An Ogden hyperelastic 3D micromechanical model to depict Poynting effect in brain white matter.

Shear and torsional load on soft solids such as brain white matter purportedly exhibits the Poynting Effect. It is a typical nonlinear phenomenon associated with soft materials whereby they tend to elongate (positive Poynting effect) or contract (negative Poynting effect) in a direction perpendicular to the shearing or twisting plane. In this research, a novel 3D micromechanical Finite Element Model (FEM) has been formulated to describe the Poynting effect in bi-phasic modeled brain white matter (BWM) representative volume element (RVE) with axons tracts embedded in surrounding extracellular matrix (ECM) for simulating brain matter's response to pure and simple shear. In the presented BWM 3D FEM, nonlinear Ogden hyper-elastic material model is deployed to interpret axons and ECM material phases. The modeled bi-phasic RVEs have axons tied to the surrounding ECM. In this proof-of-concept (POC) FEM, three simple shear loading configurations and a pure shear case were analyzed. Root mean square deviation (RMSD) was calculated for stress and deformation response plots to understand the effect of axon-ECM orientations and loading conditions on the degree of Poynting behavior. Variations in normal stresses (S11, S22, or S33) perpendicular to the shear plane underscored the significance of axonal fiber-matrix interactions. From the simulated ensemble of cases, a transitional dominance trend was noticed, as simple sheared axons showed pronounced Poynting behavior, but shear deformation build-up in the purely sheared brain model exhibited the highest Poynting behavior at higher strain % limits. At lower strain limits, simple shear imparted across and perpendicular to axonal tract directions emerged as the dominant Poynting effect configurations. At high strains, the stress-strain% plots manifested mild strain stiffening effects and bending stresses in purely sheared axons, substantiated the strong non-linearity in brain tissues' response.

Harmonic viscoelastic response of 3D histology-informed white matter model.

White matter (WM) consists of bundles of long axons embedded in a glial matrix, which lead to anisotropic mechanical properties of brain tissue, and this complicates direct numerical simulations of WM viscoelastic response. The detailed axonal geometry contains scales that range from axonal diameter (microscale) to many diameters (mesoscale) imposing an additional challenge to numerical simulations. Here we describe the development of a 3D homogenization model for the central nervous system (CNS) that accounts for the anisotropy introduced by the axon/neuroglia composite, the axonal trace curvature, and the tissue dynamic response in the frequency domain. Homogenized models that allow the incorporation of all the above factors are important for accurately simulating the tissue's mechanical behavior, and this in turn is essential in interpreting non-invasive elastography measurements. Geometric and material parameters affect the material properties and thus the response of the brain tissue. More complex, orthotropic, or anisotropic material properties are to be considered as necessitated by the 3D tissue structure. An assembly of micro-scale 3D representative elemental volumes (REVs) is constructed, leading to an integrated mesoscale WM finite element model. Assemblies of microscopic REVs, with orientations based on geometrical reconstructions driven by confocal microscopy data are employed to form the elements of the WM model. Each REV carries local material properties based on a finite element model of biphasic (axon-glial matrix) unidirectional composite. The viscoelastic response of the microscopic REVs is extracted based on geometric information and fiber volume fractions calculated from the relative distance between the local elements and global axonal trace. The response of the WM tissue model is homogenized by averaging the shear moduli over the total volume (thus deriving effective properties) under realistic external loading conditions. Under harmonic shear loading, it is proven that that the effective transverse shear moduli are higher than the axial moduli when the axon moduli are higher than the glial. Methodologically, the process of using micro-scale 3D REVs to describe more complex axon geometries avoids the partition process in traditional composite finite element methods (based on partition of finite element grids) and constitutes a robust algorithm to automatically build a WM model based on available axonal trace information. Analytically, the model provides unmatched simulation flexibility and computational power as the position, orientation, and the magnitude of each tissue building block is calculated using real tissue data, as are the training and testing processes at each level of the multiscale WM tissue.

Oligodendrocyte tethering effect on hyperelastic 3D response of axons in white matter.

