Generalized anisotropic inverse mechanics for soft tissues

On the soft tissue ultrasound elastography using FEM based inversion approach

Elastography is a medical imaging modality that enables visualization of tissue stiffness. It involves quasi-static or harmonic mechanical stimulation of the tissue to generate a displacement field which is used as input in an inversion algorithm to reconstruct tissue elastic modulus. This paper considers quasi-static stimulation and presents a novel inversion technique for elastic modulus reconstruction. The technique follows an inverse finite element framework. Reconstructed elastic modulus maps produced in this technique do not depend on the initial guess, while it is computationally less involved than iterative reconstruction approaches. The method was first evaluated using simulated data (in-silico) where modulus reconstruction's sensitivity to displacement noise and elastic modulus was assessed. To demonstrate the method's performance, displacement fields of two tissue mimicking phantoms determined using three different motion tracking techniques were used as input to the developed elastography method to reconstruct the distribution of relative elastic modulus of the inclusion to background tissue. In the next stage, the relative elastic modulus of three clinical cases pertaining to liver cancer patient were determined. The obtained results demonstrate reasonably high elastic modulus reconstruction accuracy in comparison with similar direct methods. Also it is associated with reduced computational cost in comparison with iterative techniques, which suffer from convergence and uniqueness issues, following the same formulation concept. Moreover, in comparison with other methods which need initial guess, the presented method does not require initial guess while it is easy to understand and implement.

Effect of the apron in the mechanical characterisation of hyperelastic materials by means of biaxial testing: A new method to improve accuracy

Biological soft tissues and polymers used in biomedical applications (e.g. in the cardiovascular area) are hyperelastic incompressible materials that commonly operate under multi-axial large deformation fields. Their characterisation requires biaxial tensile testing. Due to the typically small sample size, the gripping of the specimens commonly relies on rakes or sutures, where the specimen is punctured at the edges of the gauge area. This approach necessitates of an apron, excess of material around the gauge region. This work analyses the apron influence on the estimated mechanical response of biaxial tests performed by using a rakes gripping system, with the aim of verifying the test accuracy and propose improved solutions. In order to isolate the effect of the apron, avoiding the influence of anisotropy and inhomogeneity typical of most soft tissues, homogeneous and isotropic hyperplastic samples made from a uniform sheet of casted silicone were tested. The stress-strain response of specimens with different apron sizes/shapes was measured experimentally by means of biaxial testing and digital image correlation. Tests were replicated numerically, to interpret the experimental findings. The apron surrounding the gauge area acts as an additional annular constraint which stiffens the system, resulting in a significant overestimate in the stress values. This error can be avoided by introducing specific cuts in the apron. The study quantifies, for the first time, the correlation between the apron size/shape and the experimental stress overestimation, proposing a research protocol which, although identified on homogeneous hyperelastic materials, can be useful in providing more accurate characterisation of both, synthetic polymers and soft tissues.

Effects of Porosity on Piezoelectric Characteristics of Polyvinylidene Fluoride Films for Biomedical Applications

Objective: The objective of this work is to study the effects of porosity on mechanical and piezoelectric properties of polyvinylidene fluoride (PVDF) films for biomedical applications. Impact Statement: By investigating the piezoelectric properties of PVDF and the porosity effect on its electromechanical performance, there is potential for further development of PVDF as a hemodynamic sensor that can lead to further technological advancements in the biomedical field, benefiting patients and physicians alike. Introduction: PVDF thin films have shown potential in the application of hemodynamic flow sensing and monitoring the effects on blood flow caused by prosthetic valve implantation via the transcatheter aortic valve replacement operation. The piezoelectric performance of PVDF films can be influenced by the porosity of the material. Methods: In this study, strain tracking was performed on thin film PVDF specimens with various levels of porosity and pore sizes to determine the mechanical properties of the specimens. The mechanical properties were used to model the PVDF material in COMSOL multiphysics software, in which compression test simulations were performed to determine the piezoelectric coefficient d33 of the PVDF. Results: A decline in the elastic modulus was found to be highly inversely correlated with porosity of the specimens and the simulation results show that elastic modulus had a much greater effect on the piezoelectric properties than Poisson's ratio. Conclusion: A combination of experimental and computational techniques was able to characterize and correlate the mechanical properties of PVDF films of varying porosities to their piezoelectric properties.

