovine discs were bisected, in the mid-sagittal plane, to produce 36 specimens. The cut surfaces were marked at the inner and outer annulus boundaries of the annulus fibrosus, both anteriorly and posteriorly, with Alcian blue stain. The sections were sealed by a transparent plate, and thawed. A compression of 1mm at a rate of 0.2mms(-1) was applied. The displacements of the Alcian blue marks were measured from the video images, recorded during the tests, using interactive image analysis software. Before removal of the nucleus, the inner boundaries of the annulus moved outwards during compression (P<0.001, anterior; P=0.01, posterior). However, after removal of the nucleus, both inner boundaries moved inwards (P<0.001, anterior and posterior). The outer boundaries moved outwards both before and after removal of the nucleus (P<0.001). The results showed that total removal of the nucleus changes the response of the annulus to compression.

Sheep lumbar intervertebral discs as models for human discs

OBJECTIVE To determine the water content, collagen content and collagen orientation angle in different regions of sheep lumbar discs. DESIGN A laboratory study of sheep discs obtained from an abattoir. METHODS A total of 21 sheep lumbar discs were obtained from three lumbar spines. Water content was determined by oven drying (60 degrees C) to constant mass. Collagen content was determined by hydroxyproline analysis. Fibre orientation angles were determined by X-ray diffraction. RESULTS Water content increased from 74% of total tissue mass in the outer annulus, to 82% in the inner annulus, to 86% in the nucleus. Collagen content decreased from 30% of total tissue mass in the outer region to 20% in the inner region of the anterior and lateral annulus; it was 16% in the posterior annulus. The orientation angle of the collagen fibres decreased from 59 degrees in the outer region to 56 degrees in the inner region of the anterior and lateral annulus; it was 51 degrees in the posterior annulus. CONCLUSIONS Sheep lumbar intervertebral discs provide a reasonable model for human lumbar intervertebral discs. RELEVANCE Sheep lumbar discs have been used to investigate the effects of removing and replacing the nucleus. These studies indicate that removal of nucleus may lead to further disc degeneration and indicate the material properties required for an implant material. The relevance of these previous studies is increased if human and sheep lumbar discs have a similar composition and structure.

Finite element analysis of the meniscus: the influence of geometry and material properties on its behaviour

A finite element model of the knee meniscus was developed to investigate the effects of various geometrical and material properties on the behaviour of the meniscus under compressive load. Factorial methods were used to determine the relative effect of varying the properties by +/-10% of their initial value. It was found that the stresses in the meniscus were more sensitive to geometry (meniscus width and radius of curvature of the femoral surface of the meniscus) than material properties. The model was also used to investigate the effect of incongruency between the radius of curvature of the femur and the femoral surface of the meniscus. It was shown that mismatch between the curvatures of the femur and meniscus has a large effect on the stresses both in the meniscus and in the underlying cartilage. The results from the study have implications for the design and development of meniscal repair devices and replacements.

Replacing the nucleus pulposus of the intervertebral disc

OBJECTIVE To determine whether replacement of the nucleus with a synthetic material would prevent the effects of nucleus removal. DESIGN Laboratory experiments on excised tissues and a finite element model. BACKGROUND Removal of the nucleus from the intervertebral disc causes the inner margins of the annulus to bulge inwards, instead of outwards, during compression. This may cause the annulus to degenerate further. METHODS Video recordings of sheep discs, sectioned in the sagittal plane, were obtained during compression in a materials testing machine; the cut face of the disc was sealed with a Perspex window. Experiments were repeated with the nucleus removed and then replaced by a synthetic implant. A finite element model of an intact disc was also used to investigate the effect of nucleus replacement. RESULTS When the nucleus of sectioned discs was replaced with the polymer materials, the inward bulging of the annulus was not observed. The predictions from the finite element model of the intact disc were consistent with this result. CONCLUSIONS Replacement of the nucleus with a synthetic material can prevent the changes in annulus behaviour that result from removal of the nucleus. RELEVANCE A suitable implant to replace the nucleus after surgical removal may help prevent inward bulging of the inner layers of the annulus.

The effect of partial removal of the nucleus pulposus from the intervertebral disc on the response of the human annulus fibrosus to compression.

