A micromechanical hyperelastic modeling of brain white matter under large deformation

From structure to mechanics: exploring the role of axons and interconnections in anisotropic behavior of brain white matter

According to various experimental studies, the role of axons in the brain's white matter (WM) is still a subject of debate: Is the role of axons in brain white matter (WM) limited to their functional significance, or do they also play a pivotal mechanical role in defining its anisotropic behavior? Micromechanics and computational models provide valuable tools for scientists to comprehend the underlying mechanisms of tissue behavior, taking into account the contribution of microstructures. In this review, we delve into the consideration of strain level, strain rates, and injury threshold to determine when WM should be regarded as anisotropic, as well as when the assumption of isotropy can be deemed acceptable. Additionally, we emphasize the potential mechanical significance of interconnections between glial cells-axons and glial cells-vessels. Moreover, we elucidate the directionality of WM stiffness under various loading conditions and define the possible roles of microstructural components in each scenario. Ultimately, this review aims to shed light on the significant mechanical contributions of axons in conjunction with glial cells, paving the way for the development of future multiscale models capable of predicting injuries and facilitating the discovery of applicable treatments.

Stress landscape of folding brain serves as a map for axonal pathfinding

Understanding the mechanics linking cortical folding and brain connectivity is crucial for both healthy and abnormal brain development. Despite the importance of this relationship, existing models fail to explain how growing axon bundles navigate the stress field within a folding brain or how this bidirectional and dynamic interaction shapes the resulting surface morphologies and connectivity patterns. Here, we propose the concept of "axon reorientation" and formulate a mechanical model to uncover the dynamic multiscale mechanics of the linkages between cortical folding and connectivity development. Simulations incorporating axon bundle reorientation and stress-induced growth reveal potential mechanical mechanisms that lead to higher axon bundle density in gyri (ridges) compared to sulci (valleys). In particular, the connectivity patterning resulting from cortical folding exhibits a strong dependence on the growth rate and mechanical properties of the navigating axon bundles. Model predictions are supported by in vivo diffusion tensor imaging of the human brain.

Toward understanding the brain tissue behavior due to preconditioning: an experimental study and RVE approach

Brain tissue under preconditioning, as a complex issue, refers to repeated loading-unloading cycles applied in mechanical testing protocols. In previous studies, only the mechanical behavior of the tissue under preconditioning was investigated; However, the link between macrostructural mechanical behavior and microstructural changes in brain tissue remains underexplored. This study aims to bridge this gap by investigating bovine brain tissue responses both before and after preconditioning. We employed a dual approach: experimental mechanical testing and computational modeling. Experimental tests were conducted to observe microstructural changes in mechanical behavior due to preconditioning, with a focus on axonal damage. Concurrently, we developed multiscale models using statistically representative volume elements (RVE) to simulate the tissue's microstructural response. These RVEs, featuring randomly distributed axonal fibers within the extracellular matrix, provide a realistic depiction of the white matter microstructure. Our findings show that preconditioning induces significant changes in the mechanical properties of brain tissue and affects axonal integrity. The RVE models successfully captured localized stresses and facilitated the microscopic analysis of axonal injury mechanisms. These results underscore the importance of considering both macro and micro scales in understanding brain tissue behavior under mechanical loading. This comprehensive approach offers valuable insights into mechanotransduction processes and improves the analysis of microstructural phenomena in brain tissue.

Systematic analysis of constitutive models of brain tissue materials based on compression tests

It's crucial to understand the biomechanical properties of brain tissue to comprehend the potential mechanisms of traumatic brain injury. This study, distinct from homogeneous models, integrates axonal coupling in both axial and transverse compressive experiments within a continuum mechanics framework to capture its intricate mechanical behaviors. Fresh porcine brains underwent unconfined compression at strain rates of 0.001/s and 0.1/s to 0.3 strain, allowing for a comprehensive statistical analysis of the directional, regional, and strain-rate-dependent mechanical properties of brain tissue. The established constitutive model, fitted to experimental data, delineates material parameters providing intuitive insights into the stiffness of gray/white matter isotropic matrices and neural fibers. Additionally, it predicts the mechanical performance of white matter matrix and axonal fibers under compressive loading. Results reveal that gray matter is insensitive to loading direction, exhibiting insignificant stiffness variations within regions. White matter, however, displays no significant differences in mechanical properties under axial and transverse loading, with an overall higher average stress than gray matter and a more pronounced strain-rate effect. Stress-strain curves indicate that, under axial compression, white matter axons primarily resist the load before transitioning to a matrix-dominated response. Under transverse loading, axonal fibers exhibit weaker resistance to lateral pressure. The mechanical behavior of brain tissue is highly dependent on loading rate, region, direction, and peak strain. This study, by combining experimentation with phenomenological modeling, elucidates certain phenomena, contributing valuable insights for the development of precise computational models.

Histology-informed multiscale modeling of human brain white matter

In this study, we propose a novel micromechanical model for the brain white matter, which is described as a heterogeneous material with a complex network of axon fibers embedded in a soft ground matrix. We developed this model in the framework of RVE-based multiscale theories in combination with the finite element method and the embedded element technique for embedding the fibers. Microstructural features such as axon diameter, orientation and tortuosity are incorporated into the model through distributions derived from histological data. The constitutive law of both the fibers and the matrix is described by isotropic one-term Ogden functions. The hyperelastic response of the tissue is derived by homogenizing the microscopic stress fields with multiscale boundary conditions to ensure kinematic compatibility. The macroscale homogenized stress is employed in an inverse parameter identification procedure to determine the hyperelastic constants of axons and ground matrix, based on experiments on human corpus callosum. Our results demonstrate the fundamental effect of axon tortuosity on the mechanical behavior of the brain's white matter. By combining histological information with the multiscale theory, the proposed framework can substantially contribute to the understanding of mechanotransduction phenomena, shed light on the biomechanics of a healthy brain, and potentially provide insights into neurodegenerative processes.