The Lumbar Facet Capsular Ligament Becomes More Anisotropic and the Fibers Become Stiffer With Intervertebral Disc and Facet Joint Degeneration

Degeneration of the lumbar spine, and especially how that degeneration may lead to pain, remains poorly understood. In particular, the mechanics of the facet capsular ligament may contribute to low back pain, but the mechanical changes that occur in this ligament with spinal degeneration are unknown. Additionally, the highly nonlinear, heterogeneous, and anisotropic nature of the facet capsular ligament makes understanding mechanical changes more difficult. Clinically, magnetic resonance imaging (MRI)-based signs of degeneration in the facet joint and the intervertebral disc (IVD) correlate. Therefore, this study examined how the nonlinear, heterogeneous mechanics of the facet capsular ligament change with degeneration of the lumbar spine as characterized using MRI. Cadaveric human spines were imaged via MRI, and the L2-L5 facet joints and IVDs were scored using the Fujiwara and Pfirrmann grading systems. Then, the facet capsular ligament was isolated and biaxially loaded. The nonlinear mechanical properties of the ligament were obtained using a nonlinear generalized anisotropic inverse mechanics analysis (nGAIM). Then a Holzapfel-Gasser-Ogden (HGO) model was fit to the stress-strain data obtained from nGAIM. The facet capsular ligament is stiffer and more anisotropic at larger Pfirrmann grades and higher Fujiwara scores than at lower grades and scores. Analysis of ligament heterogeneity showed all tissues are highly heterogeneous, but no distinct spatial patterns of heterogeneity were found. These results show that degeneration of the lumbar spine including the facet capsular ligament appears to be occurring as a whole joint phenomenon and advance our understanding of lumbar spine degeneration.

Combining Unique Planar Biaxial Testing with Full-Field Thickness and Displacement Measurement for Spatial Characterization of Soft Tissues

Soft tissues rely on the incredible complexity of their microstructure for proper function. Local variations in material properties arise as tissues develop and adapt, often in response to changes in loading. A barrier to investigating the heterogeneous nature of soft tissues is the difficulty of developing experimental protocols and analysis tools that can accurately capture spatial variations in mechanical behavior. In this article, we detail protocols enabling mechanical characterizations of anisotropic, heterogeneous soft tissues or tissue analogs. We present a series of mechanical tests designed to maximize inhomogeneous strain fields and in-plane shear forces. A customized, 3D-printable gripping system reduces tissue handling and enhances shear. High-resolution imaging and laser micrometry capture full-field displacement and thickness, respectively. As the equipment necessary to conduct these protocols is commercially available, the experimental methods presented offer an accessible route toward addressing heterogeneity. © 2022 The Authors. Current Protocols published by Wiley Periodicals LLC. Basic Protocol 1: Unique biaxial testing of soft tissues and tissue analogs Basic Protocol 2: Full-field thickness measurement of soft tissues and tissue analogs Support Protocol 1: Creating and speckling cruciform-shaped samples for mechanical testing Support Protocol 2: Creating custom gripping system to minimize sample handling.

Local variations in mechanical properties of human hamstring tendon autografts for anterior cruciate ligament reconstruction do not translate to a mechanically inferior strand