A novel finite element model is proposed to study the mechanical response of axons embedded in extracellular matrix when subjected to tensile loads under purely non-affine kinematic boundary conditions. Ogden hyperelastic material model describes the axons and the extracellular matrix material characterizations. Two axon-glia tethering scenarios in white matter are studied a single oligodendrocyte (single-OL) with multiple connections a multi-oligodendrocyte (multi-OL) one. In the multi-OL tethering configuration, resultant forces are randomly oriented as distributed glial cells arbitrarily wrap around axons in their immediate vicinity. In the single-OL setup, a centrally located oligodendrocyte myelinates multiple axons nearby. Tethering forces are directed towards this oligodendrocyte, resulting in greater directionality and farther-reaching stress distribution. The oligodendrocyte connections to axons are represented by a spring-dashpot model. The material properties of myelin served as the upper limit for the parameterization of the oligodendrocyte stiffness ("K"). The proposed FE models enable realization of connection mechanisms and their influence on axonal stiffness to determine resultant stress states accurately. Root mean square deviation analysis of stress-strain plots of different connection scenarios reveal an increasing axonal stiffness with increasing tethering, indicating the role of oligodendrocytes in stress redistribution. In single-OL submodel, for the same number of connections per axons, RMSD values increased as "K" (the oligodendrocyte spring stiffness) values were set higher. RMSD calculations reveal that for a "K" value, single-OL model yielded slightly stiffer models compared to multi-OL. The current study also addresses the potential geometrical limitations of multi-OL model by randomizing and adding connections to ensure greater responsiveness. Cyclic bending stresses noticed in both submodels suggest the risk of axonal damage accumulation and repeated load failure.

Graphene-reinforced polymer matrix composites fabricated by in situ shear exfoliation of graphite in polymer solution: processing, rheology, microstructure, and properties.

Effective methods are needed to fabricate the next generation of high-performance graphene-reinforced polymer matrix composites (G-PMCs). In this work, a versatile and fundamental process is demonstrated to produce high-quality graphene-polymethylmethacrylate (G-PMMA) composites via in situ shear exfoliation of well-crystallized graphite particles loaded in highly-viscous liquid PMMA/acetone solutions into graphene nanoflakes using a concentric-cylinder shearing device. Unlike other methods where graphene is added externally to the polymer and mixed, our technique is a single step process where as-exfoliated graphene can bond directly with the polymer with no contamination/handling. The setup also allows for the investigation of the rheology of exfoliation and dispersion, providing process understanding in the attainment of the subsequently heat injection-molded and solidified G-PMC, essential for future manufacturing scalability, optimization, and repeatability. High PMMA/acetone concentration correlates to high mixture viscosity, which at large strain rates results in very-high shear stresses, producing a large number of mechanically-exfoliated flakes, as confirmed by liquid-phase UV-visible spectral analysis. Raman spectroscopy and other imaging evince that single- and bi-layer graphene are readily achieved. Nevertheless, a limit is reached at high mixtures viscosities where the process becomes unstable as non-Newtonian fluid behavior (e.g. viscoelastic) dominates the system. Characterization of microstructure, morphology, and properties of this new class of nanostructured composites reveals interesting trends. Observations by transmission electron microscopy, scanning electron microscopy, and helium ion microscopy of the manufactured G-PMCs show uniform distributions of unadulterated, well-bonded, discontinuous, graphene nanoflakes in a PMMA matrix, which enhances stiffness and strength via a load-transfer mechanism. Elastic modulus of 5.193 GPa and hardness of 0.265 GPa are achieved through processing at 0.7 g ml-1 of acetone/PMMA for 1% wt. starting graphite loading when injected into a sample mold at 200 °C. Mechanical properties exhibit 31% and 28.6% enhancement in elastic modulus and hardness, respectively, as measured by nano-indentation.

Sensitivity analysis of effective transverse shear viscoelastic and diffusional properties of myelinated white matter.

