An anisotropic viscous hyperelastic constitutive law for brain material finite-element modeling

Cerebrospinal fluid flow dynamics in the aqueduct of Sylvius for rigid and deformable brain models

The cerebrospinal fluid (CSF) in the subarachnoid space and brain ventricles is crucial for maintaining brain health and influencing conditions such as hydrocephalus. This study explores CSF flow dynamics in the aqueduct of Sylvius (AoS) using rigid and deformable brain models based on MRI-derived 3D geometries. The rigid model captures the main characteristics of the CSF velocity waveform in the AoS, particularly in terms of waveform and timing of peaks and troughs. This model also predicts velocity increases in hydrocephalic conditions and velocity ranges in subarachnoid space. However, the rigid model falls short of representing the full range of velocity variations in the AoS. Using fluid-structure interaction simulation, the deformable model addresses the impact of brain tissue deformation on CSF flow. The deformable brain model have an excellent prediction and agrees with MRI measurements, effectively anticipating CSF flow dynamics and flow reversals, underscoring the importance of incorporating brain tissue deformation for accurate modeling. We also found out that the effect of the inlet waveform is critical on the model prediction for the velocity within AoS during a cardiac cycle.

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.

Towards constructing a generalized structural 3D breathing human lung model based on experimental volumes, pressures, and strains

Respiratory diseases represent a significant healthcare burden, as evidenced by the devastating impact of COVID-19. Biophysical models offer the possibility to anticipate system behavior and provide insights into physiological functions, advancements which are comparatively and notably nascent when it comes to pulmonary mechanics research. In this context, an Inverse Finite Element Analysis (IFEA) pipeline is developed to construct the first continuously ventilated three-dimensional structurally representative pulmonary model informed by both organ- and tissue-level breathing experiments from a cadaveric human lung. Here we construct a generalizable computational framework directly validated by pressure, volume, and strain measurements using a novel inflating apparatus interfaced with adapted, lung-specific, digital image correlation techniques. The parenchyma, pleura, and airways are represented with a poroelastic formulation to simulate pressure flows within the lung lobes, calibrating the model's material properties with the global pressure-volume response and local tissue deformations strains. The optimization yielded the following shear moduli: parenchyma (2.8 kPa), airways (0.2 kPa), and pleura (1.7 Pa). The proposed complex multi-material model with multi-experimental inputs was successfully developed using human lung data, and reproduced the shape of the inflating pressure-volume curve and strain distribution values associated with pulmonary deformation. This advancement marks a significant step towards creating a generalizable human lung model for broad applications across animal models, such as porcine, mouse, and rat lungs to reproduce pathological states and improve performance investigations regarding medical therapeutics and intervention.

A theoretical framework for predicting the heterogeneous stiffness map of brain white matter tissue

Finding the stiffness map of biological tissues is of great importance in evaluating their healthy or pathological conditions. However, due to the heterogeneity and anisotropy of biological fibrous tissues, this task presents challenges and significant uncertainty when characterized only by single-mode loading experiments. In this study, we propose a new theoretical framework to map the stiffness landscape of fibrous tissues, specifically focusing on brain white matter tissue. Initially, a finite element (FE) model of the fibrous tissue was subjected to six loading cases, and their corresponding stress-strain curves were characterized. By employing multiobjective optimization, the material constants of an equivalent anisotropic material model were inversely extracted to best fit all six loading modes simultaneously. Subsequently, large-scale FE simulations were conducted, incorporating various fiber volume fractions and orientations, to train a convolutional neural network capable of predicting the equivalent anisotropic material properties solely based on the fibrous architecture of any given tissue. The proposed method, leveraging brain fiber tractography, was applied to a localized volume of white matter, demonstrating its effectiveness in precisely mapping the anisotropic behavior of fibrous tissue. In the long-term, the proposed method may find applications in traumatic brain injury, brain folding studies, and neurodegenerative diseases, where accurately capturing the material behavior of the tissue is crucial for simulations and experiments.

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.

Multi-scale measurement of stiffness in the developing ferret brain

Cortical folding is an important process during brain development, and aberrant folding is linked to disorders such as autism and schizophrenia. Changes in cell numbers, size, and morphology have been proposed to exert forces that control the folding process, but these changes may also influence the mechanical properties of developing brain tissue. Currently, the changes in tissue stiffness during brain folding are unknown. Here, we report stiffness in the developing ferret brain across multiple length scales, emphasizing changes in folding cortical tissue. Using rheometry to measure the bulk properties of brain tissue, we found that overall brain stiffness increases with age over the period of cortical folding. Using atomic force microscopy to target the cortical plate, we found that the occipital cortex increases in stiffness as well as stiffness heterogeneity over the course of development and folding. These findings can help to elucidate the mechanics of the cortical folding process by clarifying the concurrent evolution of tissue properties.