A ruptured anterior cruciate ligament (ACL) is often reconstructed with a multiple-strand autograft of a semitendinosus tendon alone or combined with a gracilis tendon. Up to 10% of patients experience graft rupture. This potentially results from excessive local tissue strains under physiological loading which could either result in direct mechanical failure of the graft or induce mechanobiological weakening. Since the original location in the hamstring tendon cannot be traced back from an autograft rupture site, this study explored whether clinical outcome could be further improved by avoiding specific locations or regions of human semitendinosus and/or gracilis tendons in ACL grafts due to potential mechanical or biochemical inferiority. Additionally, it examined numerically which clinically relevant graft configurations experience the lowest strains - and therefore the lowest rupture risk - when loaded with equal force. Remnant full-length gracilis tendons from human ACL reconstructions and full-length semitendinosus- and ipsilateral gracilis tendons of human cadaveric specimens were subjected to a stress-relaxation test. Locations at high risk of mechanical failure were identified using particle tracking to calculate local axial strains. As biochemical properties, the water-, collagen-, glycosaminoglycan- and DNA content per tissue region (representing graft strands) were determined. A viscoelastic lumped parameter model per tendon region was calculated. These models were applied in clinically relevant virtual graft configurations, which were exposed to physiological loading. Configurations that provided lower stiffness - i.e., experiencing higher strains under equal force - were assumed to be at higher risk of failure. Suitability of the gracilis tendon proper to replace semitendinosus muscle-tendon junction strands was examined. Deviations in local axial strains from the globally applied strain were of similar magnitude as the applied strain. Locations of maximum strains were uniformly distributed over tendon lengths. Biochemical compositions varied between tissue regions, but no trends were detected. Viscoelastic parameters were not significantly different between regions within a tendon, although semitendinosus tendons were stiffer than gracilis tendons. Virtual grafts with a full-length semitendinosus tendon alone or combined with a gracilis tendon displayed the lowest strains, whereas strains increased when gracilis tendon strands were tested for their suitability to replace semitendinosus muscle-tendon junction strands. Locations experiencing high local axial strains - which could increase risk of rupture - were present, but no specific region within any of the investigated graft configurations was found to be mechanically or biochemically deviant. Consequently, no specific tendon region could be indicated to provide a higher risk of rupture for mechanical or biochemical reasons. The semitendinosus tendon provided superior stiffness to a graft compared to the gracilis tendon. Therefore, based on our results it would be recommended to use the semitendinosus tendon, and use the gracilis tendon in cases where further reinforcement of the graft is needed to attain the desired length and cross-sectional area. All these data support current clinical standards.

Characterizing Tissue Remodeling and Mechanical Heterogeneity in Cerebral Aneurysms

Accurately assessing the complex tissue mechanics of cerebral aneurysms (CAs) is critical for elucidating how CAs grow and whether that growth will lead to rupture. The factors that have been implicated in CA progression - blood flow dynamics, immune infiltration, and extracellular matrix remodeling - all occur heterogeneously throughout the CA. Thus, it stands to reason that the mechanical properties of CAs are also spatially heterogeneous. Here, we present a new method for characterizing the mechanical heterogeneity of human CAs using generalized anisotropic inverse mechanics, which uses biaxial stretching experiments and inverse analyses to determine the local Kelvin moduli and principal alignments within the tissue. Using this approach, we find that there is significant mechanical heterogeneity within a single acquired human CA. These results were confirmed using second harmonic generation imaging of the CA's fiber architecture and a correlation was observed. This approach provides a single-step method for determining the complex heterogeneous mechanics of CAs, which has important implications for future identification of metrics that can improve accuracy in prediction risk of rupture.

Mapping the spatial variation of mitral valve elastic properties using air-pulse optical coherence elastography

The mitral valve is a highly heterogeneous tissue composed of two leaflets, anterior and posterior, whose unique composition and regional differences in material properties are essential to overall valve function. While mitral valve mechanics have been studied for many decades, traditional testing methods limit the spatial resolution of measurements and can be destructive. Optical coherence elastography (OCE) is an emerging method for measuring viscoelastic properties of tissues in a noninvasive, nondestructive manner. In this study, we employed air-pulse OCE to measure the spatial variation in mitral valve elastic properties with micro-scale resolution at 1 mm increments along the radial length of the leaflets. We analyzed differences between the leaflets, as well as between regions of the valve. We found that the anterior leaflet has a higher elastic wave velocity, which is reported as a surrogate for stiffness, than the posterior leaflet, most notably at the annular edge of the sample. In addition, we found a spatial elastic gradient in the anterior leaflet, where the annular edge was found to have a greater elastic wave velocity than the free edge. This gradient was less pronounced in the posterior leaflet. These patterns were confirmed using established uniaxial tensile testing methods. Overall, the anterior leaflet was stiffer and had greater heterogeneity in its mechanical properties than the posterior leaflet. This study measures differences between the two mitral leaflets with greater resolution than previously feasible and demonstrates a method that may be suitable for assessing valve mechanics following repair or during the engineering of synthetic valve replacements.

