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Abstract A novel finite element method (FEM) is developed to study mechanical response of axons embedded in extra cellular matrix (ECM) when subjected to harmonic uniaxial stretch under purely non-affine kinematic boundary conditions. The proposed modeling approach combines hyper-elastic (such as Ogden model) and time/frequency domain viscoelastic constitutive models to evaluate the effect of parametrically varying oligodendrocyte-axon tethering under harmonic stretch at 50Hz. A hybrid hyper-viscoelastic material (HVE) model enabled the analysis of repeated uniaxial load on stress propagation and damage accumulation in white matter. In the proposed FEM, oligodendrocyte connections to axons are depicted via a spring-dashpot model. This tethering technique facilitates contact definition at various locations, parameterizes connection points and varies stiffness of connection hubs. Results from a home-grown FE submodel configuration of a single oligodendrocyte tethered to axons at various locations are presented. Root mean square deviation (RMSD) are computed between stress-strain plots to depict trends in mechanical response. Steady-state dynamic (SSD) simulations show stress relaxation in axons. Gradual axonal softening under repetitive loads is illustrated employing Prony series - HVE models. Representative von-Mises stress plots indicate that undulated axons experience bending stresses along their tortuous path, suggesting greater susceptibility to damage accumulation and fatigue failure due to repeated strains. 

Abstract White matter (WM) characterization is challenging due to its anisotropic and inhomogeneous microstructure that necessitates multiscale and multi-modality measurements. Shear elastography is one such modality that requires the accurate interpretation of 3D shear strain measurements, which hinge on developing appropriate constitutive tissue models. Finite element methods enable the development of such models by simulating the shear response of representative elemental volumes (REV). We have developed triphasic (axon, myelin, glia), 2D REVs to simulate the influence of the intrinsic viscoelastic property and volume fraction of each phase. This work constitutes the extension of 2D- to 3D-REVs, focusing on the effect of the intrinsic material properties and their 3D representation on the viscoelastic response of the tissue. By lumping the axon and myelin phases, a flexible 3D REV generation and analysis routine is then developed to allow for shear homogenization in both the axial and transverse directions. The 2D and 3D models agree on stress distribution and total deformation when 2D cross-sectional snapshots are compared. We also conclude that the ratio of transverse to axial transverse modulus is larger than one when axon fibers are stiffer than the glial phase. 

Abstract Numerical simulations using non-linear hyper-elastic material models to describe interactions between brain white matter (axons and extra cellular matrix (ECM)) have enabled high-fidelity characterization of stress-strain response. In this paper, a novel finite element model (FEM) has been developed to study mechanical response of axons embedded in ECM when subjected to tensile loads under purely non-affine kinematic boundary conditions. FEM leveraging Ogden hyper-elastic material model is deployed to understand impact of parametrically varying oligodendrocyte-axon tethering and analyze influence of aging material characteristics on stress propagation. In proposed FEM, oligodendrocyte connections to axons are represented via spring-dashpot model, such tethering technique facilitates contact definition at various locations, parameterize connection points and vary stiffness of connection hubs. Two FE submodels are discussed: 1) multiple oligodendrocytes arbitrarily tethered to the nearest axons, and 2) single oligodendrocyte tethered to all axons at various locations. Root mean square deviation (RMSD) were computed between stress-strain plots to depict trends in mechanical response. Axonal stiffness was found to rise with increasing tethering, indicating role of oligodendrocytes in stress redistribution. Finally, stress state results for aging axon material, with varying stiffnesses and number of connections in FEM ensemble have also been discussed to demonstrate gradual softening of tissues. 

Abstract Motivated by the need to interpret the results from a combined use ofin vivobrain 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 thecorpus 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. 

Abstract A numerical and experimental hybrid approach is developed to study the constitutive behavior of the central nervous system white matter. A published transversely isotropic hyperelastic strain energy function is reviewed and used to determine stress–strain relationships for three idealized, simple loading scenarios. The proposed constitutive model is simplified to a three-parameter hyperelastic model by assuming the white matter's incompressibility. Due to a lack of experimental data in all three loading scenarios, a finite element model that accounts for microstructural axons and their kinematics is developed to simulate behaviors in simple shear loading scenarios to supplement existing uniaxial tensile test data. The parameters of the transversely isotropic hyperelastic material model are determined regressively using the hybrid data. The results highlight that a hybrid numerical virtual test coupled with experimental data, can determine the transversely isotropic hyperelastic model. It is noted that the model is not limited to small strains and can be applied to large deformations. 

A new finite element approach is proposed to study the propagation of stress in axons in the central nervous system (CNS) white matter. The axons are embedded in an extra cellular matrix (ECM) and are subjected to tensile loads under purely non-affine kinematic boundary conditions. The axons and the ECM are described by the Ogden hyperelastic material model. The effect of tethering of the axons by oligodendrocytes is investigated using the finite element model. Glial cells are often thought of as the “glue” that hold the axons together. More specifically, oligodendrocytes bond multiple axons to each other and create a myelin sheath that insulates and supports axons in the brainstem. The glial cells create a scaffold that supports the axons and can potentially bind 80 axons to a single oligodendrocyte. In this study, the microstructure of the oligodendrocyte connections to axons is modeled using a spring-dashpot approximation. The model allows for the oligodendrocytes to wrap around the outer diameter of the axons at various locations, parameterizing the number of connections, distance between connection points, and the stiffness of the connection hubs. The parameterization followed the distribution of axon-oligodendrocyte connections provided by literature data in which the values were acquired through microtome of CNS white matter. We develop two models: 1) multiple oligodendrocytes arbitrarily tethered to the nearest axons, and 2) a single oligodendrocyte tethered to all the axons at various locations. The results depict stiffening of the axons, which indicates that the oligodendrocytes do aid in the redistribution of stress. We also observe the appearance of bending stresses at inflections points along the tortuous path of the axons when subjected to tensile loading. The bending stresses appear to exhibit a cyclic variation along the length of the undulated axons. This makes the axons more susceptible to damage accumulation and fatigue. Finally, the effect of multiple axon-myelin connections in the central nervous system and the effect of the distribution of these connections in the brain tissue is further investigated at present.