Anisotropy: Definition, Computational Models, Injury Response and Future Directions

The material property shows different mechanical characteristics when the load is applied in different directions.

Anisotropy is of structural importance in computational models of traumatic brain injury.


It seeks to understand that the mechanisms of the injury can help develop methods for the treatment and mitigation of traumatic brain injury (TBI for its acronym in English).

Computational head models can provide valuable information about the complexity of multiple length scales associated with the primary nature of the diffuse axonal injury.

It involves understanding how trauma to the head (on the scale of centimeters in length) is translated into white matter tissue (on the scale of millimeters in length) and even further down the axonal length scale, where the physical injury of the axons (e.g., axonal spacing) may occur.

However, to accurately represent the development of TBI, the fidelity of these computational models is of paramount importance.

A concentrated effort has been made to improve the fidelity of the computational models by including more sophisticated material definitions and the implementation of physiologically relevant damage measures.


This article summarizes the recent computational studies that have incorporated structural anisotropy both in the material definition of the white matter and in the criterion of the lesion as a means to improve the predictive capabilities of the computational models for TBI.

We discuss the role of structural anisotropy in both the mechanical response of brain tissue and the development of the lesion. We also delineate future directions in TBI computational modeling.

Computational models that incorporate high-fidelity tissue level anatomy and the mechanical response of various tissues have become a valuable tool for studying the development of traumatic brain injury (TBI).

One of the most common pathological features in mild traumatic brain injury is diffuse axonal injury. It has been hypothesized that this injury occurs due to the stretching of the axons.

The brain’s white matter contains axons, which are grouped into fiber tracts and serve as communication pathways in the brain.

The coherent orientation of the fibers in the white matter, which leads to anisotropy in the mechanical response of the white matter, plays a vital role in the development of the lesion.

Understanding, through experiments, the relationship between tissue-level loading and axonal injury is essential to develop constitutive models that incorporate the mechanical response under various load conditions and to continue using these models to create appropriate injury measurements.

Computational models provide a platform to integrate several mechanical and even biochemical models with the brain’s anatomy.

The applications of this computational platform include the ability to predict local strain and strain and the probability of primary injury (predominantly with physics-based models).

Similarly, secondary lesions (including biochemical models), the likelihood of neurological outcomes in diffuse TBI related to axonal injuries, and the development of kinematic tolerance thresholds to protect against brain injury.

A specific application of said computational platform includes the prediction of the severity of the primary lesion immediately after a TBI event.

A functional criterion of axonal injury based on stretching can predict the probability of direct damage to axons (which is instantaneous), which considers the collective effect of multiple mechanisms, for example, increased membrane permeability, altered axonal transport, and axonal separation.

A primary lesion can lead to various biochemical cascades and other secondary injury mechanisms that develop over time.

With the increasing use of portable sensors, such as football helmets equipped with accelerometers, it has become possible to acquire kinematic data in real-time.

The measured kinematic parameters can be applied as inputs in the model, which can predict the location and extent of the injury in the brain for a given injury event, serving as an objective measure of the probability of injury.

At present, the diagnosis of mild traumatic brain injury is based mainly on neurocognitive evaluations since the structural signature of the lesion is not visible in conventional medical imaging modalities.

The damage can occur predominantly at the cellular level, beyond the resolution of many commonly used image platforms.

In such cases, computer models could serve as an invaluable tool for predicting the probability of an injury. Used together with neurocognitive assessments, they could help guide the possible location of the lesion.

Despite such potential for critical applications, there has been limited success in establishing tolerance thresholds for brain injuries using computer models.

In addition, the ability of computational models to accurately predict the location and severity of axonal damage has not yet been validated.

This may partly be due to the need to improve the biofidelity of existing computational models.

Many factors contribute to the fidelity of these models, including the use of appropriate boundary conditions and material definitions, the level of anatomical detail, and the precision of the measurement of the lesion.

Recently, there has been an impulse to apply more physiologically relevant lesion criteria and account for the white matter’s substructure.

It has been hypothesized that the orientation of the fibers in the white matter plays a vital role in developing the lesion and the mechanical response of brain tissue.

To assess the importance of structural anisotropy in computational models of TBI, we offer a summary of selected studies that have considered the structural anisotropy of white matter in computational head models.

This can be through the criterion of axonal injury or the material definition of white matter.

We discuss the effect of this structural anisotropy in the prediction of the lesion and summarize what can be learned about the role of the tissue substructure in the development of diffuse axonal injury from these studies.

