DT-MRI based computation of collagen fiber deformation in human articular cartilage: a feasibility study

DTI assessment of the maturing growth plate of the knee in adolescents and young adults

PURPOSE

To assess the growth plates of the knee in a healthy population of young adults and adolescents using DTI, and to correlate the findings with chronological age and skeletal maturation.

METHODS

A prospective, cross-sectional study to assess the tibial and femoral growth plates with DTI in 155 healthy volunteers aged between 14.0 and 21 years old. Echo-planar DTI with 15 directions and b value of 0 and 600 s/mm² was performed on a 3 T whole-body scanner.

RESULTS

A relationship was observed between chronological age and most DTI metrics (fractional anisotropy, mean diffusivity, and radial diffusivity), tract length and volume. (No significant relationship could be seen for axonal diffusivity and tract length.) Subdivision according to skeletal maturation showed the greatest tract lengths and volumes seen in stage 4b and not 4a. The intra-observer agreement was significant (P = 0.01) for all the measured variables, but agreement varied (femur 0.53 - 0.98; tibia 0.58 - 0.98). Spearman's correlation showed a significant correlation for age (P = 0.05; P = 0.01) as well as for the fractional anisotropy value within all variables in both femur and tibia. Tract number and volume had a similar correlation with most variables, especially the DTI metrics, and would seem to be interchangeable.

CONCLUSION

The current study indicates that DTI metrics could be a tool to assess the skeletal maturation process of the growth plate and its activity. Tractography seems promising to assess the activity of the growth plate in a younger population but must be used with caution in the more mature growth plate.

Magnetic resonance and diffusion tensor imaging of the adolescent rabbit growth plate of the knee

PURPOSE

To assess the ability of MRI-DTI to evaluate growth plate morphology and activity compared with that of histomorphometry and micro-CT in rabbits.

METHODS

The hind limbs of female rabbits aged 16, 20, and 24 wk (n = 4 per age group) were studied using a 9.4T MRI scanner with a multi-gradient echo 3D sequence and DTI in 14 directions (b-value = 984 s/mm² ). After MRI, the right and left hind limb were processed for histological analysis and micro-CT, respectively. The Wilcoxon signed-rank test was used to evaluate the height and volume of the growth plate. Intraclass correlation and Pearson correlation coefficient were used to evaluate the association between DTI metrics and age.

RESULTS

The growth plate height and volume were similar for all modalities at each time point and age. Age was correlated with all tractography and DTI metrics in both the femur and tibia. A correlation was also observed between all the metrics at both sites. Tract number and volume declined with age; however, tract length did not show any changes. The fractional anisotropy color map showed lateral diffusion centrally in the growth plate and perpendicular diffusion in the hypertrophic zone, as verified by histology and micro-CT.

CONCLUSION

MRI-DTI may be useful for evaluating the growth plates.

Finite element modeling with subject-specific mechanical properties to assess knee osteoarthritis initiation and progression

Finite element models of the knee can be used to identify regions at risk of mechanical failure in studies of osteoarthritis. Models of the knee often implement joint geometry obtained from magnetic resonance imaging (MRI) or gait kinematics from motion capture to increase model specificity for a given subject. However, differences exist in cartilage material properties regionally as well as between subjects. This paper presents a method to create subject-specific finite element models of the knee that assigns cartilage material properties from T₂ relaxometry. We compared our T₂ -refined model to identical models with homogeneous material properties. When tested on three subjects from the Osteoarthritis Initiative data set, we found the T₂ -refined models estimated higher principal stresses and shear strains in most cartilage regions and corresponded better to increases in KL grade in follow-ups compared to their corresponding homogeneous material models. Measures of cumulative stress within regions of a T₂ -refined model also correlated better with the region's cartilage morphology MRI Osteoarthritis Knee Score as compared with the homogeneous model. We conclude that spatially heterogeneous T₂ -refined material properties improve the subject-specificity of finite element models compared to homogeneous material properties in osteoarthritis progression studies. Statement of Clinical Significance: T₂ -refined material properties can improve subject-specific finite element model assessments of cartilage degeneration.

T2 Mapping Refined Finite Element Modeling to Predict Knee Osteoarthritis Progression

This paper presents a novel method for informing cartilage material properties in finite element models from T₂ relaxometry. In the developed pipeline, T₂ relaxation values are mapped to elements in subject-specific finite element models of the cartilage and menisci. The Young's modulus for each element within the cartilage is directly calculated from its corresponding T₂ relaxation voxel value. Our model was tested on a single subject (Subject ID 9932809, Kellgren-Lawrence grade 2) from the Osteoarthritis Initiative dataset at baseline imaging. For comparison, an identical finite element model was built with homogeneous material properties. Kinematics of the stance phase of a standard gait cycle were used as model constraints. Simulation results were compared qualitatively to the MRI Osteoarthritis Knee Score (MOAKS) from the same baseline timepoint. Our T₂-refined material model showed higher maximum shear strain in regions with moderate cartilage loss as compared to the homogeneous material model, and the homogeneous model showed higher maximum principal stress and maximum shear strain in regions with no cartilage loss. These results show that a homogeneous material model likely underestimates tissue strains in regions with cartilage damage while overestimating strains in regions with healthy cartilage. This preliminary study demonstrates that T₂-refined material properties are more appropriate than assumptions of homogeneity in predictive models of cartilage damage.Clinical relevance- The proposed pipeline demonstrates a computationally efficient way to improve the subject-specificity of finite element models used for evaluation of osteoarthritis.

