Anisotropic constitutive equations and experimental tensile behavior of brain tissue

Do human brain white matter and brain stem structures show direction-dependent mechanical behavior?

Since the corpus callosum and the brain stem are both vulnerable to diffuse axonal injury during head impacts, there is a high interest in modeling the mechanical behavior of these brain structures. In recent years, different versions of fiber-reinforced material models have been proposed for the corpus callosum and other white matter regions, as well as for the brain stem, even though there is currently no consensus on whether those structures exhibit a significant direction-dependent behavior during mechanical loading. Here, we present the first large-strain, multimodal experimental study on human brain tissue that includes the corpus callosum and the lower brain stem (medulla oblongata) tested along two different directions. Additionally, we compare those two structures with other white matter (corona radiata, cerebellar white matter) and brain stem structures (pons, midbrain) to highlight differences in their material response. Cyclic compression-tension and shear tests reveal statistically significant direction-dependent material behavior in the corpus callosum. Directional differences in the brain stem are not statistically significant and do not indicate a clear directionality. Combined with histological findings, our results suggest that the mechanical behavior of white matter structures is influenced not only by axon caliber, orientation and density, but also by the architectural organization, i.e., clustering versus even distribution, of cells and tracts, and possibly vascular density. These findings highlight the need for micromechanical constitutive models for brain white matter that do not merely include axons embedded in a matrix. Statement of significance Mechanical head injuries often result in insults like diffuse axonal injury in white matter regions of the human brain. Therefore, there is a high interest in understanding and predicting the mechanical properties of those regions in order to prevent injury and advance diagnosis and treatment strategies of neurological disorders. There has been a long controversy regarding the question whether human brain white matter regions show an anisotropic, direction-dependent mechanical response. With the goal of providing experimental evidence to conclusively answer this question, we here present large-strain, multimodal experimental data and representative histological analyses of different human brain white matter regions and brain stem structures, including directional investigations for both the corpus callosum and the medulla oblongata (lower brain stem).

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.

Mechanical properties of fresh rhesus monkey brain tissue

Studying brain tissue mechanics is critical for understanding how the brain's physical properties influence its biological functions. Non-human primates, such as rhesus monkeys, are a key translational model for human neuroscience research, yet their brain tissue mechanics remain poorly understood. We report the mechanical properties of rhesus monkey white (corona radiata, CR) and gray (basal ganglia, BG) matter during compression relaxation, tension relaxation, tension-compression cycling (strain = 0.15, nCR = 21, nBG = 14), and shear cycling (strain = 0.3, nCR = 17, nBG = 9). Compression relaxation yields short and long-term time constants of 1.13 ± 0.041 s and 26.3 ± 0.68 s for CR and 1.22 ± 0.046 s and 28.3 ± 0.70 s for BG. Tension relaxation yields short and long-term time constants of 1.10 ± 0.052 s and 28.2 ± 0.82 s for CR and 1.19 ± 0.052 s and 29 ± 1.3 s for BG. Tension-compression cycling yields elastic moduli (E₁, E₂, E₃) of 36 ± 3.8 kPa, 0.61 ± 0.096 kPa, and 9.3 ± 0.90 kPa for CR and 27 ± 4.8 kPa, 0.68 ± 0.092 kPa, and 8 ± 1.0 kPa for BG. Shear cycling yields E₁, E₂, and E₃ of 3.9 ± 0.77 kPa, 0.19 ± 0.034 kPa, and 3.1 ± 0.40 kPa for CR and 2.8 ± 0.52 kPa, 0.18 ± 0.058 kPa, and 3.2 ± 0.53 kPa for BG. Hysteresis areas are also captured during tension-compression and shear cycling. These findings extend the translatability of rhesus monkey models for neuroscience. STATEMENT OF SIGNIFICANCE: While rhesus monkeys are a valuable translational model in human neuroscience research, there is a huge gap in knowledge about rhesus monkey brain tissue mechanics. This study serves to increase our understanding of rhesus monkey brain tissue mechanics and is the first to report the stiffness, time constant, and hysteresis parameters for rhesus monkey brain tissue in compression, tension, and shear for both the corona radiata and basal ganglia. The data is available in an open-source format, allowing others to fit and validate their mechanical models.

3D-printed suction clamps for tensile testing of brain tissue

The conventional mounting of ultra-soft biological tissues often involves gluing it between two plates or manually tightening grips. Both methods demand delicate handling skills and are time-consuming. This study outlines the design and practical application of 3D-printed suction clamps for uniaxial tension tests on brain samples. Successful testing was defined by the absence of relevant slippage or the sample being drawn into the clamp. A total of 112 deer brain samples underwent testing using a universal testing machine after one freeze-thaw cycle. These samples were obtained from eight different brain regions. During sample preparation, 7 out of all samples failed. Among the 105 tests, 89 (85%) were successful. Of the 16 unsuccessful tests, 15 samples (14%) slipped, while only one sample (1%) was drawn into the clamp to an extent that testing became impossible. Medulla oblongata samples exhibited exceptionally high slippage at 38%, whereas samples from the temporal cortex, external capsule, and putamen had the lowest slippage in only one single case. In conclusion, suction clamps facilitate high-throughput testing through user-friendly and rapid sample mounting. Testing success is contingent on the specific brain site, with sample slippage being the primary reason for testing failures, while sample inspiration into the clamp is negligible.