Sex Differences in Axonal Dynamic Responses Under Realistic Tension Using Finite Element Models

Existing axonal finite element models do not consider sex morphological differences or the fidelity in dynamic input. To facilitate a systematic investigation into the micromechanics of diffuse axonal injury, we develop a parameterized modeling approach for automatic and efficient generation of sex-specific axonal models according to specified geometrical parameters. Baseline female and male axonal models in the corpus callosum with random microtubule (MT) gap configurations are generated for model calibration and evaluation. They are then used to simulate a realistic tensile loading consisting of both a loading and a recovery phase (to return to an initial undeformed state) generated from dynamic corpus callosum fiber strain in a real-world head impact simulation. We find that MT gaps and the dynamic recovery phase are both critical to successfully reproduce MT undulation as observed experimentally, which has not been reported before. This strengthens confidence in model dynamic responses. A statistical approach is further employed to aggregate axonal responses from a large sample of random MT gap configurations for both female and male axonal models (n = 10,000 each). We find that peak strains in MTs and the Ranvier node and associated neurofilament failures in female axons are substantially higher than those in male axons because there are fewer MTs in the former and also because of the random nature of MT gap locations. Despite limitations in various model assumptions as a result of limited experimental data currently available, these findings highlight the need to systematically characterize MT gap configurations and to ensure a realistic model input for axonal dynamic simulations. Finally, this study may offer fresh and improved insight into the biomechanical basis of sex differences in brain injury, and sets the stage for more systematic investigations at the microscale in the future, both numerically and experimentally.

Origins of brain tissue elasticity under multiple loading modes by analyzing the microstructure-based models

Constitutive behaviors and material properties of brain tissue play an essential role in accurately modeling its mechanical responses. However, the measured mechanical behaviors of brain tissue exhibit a large variability, and the reported elastic modulus can differ by orders of magnitude. Here we develop the micromechanical models based on the actual microstructure of the longitudinally anisotropic plane of brain tissue to investigate the microstructural origins of the large variability. Specifically, axonal fiber bundles with the specified configurations are distributed in an equivalent matrix. All micromechanical models are subjected to multiple loading modes, such as tensile, compressive, and shear loading, under periodic boundary conditions. The predicted results agree well with the experimental results. Furthermore, we investigate how brain tissue elasticity varies with its microstructural features. It is revealed that the large variability in brain tissue elasticity stems from the volume fraction of axonal fiber, the aspect ratio of axonal fiber, and the distribution of axonal fiber orientation. The volume fraction has the greatest impact on the mechanical behaviors of brain tissue, followed by the distribution of axonal fiber orientation, then the aspect ratio. This study provides critical insights for understanding the microstructural origins of the large variability in brain tissue elasticity.

Bidirectional hyperelastic characterization of brain white matter tissue

Biomechanical study of brain injuries originated from mechanical damages to white matter tissue requires detailed information on mechanical characteristics of its main components, the axonal fibers and extracellular matrix, which is very limited due to practical difficulties of direct measurement. In this paper, a new theoretical framework was established based on microstructural modeling of brain white matter tissue as a soft composite for bidirectional hyperelastic characterization of its main components. First the tissue was modeled as an Ogden hyperelastic material, and its principal Cauchy stresses were formulated in the axonal and transverse directions under uniaxial and equibiaxial tension using the theory of homogenization. Upon fitting these formulae to the corresponding experimental test data, direction-dependent hyperelastic constants of the tissue were obtained. These directional properties then were used to estimate the strain energy stored in the homogenized model under each loading scenario. A new microstructural composite model of the tissue was also established using principles of composites micromechanics, in which the axonal fibers and surrounding matrix are modeled as different Ogden hyperelastic materials with unknown constants. Upon balancing the strain energies stored in the homogenized and composite models under different loading scenarios, fully coupled nonlinear equations as functions of unknown hyperelastic constants were derived, and their optimum solutions were found in a multi-parametric multi-objective optimization procedure using the response surface methodology. Finally, these solutions were implemented, in a bottom-up approach, into a micromechanical finite element model to reproduce the tissue responses under the same loadings and predict the tissue responses under unseen non-equibiaxial loadings. Results demonstrated a very good agreement between the model predictions and experimental results in both directions under different loadings. Moreover, the axonal fibers with hyperelastic characteristics stiffer than the extracellular matrix were shown to play the dominant role in directional reinforcement of the tissue.

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.

Reconstruction of ovine axonal cytoarchitecture enables more accurate models of brain biomechanics

There is an increased need and focus to understand how local brain microstructure affects the transport of drug molecules directly administered to the brain tissue, for example in convection-enhanced delivery procedures. This study reports a systematic attempt to characterize the cytoarchitecture of commissural, long association and projection fibres, namely the corpus callosum, the fornix and the corona radiata, with the specific aim to map different regions of the tissue and provide essential information for the development of accurate models of brain biomechanics. Ovine samples are imaged using scanning electron microscopy combined with focused ion beam milling to generate 3D volume reconstructions of the tissue at subcellular spatial resolution. Focus is placed on the characteristic cytological feature of the white matter: the axons and their alignment in the tissue. For each tract, a 3D reconstruction of relatively large volumes, including a significant number of axons, is performed and outer axonal ellipticity, outer axonal cross-sectional area and their relative perimeter are measured. The study of well-resolved microstructural features provides useful insight into the fibrous organization of the tissue, whose micromechanical behaviour is that of a composite material presenting elliptical tortuous tubular axonal structures embedded in the extra-cellular matrix. Drug flow can be captured through microstructurally-based models using 3D volumes, either reconstructed directly from images or generated in silico using parameters extracted from the database of images, leading to a workflow to enable physically-accurate simulations of drug delivery to the targeted tissue.