Magnetic Resonance Imaging-based biomechanical simulation of cartilage: A systematic review

MRI-based mathematical and computational modeling studies can contribute to a better understanding of the mechanisms governing cartilage's mechanical performance and cartilage disease. In addition, distinct modeling of cartilage is needed to optimize artificial cartilage production. These studies have opened up the prospect of further deepening our understanding of cartilage function. Furthermore, these studies reveal the initiation of an engineering-level approach to how cartilage disease affects material properties and cartilage function. Aimed at researchers in the field of MRI-based cartilage simulation, research articles pertinent to MRI-based cartilage modeling were identified, reviewed, and summarized systematically. Various MRI applications for cartilage modeling are highlighted, and the limitations of different constitutive models used are addressed. In addition, the clinical application of simulations and studied diseases are discussed. The paper's quality, based on the developed questionnaire, was assessed, and out of 79 reviewed papers, 34 papers were determined as high-quality. Due to the lack of the best constitutive models for various clinical conditions, researchers may consider the effect of constitutive material models on the cartilage disease simulation. In the future, research groups may incorporate various aspects of machine learning into constitutive models and MRI data extraction to further refine the study methodology. Moreover, researchers should strive for further reproducibility and rigorous model validation and verification, such as gait analysis.

Fiber Rearrangement and Matrix Compression in Soft Tissues: Multiscale Hypoelasticity and Application to Tendon

It is widely accepted that the nonlinear macroscopic mechanical behavior of soft tissue is governed by fiber straightening and re-orientation. Here, we provide a quantitative assessment of this phenomenon, by means of a continuum micromechanics approach. Given the negligibly small bending stiffness of crimped fibers, the latter are represented through a number of hypoelastic straight fiber phases with different orientations, being embedded into a hypoelastic matrix phase. The corresponding representative volume element (RVE) hosting these phases is subjected to “macroscopic” strain rates, which are downscaled to fiber and matrix strain rates on the one hand, and to fiber spins on the other hand. This gives quantitative access to the fiber decrimping (or straightening) phenomenon under non-affine conditions, i.e. in the case where the fiber orientations cannot be simply linked to the macroscopic strain state. In the case of tendinous tissue, such an RVE relates to the fascicle material with 50 μm characteristic length, made up of crimped collagen bundles and a gel-type matrix in-between. The fascicles themselves act as parallel fibers in a similar matrix at the scale of a tissue-related RVE with 500 μm characteristic length. As evidenced by a sensitivity analysis and confirmed by various mechanical tests, it is the initial crimping angle which drives both the degree of straightening and the shape of the macroscopic stress-strain curve, while the final linear portion of this curve depends almost exclusively on the collagen bundle elasticity. Our model also reveals the mechanical cooperation of the tissue’s key microstructural components: while the fibers carry tensile forces, the matrices undergo hydrostatic pressure.

Mechanical Pressure Driving Proteoglycan Expression in Mammographic Density: a Self-perpetuating Cycle?

Regions of high mammographic density (MD) in the breast are characterised by a proteoglycan (PG)-rich fibrous stroma, where PGs mediate aligned collagen fibrils to control tissue stiffness and hence the response to mechanical forces. Literature is accumulating to support the notion that mechanical stiffness may drive PG synthesis in the breast contributing to MD. We review emerging patterns in MD and other biological settings, of a positive feedback cycle of force promoting PG synthesis, such as in articular cartilage, due to increased pressure on weight bearing joints. Furthermore, we present evidence to suggest a pro-tumorigenic effect of increased mechanical force on epithelial cells in contexts where PG-mediated, aligned collagen fibrous tissue abounds, with implications for breast cancer development attributable to high MD. Finally, we summarise means through which this positive feedback mechanism of PG synthesis may be intercepted to reduce mechanical force within tissues and thus reduce disease burden.

A backward pre-stressing algorithm for efficient finite element implementation of in vivo material and geometrical parameters into fibril-reinforced mixture models of articular cartilage

Classical continuum mechanics has been widely used for implementation of the material models of articular cartilage (AC) mainly with the aid of the finite element (FE) method, which, in many cases, considers the stress-free configuration as the initial configuration. On the contrary, the AC experimental tests typically begin with the pre-stressed state of both material and geometrical properties. Indeed, imposing the initial pre-stress onto AC models with the in vivo values as the initial state would result in nonphysiologically expansion of the FE mesh due to the soft nature of AC. This change in the model configuration can also affect the material behavior kinematically in the mixture models of cartilage due to the intrinsic compressibility of the tissue. Although several different fixed-point backward algorithms, as the most straightforward pre-stressing methods, have already been developed to incorporate these initial conditions into FE models iteratively, such methods focused merely on the geometrical parameters, and they omitted the material variations of the anisotropic mixture models of AC. To address this issue, we propose an efficient algorithm generalizing the backward schemes to restore stress-free conditions by optimizing both the involving variables, and we hypothesize that it can affect the results considerably. To this end, a comparative simulation was implemented on an advanced and validated multiphasic model by the new and conventional algorithms. The results are in support of the hypothesis, as in our illustrative general AC model, the material parameters experienced a maximum error of 16% comparing to the initial in vivo data when the older algorithm was employed, and it led to a maximum variation of 44% in the recorded stresses comparing to the results of the new method. We conclude that our methodology enhanced the model fidelity, and it is applicable in most of the existing FE solvers for future mixture studies with accurate stress distributions.