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.

Design and characterization of 3-D printed hydrogel lattices with anisotropic mechanical properties

The goal of this study was to design, fabricate, and characterize hydrogel lattice structures with consistent, controllable, anisotropic mechanical properties. Lattices, based on three unit-cell types (cubic, diamond, and vintile), were printed using stereolithography (SLA) of polyethylene glycol diacrylate (PEGDA). To create structural anisotropy in the lattices, unit cell design files were scaled by a factor of two in one direction in each layer and then printed. The mechanical properties of the scaled lattices were measured in shear and compression and compared to those of the unscaled lattices. Two apparent shear moduli of each lattice were measured by dynamic shear tests in two planes: (1) parallel and (2) perpendicular to the scaling direction, or cell symmetry axis. Three apparent Young's moduli of each lattice were measured by compression in three different directions: (1) the "build" direction or direction of added layers, (2) the scaling direction, and (3) the unscaled direction perpendicular to both scaling and build directions. For shear deformation in unscaled lattices, the apparent shear moduli were similar in the two perpendicular directions. In contrast, scaled lattices exhibit clear differences in apparent shear moduli. In compression of unscaled lattices, apparent Young's moduli were independent of direction in cubic and vintile lattices; in diamond lattices Young's moduli differed in the build direction, but were similar in the other two directions. Scaled lattices in compression exhibited additional differences in apparent Young's moduli in the scaled and unscaled directions. Notably, the effects of scaling on apparent modulus differed between each lattice type (cubic, diamond, or vintile) and deformation mode (shear or compression). Scaling of 3D-printed, hydrogel lattices may be harnessed to create tunable, structures of desired shape, stiffness, and mechanical anisotropy, in both shear and compression.

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.

Use of Brain Biomechanical Models for Monitoring Impact Exposure in Contact Sports

Head acceleration measurement sensors are now widely deployed in the field to monitor head kinematic exposure in contact sports. The wealth of impact kinematics data provides valuable, yet challenging, opportunities to study the biomechanical basis of mild traumatic brain injury (mTBI) and subconcussive kinematic exposure. Head impact kinematics are translated into brain mechanical responses through physics-based computational simulations using validated brain models to study the mechanisms of injury. First, this article reviews representative legacy and contemporary brain biomechanical models primarily used for blunt impact simulation. Then, it summarizes perspectives regarding the development and validation of these models, and discusses how simulation results can be interpreted to facilitate injury risk assessment and head acceleration exposure monitoring in the context of contact sports. Recommendations and consensus statements are presented on the use of validated brain models in conjunction with kinematic sensor data to understand the biomechanics of mTBI and subconcussion. Mainly, there is general consensus that validated brain models have strong potential to improve injury prediction and interpretation of subconcussive kinematic exposure over global head kinematics alone. Nevertheless, a major roadblock to this capability is the lack of sufficient data encompassing different sports, sex, age and other factors. The authors recommend further integration of sensor data and simulations with modern data science techniques to generate large datasets of exposures and predicted brain responses along with associated clinical findings. These efforts are anticipated to help better understand the biomechanical basis of mTBI and improve the effectiveness in monitoring kinematic exposure in contact sports for risk and injury mitigation purposes.

MR Imaging of Human Brain Mechanics In Vivo: New Measurements to Facilitate the Development of Computational Models of Brain Injury

Computational models of the brain and its biomechanical response to skull accelerations are important tools for understanding and predicting traumatic brain injuries (TBIs). However, most models have been developed using experimental data collected on animal models and cadaveric specimens, both of which differ from the living human brain. Here we describe efforts to noninvasively measure the biomechanical response of the human brain with MRI-at non-injurious strain levels-and generate data that can be used to develop, calibrate, and evaluate computational brain biomechanics models. Specifically, this paper reports on a project supported by the National Institute of Neurological Disorders and Stroke to comprehensively image brain anatomy and geometry, mechanical properties, and brain deformations that arise from impulsive and harmonic skull loadings. The outcome of this work will be a publicly available dataset ( http://www.nitrc.org/projects/bbir ) that includes measurements on both males and females across an age range from adolescence to older adulthood. This article describes the rationale and approach for this study, the data available, and how these data may be used to develop new computational models and augment existing approaches; it will serve as a reference to researchers interested in using these data.