Imaging and reconstruction of sheep brain axonal cytoarchitecture provides insight for brain biomechanics models that simulate drug delivery and other biological processes governed by interstitial fluid flow and transport.

Correlating the microstructural architecture and macrostructural behaviour of the brain

The computational simulation of pathological conditions and surgical procedures, for example the removal of cancerous tissue, can contribute crucially to the future of medicine. Especially for brain surgery, these methods can be important, as the ultra-soft tissue controls vital functions of the body. However, the microstructural interactions and their effects on macroscopic material properties remain incompletely understood. Therefore, we investigated the mechanical behaviour of brain tissue under three different deformation modes, axial tension, compression, and semi-confined compression, in different anatomical regions, and for varying axon orientation. In addition, we characterised the underlying microstructure in terms of myelin, cells, glial cells and neuron area fraction, and density. The correlation of these quantities with the material parameters of the anisotropic Ogden model reveals a decrease in shear modulus with increasing myelin area fraction. Strikingly, the tensile shear modulus correlates positively with cell and neuronal area fraction (Spearman's correlation coefficient of r_s=0.40 and r_s=0.33), whereas the compressive shear modulus decreases with increasing glial cell area (r_s=-0.33). Our study finds that tissue non-linearity significantly depends on the myelin area fraction (r_s=0.47), cell density (r_s=0.41) and glial cell area (r_s=0.49). Our results provide an important step towards understanding the micromechanical load transfer that leads to the non-linear macromechanical behaviour of the brain. STATEMENT OF SIGNIFICANCE: Within this article, we investigate the mechanical behaviour of brain tissue under three different deformation modes, in different anatomical regions, and for varying axon orientation. Further, we characterise the underlying microstructure in terms of various constituents. The correlation of these quantities with the material parameters of the anisotropic Ogden model reveals a decrease in shear modulus with increasing myelin area fraction. Strikingly, the tensile shear modulus correlates positively with cell and neuronal area fraction, whereas the compressive shear modulus decreases with increasing glial cell area. Our study finds that tissue non-linearity significantly depends on the myelin area fraction, cell density, and glial cell area. Our results provide an important step towards understanding the micromechanical load transfer that leads to the non-linear macromechanical behaviour of the brain.

On the microstructurally driven heterogeneous response of brain white matter to drug infusion pressure

Delivering therapeutic agents into the brain via convection-enhanced delivery (CED), a mechanically controlled infusion method, provides an efficient approach to bypass the blood–brain barrier and deliver drugs directly to the targeted focus in the brain. Mathematical methods based on Darcy’s law have been widely adopted to predict drug distribution in the brain to improve the accuracy and reduce the side effects of this technique. However, most of the current studies assume that the hydraulic permeability and porosity of brain tissue are homogeneous and constant during the infusion process, which is less accurate due to the deformability of the axonal structures and the extracellular matrix in brain white matter. To solve this problem, a multiscale model was established in this study, which takes into account the pressure-driven deformation of brain microstructure to quantify the change of local permeability and porosity. The simulation results were corroborated using experiments measuring hydraulic permeability in ovine brain samples. Results show that both hydraulic pressure and drug concentration in the brain would be significantly underestimated by classical Darcy’s law, thus highlighting the great importance of the present multiscale model in providing a better understanding of how drugs transport inside the brain and how brain tissue responds to the infusion pressure. This new method can assist the development of both new drugs for brain diseases and preoperative evaluation techniques for CED surgery, thus helping to improve the efficiency and precision of treatments for brain diseases.

Microscale characterisation of the time-dependent mechanical behaviour of brain white matter

Brain mechanics is a topic of deep interest because of the significant role of mechanical cues in both brain function and form. Specifically, capturing the heterogeneous and anisotropic behaviour of cerebral white matter (WM) is extremely challenging and yet the data on WM at a spatial resolution relevant to tissue components are sparse. To investigate the time-dependent mechanical behaviour of WM, and its dependence on local microstructural features when subjected to small deformations, we conducted atomic force microscopy (AFM) stress relaxation experiments on corpus callosum (CC), corona radiata (CR) and fornix (FO) of fresh ovine brain. Our experimental results show a dependency of the tissue mechanical response on axons orientation, with e.g. the stiffness of perpendicular and parallel samples is different in all three regions of WM whereas the relaxation behaviour is different for the CC and FO regions. An inverse modelling approach was adopted to extract Prony series parameters of the tissue components, i.e. axons and extra cellular matrix with its accessory cells, from experimental data. Using a bottom-up approach, we developed analytical and FEA estimates that are in good agreement with our experimental results. Our systematic characterisation of sheep brain WM using a combination of AFM experiments and micromechanical models provide a significant contribution for predicting localised time-dependent mechanics of brain tissue. This information can lead to more accurate computational simulations, therefore aiding the development of surgical robotic solutions for drug delivery and accurate tissue mimics, as well as the determination of criteria for tissue injury and predict brain development and disease progression.