On incorporating osmotic prestretch/prestress in image-driven finite element simulations of cartilage

Medical imaging performed in vivo captures geometries under Donnan osmotic loading, even when the articulating joint is otherwise mechanically unloaded. Hence patient-specific finite element (FE) models constructed from such medical images of cartilage represent osmotically induced prestretched/prestressed states. When applying classical modeling approaches to patient-specific simulations of cartilage a theoretical inconsistency arises: the in-vivo imaged geometry (used to construct the model) is not an unloaded, stress-free reference configuration. Furthermore when fitting nonlinear constitutive models that include osmotic swelling (to obtain material parameters), if one assumes that experimental data-generated from osmotically loaded cartilage-begin from a stress-free reference configuration the fitted stress-stretch relationship (parameters) obtained will actually describe a different behavior. In this study we: (1) establish a practical computational method to include osmotically induced prestretch/prestress in image-driven simulations of cartilage; and (2) investigate the influence of considering the prestretched/prestressed state both when fitting fiber-reinforced, biphasic constitutive models of cartilage that include osmotic swelling and when simulating cartilage responses. Our results highlight the importance of determining the prestretched/prestressed state within cartilage induced by osmotic loading in the imaged configuration prior to solving boundary value problems of interest. With our new constitutive model and modeling methods, we aim to improve the fidelity of FE-based, patient-specific biomechanical simulations of joints and cartilage. Improved simulations can provide medical researchers with new information often unavailable in a clinical setting, information that may contribute to better insight into the pathophysiology of cartilage diseases.

3D Fiber Orientation in Atherosclerotic Carotid Plaques

Atherosclerotic plaque rupture is the primary trigger of fatal cardiovascular events. Fibrillar collagen in atherosclerotic plaques and their directionality are anticipated to play a crucial role in plaque rupture. This study aimed assessing 3D fiber orientations and architecture in atherosclerotic plaques for the first time. Seven carotid plaques were imaged ex-vivo with a state-of-the-art Diffusion Tensor Imaging (DTI) technique, using a high magnetic field (9.4Tesla) MRI scanner. A 3D spin-echo sequence with uni-polar diffusion sensitizing pulsed field gradients was utilized for DTI and fiber directions were assessed from diffusion tensor measurements. The distribution of the 3D fiber orientations in atherosclerotic plaques were quantified and the principal fiber orientations (circumferential, longitudinal or radial) were determined. Overall, 52% of the fiber orientations in the carotid plaque specimens were closest to the circumferential direction, 34% to the longitudinal direction, and 14% to the radial direction. Statistically no significant difference was measured in the amount of the fiber orientations between the concentric and eccentric plaque sites. However, concentric plaque sites showed a distinct structural organization, where the principally longitudinally oriented fibers were closer to the luminal side and the principally circumferentially oriented fibers were located more abluminally. The acquired unique information on 3D plaque fiber direction will help understanding pathobiological mechanisms of atherosclerotic plaque progression and pave the road to more realistic biomechanical plaque modeling for rupture assessment.

Real-time observation of fluid flows in tissue during stress relaxation using Raman spectroscopy

This paper outlines a technique to measure fluid levels in articular cartilage tissue during an unconfined stress relaxation test. A time series of Raman spectrum were recorded during relaxation and the changes in the specific Raman spectral bands assigned to water and protein were monitored to determine the fluid content of the tissue. After 1000s unconfined compression the fluid content of the tissue is reduced by an average of 3.9%±1.7%. The reduction in fluid content during compression varies between samples but does not significantly increase with increasing strain. Further development of this technique will allow mapping of fluid distribution and flows during dynamic testing making it a powerful tool to understand the role of interstitial fluid in the functional performance of cartilage.

T2 MR imaging vs. computational modeling of human articular cartilage tissue functionality

The detection of early stages of cartilage degeneration remains diagnostically challenging. One promising non-invasive approach is to functionally assess the tissue response to loading by serial magnetic resonance (MR) imaging in terms of T2 mapping under simultaneous mechanical loading. As yet, however, it is not clear which cartilage component contributes to the tissue functionality as assessed by quantitative T2 mapping. To this end, quantitative T2 maps of histologically intact cartilage samples (n=8) were generated using a clinical 3.0-T MR imaging system. Using displacement-controlled quasi-static indentation loading, serial T2 mapping was performed at three defined strain levels and loading-induced relative changes were determined in distinct regions-of-interest. Samples underwent conventional biomechanical testing (by unconfined compression) as well as histological assessment (by Mankin scoring) for reference purposes. Moreover, an anisotropic hyperelastic constitutive model of cartilage was implemented into a finite element (FE) code for cross-referencing. In efforts to simulate the evolution of compositional and structural intra-tissue changes under quasi-static loading, the indentation-induced changes in quantitative T2 maps were referenced to underlying changes in cartilage composition and structure. These changes were parameterized as cartilage fluid, proteoglycan and collagen content as well as collagen orientation. On a pixel-wise basis, each individual component correlation with T2 relaxation times was determined by Spearman's ρ_s and significant correlations were found between T2 relaxation times and all four tissue parameters for all indentation strain levels. Thus, the biological changes in functional MR Imaging parameters such as T2 can further be characterized to strengthen the scientific basis of functional MRI techniques with regards to their perspective clinical applications.