How growing tumour impacts intracranial pressure and deformation mechanics of brain

Brain is an actuator for control and coordination. When a pathology arises in cranium, it may leave a degenerative, disfiguring and destabilizing impact on brain physiology. However, the leading consequences of the same may vary from case to case. Tumour, in this context, is a special type of pathology which deforms brain parenchyma permanently. From translational perspective, deformation mechanics and pressures, specifically the intracranial cerebral pressure (ICP) in a tumour-housed brain, have not been addressed holistically in literature. This is an important area to investigate in neuropathy prognosis. To address this, we aim to solve the pressure mystery in a tumour-based brain in this study and present a fairly workable methodology. Using image-based finite-element modelling, we reconstruct a tumour-based brain and probe resulting deformations and pressures (ICP). Tumour is grown by dilating the voxel region by 16 and 30 mm uniformly. Cumulatively three cases are studied including an existing stage of the tumour. Pressures of cerebrospinal fluid due to its flow inside the ventricle region are also provided to make the model anatomically realistic. Comparison of obtained results unequivocally shows that as the tumour region increases its area and size, deformation pattern changes extensively and spreads throughout the brain volume with a greater concentration in tumour vicinity. Second, we conclude that ICP pressures inside the cranium do increase substantially; however, they still remain under the normal values (15 mmHg). In the end, a correlation relationship of ICP mechanics and tumour is addressed. From a diagnostic purpose, this result also explains why generally a tumour in its initial stage does not show symptoms because the required ICP threshold has not been crossed. We finally conclude that even at low ICP values, substantial deformation progression inside the cranium is possible. This may result in plastic deformation, midline shift etc. in the brain.

Brain Strain: Computational Model-Based Metrics for Head Impact Exposure and Injury Correlation

Athletes participating in contact sports are exposed to repetitive subconcussive head impacts that may have long-term neurological consequences. To better understand these impacts and their effects, head impacts are often measured during football to characterize head impact exposure and estimate injury risk. Despite widespread use of kinematic-based metrics, it remains unclear whether any single metric derived from head kinematics is well-correlated with measurable changes in the brain. This shortcoming has motivated the increasing use of finite element (FE)-based metrics, which quantify local brain deformations. Additionally, quantifying cumulative exposure is of increased interest to examine the relationship to brain changes over time. The current study uses the atlas-based brain model (ABM) to predict the strain response to impacts sustained by 116 youth football athletes and proposes 36 new, or derivative, cumulative strain-based metrics that quantify the combined burden of head impacts over the course of a season. The strain-based metrics developed and evaluated for FE modeling and presented in the current study present potential for improved analytics over existing kinematically-based and cumulative metrics. Additionally, the findings highlight the importance of accounting for directional dependence and expand the techniques to explore spatial distribution of the strain response throughout the brain.

White Matter Tract-Oriented Deformation Is Dependent on Real-Time Axonal Fiber Orientation

Traumatic axonal injury (TAI) is a critical public health issue with its pathogenesis remaining largely elusive. Finite element (FE) head models are promising tools to bridge the gap between mechanical insult, localized brain response, and resultant injury. In particular, the FE-derived deformation along the direction of white matter (WM) tracts (i.e., tract-oriented strain) has been shown to be an appropriate predictor for TAI. The evolution of fiber orientation in time during the impact and its potential influence on the tract-oriented strain remains unknown, however. To address this question, the present study leveraged an embedded element approach to track real-time fiber orientation during impacts. A new scheme to calculate the tract-oriented strain was proposed by projecting the strain tensors from pre-computed simulations along the temporal fiber direction instead of its static counterpart directly obtained from diffuse tensor imaging. The results revealed that incorporating the real-time fiber orientation not only altered the direction but also amplified the magnitude of the tract-oriented strain, resulting in a generally more extended distribution and a larger volume ratio of WM exposed to high deformation along fiber tracts. These effects were exacerbated with the impact severities characterized by the acceleration magnitudes. Results of this study provide insights into how best to incorporate fiber orientation in head injury models and derive the WM tract-oriented deformation from computational simulations, which is important for furthering our understanding of the underlying mechanisms of TAI.

Hyperelastic Ex Vivo Cervical Tissue Mechanical Characterization

This paper presents the results of the comparison between a proposed Fourth Order Elastic Constants (FOECs) nonlinear model defined in the sense of Landau's theory, and the two most contrasted hyperelastic models in the literature, Mooney-Rivlin, and Ogden models. A mechanical testing protocol is developed to investigate the large-strain response of ex vivo cervical tissue samples in uniaxial tension in its two principal anatomical locations, the epithelial and connective layers. The final aim of this work is to compare the reconstructed shear modulus of the epithelial and connective layers of cervical tissue. According to the obtained results, the nonlinear parameter A from the proposed FOEC model could be an important biomarker in cervical tissue diagnosis. In addition, the calculated shear modulus depended on the anatomical location of the cervical tissue (μepithelial = 1.29 ± 0.15 MPa, and μconnective = 3.60 ± 0.63 MPa).

