Statistical finite element model for bone shape and biomechanical properties

We present a framework for statistical finite element analysis combining shape and material properties, and allowing performing statistical statements of biomechanical performance across a given population. In this paper, we focus on the design of orthopaedic implants that fit a maximum percentage of the target population, both in terms of geometry and biomechanical stability. CT scans of the bone under consideration are registered non-rigidly to obtain correspondences in position and intensity between them. A statistical model of shape and intensity (bone density) is computed by means of principal component analysis. Afterwards, finite element analysis (FEA) is performed to analyse the biomechanical performance of the bones. Realistic forces are applied on the bones and the resulting displacement and bone stress distribution are calculated. The mechanical behaviour of different PCA bone instances is compared.

Abnormal bone geometry at the metacarpal bone shaft of rheumatoid arthritis patients with maintained muscle-bone relationship

OBJECTIVES:: Metacarpal juxta-articular bone is altered in Rheumatoid Arthritis (RA). However, a detailed analysis of disease related geometrical adaptations of the metacarpal shaft is missing. The aim of the present study was to assess the role of RA disease, forearm muscle cross-sectional area (CSA), age and sex on bone geometry at the metacarpal shaft. METHODS:: In 64 RA patients and 128 control subjects geometric properties of the third metacarpal bone mid-shaft and forearm muscle CSA were measured by peripheral quantitative computed tomography (pQCT). Linear models were performed for cortical CSA, total bone CSA, polar stress-strain Index (polar SSI, a surrogate for bone's resistance to bending and torsion), cortical thickness and Metacarpal Index (MI=cortical CSA/total CSA) with explanatory variables muscle CSA, age, RA status and sex. RESULTS:: Forearm muscle CSA was associated with cortical and total metacarpal CSA, and polar SSI. RA group status was associated with all bone parameters except cortical CSA. There was a significant interaction between RA status and age, indicating that the RA group had a greater age-related decrease in cortical CSA, cortical thickness and MI. CONCLUSIONS:: Bone geometry of the metacarpal shaft is altered in RA patients compared to healthy controls. While bone mass of the metacarpal shaft is adapted to forearm muscle mass, cortical thickness and MI are reduced but outer bone shaft circumference and polar SSI increased in RA patients. These adaptations correspond to an enhanced aging pattern in RA patients.

Site-1 protease is essential for endochondral bone formation in mice

Site-1 protease (S1P) has an essential function in the conversion of latent, membrane-bound transcription factors to their free, active form. In mammals, abundant expression of S1P in chondrocytes suggests an involvement in chondrocyte function. To determine the requirement of S1P in cartilage and bone development, we have created cartilage-specific S1P knockout mice (S1P(cko)). S1P(cko) mice exhibit chondrodysplasia and a complete lack of endochondral ossification even though Runx2 expression, Indian hedgehog signaling, and osteoblastogenesis is intact. However, there is a substantial increase in chondrocyte apoptosis in the cartilage of S1P(cko) mice. Extraction of type II collagen is substantially lower from S1P(cko) cartilage. In S1P(cko) mice, the collagen network is disorganized and collagen becomes entrapped in chondrocytes. Ultrastructural analysis reveals that the endoplasmic reticulum (ER) in S1P(cko) chondrocytes is engorged and fragmented in a manner characteristic of severe ER stress. These data suggest that S1P activity is necessary for a specialized ER stress response required by chondrocytes for the genesis of normal cartilage and thus endochondral ossification.

Surviving endoplasmic reticulum stress is coupled to altered chondrocyte differentiation and function

In protein folding and secretion disorders, activation of endoplasmic reticulum (ER) stress signaling (ERSS) protects cells, alleviating stress that would otherwise trigger apoptosis. Whether the stress-surviving cells resume normal function is not known. We studied the in vivo impact of ER stress in terminally differentiating hypertrophic chondrocytes (HCs) during endochondral bone formation. In transgenic mice expressing mutant collagen X as a consequence of a 13-base pair deletion in Col10a1 (13del), misfolded alpha1(X) chains accumulate in HCs and elicit ERSS. Histological and gene expression analyses showed that these chondrocytes survived ER stress, but terminal differentiation is interrupted, and endochondral bone formation is delayed, producing a chondrodysplasia phenotype. This altered differentiation involves cell-cycle re-entry, the re-expression of genes characteristic of a prehypertrophic-like state, and is cell-autonomous. Concomitantly, expression of Col10a1 and 13del mRNAs are reduced, and ER stress is alleviated. ERSS, abnormal chondrocyte differentiation, and altered growth plate architecture also occur in mice expressing mutant collagen II and aggrecan. Alteration of the differentiation program in chondrocytes expressing unfolded or misfolded proteins may be part of an adaptive response that facilitates survival and recovery from the ensuing ER stress. However, the altered differentiation disrupts the highly coordinated events of endochondral ossification culminating in chondrodysplasia.