MRI magic-angle effect in femorotibial cartilages of the red kangaroo

OBJECTIVE

Kangaroo knee cartilages are robust tissues that can support knee flexion and endure high levels of compressive stress. This study aimed to develop a detailed understanding of the collagen architecture in kangaroo knee cartilages and thus obtain insights into the biophysical basis of their function.

DESIGN

Cylindrical/square plugs from femoral and tibial hyaline cartilage and tibial fibrocartilage were excised from the knees of three adult red kangaroos. Multi-slice, multi-echo MR images were acquired at the sample orientations 0° and 55° ("magic angle") with respect to the static magnetic field. Maps of the transverse relaxation rate constant (R₂) and depth profiles of R₂ and its anisotropic component (R₂^A) were constructed from the data.

RESULTS

The R₂^A profiles confirmed the classic three-zone organisation of all cartilage samples. Femoral hyaline cartilage possessed a well-developed, thick superficial zone. Tibial hyaline cartilage possessed a very thick radial zone (80% relative thickness) that exhibited large R₂^A values consistent with highly ordered collagen. The R₂^A profile of tibial fibrocartilage exhibited a unique region near the bone (bottom 5-10%) consistent with elevated proteoglycan content ("attachment sub-zone").

CONCLUSIONS

Our observations suggest that the well-developed superficial zone of femoral hyaline cartilage is suitable for supporting knee flexion; the thick and well-aligned radial zone of tibial hyaline cartilage is adapted to endure high compressive stress; while the innermost part of the radial zone of tibial fibrocartilage may facilitate anchoring of the collagen fibres to withstand high shear deformation. These findings may inspire new designs for cartilage tissue engineering.

Diffusion tensor MR microscopy of tissues with low diffusional anisotropy

Background

Diffusion tensor imaging exploits preferential diffusional motion of water molecules residing within tissue compartments for assessment of tissue structural anisotropy. However, instrumentation and post-processing errors play an important role in determination of diffusion tensor elements. In the study, several experimental factors affecting accuracy of diffusion tensor determination were analyzed.

Materials and methods

Effects of signal-to-noise ratio and configuration of the applied diffusion-sensitizing gradients on fractional anisotropy bias were analyzed by means of numerical simulations. In addition, diffusion tensor magnetic resonance microscopy experiments were performed on a tap water phantom and bovine articular cartilage-on-bone samples to verify the simulation results.

Results

In both, the simulations and the experiments, the multivariate linear regression of the diffusion-tensor analysis yielded overestimated fractional anisotropy with low SNRs and with low numbers of applied diffusion-sensitizing gradients.

Conclusions

An increase of the apparent fractional anisotropy due to unfavorable experimental conditions can be overcome by applying a larger number of diffusion sensitizing gradients with small values of the condition number of the transformation matrix. This is in particular relevant in magnetic resonance microscopy, where imaging gradients are high and the signal-to-noise ratio is low.

Modelling non-symmetric collagen fibre dispersion in arterial walls

New experimental results on collagen fibre dispersion in human arterial layers have shown that the dispersion in the tangential plane is more significant than that out of plane. A rotationally symmetric dispersion model is not able to capture this distinction. For this reason, we introduce a new non-symmetric dispersion model, based on the bivariate von Mises distribution, which is used to construct a new structure tensor. The latter is incorporated in a strain-energy function that accommodates both the mechanical and structural features of the material, extending our rotationally symmetric dispersion model (Gasser et al. 2006 J. R. Soc. Interface 3, 15-35. (doi:10.1098/rsif.2005.0073)). We provide specific ranges for the dispersion parameters and show how previous models can be deduced as special cases. We also provide explicit expressions for the stress and elasticity tensors in the Lagrangian description that are needed for a finite-element implementation. Material and structural parameters were obtained by fitting predictions of the model to experimental data obtained from human abdominal aortic adventitia. In a finite-element example, we analyse the influence of the fibre dispersion on the homogeneous biaxial mechanical response of aortic strips, and in a final example the non-homogeneous stress distribution is obtained for circumferential and axial strips under fixed extension. It has recently become apparent that this more general model is needed for describing the mechanical behaviour of a variety of fibrous tissues.

Techniques and applications of in vivo diffusion imaging of articular cartilage

Early in the process of osteoarthritis (OA) the composition (water, proteoglycan [PG], and collagen) and structure of articular cartilage is altered leading to changes in its mechanical properties. A technique that can assess the composition and structure of the cartilage in vivo can provide insight in the mechanical integrity of articular cartilage and become a powerful tool for the early diagnosis of OA. Diffusion tensor imaging (DTI) has been proposed as a biomarker for cartilage composition and structure. DTI is sensitive to the PG content through the mean diffusivity and to the collagen architecture through the fractional anisotropy. However, the acquisition of DTI of articular cartilage in vivo is challenging due to the short T2 of articular cartilage (∼40 ms at 3 Tesla) and the high resolution needed (0.5-0.7 mm in plane) to depict the cartilage anatomy. We describe the pulse sequences used for in vivo DTI of articular cartilage and discus general strategies for protocol optimization. We provide a comprehensive review of measurements of DTI of articular cartilage from ex vivo validation experiments to its recent clinical applications.