Embedded axonal fiber tracts improve finite element model predictions of traumatic brain injury

With the growing rate of traumatic brain injury (TBI), there is an increasing interest in validated tools to predict and prevent brain injuries. Finite element models (FEM) are valuable tools to estimate tissue responses, predict probability of TBI, and guide the development of safety equipment. In this study, we developed and validated an anisotropic pig brain multi-scale FEM by explicitly embedding the axonal tract structures and utilized the model to simulate experimental TBI in piglets undergoing dynamic head rotations. Binary logistic regression, survival analysis with Weibull distribution, and receiver operating characteristic curve analysis, coupled with repeated k-fold cross-validation technique, were used to examine 12 FEM-derived metrics related to axonal/brain tissue strain and strain rate for predicting the presence or absence of traumatic axonal injury (TAI). All 12 metrics performed well in predicting of TAI with prediction accuracy rate of 73-90%. The axonal-based metrics outperformed their rival brain tissue-based metrics in predicting TAI. The best predictors of TAI were maximum axonal strain times strain rate (MASxSR) and its corresponding optimal fraction-based metric (AF-MASxSR7.5) that represents the fraction of axonal fibers exceeding MASxSR of 7.5 s-1. The thresholds compare favorably with tissue tolerances found in in-vitro/in-vivo measurements in the literature. In addition, the damaged volume fractions (DVF) predicted using the axonal-based metrics, especially MASxSR (DVF = 0.05-4.5%), were closer to the actual DVF obtained from histopathology (AIV = 0.02-1.65%) in comparison with the DVF predicted using the brain-related metrics (DVF = 0.11-41.2%). The methods and the results from this study can be used to improve model prediction of TBI in humans.

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.

Identification of the visco-hyperelastic properties of brain white matter based on the combination of inverse method and experiment

To fully understand the brain injury mechanism and develop effective protective approaches, an accurate constitutive model of brain tissue is firstly required. Generally, the brain tissue is regarded as a kind of viscoelastic material and is simply used in the simulation of brain injury. In fact, the brain tissue has the behavior of the visco-hyperelastic property. Therefore, this paper presents an effective computational inverse method to determine the material parameters of visco-hyperelastic constitutive model of brain white matter through compression experiments. First, with the help of 3D hand scanner, 3D geometries of brain white matter specimens are obtained to make it possible to establish the accurate simulation models of the specific specimens. Then, the global sensitivity analysis is adopted to evaluate the importance of the material parameters and further determine the parameters which may be identified. Subsequently, based on the genetic algorithm, the optimal material parameters of brain white matter can be identified by minimizing the match error between the experimental and simulated responses. Finally, by comparing the experiment and simulation results on the other specific specimen, and the simulation results with the material parameters from the references, respectively, the accuracy and reliability of the constitutive model parameters of brain white matter are demonstrated. Graphical abstract The main flowchart of the computational inverse technique for determining the material parameters of specimen-specific on brain white matter. Generalization: Combining the computational inverse method and unconfined uniaxial compression experiment of the specific specimen, an effective identification method is presented to accurately determine the hyperelastic and viscoelastic parameters of brain white matter in this paper.

Effect of Tissue Material Properties in Blast Loading: Coupled Experimentation and Finite Element Simulation

Computational models of blast-induced traumatic brain injury (bTBI) require a robust definition of the material models of the brain. The mechanical constitutive models of these tissues are difficult to characterize, leading to a wide range of values reported in literature. Therefore, the sensitivity of the intracranial pressure (ICP) and maximum principal strain to variations in the material model of the brain was investigated through a combined computational and experimental approach. A finite element model of a rat was created to simulate a shock wave exposure, guided by the experimental measurements of rats subjected to shock loading conditions corresponding to that of mild traumatic brain injury in a field-validated shock tube. In the numerical model, the properties of the brain were parametrically varied. A comparison of the ICP measured at two locations revealed that experimental and simulated ICP were higher in the cerebellum (p < 0.0001), highlighting the significance of pressure sensor locations within the cranium. The ICP and strain were correlated with the long-term bulk (p < 0.0001) and shear moduli (p < 0.0001), with an 80 MPa effective bulk modulus value matching best with experimental measurements. In bTBI, the solution is sensitive to the brain material model, necessitating robust validation methods.