Motivated by the need to interpret the results from a combined use of in vivo brain Magnetic Resonance Elastography (MRE) and Diffusion Tensor Imaging (DTI), we developed a computational framework to study the sensitivity of single-frequency MRE and DTI metrics to white matter microstructure and cell-level mechanical and diffusional properties. White matter was modeled as a triphasic unidirectional composite, consisting of parallel cylindrical inclusions (axons) surrounded by sheaths (myelin), and embedded in a matrix (glial cells plus extracellular matrix). Only 2D mechanics and diffusion in the transverse plane (perpendicular to the axon direction) was considered, and homogenized (effective) properties were derived for a periodic domain containing a single axon. The numerical solutions of the MRE problem were performed with ABAQUS and by employing a sophisticated boundary-conforming grid generation scheme. Based on the linear viscoelastic response to harmonic shear excitation and steady-state diffusion in the transverse plane, a systematic sensitivity analysis of MRE metrics (effective transverse shear storage and loss moduli) and DTI metric (effective radial diffusivity) was performed for a wide range of microstructural and intrinsic (phase-based) physical properties. The microstructural properties considered were fiber volume fraction, and the myelin sheath/axon diameter ratio. The MRE and DTI metrics are very sensitive to the fiber volume fraction, and the intrinsic viscoelastic moduli of the glial phase. The MRE metrics are nonlinear functions of the fiber volume fraction, but the effective diffusion coefficient varies linearly with it. Finally, the transverse metrics of both MRE and DTI are insensitive to the axon diameter in steady state. Our results are consistent with the limited anisotropic MRE and co-registered DTI measurements, mainly in the corpus callosum, available in the literature. We conclude that isotropic MRE and DTI constitutive models are good approximations for myelinated white matter in the transverse plane. The unidirectional composite model presented here is used for the first time to model harmonic shear stress under MRE-relevant frequency on the cell level. This model can be extended to 3D in order to inform the solution of the inverse problem in MRE, establish the biological basis of MRE metrics, and integrate MRE/DTI with other modalities towards increasing the specificity of neuroimaging.

Tunable Electrical Properties of Embossed, Cellulose-Based Paper for Skin-like Sensing.

This article describes a process of fabricating highly porous paper from cellulosic fibers and carbon black (CB) with tunable conductivity. By embossing such paper, its porosity decreases while its conductivity increases. Tuning the porosity of composite paper alters the magnitude and trend of conductivity over a spectrum of concentrations of conductive particles. The largest increase in conductivity from 8.38 × 10-6 to 2.5 × 10-3 S/m by a factor of ∼300 occurred at a percolation threshold of 3.8 wt % (or 0.36 vol %) with the composite paper plastically compressed by 410 MPa, which caused a decrease of porosity from 88% to 42% on average. Our composite paper showed stable piezoresistive responses within a broad pressure range from 1 kPa up to 5.5 MPa for 800 cycles. The piezoresistive sensitivities of the composite paper were dependent on concentration and decreased with pressure. Composite paper with 7.5 wt % CB had sensitivities of -0.514 kPa-1 over applied pressures ranging from 1 to 50 kPa and -0.215 kPa-1 from 1 to 250 kPa. This piezoresistive paper with embossed patterns enabled touch sensing and detection of damage from darts and punches. Understanding the percolation behavior of three-phase composites (cellulosic fibers/conductive particles/air) and their response to damage, pressure, and processing conditions has the potential to enable scalable applications in prosthetics and robotics, haptic feedback, or structural health monitoring on expansive surfaces of buildings and vehicles.

Estimating axonal strain and failure following white matter stretch using contactin-associated protein as a fiduciary marker.

Axonal injury occurs during trauma when tissue-scale loads are transferred to individual axons. Computational models are used to understand this transfer and predict the circumstances that cause injury. However, these findings are limited by a lack of validating experimental work examining the mechanics of axons in their in situ state. As a first step towards validation for dynamic stretch, we use contactin-associated protein (Caspr), expressed at the nodes of Ranvier, as a fiduciary marker of quasistatic axonal stretch. We measured changes in the distance between immunolabled Caspr pairs along axons as a function of tissue-level stretch in chick embryo spinal cords harvested from different developmental periods. We then identified and characterized broken axons and adapted a kinematic model published previously by our group (Singh et al., 2015) to estimate average strain thresholds for axon mechanical failure. The distance between Caspr pairs increased with stretch, though not as much as predicted by simple continuum mechanics. For equivalent tissue stretch, greater numbers of broken axons were found at later stages of development. In adapting our kinematic model to predict a breaking threshold strain, we found that breaking thresholds decrease with development stage. When thresholds were split and classified based on kinematic behavior, non-affine, uncoupled axons had higher strain thresholds than affine, coupled axons, corroborating thresholds predicted from in vitro and in vivo preparations. These results provide a valuable launching point for generating more accurate multi-scale models in primary central nervous system injury.