Since the inclusion of structural anisotropy can increase the computational cost and complexity of a model, it is essential to evaluate its impact on the biofidelity and predictive capabilities of the model.

Structural anisotropy in TBI computational head models

The advent of the diffusion tensor image (DTI) has made it possible to incorporate the structural anisotropy of white matter in computational models of TBI. DTI measures the diffusion of water molecules in the brain.

Since water molecules diffuse faster along the fibers than perpendicular to them, the technique can be used to characterize the orientation of axons within a given region of the brain.

Inclusion of structural anisotropy in the definition of material

The white matter is anisotropic due to the coherent orientation of the fibers, and this structural anisotropy affects the mechanical behavior of the tissue.

It has been shown that the material properties of the white matter depend on the relative direction of charge concerning the orientation of the fiber.

Quasi-static stretching experiments in white matter brain tissue and data analysis to characterize the effect of fiber reinforcement suggest that the shear modulus along the direction of the fiber is approximately 42% higher compared to the module perpendicular to the direction of the thread.

In a computational model, diffusion tensor imaging (DTI) data can be used to assign the local orientation of the fiber within the white matter.

The DTI is a magnetic resonance imaging (MRI) technique that provides a measure of the average orientation of the axons and the degree of fiber dispersion for a local volume element (usually at a resolution of several cubic millimeters).

By jointly recording a diffusion tensor image of the brain with magnetic resonance anatomy data, a volume-averaged fiber orientation can be defined for each finite element within a computational model of the brain.

This, in turn, also integrates the data associated with the mechanical response of the white matter.

Then, an anisotropic material model can be applied to define the response of the local material based on this volume-averaged fiber direction for the white matter of the brain. The mechanical response of the gray matter can be assumed to be predominantly isotropic.

Effect of anisotropy on the response of the lesion

It has been shown that the structural anisotropy of the white matter has a significant effect on the prediction of possible injuries.

Several computational studies have compared the mechanical response of brain tissue with the anisotropic and anisotropic definition of white matter.

Sahoo et al. found that the inclusion of anisotropy significantly influenced the local brain movement that developed in a sham head impact.

It was found that the inclusion of anisotropy has a significant effect on the magnitude and direction of the main strains developed and on the importance of the shear stress created in some tracts of white matter fibers.

Several studies of the computational head model have compared injury predictions for an injury criterion to study the impact of a structurally based injury criterion on the predicted lesion.

It is structurally based on traditionally used tissue-level injury criteria, such as the first (i.e., maximum) primary strain, von Mises stress, and shear stress.

Simulated head impacts sustained in two well-documented motorcycle accidents. In one of the accidents, severe diffuse axonal injury and subdural hematoma occurred, while the other accident did not cause severe damage.

It was found that the magnitudes of the axonal strain, the von Mises strain, and the first principal strain were 100% higher for the impact on the head that resulted in injuries compared to the effects of the non-injurious head.

For both simulated cases, the axonal strains were significantly smaller (approximately 30%) and closer to the experimentally determined axonal injury thresholds than the magnitudes predicted by the von Mises strain and the first significant strain.

Furthermore, if sites susceptible to injury were defined as a function of the regions experiencing the most significant magnitude of stress, then differences in the resulting predictions of the injured areas were observed between measures of personal stress.

The first simulation and the von Mises strain predicted injury in the periphery of the brain, and the axonal strain criterion predicted damage to the white matter tracts, such as the corpus callosum, which is where the diffuse axonal injury is commonly found.

The impact on the head of a professional ice hockey player that resulted in a concussion injury was simulated.

Applying the tolerance thresholds commonly used for the injury, a much higher degree of damage was predicted with the shear stress and the main stress injury criteria than with the standard of axonal tension.

Fragment tracts with higher axonal stress correlated with the predominant damage regions identified in concussion injury studies.

When using a structurally based measure of tissue, such as axonal tension, Wright found that the definitions of isotropic white and anisotropic white matter materials produced similar regions of high axonal stress.

It was hypothesized that this similarity was because the substructure of the tissue was captured through the measurement of the structurally based lesion.

However, for other measures of injury at the tissue level, such as von Mises stress, shear stress, and maximum principal strain, significant differences were found between the isotropic and anisotropic material models.

These sensitivity studies illustrate that when strain-based criteria are applied to predict functional damage in axons, consideration of the strain component and the fiber’s orientation in the local region of the white matter is crucial.

He simulated 11 football and hockey impacts with concussions using helmet acceleration data instrumented as inputs in his computational head model and considered deformations along the fiber direction to measure the injury.