Regional Neurodegeneration in vitro: The Protective Role of Neural Activity

Traumatic brain injury is a devastating public health problem, the eighth leading cause of death across the world. To improve our understanding of how injury at the cellular scale affects neural circuit function, we developed a protocol to precisely injure individual neurons within an in vitro neural network. We used high speed calcium imaging to estimate alterations in neural activity and connectivity that occur followed targeted microtrauma. Our studies show that mechanically injured neurons inactivate following microtrauma and eventually re-integrate into the network. Single neuron re-integration is dependent on its activity prior to injury and initial connections in the network: more active and integrated neurons are more resistant to microtrauma and more likely to re-integrate into the network. Micromechanical injury leads to neuronal death 6 h post-injury in a subset of both injured and uninjured neurons. Interestingly, neural activity and network participation after injury were associated with survival in linear discriminate analysis (77.3% correct prediction, Wilks' Lambda = 0.838). Based on this observation, we modulated neuronal activity to rescue neurons after microtrauma. Inhibition of neuronal activity provided much greater survivability than did activation of neurons (ANOVA, p < 0.01 with post-hoc Tukey HSD, p < 0.01). Rescue of neurons by blocking activity in the post-acute period is partially mediated by mitochondrial energetics, as we observed silencing neurons after micromechanical injury led to a significant reduction in mitochondrial calcium accumulation. Overall, the present study provides deeper insight into the propagation of injury within networks, demonstrating that together the initial activity, network structure, and post-injury activity levels contribute to the progressive changes in a neural circuit after mechanical trauma.

On the Transversely Isotropic, Hyperelastic Response of Central Nervous System White Matter Using a Hybrid Approach

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.

Visco-hyperelastic characterization of human brain white matter micro-level constituents in different strain rates

In this study, we propose a computational characterization technique for obtaining the material properties of axons and extracellular matrix (ECM) in human brain white matter. To account for the dynamic behavior of the brain tissue, data from time-dependent relaxation tests of human brain white matter in different strain rates are extracted and formulated by a visco-hyperelastic constitutive model consisting of the Ogden hyperelastic model and the Prony series expansion. Through micromechanical finite element simulation, a derivative-free optimization framework designed to minimize the difference between the numerical and experimental data is used to identify the material properties of the axons and ECM. The Prony series expansion parameters of axons and ECM are found to be highly affected by the Prony series expansion coefficients of the brain white matter. The optimal parameters of axons and ECM are verified through micromechanical simulation by comparing the averaged numerical response with that of the experimental data. Moreover, the initial shear modulus and the reduced shear modulus of the axons are found for different strain rates of 0.0001, 0.01, and 1 s^-1. Consequently, first- and second-order regressions are used to find relations for the prediction of the shear modulus at the intermediate strain rates. Graphical Abstract The applied procedure for characterization of brain white matter micro-level constituents. The macro-level experimental data in different strain rates are used in the context of simulation-based optimization to obtain the properties of axons and extracellular matrix material.

Investigate the Variations of the Head and Brain Response in a Rodent Head Impact Acceleration Model by Finite Element Modeling

Diffuse axonal injury (DAI) is a severe form of traumatic brain injury and often induced by blunt trauma. The closed head impact acceleration (IA) model is the most widely used rodent DAI model. However, this model results in large variations of injury severity. Recently, the impact device/system was modified to improve the consistency of the impact energy, but variations of the head kinematics and subsequent brain injuries were still observed. This study was aimed to utilize a Finite Element (FE) model of a rat head/body and simulation to investigate the potential biomechanical factors influencing the impact energy transfer to the head. A detailed FE rat head model containing detailed skull and brain anatomy was developed based on the MRI, microCT and atlas data. The model consists of over 722,000 elements, of which 310,000 are in the brain. The white matter structures consisting of highly aligned axonal fibers were simulated with transversely isotropic material. The rat body was modeled to provide a realistic boundary at the spine-medulla junction. Rodent experiments including dynamic cortical deformation, brain-skull displacement, and IA kinematics were simulated to validate the FE model. The model was then applied to simulate the rat IA experiments. Parametric studies were conducted to investigate the effect of the helmet inclination angles (0°-5°) and skull stiffness (varied 20%) on the resulting head kinematics and maximum principal strain in the brain. The inclination angle of the helmet at 5° could vary head linear acceleration by 8-31%. The change in head rotational velocity was inversely related to the change in linear acceleration. Varying skull stiffness resulted in changes in head linear acceleration by 3% but with no effect on rotational velocity. The brain strain in the corpus callosum was only affected by head rotation while the strain in the brainstem was influenced by the combined head kinematics, local skull deformation, and head-neck position. Validated FE models of rat impact head injury can assist in exploring various biomechanical factors influencing the head impact response and internal brain response. Identification of these variables may help explain the variability of injury severity observed among experiments and across different labs.