Brain modelling in the framework of anisotropic hyperelasticity with time fractional damage evolution governed by the Caputo-Almeida fractional derivative

In this paper the human brain tissue constitutive model for monotonic loading is developed. The model in this work is based on the anisotropic hyperelasticity assumption (the transversely isotropic case) together with modelling of the evolving load-carrying capacity (scalar damage) whose change is governed by the Caputo-Almeida fractional derivative. This allows the brain constitutive law to include the memory during progressive damage, due to the characteristic time length scale which is an inherent attribute of the fractional operator. Furthermore, the rate dependence of the overall brain tissue model is included as well. The theoretical model is finally calibrated and validated with a set of experimental data.

A modified formulation of quasi-linear viscoelasticity for transversely isotropic materials under finite deformation

The theory of quasi-linear viscoelasticity (QLV) is modified and developed for transversely isotropic (TI) materials under finite deformation. For the first time, distinct relaxation responses are incorporated into an integral formulation of nonlinear viscoelasticity, according to the physical mode of deformation. The theory is consistent with linear viscoelasticity in the small strain limit and makes use of relaxation functions that can be determined from small-strain experiments, given the time/deformation separability assumption. After considering the general constitutive form applicable to compressible materials, attention is restricted to incompressible media. This enables a compact form for the constitutive relation to be derived, which is used to illustrate the behaviour of the model under three key deformations: uniaxial extension, transverse shear and longitudinal shear. Finally, it is demonstrated that the Poynting effect is present in TI, neo-Hookean, modified QLV materials under transverse shear, in contrast to neo-Hookean elastic materials subjected to the same deformation. Its presence is explained by the anisotropic relaxation response of the medium.

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.

White Matter Anisotropy for Impact Simulation and Response Sampling in Traumatic Brain Injury

Advanced neuroimaging provides new opportunities to enhance head injury models, including the incorporation of white matter (WM) structural anisotropy. Information from high-resolution neuroimaging, however, usually has to be "down-sampled" to match a typically coarse brain mesh. To understand how this mesh-image resolution mismatch affects impact simulation and subsequent response sampling, we compared three competing anisotropy implementations (using either voxels, tractography, or a multiscale submodeling) and two response sampling strategies (element-wise or tractography-based, using brain mesh or neuroimaging for region segmentation, respectively). Using the combination of high resolution options as a baseline, we studied how the choice in each individual category affected the resulting injury metrics. By simulating a recorded loss of consciousness head impact, we found that injury metrics including peak strain and injury susceptibility in the deep WM regions based on fiber strain, but not on maximum principal strain, were sensitive to the anisotropy implementation, response sampling, and region segmentation. Overall, it was recommended to use tractography for anisotropy implementation and response sampling, and to employ neuroimaging for region segmentation, because they led to more accurate injury metrics. Further refining mesh locally via submodeling was unnecessary. Brain strain responses were also parametrically found to be closer to that from minimum fiber reinforcement, consistent with the fact that the majority of WM had a rather high degree of fiber dispersion. Finally, the upgraded Worcester Head Injury Model incorporating WM anisotropy was successfully re-validated against cadaveric impacts and an in vivo head rotation ("good" to "excellent" validation with an average Correlation Analysis score of 0.437 and 0.509, respectively). These investigations may facilitate further continual development of head injury models to better study traumatic brain injury.

Utilizing a Structural Mechanics Approach to Assess the Primary Effects of Injury Loads Onto the Axon and Its Components

Diffuse axonal injury (DAI) occurs as a result of the transmission of rapid dynamic loads from the head to the brain and specifically to its neurons. Despite being one of the most common and most deleterious types of traumatic brain injury (TBI), the inherent cell injury mechanism is yet to be understood. Experimental observations have led to the formulation of different hypotheses, such as mechanoporation of the axolemma and microtubules (MTs) breakage. With the goal of singling out the mechanical aspect of DAI and to resolve the ambiguity behind its injury mechanism, a composite finite element (FE) model of a representative volume of an axon was developed. Once calibrated and validated against published experimental data, the axonal model was used to simulate injury scenarios. The resulting strain distributions along its components were then studied in dependence of strain rate and of typical modeling choices such as the applied MT constraints and tau proteins failure. Results show that oversimplifying the MT bundle kinematic entails a systematic attenuation (cf = 2.33) of the computed maximum MT strain. Nevertheless, altogether, results support the hypothesis of axolemma mechanoporation as a cell-injury trigger. Moreover, for the first time the interconnection between the axolemma and the MT bundle is shown to play a role in damage localization. The proposed FE axonal model is a valuable tool to understand the axonal injury mechanism from a mechanical perspective and could be used in turn for the definition of well-informed injury criteria at the tissue and organ level.

