Advanced morphological 3D magnetic resonance observation of cartilage repair tissue (MOCART) scoring using a new isotropic 3D proton-density, turbo spin echo sequence with variable flip angle distribution (PD-SPACE) compared to an isotropic 3D steady-state free precession sequence (True-FISP) and standard 2D sequences

To evaluate a new isotropic 3D proton-density, turbo-spin-echo sequence with variable flip-angle distribution (PD-SPACE) sequence compared to an isotropic 3D true-fast-imaging with steady-state-precession (True-FISP) sequence and 2D standard MR sequences with regard to the new 3D magnetic resonance observation of cartilage repair tissue (MOCART) score.

Three-dimensional magnetic resonance observation of cartilage repair tissue (MOCART) score assessed with an isotropic three-dimensional true fast imaging with steady-state precession sequence at 3.0 Tesla

INTRODUCTION: Cartilage defects are common pathologies and surgical cartilage repair shows promising results. In its postoperative evaluation, the magnetic resonance observation of cartilage repair tissue (MOCART) score, using different variables to describe the constitution of the cartilage repair tissue and the surrounding structures, is widely used. High-field magnetic resonance imaging (MRI) and 3-dimensional (3D) isotropic sequences may combine ideal preconditions to enhance the diagnostic performance of cartilage imaging.Aim of this study was to introduce an improved 3D MOCART score using the possibilities of an isotropic 3D true fast imaging with steady-state precession (True-FISP) sequence in the postoperative evaluation of patients after matrix-associated autologous chondrocyte transplantation (MACT) as well as to compare the results to the conventional 2D MOCART score using standard MR sequences. MATERIAL AND METHODS: The study had approval by the local ethics commission. One hundred consecutive MR scans in 60 patients at standard follow-up intervals of 1, 3, 6, 12, 24, and 60 months after MACT of the knee joint were prospectively included. The mean follow-up interval of this cross-sectional evaluation was 21.4 +/- 20.6 months; the mean age of the patients was 35.8 +/- 9.4 years. MRI was performed at a 3.0 Tesla unit. All variables of the standard 2D MOCART score where part of the new 3D MOCART score. Furthermore, additional variables and options were included with the aims to use the capabilities of isotropic MRI, to include the results of recent studies, and to adapt to the needs of patients and physician in a clinical routine examination. A proton-density turbo spin-echo sequence, a T2-weighted dual fast spin-echo (dual-FSE) sequence, and a T1-weighted turbo inversion recovery magnitude (TIRM) sequence were used to assess the standard 2D MOCART score; an isotropic 3D-TrueFISP sequence was prepared to evaluate the new 3D MOCART score. All 9 variables of the 2D MOCART score were compared with the corresponding variables obtained by the 3D MOCART score using the Pearson correlation coefficient; additionally the subjective quality and possible artifacts of the MR sequences were analyzed. RESULTS: The correlation between the standard 2D MOCART score and the new 3D MOCART showed for the 8 variables "defect fill," "cartilage interface," "surface," "adhesions," "structure," "signal intensity," "subchondral lamina," and "effusion"-a highly significant (P < 0.001) correlation with a Pearson coefficient between 0.566 and 0.932. The variable "bone marrow edema" correlated significantly (P < 0.05; Pearson coefficient: 0.257). The subjective quality of the 3 standard MR sequences was comparable to the isotropic 3D-TrueFISP sequence. Artifacts were more frequently visible within the 3D-TrueFISP sequence. CONCLUSION: In the clinical routine follow-up after cartilage repair, the 3D MOCART score, assessed by only 1 high-resolution isotropic MR sequence, provides comparable information than the standard 2D MOCART score. Hence, the new 3D MOCART score has the potential to combine the information of the standard 2D MOCART score with the possible advantages of isotropic 3D MRI at high-field. A clear limitation of the 3D-TrueFISP sequence was the high number of artifacts. Future studies have to prove the clinical benefits of a 3D MOCART score.

Convolution roots and differentiability of isotropic positive definite functions on spheres

We prove that any isotropic positive definite function on the sphere can be written as the spherical self-convolution of an isotropic real-valued function. It is known that isotropic positive definite functions on d-dimensional Euclidean space admit a continuous derivative of order [(d − 1)/2]. We show that the same holds true for isotropic positive definite functions on spheres and prove that this result is optimal for all odd dimensions.

The hp-adaptive FEM based on continuous Sobolev embeddings: isotropic refinements

The aim of this paper is to present a new class of smoothness testing strategies in the context of hp-adaptive refinements based on continuous Sobolev embeddings. In addition to deriving a modified form of the 1d smoothness indicators introduced in [26], they will be extended and applied to a higher dimensional framework. A few numerical experiments in the context of the hp-adaptive FEM for a linear elliptic PDE will be performed.