A Three-Dimensional Statistical Volume Element for Histology Informed Micromechanical Modeling of Brain White Matter

This study presents a novel statistical volume element (SVE) for micromechanical modeling of the white matter structures, with histology-informed randomized distribution of axonal tracts within the extracellular matrix. The model was constructed based on the probability distribution functions obtained from the results of diffusion tensor imaging as well as the histological observations of scanning electron micrograph, at two structures of white matter susceptible to traumatic brain injury, i.e. corpus callosum and corona radiata. A simplistic representative volume element (RVE) with symmetrical arrangement of fully alligned axonal fibers was also created as a reference for comparison. A parametric study was conducted to find the optimum grid and edge size which ensured the periodicity and ergodicity of the SVE and RVE models. A multi-objective evolutionary optimization procedure was used to find the hyperelastic and viscoelastic material constants of the constituents, based on the experimentally reported responses of corpus callosum to axonal and transverse loadings. The optimal material properties were then used to predict the homogenized and localized responses of corpus callosum and corona radiata. The results indicated similar homogenized responses of the SVE and RVE under quasi-static extension, which were in good agreement with the experimental data. Under shear strain, however, the models exhibited different behaviors, with the SVE model showing much closer response to the experimental observations. Moreover, the SVE model displayed a significantly better agreement with the reports of the experiments at high strain rates. The results suggest that the randomized fiber architecture has a great influence on the validity of the micromechanical models of white matter, with a distinguished impact on the model's response to shear deformation and high strain rates. Moreover, it can provide a more detailed presentation of the localized responses of the tissue substructures, including the stress concentrations around the low caliber axonal tracts, which is critical for studying the axonal injury mechanisms.

Nonlinear mechanics of soft composites: hyperelastic characterization of white matter tissue components

This paper presents a bi-directional closed-form analytical solution, in the framework of nonlinear soft composites mechanics, for top-down hyperelastic characterization of brain white matter tissue components, based on the directional homogenized responses of the tissue in the axial and transverse directions. The white matter is considered as a transversely isotropic neo-Hookean composite made of unidirectional distribution of axonal fibers within the extracellular matrix. First, two homogenization formulations are derived for the homogenized axial and transverse shear moduli of the tissue, based on definition of the strain energy density function. Next, the rule of mixtures and Hashin-Shtrikman theories are used to derive two coupled nonlinear equations which correlates the tissue shear moduli to these of its components. Closed-form solutions for shear moduli of the components are then obtained by solving these equations simultaneously. In order to validate the hyperelastic characteristics of components obtained in previous step, they are used in a bottom-up approach in a micromechanical model of the tissue with the aim of predicting the directional homogenized responses of the tissue. Comparison of model predictions with the experimental test results reported for corona radiata and corpus callosum white matter structures reveals very good agreements with the experimental results in both directions. The model predictions are also in good agreement with the analytical solution obtained by the iterated homogenization technique. Results indicate that axonal fibers are almost ten times stiffer than the extracellular matrix under large deformations.

Axons Embedded in a Tissue May Withstand Larger Deformations Than Isolated Axons Before Mechanoporation Occurs

Diffuse axonal injury (DAI) is the pathological consequence of traumatic brain injury (TBI) that most of all requires a multiscale approach in order to be, first, understood and then possibly prevented. While in fact the mechanical insult usually happens at the head (or macro) level, the consequences affect structures at the cellular (or microlevel). The quest for axonal injury tolerances has so far been addressed both with experimental and computational approaches. On one hand, the experimental approach presents challenges connected to both temporal and spatial resolution in the identification of a clear axonal injury trigger after the application of a mechanical load. On the other hand, computational approaches usually consider axons as homogeneous entities and therefore are unable to make inferences about their viability, which is thought to depend on subcellular damages. Here, we propose a computational multiscale approach to investigate the onset of axonal injury in two typical experimental scenarios. We simulated single-cell and tissue stretch injury using a composite finite element axonal model in isolation and embedded in a matrix, respectively. Inferences on axonal damage are based on the comparison between axolemma strains and previously established mechanoporation thresholds. Our results show that, axons embedded in a tissue could withstand higher deformations than isolated axons before mechanoporation occurred and this is exacerbated by the increase in strain rate from 1/s to 10/s.

Homogenization of heterogeneous brain tissue under quasi-static loading: a visco-hyperelastic model of a 3D RVE

Researches, in the recent years, reveal the utmost importance of brain tissue assessment regarding its mechanical properties, especially for automatic robotic tools, surgical robots and helmet producing. For this reason, experimental and computational investigation of the brain behavior under different conditions seems crucial. However, experiments do not normally show the distribution of stress and injury in microscopic scale, and due to various factors are costly. Development of micromechanical methods, which could predict the brain behavior more appropriately, could highly be helpful in reducing these costs. This study presents computational analysis of heterogeneous part of the brain tissue under quasi-static loading. Heterogeneity is created by irregular distribution of neurons in a representative volume element (RVE). Considering time-dependent behavior of the tissue, a visco-hyperelastic constitutive model is developed to predict the RVE behavior more realistically. The RVE is studied in different loads and load rates; 1, 2, 3, 10 and 15% strain load are applied at 0.03 and 0.2 s on the RVE as tensile and shear loads. Due to complexity in geometry, self-consistent approximation method is employed to increase the volume fraction of neurons and analyze RVE behavior in various NVFs. The results show increasing the load rate leads to a raise in the maximum stress that indicates the tissue is more vulnerable at higher rates. Moreover, stiffness of the tissue is enhanced in higher NVFs. Additionally, it is found that axons undergo higher stresses; hence, they are more sensitive in accidents which lead to axonal death and would cause TBI and DAI.

A 3D Computational Head Model Under Dynamic Head Rotation and Head Extension Validated Using Live Human Brain Data, Including the Falx and the Tentorium

We employ an advanced 3D computational model of the head with high anatomical fidelity, together with measured tissue properties, to assess the consequences of dynamic loading to the head in two distinct modes: head rotation and head extension. We use a subject-specific computational head model, using the material point method, built from T1 magnetic resonance images, and considering the anisotropic properties of the white matter which can predict strains in the brain under large rotational accelerations. The material model now includes the shear anisotropy of the white matter. We validate the model under head rotation and head extension motions using live human data, and advance a prior version of the model to include biofidelic falx and tentorium. We then examine the consequences of incorporating the falx and tentorium in terms of the predictions from the computational head model.