Molecular Dynamics Study of Cubic Boron Nitride Nanoparticles: Decomposition with Phase Segregation during Melting.

The relative stability and melting of cubic boron nitride (c-BN) nanoparticles of varying shapes and sizes are studied using classical molecular dynamics (MD) simulation. Focusing on the melting of octahedral c-BN nanoparticles, which consist solely of the most stable {111} facets, decomposition is observed to occur during melting, along with the formation of phase segregated boron clusters inside the c-BN nanoparticles, concurrent with vaporization of surface nitrogen atoms. To assess this MD prediction, a laser-heating experiment of c-BN powders is conducted, manifesting boron clusters for the post-treated powders. A general analysis of the geometrical and surface dependence of the nanoparticle ground-state energy using a Stillinger-Weber potential determines the relative stability of cube-shaped, octahedral, cuboctahedral, and truncated-octahedral c-BN nanoparticles. This stability is further examined using transient MD simulations of the melting behavior of the differently shaped nanoparticles, providing insights and revealing the key roles played by corner and edge initiated disorder as well as surface reconstruction from {100} to the more stable {111} facets in the melting process. Finally, the size dependence of the melting point of octahedral c-BN nanoparticles is investigated, showing the well-known qualitative trend of depression of melting temperature with decreasing size, albeit with different quantitative behavior from that predicted by existing analytical models.

A Bayesian approach for characterization of soft tissue viscoelasticity in acoustic radiation force imaging.

Biomechanical imaging techniques based on acoustic radiation force (ARF) have been developed to characterize the viscoelasticity of soft tissue by measuring the motion excited by ARF non-invasively. The unknown stress distribution in the region of excitation limits an accurate inverse characterization of soft tissue viscoelasticity, and single degree-of-freedom simplified models have been applied to solve the inverse problem approximately. In this study, the ARF-induced creep imaging is employed to estimate the time constant of a Voigt viscoelastic tissue model, and an inverse finite element (FE) characterization procedure based on a Bayesian formulation is presented. The Bayesian approach aims to estimate a reasonable quantification of the probability distributions of soft tissue mechanical properties in the presence of measurement noise and model parameter uncertainty. Gaussian process metamodeling is applied to provide a fast statistical approximation based on a small number of computationally expensive FE model runs. Numerical simulation results demonstrate that the Bayesian approach provides an efficient and practical estimation of the probability distributions of time constant in the ARF-induced creep imaging. In a comparison study with the single degree of freedom models, the Bayesian approach with FE models improves the estimation results even in the presence of large uncertainty levels of the model parameters.

Characterization of the three-dimensional kinematic behavior of axons in central nervous system white matter.

Traumatic injury to axons in white matter of the brain and spinal cord occurs primarily via tensile stretch. During injury, the stress and strain experienced at the tissue level is transferred to the microscopic axons. How this transfer occurs, and the primary constituents dictating this transfer must be better understood to develop more accurate multi-scale models of injury. Previous studies have characterized axon tortuosity and kinematic behavior in 2-dimensions (2-D), where axons have been modeled to exhibit non-affine (discrete), affine (composite-like), or switching behavior. In this study, we characterize axon tortuosity and model axon kinematic behavior in 3-dimensions (3-D). Embryonic chick spinal cords at different development stages were excised and stretched. Cords were then fixed, transversely sectioned, stained, and imaged. 3-D axon tortuosity was measured from confocal images using a custom-built MATLAB script. 2-D kinematic models previously described in Bain et al. (J Biomech Eng 125(6):798, 2003) were extended, re-derived, and validated for the 3-D case. Results showed that 3-D tortuosity decreased with stretch, exhibiting similar trends with changes in development as observed in the 2-D studies. Kinematic parameters also displayed similar general trends. Axons demonstrated more affine behavior with increasing stretch and development. In comparison with 2-D results, a smaller percentage of the populations of 3-D axons were predicted to follow pure non-affine behavior. The data and kinematic models presented herein can be incorporated into multi-scale CNS injury models, which can advance the accuracy of the models and improve the potential to identify axonal injury thresholds.