Anatomically-Driven Soft-Tissue Simulation Strategy for Cranio-Maxillofacial Surgery Using Facial Muscle Template Model

We propose a computationally efficient and biomechanically relevant soft-tissue simulation method for cranio-maxillofacial (CMF) surgery. A template-based facial muscle reconstruction was introduced to minimize the efforts on preparing a patient-specific model. A transversely isotropic mass-tensor model (MTM) was adopted to realize the effect of directional property of facial muscles in reasonable computation time. Additionally, sliding contact around teeth and mucosa was considered for more realistic simulation. Retrospective validation study with postoperative scan of a real patient showed that there were considerable improvements in simulation accuracy by incorporating template-based facial muscle anatomy and sliding contact.

The structural architecture of adult mammalian articular cartilage evolves by a synchronized process of tissue resorption and neoformation during postnatal development

OBJECTIVE: During postnatal development, mammalian articular cartilage acts as a surface growth plate for the underlying epiphyseal bone. Concomitantly, it undergoes a fundamental process of structural reorganization from an immature isotropic to a mature (adult) anisotropic architecture. However, the mechanism underlying this structural transformation is unknown. It could involve either an internal remodelling process, or complete resorption followed by tissue neoformation. The aim of this study was to establish which of these two alternative tissue reorganization mechanisms is physiologically operative. We also wished to pinpoint the articular cartilage source of the stem cells for clonal expansion and the zonal location of the chondrocyte pool with high proliferative activity. METHODS: The New Zealand white rabbit served as our animal model. The analysis was confined to the high-weight-bearing (central) areas of the medial and lateral femoral condyles. After birth, the articular cartilage layer was evaluated morphologically at monthly intervals from the first to the eighth postnatal month, when this species attains skeletal maturity. The overall height of the articular cartilage layer at each juncture was measured. The growth performance of the articular cartilage layer was assessed by calcein labelling, which permitted an estimation of the daily growth rate of the epiphyseal bone and its monthly length-gain. The slowly proliferating stem-cell pool was identified immunohistochemically (after labelling with bromodeoxyuridine), and the rapidly proliferating chondrocyte population by autoradiography (after labelling with (3)H-thymidine). RESULTS: The growth activity of the articular cartilage layer was highest 1 month after birth. It declined precipitously between the first and third months, and ceased between the third and fourth months, when the animal enters puberty. The structural maturation of the articular cartilage layer followed a corresponding temporal trend. During the first 3 months, when the articular cartilage layer is undergoing structural reorganization, the net length-gain in the epiphyseal bone exceeded the height of the articular cartilage layer. This finding indicates that the postnatal reorganization of articular cartilage from an immature isotropic to a mature anisotropic structure is not achieved by a process of internal remodelling, but by the resorption and neoformation of all zones except the most superficial (stem-cell) one. The superficial zone was found to consist of slowly dividing stem cells with bidirectional mitotic activity. In the horizontal direction, this zone furnishes new stem cells that replenish the pool and effect a lateral expansion of the articular cartilage layer. In the vertical direction, the superficial zone supplies the rapidly dividing, transit-amplifying daughter-cell pool that feeds the transitional and upper radial zones during the postnatal growth phase of the articular cartilage layer. CONCLUSIONS: During postnatal development, mammalian articular cartilage fulfils a dual function, viz., it acts not only as an articulating layer but also as a surface growth plate. In the lapine model, this growth activity ceases at puberty (3-4 months of age), whereas that of the true (metaphyseal) growth plate continues until the time of skeletal maturity (8 months). Hence, the two structures are regulated independently. The structural maturation of the articular cartilage layer coincides temporally with the cessation of its growth activity - for the radial expansion and remodelling of the epiphyseal bone - and with sexual maturation. That articular cartilage is physiologically reorganized by a process of tissue resorption and neoformation, rather than by one of internal remodelling, has important implications for the functional engineering and repair of articular cartilage tissue.

Advanced morphological and biochemical magnetic resonance imaging of cartilage repair procedures in the knee joint at 3 Tesla

Morphological and biochemical magnetic resonance imaging (MRI) is due to high field MR systems, advanced coil technology, and sophisticated sequence protocols capable of visualizing articular cartilage in vivo with high resolution in clinical applicable scan time. Several conventional two-dimensional (2D) and three-dimensional (3D) approaches show changes in cartilage structure. Furthermore newer isotropic 3D sequences show great promise in improving cartilage imaging and additionally in diagnosing surrounding pathologies within the knee joint. Functional MR approaches are additionally able to provide a specific measure of the composition of cartilage. Cartilage physiology and ultra-structure can be determined, changes in cartilage macromolecules can be detected, and cartilage repair tissue can thus be assessed and potentially differentiated. In cartilage defects and following nonsurgical and surgical cartilage repair, morphological MRI provides the basis for diagnosis and follow-up evaluation, whereas biochemical MRI provides a deeper insight into the composition of cartilage and cartilage repair tissue. A combination of both, together with clinical evaluation, may represent a desirable multimodal approach in the future, also available in routine clinical use.