In vivo modeling of interstitial pressure in a porcine model: approximation of poroelastic properties and effects of enhanced anatomical structure modeling

The purpose of this investigation is to test whether a poroelastic model with enhanced structure can capture in vivo interstitial pressure dynamics in a brain undergoing mock surgical loads. Using interstitial pressure data from a porcine study, we use an inverse model to reconstruct material properties in an effort to capture these in vivo brain tissue dynamics. Four distinct models for the reconstruction of parameters are investigated (full anatomical condition description, condition without dural septa description, condition without ventricle boundary description, and the conventional fully saturated model). These models are systematic in their development to isolate the influence of three model characteristics: the dural septa, the treatment of the ventricles, and the treatment of the brain as a saturated media. This study demonstrates that to capture appropriate pressure compartmentalization, interstitial pressure gradients, pressure transient effects, and deformations within the brain, the proposed boundary conditions and structural enhancement coupled with a heterogeneous description invoking partial saturation are needed in a biphasic poroelastic model. These findings suggest that with enhanced anatomical modeling and appropriate model assumptions, poroelastic models can be used to capture quite complex brain deformations and interstitial pressure dynamics.

A three-dimensional micromechanical model of brain white matter with histology-informed probabilistic distribution of axonal fibers

This paper presents a three-dimensional micromechanical model of brain white matter tissue as a transversely isotropic soft composite described by the generalized Ogden hyperelastic model. The embedded element technique, with corrected stiffness redundancy in large deformations, was used for the embedment of a histology-informed probabilistic distribution of the axonal fibers in the extracellular matrix. The model was linked to a multi-objective, multi-parametric optimization algorithm, using the response surface methodology, for characterization of material properties of the axonal fibers and extracellular matrix in an inverse finite element analysis. The optimum hyperelastic characteristics of the tissue constituents, obtained based on the axonal and transverse direction test results of the corona radiata tissue samples, indicated that the axonal fibers were almost thirteen times stiffer than the extracellular matrix under large deformations. Simulation of the same tissue under a different loading condition, as well as that of another white matter tissue, i.e., the corpus callosum, in the axonal and transverse directions, using the optimized hyperelastic characteristics revealed tissue responses very close to those of the experiments. The results of the model at the sub-tissue level indicated that the stress concentrations were considerably large around the small axons, which might contribute into the brain injury.

Micromechanics of brain white matter tissue: A fiber-reinforced hyperelastic model using embedded element technique

A transverse-plane hyperelastic micromechanical model of brain white matter tissue was developed using the embedded element technique (EET). The model consisted of a histology-informed probabilistic distribution of axonal fibers embedded within an extracellular matrix, both described using the generalized Ogden hyperelastic material model. A correcting method, based on the strain energy density function, was formulated to resolve the stiffness redundancy problem of the EET in large deformation regime. The model was then used to predict the homogenized tissue behavior and the associated localized responses of the axonal fibers under quasi-static, transverse, large deformations. Results indicated that with a sufficiently large representative volume element (RVE) and fine mesh, the statistically randomized microstructure implemented in the RVE exhibits directional independency in transverse plane, and the model predictions for the overall and local tissue responses, characterized by the normalized strain energy density and Cauchy and von Mises stresses, are independent from the modeling parameters. Comparison of the responses of the probabilistic model with that of a simple uniform RVE revealed that only the first one is capable of representing the localized behavior of the tissue constituents. The validity test of the model predictions for the corona radiata against experimental data from the literature indicated a very close agreement. In comparison with the conventional direct meshing method, the model provided almost the same results after correcting the stiffness redundancy, however, with much less computational cost and facilitated geometrical modeling, meshing, and boundary conditions imposing. It was concluded that the EET can be used effectively for detailed probabilistic micromechanical modeling of the white matter in order to provide more accurate predictions for the axonal responses, which are of great importance when simulating the brain trauma or tumor growth.

Micromechanical analysis of brain's diffuse axonal injury

Computational models are important tools which help researchers understand traumatic brain injury (TBI). A mechanistic multi-scale numerical approach is introduced to quantify diffuse axonal injury (DAI), the most important mechanism of TBI, induced by a mechanical insult at micro-scale regions of the white matter or voxels where fiber orientations are the same. Using the mechanical properties of a single axon with a viscoelastic constitutive relation and its functional failure in terms of electrophysiological impairment, a numerical 2D micro-level lattice method is implemented to directly analyze the percentage of injured axons in a voxel containing a bundle of axons all with the same orientation under biaxial stretches. Reference micro-injury maps are then developed with the input parameters based on the principal strain or stretch values and their direction with respect to axons, which provide the percentage of injured axons in the voxel of interest as the output. The methodology is independent of any statistical analyses of the accident data and medical reports to derive probabilistic injury risk curves for DAI. Avoiding a structurally detailed full finite element head model, this study proposes a micro-mechanical approach which considers the anatomical structure of neural axons in the white matter together with their mechanical properties using a numerical lattice method to analyze the brain's diffuse axonal injury. This work has the potential to help develop safer prevention tools and more effective diagnosis methods for DAI.

Brain shift in neuronavigation of brain tumors: A review

PURPOSE

Neuronavigation based on preoperative imaging data is a ubiquitous tool for image guidance in neurosurgery. However, it is rendered unreliable when brain shift invalidates the patient-to-image registration. Many investigators have tried to explain, quantify, and compensate for this phenomenon to allow extended use of neuronavigation systems for the duration of surgery. The purpose of this paper is to present an overview of the work that has been done investigating brain shift.