A model of human skin under large amplitude oscillatory shear

Skin mechanics is of importance in various fields of research when accurate predictions of the mechanical response of skin is essential. This study aims to develop a new constitutive model for human skin that is capable of describing the heterogeneous, nonlinear viscoelastic mechanical response of human skin under shear deformation. This complex mechanical response was determined by performing large amplitude oscillatory shear (LAOS) experiments on ex vivo human skin samples. It was combined with digital image correlation (DIC) on the cross-sectional area to assess heterogeneity. The skin is modeled as a one-dimensional layered structure, with every sublayer behaving as a nonlinear viscoelastic material. Heterogeneity is implemented by varying the stiffness with skin depth. Using an iterative parameter estimation method all model parameters were optimized simultaneously. The model accurately captures strain stiffening, shear thinning, softening effect and nonlinear viscous dissipation, as experimentally observed in the mechanical response to LAOS. The heterogeneous properties described by the model were in good agreement with the experimental DIC results. The presented mathematical description forms the basis for a future constitutive model definition that, by implementation in a finite element method, has the capability of describing the full 3D mechanical behavior of human skin.

A Novel Small-Specimen Planar Biaxial Testing System With Full In-Plane Deformation Control

Simulations of soft tissues require accurate and robust constitutive models, whose form is derived from carefully designed experimental studies. For such investigations of membranes or thin specimens, planar biaxial systems have been used extensively. Yet, all such systems remain limited in their ability to: (1) fully prescribe in-plane deformation gradient tensor F2D, (2) ensure homogeneity of the applied deformation, and (3) be able to accommodate sufficiently small specimens to ensure a reasonable degree of material homogeneity. To address these issues, we have developed a novel planar biaxial testing device that overcomes these difficulties and is capable of full control of the in-plane deformation gradient tensor F2D and of testing specimens as small as ∼4 mm × ∼4 mm. Individual actuation of the specimen attachment points, combined with a robust real-time feedback control, enabled the device to enforce any arbitrary F2D with a high degree of accuracy and homogeneity. Results from extensive device validation trials and example tissues illustrated the ability of the device to perform as designed and gather data needed for developing and validating constitutive models. Examples included the murine aortic tissues, allowing for investigators to take advantage of the genetic manipulation of murine disease models. These capabilities highlight the potential of the device to serve as a platform for informing and verifying the results of inverse models and for conducting robust, controlled investigation into the biomechanics of very local behaviors of soft tissues and membrane biomaterials.

Anisotropic linear elastic parameter estimation using error in the constitutive equation functional

A modified error in the constitutive equation-based approach for identification of heterogeneous and linear anisotropic elastic parameters involving static measurements is proposed and explored. Following an alternating minimization procedure associated with the underlying optimization problem, the new strategy results in an explicit material parameter update formula for general anisotropic material. This immediately allows us to derive the necessary constraints on measured data and thus restrictions on physical experimentation to achieve the desired reconstruction. We consider a few common materials to derive such conditions. Then, we exploit the invariant relationships of the anisotropic constitutive tensor to propose an identification procedure for space-dependent material orientations. Finally, we assess the numerical efficacy of the developed tools against a few parameter identification problems of engineering interest.

A nonlinear anisotropic inverse method for computational dissection of inhomogeneous planar tissues

Quantification of the mechanical behavior of soft tissues is challenging due to their anisotropic, heterogeneous, and nonlinear nature. We present a method for the 'computational dissection' of a tissue, by which we mean the use of computational tools both to identify and to analyze regions within a tissue sample that have different mechanical properties. The approach employs an inverse technique applied to a series of planar biaxial experimental protocols. The aggregated data from multiple protocols provide the basis for (1) segmentation of the tissue into regions of similar properties, (2) linear analysis for the small-strain behavior, assuming uniform, linear, anisotropic behavior within each region, (3) subsequent nonlinear analysis following each individual experimental protocol path and using local linear properties, and (4) construction of a strain energy data set W(E) at every point in the material by integrating the differential stress-strain functions along each strain path. The approach has been applied to simulated data and captures not only the general nonlinear behavior but also the regional differences introduced into the simulated tissue sample.