The effects of impingement and dysplasia on stress distributions in the hip joint during sitting and walking: a finite element analysis

Soft tissue damage has been observed in hip joints with pathological geometries. Our primary goal was to study the relationship between morphological variations of the bony components of the hip and resultant stresses within the soft tissues of the joint during routine daily activities. The secondary goal was to find the range of morphological parameters in which stresses are minimized. Computational models of normal and pathological joints were developed based on variations of morphological parameters of the femoral head (Alpha angle) and acetabulum (CE angle). The Alpha angle was varied between 40 degrees (normal joint) and 80 degrees (cam joint). The CE angle was varied between 0 degrees (dysplastic joint) and 40 degrees (pincer joint). Dynamic loads and motions for walking and standing to sitting were applied to all joint configurations. Contact pressures and stresses were calculated and crosscompared to evaluate the influence of morphology. The stresses in the soft tissues depended strongly on the head and acetabular geometry. For the dysplastic joint, walking produced high acetabular rim stresses. Conversely, for impinging joints, standing-to-sitting activities that involved extensive motion were critical, inducing excessive distortion and shearing of the tissue-bone interface. Zones with high von Mises stresses corresponded with clinically observed damage zones in the acetabular cartilage and labrum. Hip joint morphological parameters that minimized were 20 degrees

Overweight children have a greater proportion of fat mass relative to muscle mass in the upper limbs than in the lower limbs: implications for bone strength at the distal forearm

BACKGROUND: The influence of adiposity on upper-limb bone strength has rarely been studied in children, despite the high incidence of forearm fractures in this population. OBJECTIVE: The objective was to compare the influence of muscle and fat tissues on bone strength between the upper and lower limbs in prepubertal children. DESIGN: Bone mineral content, total bone cross-sectional area, cortical bone area (CoA), cortical thickness (CoTh) at the radius and tibia (4% and 66%, respectively), trabecular density (TrD), bone strength index (4% sites), cortical density (CoD), stress-strain index, and muscle and fat areas (66% sites) were measured by using peripheral quantitative computed tomography in 427 children (206 boys) aged 7-10 y. RESULTS: Overweight children (n = 93) had greater values for bone variables (0.3-1.3 SD; P < 0.0001) than did their normal-weight peers, except for CoD 66% and CoTh 4%. The between-group differences were 21-87% greater at the tibia than at the radius. After adjustment for muscle cross-sectional area, TrD 4%, bone mineral content, CoA, and CoTh 66% at the tibia remained greater in overweight children, whereas at the distal radius total bone cross-sectional area and CoTh were smaller in overweight children (P < 0.05). Overweight children had a greater fat-muscle ratio than did normal-weight children, particularly in the forearm (92 +/- 28% compared with 57 +/- 17%). Fat-muscle ratio correlated negatively with all bone variables, except for TrD and CoD, after adjustment for body weight (r = -0.17 to -0.54; P < 0.0001). CONCLUSIONS: Overweight children had stronger bones than did their normal-weight peers, largely because of greater muscle size. However, the overweight children had a high proportion of fat relative to muscle in the forearm, which is associated with reduced bone strength.

Postpartum osteoporosis associated with proximal tibial stress fracture

A 33-year-old woman presented with acute nonspecific knee pain, 6 months postpartum. MR imaging, computed tomography and radiography were performed and a proximal tibia plateau insufficiency fracture was detected. Bone densitometry demonstrated mild postpartum osteoporosis. To our knowledge these findings have not been described in this location and in this clinical setting. The etiology of the atraumatic fracture of the tibia is presumed to be due to a low bone mineral density. The bone loss was probably due to pregnancy, lactation and postpartum hormonal changes. There were no other inciting causes and the patient was normocalcemic. We discuss the presence of a postpartum stress fracture in a hitherto undescribed site in a patient who had lactated following an uncomplicated pregnancy and had no other identifiable cause for a stress fracture.