METHODS

A review of the literature dealing with the explanation, quantification and compensation of brain shift is presented. The review is based on a systematic search using relevant keywords and phrases in PubMed. The review is organized based on a developed taxonomy that classifies brain shift as occurring due to physical, surgical or biological factors.

RESULTS

This paper gives an overview of the work investigating, quantifying, and compensating for brain shift in neuronavigation while describing the successes, setbacks, and additional needs in the field. An analysis of the literature demonstrates a high variability in the methods used to quantify brain shift as well as a wide range in the measured magnitude of the brain shift, depending on the specifics of the intervention. The analysis indicates the need for additional research to be done in quantifying independent effects of brain shift in order for some of the state of the art compensation methods to become useful.

CONCLUSION

This review allows for a thorough understanding of the work investigating brain shift and introduces the needs for future avenues of investigation of the phenomenon.

An anisotropic hyperelastic constitutive model of brain white matter in biaxial tension and structural-mechanical relationships

Computational models of the brain require accurate and robust constitutive models to characterize the mechanical behavior of brain tissue. The anisotropy of white matter has been previously demonstrated; however, there is a lack of data describing the effects of multi-axial loading, even though brain tissue experiences multi-axial stress states. Therefore, a biaxial tensile experiment was designed to more fully characterize the anisotropic behavior of white matter in a quasi-static loading state, and the mechanical data were modeled with an anisotropic hyperelastic continuum model. A probabilistic analysis was used to quantify the uncertainty in model predictions because the mechanical data of brain tissue can show a high degree of variability, and computational studies can benefit from reporting the probability distribution of model responses. The axonal structure in white matter can be heterogeneous and regionally dependent, which can affect computational model predictions. Therefore, corona radiata and corpus callosum regions were tested, and histology and transmission electron microscopy were performed on tested specimens to relate the distribution of axon orientations and the axon volume fraction to the mechanical behavior. These measured properties were implemented into a structural constitutive model. Results demonstrated a significant, but relatively low anisotropic behavior, yet there were no conclusive mechanical differences between the two regions tested. The inclusion of both biaxial and uniaxial tests in model fits improved the accuracy of model predictions. The mechanical anisotropy of individual specimens positively correlated with the measured axon volume fraction, and, accordingly, the structural model exhibited slightly decreased uncertainty in model predictions compared to the model without structural properties.

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.

Cellular biomechanics of central nervous system injury

With a growing interest in how the brain responds and remodels itself following a traumatic injury, this chapter outlines the major organizing principles of how to study these injuries in the laboratory and extend these findings back into the clinic. A new repertoire of models is available to examine the response of isolated circuits of the brain in vitro, and to study precisely how mechanical forces applied to even small regions of these circuits can disrupt the entire circuit dysfunction. We review the existing knowledge garnered from these models and our current understanding of mechanically sensitive receptors and channels activated immediately following trauma. In turn, we point to the emergence of in silico models of network function that will lead to an improved understanding of the principles for the remodeling of circuit structure after traumatic, possibly pointing out new biological rules for circuit reassembly that would help guide new therapies for reconstructing brain circuits after trauma.

The mechanics of traumatic brain injury: a review of what we know and what we need to know for reducing its societal burden

Traumatic brain injury (TBI) is a significant public health problem, on pace to become the third leading cause of death worldwide by 2020. Moreover, emerging evidence linking repeated mild traumatic brain injury to long-term neurodegenerative disorders points out that TBI can be both an acute disorder and a chronic disease. We are at an important transition point in our understanding of TBI, as past work has generated significant advances in better protecting us against some forms of moderate and severe TBI. However, we still lack a clear understanding of how to study milder forms of injury, such as concussion, or new forms of TBI that can occur from primary blast loading. In this review, we highlight the major advances made in understanding the biomechanical basis of TBI. We point out opportunities to generate significant new advances in our understanding of TBI biomechanics, especially as it appears across the molecular, cellular, and whole organ scale.

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 micromechanical procedure for viscoelastic characterization of the axons and ECM of the brainstem

In this study, the optimal viscoelastic material parameters of axon and extracellular matrix (ECM) in porcine brain white matter were identified using a genetic algorithm (GA) optimization procedure. The procedure was combined with micromechanical finite element analysis (FEA) of brain tissue and experimental stress relaxation tests on brainstem specimens to find the optimal material coefficients of axon and ECM. The stress relaxation tests were performed in tension on 10 brainstem specimens at 3% strain level. The axonal volume fraction in brainstem was measured from the Scanning Electron Microscopy images of the brain tissue. A square periodic volume element was selected to represent the microscale homogenized brainstem tissue. Periodic boundary conditions were applied on the square volume element to mimics the repetitive nature of the volume element. Linear viscoelastic material properties were assumed for the brain tissue constituents under small deformation. The constitutive behavior was expressed in terms of Prony series. The GA procedure searched for the optimal material parameters by fitting the time-dependent tissue stresses of brain tissue FEA to the stresses of relaxation tests under the same loading conditions. The optimization procedure converged after 60 iterations. The initial elastic modulus of axon was found to be 12.86kPa, three times larger than that of ECM. The long-term elastic modulus of axon was 3.7kPa, while for ECM this value was 1.03kPa. The concordance correlation coefficient between FEA estimated elastic modulus of brainstem tissue using the optimal material properties and the experimental elastic modulus of brainstem specimens was 0.952, showing a strong agreement. The optimal material properties of brain tissue constituents can find applications in micromechanical analysis of brain tissue to gain insight into diffuse axonal injures (DAIs).