Cognition based bTBI mechanistic criteria; a tool for preventive and therapeutic innovations

Blast-induced traumatic brain injury has been associated with neurodegenerative and neuropsychiatric disorders. To date, although damage due to oxidative stress appears to be important, the specific mechanistic causes of such disorders remain elusive. Here, to determine the mechanical variables governing the tissue damage eventually cascading into cognitive deficits, we performed a study on the mechanics of rat brain under blast conditions. To this end, experiments were carried out to analyse and correlate post-injury oxidative stress distribution with cognitive deficits on a live rat exposed to blast. A computational model of the rat head was developed from imaging data and validated against in vivo brain displacement measurements. The blast event was reconstructed in silico to provide mechanistic thresholds that best correlate with cognitive damage at the regional neuronal tissue level, irrespectively of the shape or size of the brain tissue types. This approach was leveraged on a human head model where the prediction of cognitive deficits was shown to correlate with literature findings. The mechanistic insights from this work were finally used to propose a novel protective device design roadmap and potential avenues for therapeutic innovations against blast traumatic brain injury.

Compressive mechanical characterization of non-human primate spinal cord white matter

The goal of developing computational models of spinal cord injury (SCI) is to better understand the human injury condition. However, finite element models of human SCI have used rodent spinal cord tissue properties due to a lack of experimental data. Central nervous system tissues in non human primates (NHP) closely resemble that of humans and therefore, it is expected that material constitutive models obtained from NHPs will increase the fidelity and the accuracy of human SCI models. Human SCI most often results from compressive loading and spinal cord white matter properties affect FE predicted patterns of injury; therefore, the objectives of this study were to characterize the unconfined compressive response of NHP spinal cord white matter and present an experimentally derived, finite element tractable constitutive model for the tissue. Cervical spinal cords were harvested from nine male adult NHPs (Macaca mulatta). White matter biopsy samples (3 mm in diameter) were taken from both lateral columns of the spinal cord and were divided into four strain rate groups for unconfined dynamic compression and stress relaxation (post-mortem <1-hour). The NHP spinal cord white matter compressive response was sensitive to strain rate and showed substantial stress relaxation confirming the viscoelastic behavior of the material. An Ogden 1st order model best captured the non-linear behavior of NHP white matter in a quasi-linear viscoelastic material model with 4-term Prony series. This study is the first to characterize NHP spinal cord white matter at high (>10/sec) strain rates typical of traumatic injury. The finite element derived material constitutive model of this study will increase the fidelity of SCI computational models and provide important insights for transferring pre-clinical findings to clinical treatments.

STATEMENT OF SIGNIFICANCE

Spinal cord injury (SCI) finite element (FE) models provide an important tool to bridge the gap between animal studies and human injury, assess injury prevention technologies (e.g. helmets, seatbelts), and provide insight into the mechanisms of injury. Although, FE model outcomes depend on the assumed material constitutive model, there is limited experimental data for fresh spinal cords and all was obtained from rodent, porcine or bovine tissues. Central nervous system tissues in non human primates (NHP) more closely resemble humans. This study characterizes fresh NHP spinal cord material properties at high strains rates and large deformations typical of SCI for the first time. A constitutive model was defined that can be readily implemented in finite strain FE analysis of SCI.

Do blast induced skull flexures result in axonal deformation?

Subject-specific computer models (male and female) of the human head were used to investigate the possible axonal deformation resulting from the primary phase blast-induced skull flexures. The corresponding axonal tractography was explicitly incorporated into these finite element models using a recently developed technique based on the embedded finite element method. These models were subjected to extensive verification against experimental studies which examined their pressure and displacement response under a wide range of loading conditions. Once verified, a parametric study was developed to investigate the axonal deformation for a wide range of loading overpressures and directions as well as varying cerebrospinal fluid (CSF) material models. This study focuses on early times during a blast event, just as the shock transverses the skull (< 5 milliseconds). Corresponding boundary conditions were applied to eliminate the rotation effects and the resulting axonal deformation. A total of 138 simulations were developed– 128 simulations for studying the different loading scenarios and 10 simulations for studying the effects of CSF material model variance–leading to a total of 10,702 simulation core hours. Extreme strains and strain rates along each of the fiber tracts in each of these scenarios were documented and presented here. The results suggest that the blast-induced skull flexures result in strain rates as high as 150–378 s^-1. These high-strain rates of the axonal fiber tracts, caused by flexural displacement of the skull, could lead to a rate dependent micro-structural axonal damage, as pointed by other researchers.