Effect of partial H2O-D2O replacement on the anisotropy of transverse proton spin relaxation in bovine articular cartilage

Anisotropy of transverse proton spin relaxation in collagen-rich tissues like cartilage and tendon is a well-known phenomenon that manifests itself as the "magic-angle" effect in magnetic resonance images of these tissues. It is usually attributed to the non-zero averaging of intra-molecular dipolar interactions in water molecules bound to oriented collagen fibers. One way to manipulate the contributions of these interactions to spin relaxation is by partially replacing the water in the cartilage sample with deuterium oxide. It is known that dipolar interactions in deuterated solutions are weaker, resulting in a decrease in proton relaxation rates. In this work, we investigate the effects of deuteration on the longitudinal and the isotropic and anisotropic contributions to transverse relaxation of water protons in bovine articular cartilage. We demonstrate that the anisotropy of transverse proton spin relaxation in articular cartilage is independent of the degree of deuteration, bringing into question some of the assumptions currently held over the origins of relaxation anisotropy in oriented tissues.

Accurate Prediction of Stress in Fibers with Distributed Orientations Using Generalized High-Order Structure Tensors

The orientation of collagen fibers plays an important role on the mechanics of connective tissues. Connective tissues have fibers with different orientation distributions. The angular integration formulation used to model the mechanics of fibers with distributed orientation is accurate, but computationally expensive for numerical methods such as finite elements. This study presents a formulation based on pre-integrated Generalized High-Order Structure Tensors (GHOST) which greatly improves the accuracy of the predicted stress. Simplifications of the GHOST formulation for transversely-isotropic and planar fiber distributions are also presented. Additionally, the GHOST and the angular integration formulations are compared for different loading conditions, fiber orientation functions, strain energy functions and degrees of fiber non-linearity. It was found that the GHOST formulation predicted the stress of the fibers with an error lower than 10% for uniaxial and biaxial tension. Fiber non-linearity increased the error of the GHOST formulation; however, the error was reduced to negligible values by considering higher order structure tensors. The GHOST formulation produced lower errors when used with an elliptical fiber density function and a binomial strain energy function. In conclusion, the GHOST formulation is able to accurately predict the stress of fibers with distributed orientation without requiring numerous integral calculations. Consequently, the GHOST formulation may reduce the computational effort needed to analyze the mechanics of fibrous tissues with distributed orientations.

Simultaneous magnetic resonance imaging and consolidation measurement of articular cartilage

Magnetic resonance imaging (MRI) offers the opportunity to study biological tissues and processes in a non-disruptive manner. The technique shows promise for the study of the load-bearing performance (consolidation) of articular cartilage and changes in articular cartilage accompanying osteoarthritis. Consolidation of articular cartilage involves the recording of two transient characteristics: the change over time of strain and the hydrostatic excess pore pressure (HEPP). MRI study of cartilage consolidation under mechanical load is limited by difficulties in measuring the HEPP in the presence of the strong magnetic fields associated with the MRI technique. Here we describe the use of MRI to image and characterize bovine articular cartilage deforming under load in an MRI compatible consolidometer while monitoring pressure with a Fabry-Perot interferometer-based fiber-optic pressure transducer.

Cartilage repair surgery: outcome evaluation by using noninvasive cartilage biomarkers based on quantitative MRI techniques?

BACKGROUND

New quantitative magnetic resonance imaging (MRI) techniques are increasingly applied as outcome measures after cartilage repair.

OBJECTIVE

To review the current literature on the use of quantitative MRI biomarkers for evaluation of cartilage repair at the knee and ankle.

METHODS

Using PubMed literature research, studies on biochemical, quantitative MR imaging of cartilage repair were identified and reviewed.

RESULTS

Quantitative MR biomarkers detect early degeneration of articular cartilage, mainly represented by an increasing water content, collagen disruption, and proteoglycan loss. Recently, feasibility of biochemical MR imaging of cartilage repair tissue and surrounding cartilage was demonstrated. Ultrastructural properties of the tissue after different repair procedures resulted in differences in imaging characteristics. T2 mapping, T1rho mapping, delayed gadolinium-enhanced MRI of cartilage (dGEMRIC), and diffusion weighted imaging (DWI) are applicable on most clinical 1.5 T and 3 T MR scanners. Currently, a standard of reference is difficult to define and knowledge is limited concerning correlation of clinical and MR findings. The lack of histological correlations complicates the identification of the exact tissue composition.

CONCLUSIONS

A multimodal approach combining several quantitative MRI techniques in addition to morphological and clinical evaluation might be promising. Further investigations are required to demonstrate the potential for outcome evaluation after cartilage repair.

Modeling the dispersion in electromechanically coupled myocardium

We present an approach to model the dispersion of fiber and sheet orientations in the myocardium. By utilizing structure parameters, an existing orthotropic and invariant-based constitutive model developed to describe the passive behavior of the myocardium is augmented. Two dispersion parameters are fitted to experimentally observed angular dispersion data of the myocardial tissue. Computations are performed on a unit myocardium tissue cube and on a slice of the left ventricle indicating that the dispersion parameter has an effect on the myocardial deformation and stress development. The use of fiber dispersions relating to a pathological myocardium had a rather big effect. The final example represents an ellipsoidal model of the left ventricle indicating the influence of fiber and sheet dispersions upon contraction over a cardiac cycle. Although only a minor shift in the pressure-volume (PV) loops between the cases with no dispersions and with fiber and sheet dispersions for a healthy myocardium was observed, a remarkably different behavior is obtained with a fiber dispersion relating to a diseased myocardium. In future simulations, this dispersion model for myocardial tissue may advantageously be used together with models of, for example, growth and remodeling of various cardiac diseases.