The White Matter Fiber Tract Deforms Most in the Perpendicular Direction During In Vivo Volunteer Impacts

White matter (WM) tract-related strains are increasingly used to quantify brain mechanical responses, but their dynamics in live human brains during in vivo impact conditions remain largely unknown. Existing research primarily looked into the normal strain along the WM fiber tracts (i.e., tract-oriented normal strain), but it is rarely the case that the fiber tract only endures tract-oriented normal strain during impacts. In this study, we aim to extend the in vivo measurement of WM fiber deformation by quantifying the normal strain perpendicular to the fiber tract (i.e., tract-perpendicular normal strain) and the shear strain along and perpendicular to the fiber tract (i.e., tract-oriented shear strain and tract-perpendicular shear strain, respectively). To achieve this, we combine the three-dimensional strain tensor from the tagged magnetic resonance imaging with the diffusion tensor imaging (DTI) from an open-access dataset, including 44 volunteer impacts under two head loading modes, i.e., neck rotations (N = 30) and neck extensions (N = 14). The strain tensor is rotated to the coordinate system with one axis aligned with DTI-revealed fiber orientation, and then four tract-related strain measures are calculated. The results show that tract-perpendicular normal strain peaks are the largest among the four strain types (p < 0.05, Friedman's test). The distribution of tract-related strains is affected by the head loading mode, of which laterally symmetric patterns with respect to the midsagittal plane are noted under neck extensions, but not under neck rotations. Our study presents a comprehensive in vivo strain quantification toward a multifaceted understanding of WM dynamics. We find that the WM fiber tract deforms most in the perpendicular direction, illuminating new fundamentals of brain mechanics. The reported strain images can be used to evaluate the fidelity of computational head models, especially those intended to predict fiber deformation under noninjurious conditions.

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.

Machine learning reveals correlations between brain age and mechanics

Our brain undergoes significant micro- and macroscopic changes throughout its life cycle. It is therefore crucial to understand the effect of aging on the mechanical properties of the brain in order to develop accurate personalized simulations and diagnostic tools. Here we systematically probed the mechanical behavior of n=439 brain tissue samples in tension and compression, in different anatomical regions, for different axon orientations, across five age groups. We used Bayesian statistics to characterize the relation between brain age and mechanical properties and quantify uncertainties. Our results, based on our experimental data and material parameters for the isotropic Ogden and the anisotropic Gasser-Ogden-Holzapfel models, reveal a non-linear relationship between age and mechanics across the life cycle of the porcine brain. Both tensile and compressive shear moduli reached peak values ranging from 0.4-1.0 kPa in tension to 0.16-0.32 kPa in compression at three years of age. Anisotropy was most pronounced at six months, and then decreased. These results represent an important step in understanding age-dependent changes in the mechanical properties of brain tissue and provide the scientific basis for more accurate and realistic computational brain simulations. STATEMENT OF SIGNIFICANCE: In this paper, we investigate the age-dependent mechanical properties of brain tissue based on different deformation modes, anatomical regions, and axon orientations. Hierarchical Bayesian modeling was used to identify isotropic and anisotropic material parameters. The study reveals a nonlinear relationship between shear modulus, degree of anisotropy, and tension-compression asymmetry over the life cycle of the brain. By demonstrating the non-linearity of these relationships, the study fills a significant knowledge gap in current research. This work is a fundamental step in accurately characterizing the complex relationship between brain aging and mechanical properties.

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.

Embedded finite element modeling of the mechanics of brain axonal fiber tracts under head impact conditions

Investigating and understanding the biomechanical kinematics and kinetics of human brain axonal fibers during head impact process is crucial to study the mechanisms of Traumatic Axonal Injury (TAI). Such a study may require the explicit incorporation of brain fiber tracts into the host brain in order to distinguish the mechanical states of axonal fibers and brain tissue. Herein we extend our previously developed human head model by using an embedded element method to include fiber tracts reconstructed from diffusion tensor images in a host brain with the purpose of numerically tracking the deformation state of axonal fiber tracts during a head impact simulation. The updated model is validated by comparing its prediction of intracranial pressures with experimental data, followed by a thorough study of the effects of element types used for fiber tracts and the stiffness ratios of fiber to host brain. The validated model is also used to predict and visualize the damaged region of fiber tracts during the head impact process based on different injury criteria. The model is promising in tracking the state of fiber tracts and can add more objective functions such as axonal fiber deformation if used in the future design optimization of head protective equipment such as a football helmet.

Region specific anisotropy and rate dependence of Göttingen minipig brain tissue

Traumatic brain injury is a major cause of morbidity in civilian as well as military populations. Computational simulations of injurious events are an important tool to understanding the biomechanics of brain injury and evaluating injury criteria and safety measures. However, these computational models are highly dependent on the material parameters used to represent the brain tissue. Reported material properties of tissue from the cerebrum and cerebellum remain poorly defined at high rates and with respect to anisotropy. In this work, brain tissue from the cerebrum and cerebellum of male Göttingen minipigs was tested in one of three directions relative to axon fibers in oscillatory simple shear over a large range of strain rates from 0.025 to 250 s-1. Brain tissue showed significant direction dependence in both regions, each with a single preferred loading direction. The tissue also showed strong rate dependence over the full range of rates considered. Transversely isotropic hyper-viscoelastic constitutive models were fit to experimental data using dynamic inverse finite element models to account for wave propagation observed at high strain rates. The fit constitutive models predicted the response in all directions well at rates below 100 s-1, after which they adequately predicted the initial two loading cycles, with the exception of the 250 s-1 rate, where models performed poorly. These constitutive models can be readily implemented in finite element packages and are suitable for simulation of both conventional and blast injury in porcine, especially Göttingen minipig, models.

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.