High-resolution morphological and biochemical imaging of articular cartilage of the ankle joint at 3.0 T using a new dedicated phased array coil: in vivo reproducibility study

OBJECTIVE: The objective of this study was to evaluate the feasibility and reproducibility of high-resolution magnetic resonance imaging (MRI) and quantitative T2 mapping of the talocrural cartilage within a clinically applicable scan time using a new dedicated ankle coil and high-field MRI. MATERIALS AND METHODS: Ten healthy volunteers (mean age 32.4 years) underwent MRI of the ankle. As morphological sequences, proton density fat-suppressed turbo spin echo (PD-FS-TSE), as a reference, was compared with 3D true fast imaging with steady-state precession (TrueFISP). Furthermore, biochemical quantitative T2 imaging was prepared using a multi-echo spin-echo T2 approach. Data analysis was performed three times each by three different observers on sagittal slices, planned on the isotropic 3D-TrueFISP; as a morphological parameter, cartilage thickness was assessed and for T2 relaxation times, region-of-interest (ROI) evaluation was done. Reproducibility was determined as a coefficient of variation (CV) for each volunteer; averaged as root mean square (RMSA) given as a percentage; statistical evaluation was done using analysis of variance. RESULTS: Cartilage thickness of the talocrural joint showed significantly higher values for the 3D-TrueFISP (ranging from 1.07 to 1.14 mm) compared with the PD-FS-TSE (ranging from 0.74 to 0.99 mm); however, both morphological sequences showed comparable good results with RMSA of 7.1 to 8.5%. Regarding quantitative T2 mapping, measurements showed T2 relaxation times of about 54 ms with an excellent reproducibility (RMSA) ranging from 3.2 to 4.7%. CONCLUSION: In our study the assessment of cartilage thickness and T2 relaxation times could be performed with high reproducibility in a clinically realizable scan time, demonstrating new possibilities for further investigations into patient groups.

High resolution fast T1 mapping technique for dGEMRIC

PURPOSE: To determine the feasibility of using a high resolution isotropic three-dimensional (3D) fast T1 mapping sequence for delayed gadolinium-enhanced MRI of cartilage (dGEMRIC) to assess osteoarthritis in the hip. MATERIALS AND METHODS: T1 maps of the hip were acquired using both low and high resolution techniques following the administration of 0.2 mmol/kg Gd-DTPA(2-) in 35 patients. Both T1 maps were generated from two separate spoiled GRE images. The high resolution T1 map was reconstructed in the anatomically equivalent plane as the low resolution map. T1 values from the equivalent anatomic regions containing femoral and acetabular cartilages were measured on the low and high resolution maps and compared using regression analysis. RESULTS: In vivo T1 measurements showed a statistically significant correlation between the low and high resolution acquisitions at 1.5 Tesla (R(2) = 0.958, P < 0.001). These results demonstrate the feasibility of using a fast two-angle T1 mapping (F2T1) sequence with isotropic spatial resolution (0.8 x 0.8 x 0.8 mm) for quantitative assessment of biochemical status in articular cartilage of the hip. CONCLUSION: The high resolution 3D F2T1 sequence provides accurate T1 measurements in femoral and acetabular cartilages of the hip, which enables the biochemical assessment of articular cartilage in any plane through the joint. It is a powerful tool for researchers and clinicians to acquire high resolution data in a reasonable scan time (< 30 min).

Mechanical characterization of brushite and hydroxyapatite cements

Compression, tension and torsion tests were designed and completed successfully on a brushite and a precipitated hydroxyapatite cement in moist condition. Elastic and strength properties were measured for these three loading cases. For each cement, the full set of strength data was fitted to an isotropic Tsai-Wu criterion and the associated coefficients identified. Since the compressive Young's moduli were about 10% larger than the tensile moduli, the full set of elastic data of each cement was fitted to a conewise linear elastic model. Hysteresis of the stress-strain curves was also observed, indicating dissipation mechanisms within these cement microstructures. A comparison of the measured mechanical properties with human cancellous bone confirmed the indication of brushite as a bone filling material and the potential of the hydroxyapatite cement as a structural biomaterial.

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.

