Daniel H. Cortes

Characterizing the mechanical contribution of fiber angular distribution in connective tissue: comparison of two modeling approaches

Modeling of connective tissues often includes collagen fibers explicitly as one of the components. These fibers can be oriented in many directions; therefore, several studies have considered statistical distributions to describe the fiber arrangement. One approach to formulate a constitutive framework for distributed fibers is to express the mechanical parameters, such as strain energy and stresses, in terms of angular integrals. These integrals represent the addition of the contribution of infinitesimal fractions of fibers oriented in a given direction. This approach leads to accurate results; however, it requires lengthy calculations. Recently, the use of generalized structure tensors has been proposed to represent the angular distribution in the constitutive equations of the fibers. Although this formulation is much simpler and fewer calculations are required, such structure tensors can only be used when all the fibers are in tension and the angular distribution is small. However, the amount of error introduced in these cases of non-tensile fiber loading and large angular distributions have not been quantified. Therefore, the objective of this study is to determine the range of values of angular distribution for which acceptable differences (less than 10%) between these two formulations are obtained. It was found, analytically and numerically, that both formulations are equivalent for planar distributions under equal-biaxial stretch. The comparison also showed, for other loading conditions, that the differences decrease when the fiber distribution is very small. Differences of less than 10% were usually obtained when the fiber distribution was very low (κ ≈ 0.03; κ ranges between 0 and 1/3, for aligned and isotropic distributed fibers, respectively). This range of angular distribution greatly limits the types of tissue that can be accurately analyzed using generalized structure tensors. It is expected that the results from this study guide the selection of a proper approach to analyze a particular tissue under a particular loading condition.

Elastic, permeability and swelling properties of human intervertebral disc tissues: A benchmark for tissue engineering.

The aim of functional tissue engineering is to repair and replace tissues that have a biomechanical function, i.e., connective orthopaedic tissues. To do this, it is necessary to have accurate benchmarks for the elastic, permeability, and swelling (i.e., biphasic-swelling) properties of native tissues. However, in the case of the intervertebral disc, the biphasic-swelling properties of individual tissues reported in the literature exhibit great variation and even span several orders of magnitude. This variation is probably caused by differences in the testing protocols and the constitutive models used to analyze the data. Therefore, the objective of this study was to measure the human lumbar disc annulus fibrosus (AF), nucleus pulposus (NP), and cartilaginous endplates (CEP) biphasic-swelling properties using a consistent experimental protocol and analyses. The testing protocol was composed of a swelling period followed by multiple confined compression ramps. To analyze the confined compression data, the tissues were modeled using a biphasic-swelling model, which augments the standard biphasic model through the addition of a deformation-dependent osmotic pressure term. This model allows considering the swelling deformations and the contribution of osmotic pressure in the analysis of the experimental data. The swelling stretch was not different between the disc regions (AF: 1.28±0.16; NP: 1.73±0.74; CEP: 1.29±0.26), with a total average of 1.42. The aggregate modulus (Ha) of the extra-fibrillar matrix was higher in the CEP (390kPa) compared to the NP (100kPa) or AF (30kPa). The permeability was very different across tissue regions, with the AF permeability (64 E(-16)m(4)/Ns) higher than the NP and CEP (~5.5 E(-16)m(4)/Ns). Additionally, a normalized time-constant (3000s) for the stress relaxation was similar for all the disc tissues. The properties measured in this study are important as benchmarks for tissue engineering and for modeling the discs mechanical behavior and transport.

Biaxial Tensile Testing and Constitutive Modeling of Human Supraspinatus Tendon

The heterogeneous composition, collagen fiber organization and mechanical properties of the supraspinatus tendon (SST) offer an opportunity for studying the structure-function relationships of fibrous musculoskeletal connective tissues. The objective of this study was to evaluate the contribution of collagen fiber organization to the planar tensile mechanics of the human SST. This was accomplished by fitting biaxial tensile data with a structural constitutive model that incorporates a sample-specific angular distribution of nonlinear fibers. Biaxial testing was employed to avoid the limitation of non-physiologic traction-free boundary conditions present during uniaxial testing. Samples were tested under a range of boundary conditions with simultaneous monitoring of collagen fiber orientation via polarized light imaging. The experimental data were input into a hyperelastic constitutive model incorporating the contributions of the uncrimped fibers. The model fit the longitudinal stresses well and was successfully validated. The transverse stresses were fit less well with greater errors observed for less aligned samples. Additional strain energy terms representing fiber-fiber interactions are likely necessary to provide closer approximation of the transverse stresses. This approach demonstrated that the longitudinal tensile mechanics of the SST are primarily dependent on the moduli, crimp, and angular distribution of its collagen fibers.