A comprehensive review on modeling aspects of infusion-based drug delivery in the brain

Brain disorders represent an ever-increasing health challenge worldwide. While conventional drug therapies are less effective due to the presence of the blood-brain barrier, infusion-based methods of drug delivery to the brain represent a promising option. Since these methods are mechanically controlled and involve multiple physical phases ranging from the neural and molecular scales to the brain scale, highly efficient and precise delivery procedures can significantly benefit from a comprehensive understanding of drug-brain and device-brain interactions. Behind these interactions are principles of biophysics and biomechanics that can be described and captured using mathematical models. Although biomechanics and biophysics have received considerable attention, a comprehensive mechanistic model for modeling infusion-based drug delivery in the brain has yet to be developed. Therefore, this article reviews the state-of-the-art mechanistic studies that can support the development of next-generation models for infusion-based brain drug delivery from the perspective of fluid mechanics, solid mechanics, and mathematical modeling. The supporting techniques and database are also summarized to provide further insights. Finally, the challenges are highlighted and perspectives on future research directions are provided. STATEMENT OF SIGNIFICANCE: Despite the immense potential of infusion-based drug delivery methods for bypassing the blood-brain barrier and efficiently delivering drugs to the brain, achieving optimal drug distribution remains a significant challenge. This is primarily due to our limited understanding of the complex interactions between drugs and the brain that are governed by principles of biophysics and biomechanics, and can be described using mathematical models. This article provides a comprehensive review of state-of-the-art mechanistic studies that can help to unravel the mechanism of drug transport in the brain across the scales, which underpins the development of next-generation models for infusion-based brain drug delivery. More broadly, this review will serve as a starting point for developing more effective treatments for brain diseases and mechanistic models that can be used to study other soft tissue and biomaterials.

Frequency and time dependent viscoelastic characterization of pediatric porcine brain tissue in compression

Understanding the viscoelastic behavior of pediatric brain tissue is critical to interpret how external mechanical forces affect head injury in children. However, knowledge of the viscoelastic properties of pediatric brain tissue is limited, and this reduces the biofidelity of developed numeric simulations of the pediatric head in analysis of brain injury. Thus, it is essential to characterize the viscoelastic behavior of pediatric brain tissue in various loading conditions and to identify constitutive models. In this study, the pediatric porcine brain tissue was investigated in compression with determine the viscoelasticity under small and large strain, respectively. A range of frequencies between 0.1 and 40 Hz was applied to determine frequency-dependent viscoelastic behavior via dynamic mechanical analysis, while brain samples were divided into three strain rate groups of 0.01/s, 1/s and 10/s for compression up to 0.3 strain level and stress relaxation to obtain time-dependent viscoelastic properties. At frequencies above 20 Hz, the storage modulus did not increase, while the loss modulus increased continuously. With strain rate increasing from 0.01/s to 10/s, the mean stress at 0.1, 0.2 and 0.3 strain increased to approximate 6.8, 5.6 and 4.4 times, respectively. The brain compressive response was sensitive to strain rate and frequency. The characterization of brain tissue will be valuable for development of head protection systems and prediction of brain injury.

Region-Dependent Mechanical Properties of Human Brain Tissue Under Large Deformations Using Inverse Finite Element Modeling

This study aims to facilitate intracranial simulation of traumatic events by determining the mechanical properties of different anatomical structures of the brain. Our experimental indentation paradigm used fresh, post-operative human tissue, which is highly advantageous in determining mechanical properties without being affected by postmortem time. This study employed an inverse finite element approach coupled with experimental indentation data to characterize mechanical properties of the human hippocampus (CA1, CA3, dentate gyrus), cortex white matter, and cortex grey matter. We determined that an uncoupled viscoelastic Ogden constitutive formulation was most appropriate to represent the mechanical behavior of these different regions of brain. Anatomical regions were significantly different in their mechanical properties. The cortex white matter was stiffer than cortex grey matter, and the CA1 and dentate gyrus were both stiffer than cortex grey matter. Although no sex dependency was observed, there were trends indicating that male brain regions were generally stiffer than corresponding female regions. In addition, there were no statistically significant age dependent differences. This study provides a structure-specific description of fresh human brain tissue mechanical properties, which will be an important step toward explicitly modeling the heterogeneity of brain tissue deformation during TBI through finite element modeling.

Simultaneous High-Frame-Rate Acoustic Plane-Wave and Optical Imaging of Intracranial Cavitation in Polyacrylamide Brain Phantoms during Blunt Force Impact

Blunt and blast impacts occur in civilian and military personnel, resulting in traumatic brain injuries necessitating a complete understanding of damage mechanisms and protective equipment design. However, the inability to monitor in vivo brain deformation and potential harmful cavitation events during collisions limits the investigation of injury mechanisms. To study the cavitation potential, we developed a full-scale human head phantom with features that allow a direct optical and acoustic observation at high frame rates during blunt impacts. The phantom consists of a transparent polyacrylamide material sealed with fluid in a 3D-printed skull where windows are integrated for data acquisition. The model has similar mechanical properties to brain tissue and includes simplified yet key anatomical features. Optical imaging indicated reproducible cavitation events above a threshold impact energy and localized cavitation to the fluid of the central sulcus, which appeared as high-intensity regions in acoustic images. An acoustic spectral analysis detected cavitation as harmonic and broadband signals that were mapped onto a reconstructed acoustic frame. Small bubbles trapped during phantom fabrication resulted in cavitation artifacts, which remain the largest challenge of the study. Ultimately, acoustic imaging demonstrated the potential to be a stand-alone tool, allowing observations at depth, where optical techniques are limited.

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.