Modeling sample/patient-specific structural and diffusional responses of cartilage using DT-MRI

We propose a new 3D biphasic constitutive model designed to incorporate structural data on the sample/patient-specific collagen fiber network. The finite strain model focuses on the load-bearing morphology, that is, an incompressible, poroelastic solid matrix, reinforced by an inhomogeneous, dispersed fiber fabric, saturated with an incompressible fluid at constant electrolytic conditions residing in strain-dependent pores of the collagen-proteoglycan solid matrix. In addition, the fiber network of the solid influences the fluid permeability and an intrafibrillar portion that cannot be 'squeezed out' from the tissue. We implement the model into a finite element code. To demonstrate the utility of our proposed modeling approach, we test two hypotheses by simulating an indentation experiment for a human tissue sample. The simulations use ultra-high field diffusion tensor magnetic resonance imaging that was performed on the tissue sample. We test the following hypotheses: (i) the through-thickness structural arrangement of the collagen fiber network adjusts fluid permeation to maintain fluid pressure (Biomech. Model. Mechanobiol. 7:367-378, 2008); and (ii) the inhomogeneity of mechanical properties through the cartilage thickness acts to maintain fluid pressure at the articular surface (J. Biomech. Eng. 125:569-577, 2003). For the tissue sample investigated, both through-thickness inhomogeneities of the collagen fiber distribution and of the material properties serve to influence the interstitial fluid pressure distribution and maintain fluid pressure underneath the indenter at the cartilage surface. Tissue inhomogeneity appears to have a larger effect on fluid pressure retention in this tissue sample and on the advantageous pressure distribution.

Implementation of a gait cycle loading into healthy and meniscectomised knee joint models with fibril-reinforced articular cartilage

Computational models can be used to evaluate the functional properties of knee joints and possible risk locations within joints. Current models with fibril-reinforced cartilage layers do not provide information about realistic human movement during walking. This study aimed to evaluate stresses and strains within a knee joint by implementing load data from a gait cycle in healthy and meniscectomised knee joint models with fibril-reinforced cartilages. A 3D finite element model of a knee joint with cartilages and menisci was created from magnetic resonance images. The gait cycle data from varying joint rotations, translations and axial forces were taken from experimental studies and implemented into the model. Cartilage layers were modelled as a fibril-reinforced poroviscoelastic material with the menisci considered as a transversely isotropic elastic material. In the normal knee joint model, relatively high maximum principal stresses were specifically predicted to occur in the medial condyle of the knee joint during the loading response. Bilateral meniscectomy increased stresses, strains and fluid pressures in cartilage on the lateral side, especially during the first 50% of the stance phase of the gait cycle. During the entire stance phase, the superficial collagen fibrils modulated stresses of cartilage, especially in the medial tibial cartilage. The present computational model with a gait cycle and fibril-reinforced biphasic cartilage revealed time- and location-dependent differences in stresses, strains and fluid pressures occurring in cartilage during walking. The lateral meniscus was observed to have a more significant role in distributing loads across the knee joint than the medial meniscus, suggesting that meniscectomy might initiate a post-traumatic process leading to osteoarthritis at the lateral compartment of the knee joint.

A review of the combination of experimental measurements and fibril-reinforced modeling for investigation of articular cartilage and chondrocyte response to loading

The function of articular cartilage depends on its structure and composition, sensitively impaired in disease (e.g. osteoarthritis, OA). Responses of chondrocytes to tissue loading are modulated by the structure. Altered cell responses as an effect of OA may regulate cartilage mechanotransduction and cell biosynthesis. To be able to evaluate cell responses and factors affecting the onset and progression of OA, local tissue and cell stresses and strains in cartilage need to be characterized. This is extremely challenging with the presently available experimental techniques and therefore computational modeling is required. Modern models of articular cartilage are inhomogeneous and anisotropic, and they include many aspects of the real tissue structure and composition. In this paper, we provide an overview of the computational applications that have been developed for modeling the mechanics of articular cartilage at the tissue and cellular level. We concentrate on the use of fibril-reinforced models of cartilage. Furthermore, we introduce practical considerations for modeling applications, including also experimental tests that can be combined with the modeling approach. At the end, we discuss the prospects for patient-specific models when aiming to use finite element modeling analysis and evaluation of articular cartilage function, cellular responses, failure points, OA progression, and rehabilitation.

Quantitative parametric MRI of articular cartilage: a review of progress and open challenges

With increasing life expectancies and the desire to maintain active lifestyles well into old age, the impact of the debilitating disease osteoarthritis (OA) and its burden on healthcare services is mounting. Emerging regenerative therapies could deliver significant advances in the effective treatment of OA but rely upon the ability to identify the initial signs of tissue damage and will also benefit from quantitative assessment of tissue repair in vivo. Continued development in the field of quantitative MRI in recent years has seen the emergence of techniques able to probe the earliest biochemical changes linked with the onset of OA. Quantitative MRI measurements including T(1), T(2) and T(1ρ) relaxometry, diffusion weighted imaging and magnetisation transfer have been studied and linked to the macromolecular structure of cartilage. Delayed gadolinium-enhanced MRI of cartilage, sodium MRI and glycosaminoglycan chemical exchange saturation transfer techniques are sensitive to depletion of cartilage glycosaminoglycans and may allow detection of the earliest stages of OA. We review these current and emerging techniques for the diagnosis of early OA, evaluate the progress that has been made towards their implementation in the clinic and identify future challenges in the field.