Dynamic simulation of viscoelastic soft tissue in acoustic radiation force creep imaging.

Acoustic radiation force (ARF) creep imaging applies step ARF excitation to induce creep displacement of soft tissue, and the corresponding time-dependent responses are used to estimate soft tissue viscoelasticity or its contrast. Single degree of freedom (SDF) and homogeneous analytical models have been used to characterize soft tissue viscoelasticity in ARF creep imaging. The purpose of this study is to investigate the fundamental limitations of the commonly used SDF and homogeneous assumptions in ARF creep imaging. In this paper, finite element (FE) models are developed to simulate the dynamic behavior of viscoelastic soft tissue subjected to step ARF. Both homogeneous and heterogeneous models are studied with different soft tissue viscoelasticity and ARF configurations. The results indicate that the SDF model can provide good estimations for homogeneous soft tissue with high viscosity, but exhibits poor performance for low viscosity soft tissue. In addition, a smaller focal region of the ARF is desirable to reduce the estimation error with the SDF models. For heterogeneous media, the responses of the focal region are highly affected by the local heterogeneity, which results in deterioration of the effectiveness of the SDF and homogeneous simplifications.

Modelling of global boundary effects on harmonic motion imaging of soft tissues.

Biomechanical imaging techniques have been developed for soft tissue characterisation and detection of breast tumours. Harmonic motion imaging (HMI) uses a focused ultrasound technology to generate a harmonic radiation force in a localised region inside a soft tissue. The resulting dynamic response is used to map the local distribution of the mechanical properties of the tissue. In this study, a finite element (FE) model is developed to investigate the effect of global boundary conditions on the dynamic response of a soft tissue during HMI. The direct-solution steady-state dynamic analysis procedure is used to compute the harmonic displacement amplitude in FE simulations. The model is parameterised in terms of boundary conditions and viscoelastic properties, and the corresponding raster-scan displacement amplitudes are captured to examine its response. The effect of the model's global dimensions on the harmonic response is also investigated. It is observed that the dynamic response of soft tissue with high viscosity is independent of the global boundary conditions for regions remote to the boundary; thus, it can be subjected to local analysis to estimate the underlying mechanical properties. However, the dynamic response is sensitive to global boundary conditions for tissue with low viscosity or regions located near to the boundary.

Finite Element Modeling of CNS White Matter Kinematics: Use of a 3D RVE to Determine Material Properties.

Axonal injury represents a critical target area for the prevention and treatment of traumatic brain and spinal cord injuries. Finite element (FE) models of the head and/or brain are often used to predict brain injury caused by external mechanical loadings, such as explosive waves and direct impact. The accuracy of these numerical models depends on correctly determining the material properties and on the precise depiction of the tissues' microstructure (microscopic level). Moreover, since the axonal microstructure for specific regions of the brain white matter is locally oriented, the stress, and strain fields are highly anisotropic and axon orientation dependent. Additionally, mechanical strain has been identified as the proximal cause of axonal injury, which further demonstrates the importance of this multi-scale relationship. In this study, our previously developed FE and kinematic axonal models are coupled and applied to a pseudo 3-dimensional representative volume element of central nervous system white matter to investigate the multi-scale mechanical behavior. An inverse FE procedure was developed to identify material parameters of spinal cord white matter by combining the results of uniaxial testing with FE modeling. A satisfactory balance between simulation and experiment was achieved via optimization by minimizing the squared error between the simulated and experimental force-stretch curve. The combination of experimental testing and FE analysis provides a useful analysis tool for soft biological tissues in general, and specifically enables evaluations of the axonal response to tissue-level loading and subsequent predictions of axonal damage.