Biaxial Tension of Fibrous Tissue: Using Finite Element Methods to Address Experimental Challenges Arising From Boundary Conditions and Anisotropy

Planar biaxial tension remains a critical loading modality for fibrous soft tissue and is widely used to characterize tissue mechanical response, evaluate treatments, develop constitutive formulas, and obtain material properties for use in finite element studies. Although the application of tension on all edges of the test specimen represents the in situ environment, there remains a need to address the interpretation of experimental results. Unlike uniaxial tension, in biaxial tension the applied forces at the loading clamps do not transmit fully to the region of interest (ROI), which may lead to improper material characterization if not accounted for. In this study, we reviewed the tensile biaxial literature over the last ten years, noting experimental and analysis challenges. In response to these challenges, we used finite element simulations to quantify load transmission from the clamps to the ROI in biaxial tension and to formulate a correction factor that can be used to determine ROI stresses. Additionally, the impact of sample geometry, material anisotropy, and tissue orientation on the correction factor were determined. Large stress concentrations were evident in both square and cruciform geometries and for all levels of anisotropy. In general, stress concentrations were greater for the square geometry than the cruciform geometry. For both square and cruciform geometries, materials with fibers aligned parallel to the loading axes reduced stress concentrations compared to the isotropic tissue, resulting in more of the applied load being transferred to the ROI. In contrast, fiber-reinforced specimens oriented such that the fibers aligned at an angle to the loading axes produced very large stress concentrations across the clamps and shielding in the ROI. A correction factor technique was introduced that can be used to calculate the stresses in the ROI from the measured experimental loads at the clamps. Application of a correction factor to experimental biaxial results may lead to more accurate representation of the mechanical response of fibrous soft tissue.

Continuous Shear Wave Elastography: A New Method to Measure Viscoelastic Properties of Tendons in Vivo.

Viscoelastic mechanical properties are frequently altered after tendon injuries and during recovery. Therefore, non-invasive measurements of shear viscoelastic properties may help evaluate tendon recovery and compare the effectiveness of different therapies. The objectives of this study were to describe an elastography method for measuring localized viscoelastic properties of tendons and to discuss the initial results in healthy and injured human Achilles and semitendinosus tendons. The technique used an external actuator to generate the shear waves in the tendon at different frequencies and plane wave imaging to measure shear wave displacements. For each of the excitation frequencies, maps of direction-specific wave speeds were calculated using local frequency estimation. Maps of viscoelastic properties were obtained using a pixel-wise curve fit of wave speed and frequency. The method was validated by comparing measurements of wave speed in agarose gels with those obtained using magnetic resonance elastography. Measurements in human healthy Achilles tendons revealed a pronounced increase in wave speed as a function of frequency, which highlights the importance of tendon viscoelasticity. Additionally, the viscoelastic properties of the Achilles tendon were larger than those reported for other tissues. Measurements in a tendinopathic Achilles tendon indicated that it is feasible to quantify local viscoelastic properties. Similarly, measurement in the semitendinosus tendon revealed substantial differences in viscoelastic properties between the healthy and contralateral tendons. Consequently, this technique has the potential to evaluate localized changes in tendon viscoelastic properties caused by injury and during recovery in a clinical setting.

Validation and application of an intervertebral disc finite element model utilizing independently constructed tissue-level constitutive formulations that are nonlinear, anisotropic, and time-dependent.

Finite element (FE) models are advantageous in the study of intervertebral disc mechanics as the stress-strain distributions can be determined throughout the tissue and the applied loading and material properties can be controlled and modified. However, the complicated nature of the disc presents a challenge in developing an accurate and predictive disc model, which has led to limitations in FE geometry, material constitutive models and properties, and model validation. The objective of this study was to develop a new FE model of the intervertebral disc, to validate the models nonlinear and time-dependent responses without tuning or calibration, and to evaluate the effect of changes in nucleus pulposus (NP), cartilaginous endplate (CEP), and annulus fibrosus (AF) material properties on the disc mechanical response. The new FE disc model utilized an analytically-based geometry. The model was created from the mean shape of human L4/L5 discs, measured from high-resolution 3D MR images and averaged using signed distance functions. Structural hyperelastic constitutive models were used in conjunction with biphasic-swelling theory to obtain material properties from recent tissue tests in confined compression and uniaxial tension. The FE disc model predictions fit within the experimental range (mean ± 95% confidence interval) of the discs nonlinear response for compressive slow loading ramp, creep, and stress-relaxation simulations. Changes in NP and CEP properties affected the neutral-zone displacement but had little effect on the final stiffness during slow-ramp compression loading. These results highlight the need to validate FE models using the discs full nonlinear response in multiple loading scenarios.