A Bayesian method with nonlinear noise model to calibrate constitutive parameters of soft tissue

The measured mechanical responses of soft tissue exhibit large variability and errors, especially for the softest brain tissue, while calibrating its constitutive parameters in a deterministic way remains a common practice. Here we implement a Bayesian method considering the nonlinear noise model to calibrate constitutive parameters of brain tissue. A probability model is first developed based on the measured experimental data, likelihood function, and prior function, from which the posterior distributions of model parameters are formulated. The likelihood function considers the nonlinear behaviors of the constitutive response and noise distribution of the experimentally measured data. Meanwhile, the prior predictive distribution is computed to check the probability model preliminarily. Secondly, the Markov Chain Monte Carlo (MCMC) method is used to compute the posterior distributions of model parameters, enabling assessment of parameter uncertainty, correlation, and model calibration error. Finally, the posterior predictive distributions of the overall response, constitutive response, and noise response are computed to validate the probabilistic model, all of which are consistent with the corresponding data. Furthermore, the effect of the prior distribution, experimental data, and noise model on posterior distribution is studied. Our study provides a general approach to calibrating constitutive parameters of soft tissue despite errors and large variability in experimental data.

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.

Mechanisms of mechanical load transfer through brain tissue

Brain injuries are often characterized by diffusely distributed axonal and vascular damage invisible to medical imaging techniques. The spatial distribution of mechanical stresses and strains plays an important role, but is not sufficient to explain the diffuse distribution of brain lesions. It remains unclear how forces are transferred from the organ to the cell scale and why some cells are damaged while neighboring cells remain unaffected. To address this knowledge gap, we subjected histologically stained fresh human and porcine brain tissue specimens to compressive loading and simultaneously tracked cell and blood vessel displacements. Our experiments reveal different mechanisms of load transfer from the organ or tissue scale to single cells, axons, and blood vessels. Our results show that cell displacement fields are inhomogeneous at the interface between gray and white matter and in the vicinity of blood vessels-locally inducing significant deformations of individual cells. These insights have important implications to better understand injury mechanisms and highlight the importance of blood vessels for the local deformation of the brain's cellular structure during loading.

A position- and time-dependent pressure profile to model viscoelastic mechanical behavior of the brain tissue due to tumor growth

This study proposed a computational framework to calculate the resultant position- and time-dependent pressure profile on the brain tissue due to tumor growth. A finite element (FE) patch of the brain tissue was constructed and an inverse dynamic FE-optimization algorithm was used to calculate its viscoelastic mechanical properties under compressive uniaxial loading. Two patient-specific post-tumor resection FE models were input to the FE-optimization algorithm to calculate the optimized 3rd-order position-dependent and normal distribution time-dependent pressure profile parameters. The optimized viscoelastic material properties, the most suitable simulation time, and the optimized 3rd-order position- and -time-dependent pressure profiles were calculated.

Development, validation and a case study: The female finite element head model (FeFEHM)

BACKGROUND AND OBJECTIVE

Traumatic brain injuries are one of the leading causes of death and disability in the world. To better understand the interactions and forces applied in different constituents of the human head, several finite element head models have been developed throughout the years, for offering a good cost-effective and ethical approach compared to experimental tests. Once validated, the female finite element head model (FeFEHM) will allow a better understanding of injury mechanisms resulting in neuronal damage, which can later evolve into neurodegenerative diseases.

METHODS

This work encompasses the approached methodology starting from medical images and finite element modelling until the validation process using novel experimental data of brain displacements conducted on human cadavers. The material modelling of the brain is performed using an age-specific characterization of the brain using microindentation at dynamic rates and under large deformation, with a similar age to the patient used to model the FeFEHM.

RESULTS

The numerical displacement curves are in good accordance with the experimental data, displaying similar peak times and values, in all three anatomical planes. The case study result shows a similarity between the pressure fields of the FeFEHM compared to another model, highlighting the future potential of the model.

CONCLUSIONS

The initial objective was met, and a new female finite element head model has been developed with biofidelic brain motion. This model will be used for the assessment of repetitive impact scenarios and its repercussions on the female brain.

A transversely isotropic viscohyperelastic-damage model for the brain tissue with strain rate sensitivity

Understanding the mechanical behaviors and properties of brain tissue are crucial to study the mechanisms of traumatic brain injury (TBI). Such injury may be associated with high rate loading conditions and the large deformation of brain tissue. Thus, constitutive models that consider the rate dependent large deformation of brain tissue and its possible damage initiation and evolution may help uncover the related mechanisms of TBI. Motivated from this, in this paper we present a fully three-dimensional large strain viscohyperelastic-damage model with the purpose of reproducing the experimentally observed rate sensitive elastic and damage-induced stress softening behaviors of brain tissue. The parameters of the proposed model can be identified using the experimental data from simple monotonic tests such as uniaxial tension, compression and simple shear. The proposed model is validated by comparing its prediction with experimental data. Good agreement between predictive results and experimental data is achieved indicating the potential of the proposed model in characterizing the mechanical behaviors of brain tissue considering rate dependence and damage effect.

Modeling the damage-induced softening behavior of brain white matter using a coupled hyperelasticty-damage model

White matter in the brain is structurally anisotropic consisting of large bundle of aligned axonal fibers. Hyperelastic, transversely isotropic constitutive models are typically used in the modeling and simulation of such tissues. However, most studies constrain the material models to describe the mechanical behavior of white matter in the limit of small deformation, without considering the experimentally observed damage initiation and damage-induced material softening in large strain regime. In this study, we extend a previously developed transversely isotropic hyperelasticity model for white matter by coupling it with damage equations within the framework of thermodynamics and using continuum damage mechanics method. Two homogeneous deformation cases are used to demonstrate the proposed model's capability in capturing the damage-induced softening behaviors of white matter under uniaxial loading and simple shear, along with the investigation of fiber orientation effect on such behaviors and material stiffness. As a demonstration case of inhomogeneous deformation, the proposed model is also implemented into finite element codes to reproduce the experimental data (nonlinear material behavior and damage initiation) from an indentation configuration of porcine white matter. Good agreement between numerical results and experimental data is achieved indicating the potential of the proposed model in characterizing the mechanical behaviors of white matter considering damage at large strain.