A transition model for finite element simulation of kinematics of central nervous system white matter.

Mechanical damage to axons is a proximal cause of deficits following traumatic brain injury and spinal cord injury. Axons are injured predominantly by tensile strain, and identifying the strain experienced by axons is a critical step toward injury prevention. White matter demonstrates complex nonlinear mechanical behavior at the continuum level that evolves from even more complex, dynamic, and composite behavior between axons and the "glial matrix" at the microlevel. In situ, axons maintain an undulated state that depends on the location of the white matter and the stage of neurodevelopment. When exposed to tissue strain, axons do not demonstrate pure affine or non-affine behavior, but instead transition from non-affine-dominated kinematics at low stretch levels to affine kinematics at high stretch levels. This transitional and predominant kinematic behavior has been linked to the natural coupling of axons to each other via the glial matrix. In this paper, a transitional kinematic model is applied to a micromechanics finite element model to simulate the axonal behavior within a white matter tissue subjected to uniaxial tensile stretch. The effects of the transition parameters and the volume fraction of axons on axonal behavior is evaluated and compared to previous experimental data and numerical simulations.

Finite element dynamic analysis of soft tissues using state-space model.

A finite element (FE) model is employed to investigate the dynamic response of soft tissues under external excitations, particularly corresponding to the case of harmonic motion imaging. A solid 3D mixed 'u-p' element S8P0 is implemented to capture the near-incompressibility inherent in soft tissues. Two important aspects in structural modelling of these tissues are studied; these are the influence of viscous damping on the dynamic response and, following FE-modelling, a developed state-space formulation that valuates the efficiency of several order reduction methods. It is illustrated that the order of the mathematical model can be significantly reduced, while preserving the accuracy of the observed system dynamics. Thus, the reduced-order state-space representation of soft tissues for general dynamic analysis significantly reduces the computational cost and provides a unitary framework for the 'forward' simulation and 'inverse' estimation of soft tissues. Moreover, the results suggest that damping in soft-tissue is significant, effectively cancelling the contribution of all but the first few vibration modes.

Dynamic simulation of viscoelastic soft tissues in harmonic motion imaging application.

A finite element model was built to simulate the dynamic behavior of soft tissues subjected to sinusoidal excitation during harmonic motion imaging. In this study, soft tissues and tissue-like phantoms were modeled as isotropic, viscoelastic, and nearly incompressible media. A 3D incompressible mixed u-p element of eight nodes, S1P0, was developed to accurately calculate the stiffness matrix for soft tissues. The finite element equations of motion were solved using the Newmark method. The Voigt description for tissue viscosity was applied to estimate the relative viscous coefficient from the phase shift between the response and excitation in a harmonic case. After validating our model via ANSYS simulation and experiments, a MATLAB finite element program was then employed to explore the effect of excitation location, viscosity, and multiple frequencies on the dynamic displacement at the frequency of interest.

A mechanical model to compute elastic modulus of tissues for harmonic motion imaging.

Numerous experimental and computational methods have been developed to estimate tissue elasticity. The existing testing techniques are generally classified into in vitro, invasive in vivo and non-invasive in vivo. For each experimental method, a computational scheme is accordingly proposed to calculate mechanical properties of soft biological tissues. Harmonic motion imaging (HMI) is a new technique that performs radio frequency (RF) signal tracking to estimate the localized oscillatory motion resulting from a radiation force produced by focused ultrasound. A mechanical model and computational scheme based on the superposition principle are developed in this paper to estimate the Young's modulus of a tissue mimicking phantom and bovine liver in vitro tissue from the harmonic displacement measured by HMI. The simulation results are verified by two groups of measurement data, and good agreement is shown in each comparison. Furthermore, an inverse function is observed to correlate the elastic modulus of uniform phantoms with amplitude of displacement measured in HMI. The computational scheme is also implemented to estimate 3D elastic modulus of bovine liver in vitro.