Material properties of the brain in injury-relevant conditions - Experiments and computational modeling

Material properties of the brain have been extensively studied but remain poorly characterized. The vast variations in constitutive models and material constants are well documented. However, no study exists to translate the variations into disparities in impact-induced brain strains most relevant to brain injury. Here, we reviewed a subset of injury-relevant brain material properties either characterized in experiments or adopted in recent head injury models. To highlight how variations in measured brain material properties manifested in simulated brain strains, we selected six experiments that have provided a complete set of brain material model and constants to implement a common head injury model. Responses resulting from two extreme events representing a high-rate cadaveric head impact and a low-rate in vivo head rotation, respectively, varied substantially. We hypothesized, and further confirmed, that the time-varying shear moduli at the appropriate time scales (e.g., ~5 ms and ~40 ms corresponding to the impulse durations of the major acceleration peaks for the two impacts, respectively), rather than the initial or long-term shear moduli, were the most indicative of impact-induced brain strains. These results underscored the need to implement measured brain material properties into an actual head injury model for evaluation. They may also provide guidelines to better characterize brain material properties in future experiments and head injury models. Finally, our finding provided a practical solution to satisfy head injury model validation requirements at both ends of the impact severity spectrum. This would improve the confidence in model simulation performance across a range of time scales relevant to concussion and sub-concussion in the real-world.

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.

Modeling the mechanics of axonal fiber tracts using the embedded finite element method

A subject-specific human head finite element model with embedded axonal fiber tractography obtained from diffusion tensor imaging was developed. The axonal fiber tractography finite element model was coupled with the volumetric elements in the head model using the embedded element method. This technique enables the calculation of axonal strains and real-time tracking of the mechanical response of the axonal fiber tracts. The coupled model was then verified using pressure and relative displacement-based (between skull and brain) experimental studies and was employed to analyze a head impact, demonstrating the applicability of this method in studying axonal injury. Following this, a comparison study of different injury criteria was performed. This model was used to determine the influence of impact direction on the extent of the axonal injury. The results suggested that the lateral impact loading is more dangerous compared to loading in the sagittal plane, a finding in agreement with previous studies. Through this analysis, we demonstrated the viability of the embedded element method as an alternative numerical approach for studying axonal injury in patient-specific human head models.

Head injury assessment of non-lethal projectile impacts: A combined experimental/computational method

The main objective of this study is to develop a methodology to assess this risk based on experimental tests versus numerical predictive head injury simulations. A total of 16 non-lethal projectiles (NLP) impacts were conducted with rigid force plate at three different ranges of impact velocity (120, 72 and 55m/s) and the force/deformation-time data were used for the validation of finite element (FE) NLP. A good accordance between experimental and simulation data were obtained during validation of FE NLP with high correlation value (>0.98) and peak force discrepancy of less than 3%. A state-of-the art finite element head model with enhanced brain and skull material laws and specific head injury criteria was used for numerical computation of NLP impacts. Frontal and lateral FE NLP impacts to the head model at different velocities were performed under LS-DYNA. It is the very first time that the lethality of NLP is assessed by axonal strain computation to predict diffuse axonal injury (DAI) in NLP impacts to head. In case of temporo-parietal impact the min-max risk of DAI is 0-86%. With a velocity above 99.2m/s there is greater than 50% risk of DAI for temporo-parietal impacts. All the medium- and high-velocity impacts are susceptible to skull fracture, with a percentage risk higher than 90%. This study provides tool for a realistic injury (DAI and skull fracture) assessment during NLP impacts to the human head.

The Transverse Isotropy of Spinal Cord White Matter Under Dynamic Load

The rostral-caudally aligned fiber-reinforced structure of spinal cord white matter (WM) gives rise to transverse isotropy in the material. Stress and strain patterns generated in the spinal cord parenchyma following spinal cord injury (SCI) are multidirectional and dependent on the mechanism of the injury. Our objective was to develop a WM constitutive model that captures the material transverse isotropy under dynamic loading. The WM mechanical behavior was extracted from the published tensile and compressive experiments. Combinations of isotropic and fiber-reinforcing models were examined in a conditional quasi-linear viscoelastic (QLV) formulation to capture the WM mechanical behavior. The effect of WM transverse isotropy on SCI model outcomes was evaluated by simulating a nonhuman primate (NHP) contusion injury experiment. A second-order reduced polynomial hyperelastic energy potential conditionally combined with a quadratic reinforcing function in a four-term Prony series QLV model best captured the WM mechanical behavior (0.89 < R2 < 0.99). WM isotropic and transversely isotropic material models combined with discrete modeling of the pia mater resulted in peak impact forces that matched the experimental outcomes. The transversely isotropic WM with discrete pia mater resulted in maximum principal strain (MPS) distributions which effectively captured the combination of ipsilateral peripheral WM sparing, ipsilateral injury and contralateral sparing, and the rostral/caudal spread of damage observed in in vivo injuries. The results suggest that the WM transverse isotropy could have an important role in correlating tissue damage with mechanical measures and explaining the directional sensitivity of the spinal cord to injury.