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.

An introduction to the Ogden model in biomechanics: benefits, implementation tools and limitations

Constitutive models are important to biomechanics for two key reasons. First, constitutive modelling is an essential component of characterizing tissues' mechanical properties for informing theoretical and computational models of biomechanical systems. Second, constitutive models can be used as a theoretical framework for extracting and comparing key quantities of interest from material characterization experiments. Over the past five decades, the Ogden model has emerged as a popular constitutive model in soft tissue biomechanics with relevance to both informing theoretical and computational models and to comparing material characterization experiments. The goal of this short review is threefold. First, we will discuss the broad relevance of the Ogden model to soft tissue biomechanics and the general characteristics of soft tissues that are suitable for approximating with the Ogden model. Second, we will highlight exemplary uses of the Ogden model in brain tissue, blood clot and other tissues. Finally, we offer a tutorial on fitting the one-term Ogden model to pure shear experimental data via both an analytical approximation of homogeneous deformation and a finite-element model of the tissue domain. Overall, we anticipate that this short review will serve as a practical introduction to the use of the Ogden model in biomechanics. This article is part of the theme issue 'The Ogden model of rubber mechanics: Fifty years of impact on nonlinear elasticity'.

An introduction to the Ogden model in biomechanics: benefits, implementation tools and limitations

Constitutive models are important to biomechanics for two key reasons. First, constitutive modelling is an essential component of characterizing tissues' mechanical properties for informing theoretical and computational models of biomechanical systems. Second, constitutive models can be used as a theoretical framework for extracting and comparing key quantities of interest from material characterization experiments. Over the past five decades, the Ogden model has emerged as a popular constitutive model in soft tissue biomechanics with relevance to both informing theoretical and computational models and to comparing material characterization experiments. The goal of this short review is threefold. First, we will discuss the broad relevance of the Ogden model to soft tissue biomechanics and the general characteristics of soft tissues that are suitable for approximating with the Ogden model. Second, we will highlight exemplary uses of the Ogden model in brain tissue, blood clot and other tissues. Finally, we offer a tutorial on fitting the one-term Ogden model to pure shear experimental data via both an analytical approximation of homogeneous deformation and a finite-element model of the tissue domain. Overall, we anticipate that this short review will serve as a practical introduction to the use of the Ogden model in biomechanics. This article is part of the theme issue 'The Ogden model of rubber mechanics: Fifty years of impact on nonlinear elasticity'.

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.

Quantifying stability of parameter estimates forin vivonearly incompressible transversely-isotropic brain MR elastography

Easily computable quality metrics for measured medical data at point-of-care are important for imaging technologies involving offline reconstruction. Accordingly, we developed a new data quality metric forin vivotransversely-isotropic (TI) magnetic resonance elastography (MRE) based on a generalization of the widely accepted octahedral shear-strain calculation. The metric uses MRE displacement data and an estimate of the TI property field to yield a 'stability map' which predicts regions of low versus high accuracy in the resulting material property reconstructions. We can also calculate an average TI parameter stability (TIPS) score over all voxels in a region of interest for a given measurement to indicate how reliable the recovered mechanical property estimate for the region is expected to be. The calculation is rapid and places little demand on computing resources compared to the computationally intensive material property reconstruction from non-linear inversion (TI-NLI) of displacement fields, making it ideal for point-of-care evaluation of data quality. We test the predictions of the stability map for both simulated phantoms andin vivohuman brain data. We used a range of different displacement datasets from vibrations applied in the anterior-posterior (AP), left-right (LR) and combined AP + LR directions. The TIPS and variability maps (noise sensitivity or variation from the mean of repeated MRE scans) were consistently anti-correlated. Notably, Spearman correlation coefficients ∣R∣>0.6 were found between variability and TIPS score for individual white matter tracts within vivodata. These observations demonstrate the reliability and promise of this data quality metric to screen data rapidly in realistic clinical MRE applications.

Tissue-Scale Biomechanical Testing of Brain Tissue for the Calibration of Nonlinear Material Models

Brain tissue is one of the most complex and softest tissues in the human body. Due to its ultrasoft and biphasic nature, it is difficult to control the deformation state during biomechanical testing and to quantify the highly nonlinear, time-dependent tissue response. In numerous experimental studies that have investigated the mechanical properties of brain tissue over the last decades, stiffness values have varied significantly. One reason for the observed discrepancies is the lack of standardized testing protocols and corresponding data analyses. The tissue properties have been tested on different length and time scales depending on the testing technique, and the corresponding data have been analyzed based on simplifying assumptions. In this review, we highlight the advantage of using nonlinear continuum mechanics based modeling and finite element simulations to carefully design experimental setups and protocols as well as to comprehensively analyze the corresponding experimental data. We review testing techniques and protocols that have been used to calibrate material model parameters and discuss artifacts that might falsify the measured properties. The aim of this work is to provide standardized procedures to reliably quantify the mechanical properties of brain tissue and to more accurately calibrate appropriate constitutive models for computational simulations of brain development, injury and disease. Computational models can not only be used to predictively understand brain tissue behavior, but can also serve as valuable tools to assist diagnosis and treatment of diseases or to plan neurosurgical procedures. © 2022 The Authors. Current Protocols published by Wiley Periodicals LLC.

Inverse dynamic finite element-optimization modeling of the brain tumor mass-effect using a variable pressure boundary

BACKGROUND AND OBJECTIVE

Statistical atlases of brain structure can potentially contribute in the surgical and radiotherapeutic treatment planning for the brain tumor patients. However, the current brain image-registration methods lack of accuracy when it comes to the mass-effect caused by tumor growth. Numerical simulations, such as finite element method (FEM), allow us to calculate the resultant pressure and deformation in the brain tissue due to tumor growth, and to predict the mass-effect. To date, however, the pressure boundary in the brain tissue due to tumor growth has been simply presented as a constant profile throughout the entire tumor outer surface that resulted in discrepancy between the patient imaging data and brain atlases.