The shear modulus of the nucleus pulposus measured using magnetic resonance elastography: A potential biomarker for intervertebral disc degeneration

This study aims to: (1) measure the shear modulus of nucleus pulposus (NP) in intact human vertebra–disc–vertebra segments using a magnetic resonance elastography setup for a 7T whole‐body scanner, (2) quantify the effect of disc degeneration on the NP shear modulus measured using magnetic resonance elastography, and (3) compare the NP shear modulus to other magnetic resonance‐based biomarkers of dis degeneration.

Mechanical properties of the extra-fibrillar matrix of human annulus fibrosus are location and age dependent

The mechanical behavior of the annulus fibrosus (AF) of the intervertebral disc can be modeled as a mixture of fibers, extra‐fibrillar matrix (EFM), ions, and fluid. However, the properties of the EFM have not been measured directly. We measured mechanical properties of the human EFM at several locations, determined the effect of age and degeneration, and evaluated whether changes in EFM properties correspond to AF compositional changes. EFM mechanical properties were measured using a method that combines osmotic loading and confined compression. AF samples were dissected from several locations, and mechanical properties were correlated with age, degeneration, and composition. EFM modulus was found to range between 10 and 50 kPa, increasing nonlinearly with compression magnitude and being highest in the AF outer‐anterior region. EFM properties were not correlated with composition or degeneration. However, the EFM modulus, its relative contribution to tissue modulus, and model parameters were correlated with age. These measurements will result in more accurate predictions of deformations in the intervertebral disc. Additionally, parameters such as permeability and diffusivity used for biotransport analysis of glucose and other solutes depend on EFM deformation. Consequently, the accuracy of biotransport simulations will be greatly improved.

Viscoelastic Properties of Healthy Achilles Tendon are Independent of Isometric Plantar Flexion Strength and Cross-Sectional Area

Changes in tendon viscoelastic properties are observed after injuries and during healing as a product of altered composition and structure. Continuous Shear Wave Elastography is a new technique measuring viscoelastic properties of soft tissues using external shear waves. Tendon has not been studied with this technique, therefore, the aims of this study were to establish the range of shear and viscosity moduli in healthy Achilles tendons, determine bilateral differences of these parameters and explore correlations of viscoelasticity to plantar flexion strength and tendon area. Continuous Shear Wave Elastography was performed over the free portion of both Achilles tendons from 29 subjects. Isometric plantar flexion strength and cross sectional area were measured. The average shear and viscous moduli was 83.2 kPa and 141.0 Pa‐s, respectively. No correlations existed between the shear or viscous modulus and area or strength. This indicates that viscoelastic properties can be considered novel, independent biomarkers. The shear and viscosity moduli were bilaterally equivalent (p = 0.013, 0.017) which allows determining pathologies through side‐to‐side deviations. The average bilateral coefficient of variation was 7.2% and 9.4% for shear and viscosity modulus, respectively. The viscoelastic properties of the Achilles tendon may provide an unbiased, non‐subjective rating system of tendon recovery and optimizing treatment strategies.

Elastic guided wave propagation in a periodic array of multi-layered piezoelectric plates with finite cross-sections

In this study, we present a model study of guided wave dispersion and resonance behavior of an array of piezoelectric plates with arbitrary cross-sections. The objective of this work is to analyze the influence of the geometry of an element of a 1D-array ultrasound transducer on generating multi-resonance frequency so as to increase the frequency bandwidth of the transducers. A semi-analytical finite-element (SAFE) method is used to model guided wave propagation in multi-layered 1D-array ultrasound transducers. Each element of the array is composed of LiNbO3 piezoelectric material with rectangular or subdiced cross-section. Four-node bilinear finite-elements have been used to discretize the cross-section of the transducer. Dispersion curves showing the dependence of phase and group velocities on the frequency, and mode shapes of propagating modes were obtained for different geometry consurations. A parametric analysis was carried out to determine the effect of the aspect ratio, subdicing, inversion layer and matching layers on the vibrational behavior of 1D-array ultrasound transducers. It was found that the geometry with subdiced cross-section causes more vibration modes compared with the rectangular section. Modal analysis showed that the additional modes correspond to lateral modes of the piezoelectric subdiced section. In addition, some modes have strong normal displacements, which may influence the bandwidth and the pressure field in front of the transducer. In addition, the dispersion curves reveal strong coupling between waveguide modes due to the anisotropy of the piezoelectric crystal. The effect of the matching layers was to cluster extensional and flexural modes within a certain frequency range. Finally, inversion layer is found to have a minor effect on the dispersion curves. This analysis may provide a means to analyze and understand the dynamic response of 1D-array ultrasound transducers.