Brain injury tolerance limit based on computation of axonal strain

Traumatic brain injury (TBI) is the leading cause of death and permanent impairment over the last decades. In both the severe and mild TBIs, diffuse axonal injury (DAI) is the most common pathology and leads to axonal degeneration. Computation of axonal strain by using finite element head model in numerical simulation can enlighten the DAI mechanism and help to establish advanced head injury criteria. The main objective of this study is to develop a brain injury criterion based on computation of axonal strain. To achieve the objective a state-of-the-art finite element head model with enhanced brain and skull material laws, was used for numerical computation of real world head trauma. The implementation of new medical imaging data such as, fractional anisotropy and axonal fiber orientation from Diffusion Tensor Imaging (DTI) of 12 healthy patients into the finite element brain model was performed to improve the brain constitutive material law with more efficient heterogeneous anisotropic visco hyper-elastic material law. The brain behavior has been validated in terms of brain deformation against Hardy et al. (2001), Hardy et al. (2007), and in terms of brain pressure against Nahum et al. (1977) and Trosseille et al. (1992) experiments. Verification of model stability has been conducted as well. Further, 109 well-documented TBI cases were simulated and axonal strain computed to derive brain injury tolerance curve. Based on an in-depth statistical analysis of different intra-cerebral parameters (brain axonal strain rate, axonal strain, first principal strain, Von Mises strain, first principal stress, Von Mises stress, CSDM (0.10), CSDM (0.15) and CSDM (0.25)), it was shown that axonal strain was the most appropriate candidate parameter to predict DAI. The proposed brain injury tolerance limit for a 50% risk of DAI has been established at 14.65% of axonal strain. This study provides a key step for a realistic novel injury metric for DAI.

Influence of stiffness and shape of contact surface on skull fractures and biomechanical metrics of the human head of different population underlateral impacts

The objective of this study was to determine the responses of 5th-percentile female, and 50th- and 95th-percentile male human heads during lateral impacts at different velocities and determine the role of the stiffness and shape of the impacting surface on peak forces and derived skull fracture metrics. A state-of-the-art validated finite element (FE) head model was used to study the influence of different population human heads on skull fracture for lateral impacts. The mass of the FE head model was altered to match the adult size dummies. Numerical simulations of lateral head impacts for 45 cases (15 experiments×3 different population human heads) were performed at velocities ranging from 2.4 to 6.5m/s and three impacting conditions (flat and cylindrical 90D; and flat 40D padding). The entire force-time signals from simulations were compared with experimental mean and upper/lower corridors at each velocity, stiffness (40 and 90 durometer) and shapes (flat and cylindrical) of the impacting surfaces. Average deviation of peak force from the 50th male to 95th male and 5th female were 6.4% and 10.6% considering impacts on the three impactors. These results indicate hierarchy of variables which can be used in injury mitigation efforts.

Measurements of mechanical anisotropy in brain tissue and implications for transversely isotropic material models of white matter

White matter in the brain is structurally anisotropic, consisting largely of bundles of aligned, myelin-sheathed axonal fibers. White matter is believed to be mechanically anisotropic as well. Specifically, transverse isotropy is expected locally, with the plane of isotropy normal to the local mean fiber direction. Suitable material models involve strain energy density functions that depend on the I4 and I5 pseudo-invariants of the Cauchy-Green strain tensor to account for the effects of relatively stiff fibers. The pseudo-invariant I4 is the square of the stretch ratio in the fiber direction; I5 contains contributions of shear strain in planes parallel to the fiber axis. Most, if not all, published models of white matter depend on I4 but not on I5. Here, we explore the small strain limits of these models in the context of experimental measurements that probe these dependencies. Models in which strain energy depends on I4 but not I5 can capture differences in Young's (tensile) moduli, but will not exhibit differences in shear moduli for loading parallel and normal to the mean direction of axons. We show experimentally, using a combination of shear and asymmetric indentation tests, that white matter does exhibit such differences in both tensile and shear moduli. Indentation tests were interpreted through inverse fitting of finite element models in the limit of small strains. Results highlight that: (1) hyperelastic models of transversely isotropic tissues such as white matter should include contributions of both the I4 and I5 strain pseudo-invariants; and (2) behavior in the small strain regime can usefully guide the choice and initial parameterization of more general material models of white matter.