METHODS

In this study, we employed a fully-coupled inverse dynamic FE-optimization method to estimate the resultant variable pressure boundary due to tumor resection surgery. To do that, magnetic resonance imaging data of two patients' pre- and post-tumor resection surgery were registered, segmented, volume-meshed, and prepared for fully-coupled inverse dynamic FE-optimization simulations. Two different pressure boundaries were defined on the brain cavity after tumor resection including: a) a constant pressure boundary and b) a variable pressure boundary. The inverse FE-optimization algorithm was used to find the optimum constant and variable pressure boundaries that result in the least distance between the surface-nodes of the post-surgery brain cavity and pre-surgery tumor.

RESULTS

The results revealed that a variable pressure boundary causes a considerably lower mean percentage error compared to a constant pressure one; hence, it can more effectively address the realistic boundary in tumor resection surgery and predict the mass-effect.

CONCLUSIONS

The proposed variable pressure boundary can be a robust tool that allows batch processing to register the brains with tumors to statistical atlases of normal brains and construction of brain tumor atlases. This approach is also computationally inexpensive and can be coupled to any FE software to run. The findings of this study have implications for not only predicting the accurate pressure boundary and mass-effect before tumor resection surgery, but also for predicting some clinical symptoms of brain cancers and presenting useful tools for APPLICATIONs in image-guided neurosurgery.

Mechanical properties of whole-body soft human tissues: a review

The mechanical properties of soft tissues play a key role in studying human injuries and their mitigation strategies. While such properties are indispensable for computational modelling of biological systems, they serve as important references in loading and failure experiments, and also for the development of tissue simulants. To date, experimental studies have measured the mechanical properties of peripheral tissues (e.g. skin)in-vivoand limited internal tissuesex-vivoin cadavers (e.g. brain and the heart). The lack of knowledge on a majority of human tissues inhibit their study for applications ranging from surgical planning, ballistic testing, implantable medical device development, and the assessment of traumatic injuries. The purpose of this work is to overcome such challenges through an extensive review of the literature reporting the mechanical properties of whole-body soft tissues from head to toe. Specifically, the available linear mechanical properties of all human tissues were compiled. Non-linear biomechanical models were also introduced, and the soft human tissues characterized using such models were summarized. The literature gaps identified from this work will help future biomechanical studies on soft human tissue characterization and the development of accurate medical models for the study and mitigation of injuries.

Investigation of the Compressive Viscoelastic Properties of Brain Tissue Under Time and Frequency Dependent Loading Conditions

The mechanical characterization of brain tissue has been generally analyzed in the frequency and time domain. It is crucial to understand the mechanics of the brain under realistic, dynamic conditions and convert it to enable mathematical modelling in a time domain. In this study, the compressive viscoelastic properties of brain tissue were investigated under time and frequency domains with the same physical conditions and the theory of viscoelasticity was applied to estimate the prediction of viscoelastic response in the time domain based on frequency-dependent mechanical moduli through Finite Element models. Storage and loss modulus were obtained from white and grey matter, of bovine brains, using dynamic mechanical analysis and time domain material functions were derived based on a Prony series representation. The material models were evaluated using brain testing data from stress relaxation and hysteresis in the time dependent analysis. The Finite Element models were able to represent the trend of viscoelastic characterization of brain tissue under both testing domains. The outcomes of this study contribute to a better understanding of brain tissue mechanical behaviour and demonstrate the feasibility of deriving time-domain viscoelastic parameters from frequency-dependent compressive data for biological tissue, as validated by comparing experimental tests with computational simulations.

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.

Full-field MRI measurements of in-vivo positional brain shift reveal the significance of intra-cranial geometry and head orientation for stereotactic surgery

Positional brain shift (PBS), the sagging of the brain under the effect of gravity, is comparable in magnitude to the margin of error for the success of stereotactic interventions ([Formula: see text] 1 mm). This non-uniform shift due to slight differences in head orientation can lead to a significant discrepancy between the planned and the actual location of surgical targets. Accurate in-vivo measurements of this complex deformation are critical for the design and validation of an appropriate compensation to integrate into neuronavigational systems. PBS arising from prone-to-supine change of head orientation was measured with magnetic resonance imaging on 11 young adults. The full-field displacement was extracted on a voxel-basis via digital volume correlation and analysed in a standard reference space. Results showed the need for target-specific correction of surgical targets, as a significant displacement ranging from 0.52 to 0.77 mm was measured at surgically relevant structures. Strain analysis further revealed local variability in compressibility: anterior regions showed expansion (both volume and shape change), whereas posterior regions showed small compression, mostly dominated by shape change. Finally, analysis of correlation demonstrated the potential for further patient- and intervention-specific adjustments, as intra-cranial breadth and head tilt correlated with PBS reaching statistical significance.

Study on the Effect of Sample Temperature on the Uniaxial Compressive Mechanical Properties of the Brain Tissue

Craniocerebral injury has been a research focus in the field of injury biomechanics. Although experimental endeavors have made certain progress in characterizing the material behavior of the brain, the temperature dependency of brain mechanics appears to be inconclusive thus far. To partially address this knowledge gap, the current study measured the brain material behavior via unconstrained uniaxial compression tests under low strain rate (0.0083 s^-1) and high strain rate (0.83 s^-1) at four different sample temperatures (13°C, 20°C, 27°C, and 37°C). Each group has 9~12 samples. One-way analysis of variance method was used to study the influence of sample temperature on engineering stress. The results show that the effect of sample temperature on the mechanical properties of brain tissue is significant under the high strain rate, especially at low temperature (13°C), in which the hardening of the brain tissue is very obvious. At the low strain rate, no temperature dependency of brain mechanics is noted. Therefore, the current results highlight that the temperature of the brain sample should be ensured to be in accordance with the living subject when studying the biomechanical response of living tissue.