Electroelasticity of charged black branes

We present the first-order corrected dynamics of fluid branes carrying higher-form charge by obtaining the general form of their equations of motion to pole-dipole order in the absence of external forces. Assuming linear response theory, we characterize the corresponding effective theory of stationary bent charged (an)isotropic fluid branes in terms of two sets of response coefficients, the Young modulus and the piezoelectric moduli. We subsequently find large classes of examples in gravity of this effective theory, by constructing stationary strained charged black brane solutions to first order in a derivative expansion. Using solution generating techniques and bent neutral black branes as a seed solution, we obtain a class of charged black brane geometries carrying smeared Maxwell charge in Einstein-Maxwell-dilaton gravity. In the specific case of ten-dimensional space-time we furthermore use T-duality to generate bent black branes with higher-form charge, including smeared D-branes of type II string theory. By subsequently measuring the bending moment and the electric dipole moment which these geometries acquire due to the strain, we uncover that their form is captured by classical electroelasticity theory. In particular, we find that the Young modulus and the piezoelectric moduli of our strained charged black brane solutions are parameterized by a total of 4 response coefficients, both for the isotropic as well as anisotropic cases.

Morphology–elasticity relationships using decreasing fabric information of human trabecular bone from three major anatomical locations

With improving clinical CT scanning technology, the accuracy of CT-based finite element (FE) models of the human skeleton may be ameliorated by an enhanced description of apparent level bone mechanical properties. Micro-finite element (μFE) modeling can be used to study the apparent elastic behavior of human cancellous bone. In this study, samples from the femur, radius and vertebral body were investigated to evaluate the predictive power of morphology–elasticity relationships and to compare them across different anatomical regions. μFE models of 701 trabecular bone cubes with a side length of 5.3 mm were analyzed using kinematic boundary conditions. Based on the FE results, four morphology–elasticity models using bone volume fraction as well as full, limited or no fabric information were calibrated for each anatomical region. The 5 parameter Zysset–Curnier model using full fabric information showed excellent predictive power with coefficients of determination ( r2adj ) of 0.98, 0.95 and 0.94 of the femur, radius and vertebra data, respectively, with mean total norm errors between 14 and 20%. A constant orthotropy model and a constant transverse isotropy model, where the elastic anisotropy is defined by the model parameters, yielded coefficients of determination between 0.90 and 0.98 with total norm errors between 16 and 25%. Neglecting fabric information and using an isotropic model led to r2adj between 0.73 and 0.92 with total norm errors between 38 and 49%. A comparison of the model regressions revealed minor but significant (p<0.01) differences for the fabric–elasticity model parameters calibrated for the different anatomical regions. The proposed models and identified parameters can be used in future studies to compute the apparent elastic properties of human cancellous bone for homogenized FE models.

Measurement of structural anisotropy in femoral trabecular bone using clinical-resolution CT images

Discrepancies in finite-element model predictions of bone strength may be attributed to the simplified modeling of bone as an isotropic structure due to the resolution limitations of clinical-level Computed Tomography (CT) data. The aim of this study is to calculate the preferential orientations of bone (the principal directions) and the extent to which bone is deposited more in one direction compared to another (degree of anisotropy). Using 100 femoral trabecular samples, the principal directions and degree of anisotropy were calculated with a Gradient Structure Tensor (GST) and a Sobel Structure Tensor (SST) using clinical-level CT. The results were compared against those calculated with the gold standard Mean-Intercept-Length (MIL) fabric tensor using micro-CT. There was no significant difference between the GST and SST in the calculation of the main principal direction (median error=28°), and the error was inversely correlated to the degree of transverse isotropy (r=−0.34, p<0.01). The degree of anisotropy measured using the structure tensors was weakly correlated with the MIL-based measurements (r=0.2, p<0.001). Combining the principal directions with the degree of anisotropy resulted in a significant increase in the correlation of the tensor distributions (r=0.79, p<0.001). Both structure tensors were robust against simulated noise, kernel sizes, and bone volume fraction. We recommend the use of the GST because of its computational efficiency and ease of implementation. This methodology has the promise to predict the structural anisotropy of bone in areas with a high degree of anisotropy, and may improve the in vivo characterization of bone.

Harmonic Oscillator in a 1D or 2D Cavity with General Perfectly Reflecting Walls

We investigate the simple harmonic oscillator in a 1-d box, and the 2-d isotropic harmonic oscillator problem in a circular cavity with perfectly reflecting boundary conditions. The energy spectrum has been calculated as a function of the self-adjoint extension parameter. For sufficiently negative values of the self-adjoint extension parameter, there are bound states localized at the wall of the box or the cavity that resonate with the standard bound states of the simple harmonic oscillator or the isotropic oscillator. A free particle in a circular cavity has been studied for the sake of comparison. This work represents an application of the recent generalization of the Heisenberg uncertainty relation related to the theory of self-adjoint extensions in a finite volume.