OBJECTIVE To determine how partial removal of the nucleus changes the response of the annulus to compression. DESIGN The deformation of the annulus in the mid-sagittal plane, during compression, was determined from digital video images. BACKGROUND Several studies have shown that removal of the nucleus changes the external behaviour of the intervertebral disc, but few studies have investigated changes to internal behaviour. METHODS Six frozen human lumbar discs were bisected in the sagittal plane to produce 12 specimens. The cut surfaces were marked with seven dots of Alcian blue stain. The specimens were sealed, enabling their internal structure to be viewed directly by a digital video recording system, and thawed. The video system recorded the response of each specimen as it was compressed by up to 1.8 mm at a rate of 0.2 mm s(-1). The displacements of the Alcian blue marks were measured using an image analysis program. Magnetic resonance imaging was used to investigate the validity of this technique. RESULTS Partial removal of the nucleus changed the way that the disc deformed under compression. A highly significant change in direction of movement was seen in the inner posterior region of the annulus. CONCLUSIONS Partial removal of the nucleus changes the response of the annulus to compression. RELEVANCE Partial denucleation of the human intervertebral disc is shown to change the direction of bulging of the inner annulus when the disc is compressed. Increases in shear stress, arising from these changes, may lead to further disc degeneration in the form of circumferential tears.

Replacing the nucleus pulposus of the intervertebral disk: prediction of suitable properties of a replacement material using finite element analysis

An axisymmetric finite element model of a human lumbar disk was developed to investigate the properties required of an implant to replace the nucleus pulposus. In the intact disk, the nucleus was modeled as a fluid, and the annulus as an elastic solid. The Youngs modulus of the annulus was determined empirically by matching model predictions to experimental results. The model was checked for sensitivity to the input parameter values and found to give reasonable behavior. The model predicted that removal of the nucleus would change the response of the annulus to compression. This prediction was consistent with experimental results, thus validating the model. Implants to fill the cavity produced by nucleus removal were modeled as elastic solids. The Poissons ratio was fixed at 0.49, and the Youngs modulus was varied from 0.5 to 100 MPa. Two sizes of implant were considered: full size (filling the cavity) and small size (smaller than the cavity). The model predicted that a full size implant would reverse the changes to annulus behavior, but a smaller implant would not. By comparing the stress distribution in the annulus, the ideal Youngs modulus was predicted to be approximately 3 MPa. These predictions have implications for current nucleus implant designs.

The effect of axial load on the sagittal plane curvature of the upright human spine in vivo

Determining the effect of load carriage on the human spine in vivo is important for determining spinal forces and establishing potential mechanisms of back injury. Previous studies have suggested that the natural curvature of the spine straightens under load, but are based on modelling and external measurements from the surface of the back. In the current study, an upright positional MRI scanner was used to acquire sagittal images of the lumbar and lower thoracic spine of 24 subjects. The subjects were imaged in standing whilst supporting 0, 8 and 16 kg of load which was applied axially across the shoulders using an apron. An active shape model of the vertebral bodies from T10 to S1 was created and used to characterise the effect of load. The results from the shape model showed that the behaviour of the average-shaped spine was to straighten slightly. However, the shape model also showed that the effect of load exhibited systematic variation between individuals. Those who had a smaller than average curvature before loading straightened under load, whereas those who had a greater than average curvature before loading showed an increase in curvature under load. The variation in behaviour of differently shaped spines may have further implications for the effects of load in lifting manoeuvres and in understanding the aetiology of back pain.

Thermal analysis of poly(2-hydroxyethyl methacrylate) (pHEMA) hydrogels.

The influence of water on the physical properties of a hydrogel is important for understanding natural tissues and in designing synthetic materials to replace them. In this study, poly (2-hydroxyethyl methacrylate) (pHEMA) was used as a model system to understand how water interacts with the polymer of a hydrogel. Thermal analysis methods (thermogravimetric analysis coupled to mass spectrometry and differential scanning calorimetry) were used to determine: (i) the total water content of pHEMA gels; (ii) how this water was lost during heating; (iii) the relationship between water content of the gel and its glass transition temperature; and (iv) the behavior of the water in the gel on cooling. Previous researchers have invoked various models to describe the organization of water in a hydrogel. In this study, the simplest model which could explain all of the results from the different thermal analysis techniques was one which consisted of three classes of water: (i) hydration water in close proximity to the polymer; (ii) interstitial water in regions or cavities surrounded by polymer chains; and (iii) bulk water.