The importance of axonal directions in the brainstem injury during neurosurgical interventions

Brainstem, which connects the distal part of the brain and the spinal cord, contains main motor and sensory nerves and facilitates communication between the cerebrum, cerebellum, and spinal cord. Due to the complicated anatomy and neurostructure of brainstem, surgical interventions to resect brainstem tumors are particularly challenging, and new approaches to reduce the risk of surgical brain injury are of utmost importance. Although previous studies have investigated the structural anisotropy of brain white matter, the effect of axonal fibers on the mechanical properties of white matter has not yet been fully understood. The current study aims to compare the effect of axonal orientation on changes in material properties of brainstem under large deformations and failure through a novel approach. Using diffusion tensor imaging (DTI) on ex-vivo bovine brains, we determined the orientation of axons in brainstem. We extracted brainstem samples in two orthogonal directions, parallel and perpendicular to the axons, and subjected to uniaxial tension to reach the failure at loading rates of 50 mm/min and 150 mm/min. The results showed that the tearing energy and failure strain of samples with axons parallel to the force direction were approximately 1.5 times higher than the samples with axons perpendicular to the force direction. The results also revealed that as the sample's initial length increases, its failure strain decreases. These results emphasize the importance of the axon orientation in the mechanical properties of brainstem, and suggest that considering the directional-dependent behavior for this tissue could help to propose new surgical interventions for reducing the risk of injury during tumor resection.

Effect of axonal fiber architecture on mechanical heterogeneity of the white matter-a statistical micromechanical model

A diffusion tensor imaging (DTI) -based statistical micromechanical model was developed to study the effect of axonal fiber architecture on the inter- and intra-regional mechanical heterogeneity of the white matter. Three characteristic regions within the white matter, i.e., corpus callosum, brain stem, and corona radiata, were studied considering the previous observations of locations of diffuse axonal injury. The embedded element technique was used to create a fiber-reinforced model, where the fiber was characterized by a Holzapfel hyperelastic material model with variable dispersion of axonal orientations. A relationship between the fractional anisotropy and the dispersion parameter of the hyperelastic model was used to introduce the statistical DTI data into the representative volume element. The FA-informed statistical micromechanical models of three characteristic regions of white matter were developed by deriving the corresponding probabilistic measures of FA variations. Comparison of the model predictions and experimental data indicated a good agreement, suggesting that the model could reasonably capture the inter-regional heterogeneity of white matter. Moreover, the standard deviations of experimental results correlated well with the model predictions, suggesting that the model could capture the intra-regional mechanical heterogeneity for different regions of white matter.

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.

Is There a Relationship between the Elasticity of Brain Tumors, Changes in Diffusion Tensor Imaging, and Histological Findings? A Pilot Study Using Intraoperative Ultrasound Elastography

Intraoperative ultrasound elastography (IOUS-E) is a novel image modality applied in brain tumor assessment. However, the potential links between elastographic findings and other histological and neuroimaging features are unknown. This study aims to find associations between brain tumor elasticity, diffusion tensor imaging (DTI) metrics, and cell proliferation. A retrospective study was conducted to analyze consecutively admitted patients who underwent craniotomy for supratentorial brain tumors between March 2018 and February 2020. Patients evaluated by IOUS-E and preoperative DTI were included. A semi-quantitative analysis was performed to calculate the mean tissue elasticity (MTE). Diffusion coefficients and the tumor proliferation index by Ki-67 were registered. Relationships between the continuous variables were determined using the Spearman ρ test. A predictive model was developed based on non-linear regression using the MTE as the dependent variable. Forty patients were evaluated. The pathologic diagnoses were as follows: 21 high-grade gliomas (HGG); 9 low-grade gliomas (LGG); and 10 meningiomas. Cases with a proliferation index of less than 10% had significantly higher medians of MTE (110.34 vs. 79.99, p < 0.001) and fractional anisotropy (FA) (0.24 vs. 0.19, p = 0.020). We found a strong positive correlation between MTE and FA (r_s (38) = 0.91, p < 0.001). A cubic spline non-linear regression model was obtained to predict tumoral MTE from FA (R² = 0.78, p < 0.001). According to our results, tumor elasticity is associated with histopathological and DTI-derived metrics. These findings support the usefulness of IOUS-E as a complementary tool in brain tumor surgery.

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

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

The effect of large deformation on Poisson's ratio of brain white matter: An experimental study

A more Accurate description of the mechanical behavior of brain tissue could improve the results of computational models. While most studies have assumed brain tissue as an incompressible material with constant Poisson's ratio of almost 0.5 and constructed their modeling approach according to this assumption, the relationship between this ratio and levels of applied strains has not yet been studied. Since the mechanical response of the tissue is highly sensitive to the value of Poisson's ratio, this study was designed to investigate the characteristics of the Poisson's ratio of brain tissue at different levels of applied strains. Samples were extracted from bovine brain tissue and tested under unconfined compression at strain values of 5%, 10%, and 30%. Using an image processing method, the axial and transverse strains were measured over a 60-s period to calculate the Poisson's ratio for each sample. The results of this study showed that the Poisson's ratio of brain tissue at strain levels of 5% and 10% was close to 0.5, and assuming brain tissue as an incompressible material is a valid assumption at these levels of strain. For samples under 30% compression, this ratio was higher than 0.5, which could suggest that under strains higher than the brain injury threshold (approximately 18%), tissue integrity was impaired. Based on these observations, it could be concluded that for strain levels higher than the injury threshold, brain tissue could not be assumed as an incompressible material, and new material models need to be proposed to predict the material behavior of the tissue. In addition, the results showed that brain tissue under unconfined compression uniformly stretched in the transverse direction, and the bulging in the samples is negligible.