Automatic Segmentation of Mechanically Inhomogeneous Tissues Based on Deformation Gradient Jump

Variations in properties, active behavior, injury, scarring, and/or disease can all cause a tissue's mechanical behavior to be heterogeneous. Advances in imaging technology allow for accurate full-field displacement tracking of both in vitro and in vivo deformation from an applied load. While detailed strain fields provide some insight into tissue behavior, material properties are usually determined by fitting stress-strain behavior with a constitutive equation. However, the determination of the mechanical behavior of heterogeneous soft tissue requires a spatially varying constitutive equation (i.e., one in which the material parameters vary with position). We present an approach that computationally dissects the sample domain into many homogeneous subdomains, wherein subdomain boundaries are formed by applying a betweenness based graphical analysis to the deformation gradient field to identify locations with large discontinuities. This novel partitioning technique successfully determined the shape, size and location of regions with locally similar material properties for: (1) a series of simulated soft tissue samples prescribed with both abrupt and gradual changes in anisotropy strength, prescribed fiber alignment, stiffness, and nonlinearity, (2) tissue analogs (PDMS and collagen gels) which were tested biaxially and speckle tracked (3) and soft tissues which exhibited a natural variation in properties (cadaveric supraspinatus tendon), a pathologic variation in properties (thoracic aorta containing transmural plaque), and active behavior (contracting cardiac sheet). The routine enables the dissection of samples computationally rather than physically, allowing for the study of small tissues specimens with unknown and irregular inhomogeneity.

Collagen structure and mechanical properties of the human sclera: analysis for the effects of age

The objective of this study was to measure the collagen fiber structure and estimate the material properties of 7 human donor scleras, from age 53 to 91. The specimens were subjected to inflation testing, and the full-field displacement maps were measured by digital image correlation. After testing, the collagen fiber structure was mapped using wide-angle X-ray scattering. A specimen-specific inverse finite element method was applied to calculate the material properties of the collagen fibers and interfiber matrix by minimizing the difference between the experimental displacements and model predictions. Age effects on the fiber structure and material properties were estimated using multivariate models accounting for spatial autocorrelation. Older age was associated with a larger matrix stiffness (p = 0.001), a lower degree of fiber alignment in the peripapillary sclera (p = 0.01), and a lower mechanical anisotropy in the peripapillary sclera (p = 0.03).

A novel device to quantify the mechanical properties of electrospun nanofibers

Mechanical deformation of cell-seeded electrospun matrices plays an important role in cell signaling. However, electrospun biomaterials have inherently complex geometries due to the random deposition of fibers during the electrospinning process. This confounds attempts at quantifying strains exerted on adherent cells during electrospun matrix deformation. We have developed a novel mechanical test platform that allows deposition and tensile testing of electrospun fibers in a highly parallel arrangement to simplify mechanical analysis of the fibers alone and with adherent cells. The device is capable of optically recording fiber strain in a cell culture environment. Here we report on the mechanical and viscoelastic properties of highly parallel electrospun poly(ε-caprolactone) fibers. Force-strain data derived from this device will drive the development of cellular mechanotransduction studies as well as the customization of electrospun matrices for specific engineered tissue applications.

A Quasi-Nonlinear Analysis of the Anisotropic Behaviour of Human Gallbladder Wall

Estimation of biomechanical parameters of soft tissues from noninvasive measurements has clinical significance in patient-specific modeling and disease diagnosis. In this work, we present a quasi-nonlinear method that is used to estimate the elastic moduli of the human gallbladder wall. A forward approach based on a transversely isotropic membrane material model is used, and an inverse iteration is carried out to determine the elastic moduli in the circumferential and longitudinal directions between two successive ultrasound images of gallbladder. The results demonstrate that the human gallbladder behaves in an anisotropic manner, and constitutive models need to incorporate this. The estimated moduli are also nonlinear and patient dependent. Importantly, the peak stress predicted here differs from the earlier estimate from linear membrane theory. As the peak stress inside the gallbladder wall has been found to strongly correlate with acalculous gallbladder pain, reliable mechanical modeling for gallbladder tissue is crucial if this information is to be used in clinical diagnosis.