An anisotropic elastic-viscoplastic damage model for bone tissue

A new anisotropic elastic-viscoplastic damage constitutive model for bone is proposed using an eccentric elliptical yield criterion and nonlinear isotropic hardening. A micromechanics-based multiscale homogenization scheme proposed by Reisinger et al. is used to obtain the effective elastic properties of lamellar bone. The dissipative process in bone is modeled as viscoplastic deformation coupled to damage. The model is based on an orthotropic ecuntric elliptical criterion in stress space. In order to simplify material identification, an eccentric elliptical isotropic yield surface was defined in strain space, which is transformed to a stress-based criterion by means of the damaged compliance tensor. Viscoplasticity is implemented by means of the continuous Perzyna formulation. Damage is modeled by a scalar function of the accumulated plastic strain D(κ) , reducing all element s of the stiffness matrix. A polynomial flow rule is proposed in order to capture the rate-dependent post-yield behavior of lamellar bone. A numerical algorithm to perform the back projection on the rate-dependent yield surface has been developed and implemented in the commercial finite element solver Abaqus/Standard as a user subroutine UMAT. A consistent tangent operator has been derived and implemented in order to ensure quadratic convergence. Correct implementation of the algorithm, convergence, and accuracy of the tangent operator was tested by means of strain- and stress-based single element tests. A finite element simulation of nano- indentation in lamellar bone was finally performed in order to show the abilities of the newly developed constitutive model.

A generalized anisotropic quadric yield criterion and its application to bone tissue at multiple length scales

Nonlinear computational analysis of materials showing elasto-plasticity or damage relies on knowledge of their yield behavior and strengths under complex stress states. In this work, a generalized anisotropic quadric yield criterion is proposed that is homogeneous of degree one and takes a convex quadric shape with a smooth transition from ellipsoidal to cylindrical or conical surfaces. If in the case of material identification, the shape of the yield function is not known a priori, a minimization using the quadric criterion will result in the optimal shape among the convex quadrics. The convexity limits of the criterion and the transition points between the different shapes are identified. Several special cases of the criterion for distinct material symmetries such as isotropy, cubic symmetry, fabric-based orthotropy and general orthotropy are presented and discussed. The generality of the formulation is demonstrated by showing its degeneration to several classical yield surfaces like the von Mises, Drucker–Prager, Tsai–Wu, Liu, generalized Hill and classical Hill criteria under appropriate conditions. Applicability of the formulation for micromechanical analyses was shown by transformation of a criterion for porous cohesive-frictional materials by Maghous et al. In order to demonstrate the advantages of the generalized formulation, bone is chosen as an example material, since it features yield envelopes with different shapes depending on the considered length scale. A fabric- and density-based quadric criterion for the description of homogenized material behavior of trabecular bone is identified from uniaxial, multiaxial and torsional experimental data. Also, a fabric- and density-based Tsai–Wu yield criterion for homogenized trabecular bone from in silico data is converted to an equivalent quadric criterion by introduction of a transformation of the interaction parameters. Finally, a quadric yield criterion for lamellar bone at the microscale is identified from a nanoindentation study reported in the literature, thus demonstrating the applicability of the generalized formulation to the description of the yield envelope of bone at multiple length scales.

In situ micropillar compression reveals superior strength and ductility but an absence of damage in lamellar bone

Ageing societies suffer from an increasing incidence of bone fractures. Bone strength depends on the amount of mineral measured by clinical densitometry, but also on the micromechanical properties of the hierarchical organization of bone. Here, we investigate the mechanical response under monotonic and cyclic compression of both single osteonal lamellae and macroscopic samples containing numerous osteons. Micropillar compression tests in a scanning electron microscope, microindentation and macroscopic compression tests were performed on dry ovine bone to identify the elastic modulus, yield stress, plastic deformation, damage accumulation and failure mechanisms. We found that isolated lamellae exhibit a plastic behaviour, with higher yield stress and ductility but no damage. In agreement with a proposed rheological model, these experiments illustrate a transition from a ductile mechanical behaviour of bone at the microscale to a quasi-brittle response driven by the growth of cracks along interfaces or in the vicinity of pores at the macroscale.