Mathematical Models of Blast-Induced TBI: Current Status, Challenges, and Prospects

Blast-induced traumatic brain injury (TBI) has become a signature wound of recent military activities and is the leading cause of death and long-term disability among U.S. soldiers. The current limited understanding of brain injury mechanisms impedes the development of protection, diagnostic, and treatment strategies. We believe mathematical models of blast wave brain injury biomechanics and neurobiology, complemented with in vitro and in vivo experimental studies, will enable a better understanding of injury mechanisms and accelerate the development of both protective and treatment strategies. The goal of this paper is to review the current state of the art in mathematical and computational modeling of blast-induced TBI, identify research gaps, and recommend future developments. A brief overview of blast wave physics, injury biomechanics, and the neurobiology of brain injury is used as a foundation for a more detailed discussion of multiscale mathematical models of primary biomechanics and secondary injury and repair mechanisms. The paper also presents a discussion of model development strategies, experimental approaches to generate benchmark data for model validation, and potential applications of the model for prevention and protection against blast wave TBI.

Measurement of the uniaxial mechanical properties of rat brains infected by Plasmodium berghei ANKA

Degenerative and demyelinating diseases are known to alter the mechanical properties of brain tissue. While few studies have characterized these biomechanical changes, it is clear that accurate characterization of the mechanical properties of diseased brain tissue could be a substantial asset to neuronavigation and surgery simulation through haptic devices. In this study, samples of brain tissue from rats infected with Plasmodium berghei ANKA, an African murine malaria parasite, are evaluated using a uniaxial tensile test machine. Infected brains having different levels of parasitemia are mounted on the testing machine and extended until failure of the tissue. The stress–strain curve of each sample is obtained and compared to healthy rat brain tissue. Young’s modulus of each sample is extracted from the Hookean part of the stress–strain diagram. Young’s modulus of rats’ brain shows considerable difference among the samples having various levels of parasitemia compared with the controls. For instance, the brains with 0% (control), 1.5%, and 9% parasitemia showed a Young’s modulus of 46.15, 54.54, and 266.67 kPa, respectively. This suggests sequestration of the stiffened and less deformable parasitized red blood cells in the brain microvasculature.

The role of tissue microstructure and water exchange in biophysical modelling of diffusion in white matter

Biophysical models that describe the outcome of white matter diffusion MRI experiments have various degrees of complexity. While the simplest models assume equal-sized and parallel axons, more elaborate ones may include distributions of axon diameters and axonal orientation dispersions. These microstructural features can be inferred from diffusion-weighted signal attenuation curves by solving an inverse problem, validated in several Monte Carlo simulation studies. Model development has been paralleled by microscopy studies of the microstructure of excised and fixed nerves, confirming that axon diameter estimates from diffusion measurements agree with those from microscopy. However, results obtained in vivo are less conclusive. For example, the amount of slowly diffusing water is lower than expected, and the diffusion-encoded signal is apparently insensitive to diffusion time variations, contrary to what may be expected. Recent understandings of the resolution limit in diffusion MRI, the rate of water exchange, and the presence of microscopic axonal undulation and axonal orientation dispersions may, however, explain such apparent contradictions. Knowledge of the effects of biophysical mechanisms on water diffusion in tissue can be used to predict the outcome of diffusion tensor imaging (DTI) and of diffusion kurtosis imaging (DKI) studies. Alterations of DTI or DKI parameters found in studies of pathologies such as ischemic stroke can thus be compared with those predicted by modelling. Observations in agreement with the predictions strengthen the credibility of biophysical models; those in disagreement could provide clues of how to improve them. DKI is particularly suited for this purpose; it is performed using higher b-values than DTI, and thus carries more information about the tissue microstructure. The purpose of this review is to provide an update on the current understanding of how various properties of the tissue microstructure and the rate of water exchange between microenvironments are reflected in diffusion MRI measurements. We focus on the use of biophysical models for extracting tissue-specific parameters from data obtained with single PGSE sequences on clinical MRI scanners, but results obtained with animal MRI scanners are also considered. While modelling of white matter is the central theme, experiments on model systems that highlight important aspects of the biophysical models are also reviewed.

The importance of axonal undulation in diffusion MR measurements: a Monte Carlo simulation study

Many axons follow wave-like undulating courses. This is a general feature of extracranial nerve segments, but is also found in some intracranial nervous tissue. The importance of axonal undulation has previously been considered, for example, in the context of biomechanics, where it has been shown that posture affects undulation properties. However, the importance of axonal undulation in the context of diffusion MR measurements has not been investigated. Using an analytical model and Monte Carlo simulations of water diffusion, this study compared undulating and straight axons in terms of diffusion propagators, diffusion-weighted signal intensities and parameters derived from diffusion tensor imaging, such as the mean diffusivity (MD), the eigenvalues and the fractional anisotropy (FA). All parameters were strongly affected by the presence of undulation. The diffusivity perpendicular to the undulating axons increased with the undulation amplitude, thus resembling that of straight axons with larger diameters. Consequently, models assuming straight axons for the estimation of the axon diameter from diffusion MR measurements might overestimate the diameter if undulation is present. FA decreased from approximately 0.7 to 0.5 when axonal undulation was introduced into the simulation model structure. Our results indicate that axonal undulation may play a role in diffusion measurements when investigating, for example, the optic and sciatic nerves and the spinal cord. The simulations also demonstrate that the stretching or compression of neuronal tissue comprising undulating axons alters the observed water diffusivity, suggesting that posture may be of importance for the outcome of diffusion MRI measurements.