Mechanical changes in the rat right ventricle with decellularization

The stiffness, anisotropy, and heterogeneity of freshly dissected (control) and perfusion-decellularized rat right ventricles were compared using an anisotropic inverse mechanics method. Cruciform tissue samples were speckled and then tested under a series of different biaxial loading configurations with simultaneous force measurement on all four arms and displacement mapping via image correlation. Based on the displacement and force data, the sample was segmented into piecewise homogeneous partitions. Tissue stiffness and anisotropy were characterized for each partition using a large-deformation extension of the general linear elastic model. The perfusion-decellularized tissue had significantly higher stiffness than the control, suggesting that the cellular contribution to stiffness, at least under the conditions used, was relatively small. Neither anisotropy nor heterogeneity (measured by the partition standard deviation of the modulus and anisotropy) varied significantly between control and decellularized samples. We thus conclude that although decellularization produces quantitative differences in modulus, decellularized tissue can provide a useful model of the native tissue extracellular matrix. Further, the large-deformation inverse method presented herein can be used to characterize complex soft tissue behaviors.

Transversely isotropic elasticity imaging of cancellous bone

To measure spatial variations in mechanical properties of biological materials, prior studies have typically performed mechanical tests on excised specimens of tissue. Less invasive measurements, however, are preferable in many applications, such as patient-specific modeling, disease diagnosis, and tracking of age- or damage-related degradation of mechanical properties. Elasticity imaging (elastography) is a nondestructive imaging method in which the distribution of elastic properties throughout a specimen can be reconstructed from measured strain or displacement fields. To date, most work in elasticity imaging has concerned incompressible, isotropic materials. This study presents an extension of elasticity imaging to three-dimensional, compressible, transversely isotropic materials. The formulation and solution of an inverse problem for an anisotropic tissue subjected to a combination of quasi-static loads is described, and an optimization and regularization strategy that indirectly obtains the solution to the inverse problem is presented. Several applications of transversely isotropic elasticity imaging to cancellous bone from the human vertebra are then considered. The feasibility of using isotropic elasticity imaging to obtain meaningful reconstructions of the distribution of material properties for vertebral cancellous bone from experiment is established. However, using simulation, it is shown that an isotropic reconstruction is not appropriate for anisotropic materials. It is further shown that the transversely isotropic method identifies a solution that predicts the measured displacements, reveals regions of low stiffness, and recovers all five elastic parameters with approximately 10% error. The recovery of a given elastic parameter is found to require the presence of its corresponding strain (e.g., a deformation that generates ɛ₁₂ is necessary to reconstruct C₁₂₁₂), and the application of regularization is shown to improve accuracy. Finally, the effects of noise on reconstruction quality is demonstrated and a signal-to-noise ratio (SNR) of 40 dB is identified as a reasonable threshold for obtaining accurate reconstructions from experimental data. This study demonstrates that given an appropriate set of displacement fields, level of regularization, and signal strength, the transversely isotropic method can recover the relative magnitudes of all five elastic parameters without an independent measurement of stress. The quality of the reconstructions improves with increasing contrast, magnitude of deformation, and asymmetry in the distributions of material properties, indicating that elasticity imaging of cancellous bone could be a useful tool in laboratory studies to monitor the progression of damage and disease in this tissue.

Identification of regional mechanical anisotropy in soft tissue analogs

In a previous work (Raghupathy and Barocas, 2010, "Generalized Anisotropic Inverse Mechanics for Soft Tissues,"J. Biomech. Eng., 132(8), pp. 081006), a generalized anisotropic inverse mechanics method applicable to soft tissues was presented and tested against simulated data. Here we demonstrate the ability of the method to identify regional differences in anisotropy from full-field displacements and boundary forces obtained from biaxial extension tests on soft tissue analogs. Tissue heterogeneity was evaluated by partitioning the domain into homogeneous subdomains. Tests on elastomer samples demonstrated the performance of the method on isotropic materials with uniform and nonuniform properties. Tests on fibroblast-remodeled collagen cruciforms indicated a strong correlation between local structural anisotropy (measured by polarized light microscopy) and the evaluated local mechanical anisotropy. The results demonstrate the potential to quantify regional anisotropic material behavior on an intact tissue sample.