Experimental, theoretical and numerical investigation of the nonlinear micromechanical properties of bone

Aging societies suffer from an increasing incidence of bone fractures. Bone strength depends on the amount of mineral measured by clinical densitometry, but also on the micromechanical properties of the bone hierarchical organization. A good understanding has been reached for elastic properties on several length scales, but up to now there is a lack of reliable postyield data on the lower length scales. In order to be able to describe the behavior of bone at the microscale, an anisotropic elastic-viscoplastic damage model was developed using an eccentric generalized Hill criterion and nonlinear isotropic hardening. The model was implemented as a user subroutine in Abaqus and verified using single element tests. A FE simulation of microindentation in lamellar bone was finally performed show-ing that the new constitutive model can capture the main characteristics of the indentation response of bone. As the generalized Hill criterion is limited to elliptical and cylindrical yield surfaces and the correct shape for bone is not known, a new yield surface was developed that takes any convex quadratic shape. The main advantage is that in the case of material identification the shape of the yield surface does not have to be anticipated but a minimization results in the optimal shape among all convex quadrics. The generality of the formulation was demonstrated by showing its degeneration to classical yield surfaces. Also, existing yield criteria for bone at multiple length scales were converted to the quadric formulation. Then, a computational study to determine the influence of yield surface shape and damage on the in-dentation response of bone using spherical and conical tips was performed. The constitutive model was adapted to the quadric criterion and yield surface shape and critical damage were varied. They were shown to have a major impact on the indentation curves. Their influence on indentation modulus, hardness, their ratio as well as the elastic to total work ratio were found to be very well described by multilinear regressions for both tip shapes. For conical tips, indentation depth was not a significant fac-tor, while for spherical tips damage was insignificant. All inverse methods based on microindentation suffer from a lack of uniqueness of the found material properties in the case of nonlinear material behavior. Therefore, monotonic and cyclic micropillar com-pression tests in a scanning electron microscope allowing a straightforward interpretation comple-mented by microindentation and macroscopic uniaxial compression tests were performed on dry ovine bone to identify modulus, yield stress, plastic deformation, damage accumulation and failure mecha-nisms. While the elastic properties were highly consistent, the postyield deformation and failure mech-anisms differed between the two length scales. A majority of the micropillars showed a ductile behavior with strain hardening until failure by localization in a slip plane, while the macroscopic samples failed in a quasi-brittle fashion with microcracks coalescing into macroscopic failure surfaces. In agreement with a proposed rheological model, these experiments illustrate a transition from a ductile mechanical behavior of bone at the microscale to a quasi-brittle response driven by the growth of preexisting cracks along interfaces or in the vicinity of pores at the macroscale. Subsequently, a study was undertaken to quantify the topological variability of indentations in bone and examine its relationship with mechanical properties. Indentations were performed in dry human and ovine bone in axial and transverse directions and their topography measured by AFM. Statistical shape modeling of the residual imprint allowed to define a mean shape and describe the variability with 21 principal components related to imprint depth, surface curvature and roughness. The indentation profile of bone was highly consistent and free of any pile up. A few of the topological parameters, in particular depth, showed significant correlations to variations in mechanical properties, but the cor-relations were not very strong or consistent. We could thus verify that bone is rather homogeneous in its micromechanical properties and that indentation results are not strongly influenced by small de-viations from the ideal case. As the uniaxial properties measured by micropillar compression are in conflict with the current literature on bone indentation, another dissipative mechanism has to be present. The elastic-viscoplastic damage model was therefore extended to viscoelasticity. The viscoelastic properties were identified from macroscopic experiments, while the quasistatic postelastic properties were extracted from micropillar data. It was found that viscoelasticity governed by macroscale properties has very little influence on the indentation curve and results in a clear underestimation of the creep deformation. Adding viscoplasticity leads to increased creep, but hardness is still highly overestimated. It was possible to obtain a reasonable fit with experimental indentation curves for both Berkovich and spherical indenta-tion when abandoning the assumption of shear strength being governed by an isotropy condition. These results remain to be verified by independent tests probing the micromechanical strength prop-erties in tension and shear. In conclusion, in this thesis several tools were developed to describe the complex behavior of bone on the microscale and experiments were performed to identify its material properties. Micropillar com-pression highlighted a size effect in bone due to the presence of preexisting cracks and pores or inter-faces like cement lines. It was possible to get a reasonable fit between experimental indentation curves using different tips and simulations using the constitutive model and uniaxial properties measured by micropillar compression. Additional experimental work is necessary to identify the exact nature of the size effect and the mechanical role of interfaces in bone. Deciphering the micromechanical behavior of lamellar bone and its evolution with age, disease and treatment and its failure mechanisms on several length scales will help preventing fractures in the elderly in the future.