Computational investigation of in situ chondrocyte deformation and actin cytoskeleton remodelling under physiological loading

Previous experimental studies have determined local strain fields for both healthy and degenerate cartilage tissue during mechanical loading. However, the biomechanical response of chondrocytes in situ, in particular the response of the actin cytoskeleton to physiological loading conditions, is poorly understood. In the current study a three-dimensional (3-D) representative volume element (RVE) for cartilage tissue is created, comprising a chondrocyte surrounded by a pericellular matrix and embedded in an extracellular matrix. A 3-D active modelling framework incorporating actin cytoskeleton remodelling and contractility is implemented to predict the biomechanical behaviour of chondrocytes. Physiological and abnormal strain fields, based on the experimental study of Wong and Sah (J. Orthop. Res. 2010; 28: 1554-1561), are applied to the RVE. Simulations demonstrate that the presence of a focal defect significantly affects cellular deformation, increases the stress experienced by the nucleus, and alters the distribution of the actin cytoskeleton. It is demonstrated that during dynamic loading cyclic tension reduction in the cytoplasm causes continuous dissociation of the actin cytoskeleton. In contrast, during static loading significant changes in cytoplasm tension are not predicted and hence the rate of dissociation of the actin cytoskeleton is reduced. It is demonstrated that chondrocyte behaviour is affected by the stiffness of the pericellular matrix, and also by the anisotropy of the extracellular matrix. The findings of the current study are of particular importance in understanding the biomechanics underlying experimental observations such as actin cytoskeleton dissociation during the dynamic loading of chondrocytes.

Subject-specific analysis of joint contact mechanics: application to the study of osteoarthritis and surgical planning

Advances in computational mechanics, constitutive modeling, and techniques for subject-specific modeling have opened the door to patient-specific simulation of the relationships between joint mechanics and osteoarthritis (OA), as well as patient-specific preoperative planning. This article reviews the application of computational biomechanics to the simulation of joint contact mechanics as relevant to the study of OA. This review begins with background regarding OA and the mechanical causes of OA in the context of simulations of joint mechanics. The broad range of technical considerations in creating validated subject-specific whole joint models is discussed. The types of computational models available for the study of joint mechanics are reviewed. The types of constitutive models that are available for articular cartilage are reviewed, with special attention to choosing an appropriate constitutive model for the application at hand. Issues related to model generation are discussed, including acquisition of model geometry from volumetric image data and specific considerations for acquisition of computed tomography and magnetic resonance imaging data. Approaches to model validation are reviewed. The areas of parametric analysis, factorial design, and probabilistic analysis are reviewed in the context of simulations of joint contact mechanics. Following the review of technical considerations, the article details insights that have been obtained from computational models of joint mechanics for normal joints; patient populations; the study of specific aspects of joint mechanics relevant to OA, such as congruency and instability; and preoperative planning. Finally, future directions for research and application are summarized.

Integrating qPLM and biomechanical test data with an anisotropic fiber distribution model and predictions of TGF-β1 and IGF-1 regulation of articular cartilage fiber modulus

A continuum mixture model with distinct collagen (COL) and glycosaminoglycan elastic constituents was developed for the solid matrix of immature bovine articular cartilage. A continuous COL fiber volume fraction distribution function and a true COL fiber elastic modulus ([Formula: see text] were used. Quantitative polarized light microscopy (qPLM) methods were developed to account for the relatively high cell density of immature articular cartilage and used with a novel algorithm that constructs a 3D distribution function from 2D qPLM data. For specimens untreated and cultured in vitro, most model parameters were specified from qPLM analysis and biochemical assay results; consequently, [Formula: see text] was predicted using an optimization to measured mechanical properties in uniaxial tension and unconfined compression. Analysis of qPLM data revealed a highly anisotropic fiber distribution, with principal fiber orientation parallel to the surface layer. For untreated samples, predicted [Formula: see text] values were 175 and 422 MPa for superficial (S) and middle (M) zone layers, respectively. TGF-[Formula: see text]1 treatment was predicted to increase and decrease [Formula: see text] values for the S and M layers to 281 and 309 MPa, respectively. IGF-1 treatment was predicted to decrease [Formula: see text] values for the S and M layers to 22 and 26 MPa, respectively. A novel finding was that distinct native depth-dependent fiber modulus properties were modulated to nearly homogeneous values by TGF-[Formula: see text]1 and IGF-1 treatments, with modulated values strongly dependent on treatment.

Multilayer material properties of aorta determined from nanoindentation tests

In a wide range of biomechanical modeling of aorta from traumatic injury to stent grafts, the arterial wall has been considered as a single homogeneous layer vessel, ignoring the fact that arteries are composed of distinct anatomical layers with different mechanical characteristics. In this study, using a custom-made nanoindentation technique, changes in the mechanical properties of porcine thoracic aorta wall in the radial direction were characterized using a quasi-linear viscoelastic model. Two layers of equal thickness were mechanically distinguishable in descending aorta based on the radial variations in the instantaneous Young's modulus E and reduced relaxation function G(t). Overall, comparison of E and G(∞) of the outer half (70.27±2.47 kPa and 0.35±0.01) versus the inner half (60.32±1.65 kPa and 0.33±0.01) revealed that the outer half was stiffer and showed less relaxation. The results were used to explain local mechanisms of deformation, force transmission, tear propagation and failure in arteries.