Hyper-viscoelastic mechanical behavior of cranial pia mater in tension

BACKGROUND

Cranial pia mater, the innermost layer of the meninges, protects the central nervous system by tightly wrapping the brain and damping the external impact force to the brain. Accurate experimental data of the mechanical property of the cranial pia mater can enhance the theoretical prediction of traumatic brain injury or the scientific surgery design for brain disease. The aim of this study is to characterize the mechanical behavior of the cranial pia mater.

METHODS

In vitro tensile and stress-relaxation experiments of ovine cranial pia mater specimens were conducted at eight strain rates to characterize the rate-dependent viscoelastic property. The tensile and stress-relaxation experimental data were fitted by an Ogden hyper-viscoelastic model with a strain rate function to describe the mechanical behavior of the cranial pia mater.

FINDINGS

The elastic modulus and the ultimate stress are significantly increased from 5.545 MPa and 0.535 MPa at 0.00167 s^-1 to 18.345 MPa and 2.547 MPa at 0.83 s^-1 (p < .0001), respectively. The initial stress and the long-term stress (300 s) are also increased significantly with the increasing strain rates (p < .0001). A good fit of the experimental data with the Ogden hyper-viscoelastic model incorporated with a strain rate function was achieved (R² > 0.93).

INTERPRETATION

The cranial pia mater exhibits as a rate-dependent hyper-viscoelastic material in the tensile and stress-relaxation experiments. Compared with the brain, the stiffer nature of the cranial pia mater indicates its essential role in brain protection. The rate-dependent constitutive model provides a proper description of the hyper-viscoelastic characteristics of the cranial pia mater in tension and may provide a basic constitutive relationship for numerical simulations of traumatic brain injury.

Dynamic mechanical characterization and viscoelastic modeling of bovine brain tissue

Brain tissue is vulnerable and sensitive, predisposed to potential damage under various conditions of mechanical loading. Although its material properties have been investigated extensively, the frequency-dependent viscoelastic characterization is currently limited. Computational models can provide a non-invasive method by which to analyze brain injuries and predict the mechanical response of the tissue. The brain injuries are expected to be induced by dynamic loading, mostly in compression and measurement of dynamic viscoelastic properties are essential to improve the accuracy and variety of finite element simulations on brain tissue. Thus, the aim of this study was to investigate the compressive frequency-dependent properties of brain tissue and present a mathematical model in the frequency domain to capture the tissue behavior based on experimental results. Bovine brain specimens, obtained from four locations of corona radiata, corpus callosum, basal ganglia and cortex, were tested under compression using dynamic mechanical analysis over a range of frequencies between 0.5 and 35 Hz to characterize the regional and directional response of the tissue. The compressive dynamic properties of bovine brain tissue were heterogenous for regions but not sensitive to orientation showing frequency dependent statistical results, with viscoelastic properties increasing with frequency. The mean storage and loss modulus were found to be 12.41 kPa and 5.54 kPa, respectively. The material parameters were obtained using the linear viscoelastic model in the frequency domain and the numeric simulation can capture the compressive mechanical behavior of bovine brain tissue across a range of frequencies. The frequency-dependent viscoelastic characterization of brain tissue will improve the fidelity of the computational models of the head and provide essential information to the prediction and analysis of brain injuries in clinical treatments.

Localized characterization of brain tissue mechanical properties by needle induced cavitation rheology and volume controlled cavity expansion

Changes in the elastic properties of brain tissue have been correlated with injury, cancers, and neurodegenerative diseases. However, discrepancies in the reported elastic moduli of brain tissue are persistent, and spatial inhomogeneities complicate the interpretation of macroscale measurements such as rheology. Here we introduce needle induced cavitation rheology (NICR) and volume-controlled cavity expansion (VCCE) as facile methods to measure the apparent Young's modulus E of minimally manipulated brain tissue, at specific tissue locations and with sub-millimeter spatial resolution. For different porcine brain regions and sections analyzed by NICR, we found E to be 3.7 ± 0.7 kPa and 4.8 ± 1.0 kPa for gray matter, and white matter, respectively. For different porcine brain regions and sections analyzed by VCCE, we found E was 0.76 ± 0.02 kPa for gray matter and 0.92 ± 0.01 kPa for white matter. Measurements from VCCE were more similar to those obtained from macroscale shear rheology (0.75 ± 0.06 kPa) and from instrumented microindentation of white matter (0.97 ± 0.40 kPa) and gray matter (0.86 ± 0.20 kPa). We attributed the higher stiffness reported from NICR to that method's assumption of a cavitation instability due to a neo-Hookean constitutive response, which does not capture the strain-stiffening behavior of brain tissue under large strains, and therefore did not provide appropriate measurements. We demonstrate via both analytical modeling of a spherical cavity and finite element modeling of a needle geometry, that this strain stiffening may prevent a cavitation instability. VCCE measurements take this stiffening behavior into account by employing an incompressible one-term Ogden model to find the nonlinear elastic properties of the tissue. Overall, VCCE afforded rapid and facile measurement of nonlinear mechanical properties of intact, healthy mammalian brain tissue, enabling quantitative comparison among brain tissue regions and also between species. Finally, accurate estimation of elastic properties for this strain stiffening tissue requires methods that include appropriate constitutive models of the brain tissue response, which here are represented by inclusion of the Ogden model in VCCE.