They found significant differences in the distribution and extent of damage predicted between the fiber tension and the first strain criterion for the lesion.

The distribution of regions with high fiber tension was consistent with typical heterogeneous patterns of diffuse axonal injury.

In all these studies, the pattern of injury predicted with an injury criterion based on the structure that considers the structural anisotropy of the white matter was very consistent with that observed in pathological studies of TBI.

Another important conclusion that can be drawn from these studies is that the development of a diffuse axonal lesion depends significantly on the direction of the load.

He simulated two football concussion shocks and found that the first significant strain as a criterion for the injury over-predicted the extent of the damage compared to the standard of axonal stress injury.

It was found that the degree of over-prediction depends on the direction of loading concerning the orientation of the axonal fibers.

Kraft et al. applied an evolutionary injury criterion in time based on axonal tension and deformation rate.

They found that the orientation of the fibers concerning the direction of impact affected the extent of the anticipated injury.

It has been shown that the extent and degree of injury are significantly different between the linear and rotational accelerations of the head.

The hypothesis that the rotation component of acceleration is the dominant contributor to axonal damage has been formulated in general head impact conditions.

These studies highlight the importance of load direction, which has an important implication in developing kinematic tolerance criteria for diffuse axonal injury.

Kinematic tolerance criteria should not only include a tolerance threshold for the injury depending on the magnitude of the acceleration but should also take into account the direction of impact.

Future directions

The mechanics associated with biology are at the center of a predictive computational tool.

Computational models, combined with imaging techniques, such as diffusion tensor imaging, can improve our understanding of the lesion by providing a framework for integrating different physical models for a variety of injury mechanisms.

The ability of calculations to predict the complex behavior of these coupled systems needs a closer look from the point of view of the applications.

It is fundamental to be able to correlate the anticipated injuries with the actual damages and to train the framework to improve according to the quantitative measures of comparison.

It was shown that anisotropy is essential and has a significant effect on the predicted damage response.

However, it has not yet been validated if the inclusion of anisotropy is sufficient to improve the predictive capabilities of a model.

Recently, deformation in the human brain has been quantified in vivo using labeled MRI and may be a promising way to validate the predictions of the brain deformation model.

Computational models representing multiple fiber families within each volume element can also improve prediction capacity.

Current DTI methods are limited to the precise characterization of two or more families of cross fibers in a single voxel.

As DTI methods improve, higher-resolution substructure data can be incorporated into computational models.

Another future direction involves using a combination of patient-specific computational head models, event-specific kinematics, and DTI / MRI information for cross-validation of the models with actual patient data.

Observations of the secondary lesion, neurodegeneration over time, and tissue experiments demonstrating the effects of frequency are also essential considerations from a modeling point of view to accurately represent the progression of the lesion.

A noble goal in developing these models is to allow health professionals to have a greater understanding of the probability of diffuse axonal injury.

From the doctors’ point of view, the clinical diagnosis of a concussion with the tools currently available may not be enough.

One might be able to use computational tools based on physics and imaging methods to relate the predicted locations of the injury to actual physical and functional changes.

As neuroimaging methods continue to improve, this may be a possibility.

Quantifying the probability and extent of the injury remains a significant challenge.

Approaches that quantify the likelihood of injury to specific fiber tracts using a white matter atlas, which maps neural connections in the brain, can help close the gap between data obtained in a clinical setting and model results—computer-based physics.

Computational models that incorporate the structural anisotropy of the white matter and the lesion criterion at the axonal level could lead to much more rigorous and exhaustive thresholds for the probability of diffuse axonal injury due to a traumatic event brain injury.


There is still a critical need to improve the predictive capabilities of TBI computational models to provide a better understanding of the mechanisms of the injury.

With improvements in medical imaging techniques and the availability of real-time measurements of kinematic data during injury events, we have more resources available to improve the biofidelity of these computational tools.

To improve the representation of the mechanical behavior of brain tissue and the response to predicted damage, there has been an impulse to explain the structural anisotropy of white matter in computational analysis.

Recent studies have shown that this structural anisotropy can significantly affect the deformation of the brain, and the use of a structure-based injury criterion can lead to predictions of lesions that are more consistent with known patterns of axonal injury.

The results suggest that the inclusion of structural anisotropy may be a step in the right direction to improve the biofidelity of TBI computational models.

The effectiveness of this computational approach would benefit from higher-order validation using high-fidelity experimental measurements of strain in the brain and cross-validation with actual patient data.

As we continue to improve the predictive capabilities of these models, they will serve even more to understand the development of TBI.