Collagen-agarose co-gels as a model for collagen-matrix interaction in soft tissues subjected to indentation

The mechanical properties of soft tissues depend on the collagen fiber network and the surrounding non-fibrillar matrix. The mechanical role of non-fibrillar material remains poorly understood. Our recent study (Lake and Barocas, Ann Biomed Eng 2011) introduced collagen-agarose co-gels as a simple experimental model system to evaluate the mechanical contribution of non-fibrillar matrix, and evaluated co-gel properties in uniaxial tension. In this study, we utilized similar co-gels to examine collagen-matrix interaction in tissues subjected to incremental stress-relaxation indentation tests. Mechanical testing was performed using two orthogonal custom test devices, and polarized light imaging was used to quantify 3D collagen fiber kinematics under load. The addition of agarose led to concentration-dependent changes in the time-dependent mechanical response and magnitude/spread of collagen fiber reorganization of tissue analogs. Specifically, peak/relaxed loads increased, and relaxation rate decreased, with increasing agarose concentration. In addition, increasing agarose content led to larger magnitude changes in orientation direction and alignment strength that were more localized near the indenter. Results suggest that non-fibrillar material significantly contributes to the behavior of co-gels in indentation, likely by reducing permeability and resisting volume change, thereby providing insight into the properties of artificial and native tissues subjected to non-tensile loading.

Structural analysis of articular cartilage using multiphoton microscopy: input for biomechanical modeling

The 3-D morphology of chicken articular cartilage was quantified using multiphoton microscopy (MPM) for use in continuum-mechanical modeling. To motivate this morphological study we propose aspects of a new, 3-D finite strain constitutive model for articular cartilage focusing on the essential load-bearing morphology: an inhomogeneous, poro-(visco)elastic solid matrix reinforced by an anisotropic, (visco)elastic dispersed fiber fabric which is saturated by an incompressible fluid residing in strain-dependent pores. Samples of fresh chicken cartilage were sectioned in three orthogonal planes and imaged using MPM, specifically imaging the collagen fibers using second harmonic generation. Employing image analysis techniques based on Fourier analysis, we derived the principal directionality and dispersion of the collagen fiber fabric in the superficial layer. In the middle layer, objective thresholding techniques were used to extract the volume fraction occupied by extracellular collagen matrix. In conjunction with information available in the literature, or additional experimental testing, we show how this data can be used to derive a 3-D map of the initial solid volume fraction and Darcy permeability.

A constrained von Mises distribution to describe fiber organization in thin soft tissues

The semi-circular von Mises distribution is widely used to describe the unimodal planar organization of fibers in thin soft tissues. However, it cannot accurately describe the isotropic subpopulation of fibers present in such tissues, and therefore an improved mathematical description is needed. We present a modified distribution, formed as a weighted mixture of the semi-circular uniform distribution and the semi-circular von Mises distribution. It is described by three parameters: β, which weights the contribution from each mixture component; k, the fiber concentration factor; and θ ( p ), the preferred fiber orientation. This distribution was used to fit data obtained by small-angle light scattering experiments from various thin soft tissues. Initial use showed that satisfactory fits of fiber distributions could be obtained (error generally < 1%), but at the cost of non-physically meaningful values of k and β. To address this issue, an empirical constraint between the parameters k and β was introduced, resulting in a constrained 2-parameter fiber distribution. Compared to the 3-parameter distribution, the constrained 2-parameter distribution fits experimental data well (error generally < 2%) and had the advantage of producing physically meaningful parameter values. In addition, the constrained 2-parameter approach was more robust to experimental noise. The constrained 2-parameter fiber distribution can replace the semi-circular von Mises distribution to describe unimodal planar organization of fibers in thin soft tissues. Inclusion of such a function in constitutive models for finite element simulations should provide better quantitative estimates of soft tissue biomechanics under normal and pathological conditions.

Prediction of fibre architecture and adaptation in diseased carotid bifurcations

Many studies have used patient-specific finite element models to estimate the stress environment in atherosclerotic plaques, attempting to correlate the magnitude of stress to plaque vulnerability. In complex geometries, few studies have incorporated the anisotropic material response of arterial tissue. This paper presents a fibre remodelling algorithm to predict the fibre architecture, and thus anisotropic material response in four patient-specific models of the carotid bifurcation. The change in fibre architecture during disease progression and its affect on the stress environment in the plaque were predicted. The mean fibre directions were assumed to lie at an angle between the two positive principal strain directions. The angle and the degree of dispersion were assumed to depend on the ratio of principal strain values. Results were compared with experimental observations and other numerical studies. In non-branching regions of each model, the typical double helix arterial fibre pattern was predicted while at the bifurcation and in regions of plaque burden, more complex fibre architectures were found. The predicted change in fibre architecture in the arterial tissue during plaque progression was found to alter the stress environment in the plaque. This suggests that the specimen-specific anisotropic response of the tissue should be taken into account to accurately predict stresses in the plaque. Since determination of the fibre architecture in vivo is a difficult task, the system presented here provides a useful method of estimating the fibre architecture in complex arterial geometries.