The influence of tension-compression switches on brain anisotropic modelling

This study demonstrates that the choice of tension-compression switching scheme significantly influences white matter deformation predictions in finite element brain models, advocating for the use of fiber stretch as the switching parameter to improve the accuracy of anisotropic constitutive modeling.

The Gist

The Big Picture: Why Does This Matter?

Imagine your brain is a giant, wet, jiggly sponge inside a hard helmet (your skull). When you get hit in a car accident or a football game, that sponge gets squished and stretched. Sometimes, this stretching tears the tiny wires inside the sponge, causing a concussion or a brain injury.

Scientists use computer models (like a video game simulation) to predict how much the brain stretches during a crash. If the computer model is wrong, we can't design good helmets or safety rules.

This paper is about fixing a specific "glitch" in how these computer models treat the white matter inside the brain. Think of white matter as the highway system of the brain, made of millions of tiny cables (axons) that carry messages.

The Problem: The "Stretchy String" Rule

The computer models use a set of math rules (called the GOH model) to describe how these cables behave. There is one very important rule in these math equations:

"Cables are strong when you pull them (tension), but they go floppy and do nothing when you push them (compression)."

Think of a spaghetti noodle:

• If you pull it, it gets tight and resists.
• If you push it, it just bends and buckles; it doesn't push back.

To make the computer understand this, programmers use a "switch" (a tension-compression switch). This switch tells the computer: "If the cable is being pulled, count its strength. If it's being pushed, ignore it."

The Conflict: Three Different Switches

The problem is that different scientists and different computer software programs use three different ways to flip this switch. They all claim to be doing the same thing, but they are actually using different logic:

1. Switch A (The "Strain" Switch): Checks if the overall shape of the material is stretching. If the math says "stretch," it counts the cable.
2. Switch B (The "Stretch" Switch): Checks if the cable itself is getting longer. If the cable is longer, it counts it.
3. Switch C (The "Original" Switch): Checks if the cable is longer, but if it's shorter, it still counts the "sponge" part of the brain, just not the cable.

The authors of this paper asked: "Does it actually matter which switch we use?"

The Experiment: The Football Crash Test

The researchers took a realistic 3D model of a human head and simulated 38 different football hits (some causing concussions, some not). They ran the exact same crash three times, once with each of the three "switches."

They looked at two things:

1. The Element Level: How much did a tiny chunk of brain stretch?
2. The Fiber Level: How much did the specific cables inside that chunk stretch?

The Findings: It Matters A Lot!

The results were surprising. The three switches gave very different answers.

• The "Strain" Switch (Switch A) was lying. It sometimes counted the cables as "pulling" even when they were actually being "pushed" or buckling. It was like a faulty sensor on a car that thinks the engine is running even when it's off. This made the brain look stiffer and stronger than it really is.
• The "Stretch" Switches (B and C) were more accurate. They correctly identified when the cables were buckling and stopped counting them.
• The Difference was Huge: In some cases, the "wrong" switch predicted brain strain that was 25% to 33% higher than the "right" switch.

The Analogy:
Imagine you are trying to predict how much a bridge will bend when a truck drives over it.

• Switch A says: "The bridge is bending, so the steel cables inside are helping hold it up!" (Even though the cables are actually slack and useless).
• Switch B says: "The cables are slack, so they aren't helping. Only the concrete is holding the weight."

If you use Switch A, you might think the bridge is super safe. If you use Switch B, you realize it's actually much weaker. In brain injury, thinking the brain is stronger than it is could lead to unsafe helmets.

The Conclusion: What Should We Do?

The paper concludes with two main messages:

1. Stop using the "Strain" switch. The method currently used in many popular software programs (like Abaqus) is flawed for brain modeling. It needs to be replaced with the method that checks if the fiber itself is stretching (the "Stretch" switch).
2. We need real-world proof. The math says cables buckle when pushed, but we haven't actually watched a single human brain cable buckle under a microscope yet. We need more experiments to prove exactly how these tiny wires behave when squished.

In short: The computer models we use to design safety gear have been using a slightly broken calculator. By fixing the "switch" that tells the computer how brain cables behave, we can get much more accurate predictions of brain injuries and, hopefully, save more lives.

Technical Summary

1. Problem Statement

Finite Element (FE) head models are critical for investigating Traumatic Brain Injury (TBI) mechanics. White Matter (WM) is anisotropic due to aligned axonal fibers and is frequently modeled using the Gasser-Ogden-Holzapfel (GOH) hyperelastic constitutive model. A fundamental assumption of the GOH model is that fibers contribute to stiffness only in tension and buckle (contributing zero or minimal energy) in compression.

To implement this, a tension-compression switch is required to exclude compressed fibers from the stress calculation. However, the paper identifies significant inconsistencies in the literature regarding how this switch is formulated:

• Switching Parameter: Some studies use the fourth invariant ($\bar{I}_{4\alpha}$, representing fiber stretch squared), while others use a Green-Lagrange strain-like quantity ($\bar{E}_{\alpha}$).
• Treatment of Compressed Fibers: Some implementations completely eliminate the fiber contribution in compression (setting it to zero), while others retain an isotropic matrix contribution.

The lack of consensus on these formulations raises questions about their impact on the biofidelity of brain injury predictions, specifically regarding strain metrics used for injury criteria (e.g., Maximum Principal Strain).

2. Methodology

The authors conducted a systematic theoretical and numerical investigation using a high-fidelity FE head model (KTH model) and LS-DYNA software.

• Material Modeling: The brain tissue was modeled as a fiber-reinforced anisotropic material using the GOH model. WM elements were assigned fiber orientations and dispersion parameters ($\kappa$) derived from Diffusion Tensor Imaging (DTI) data (ICBM DTI-81 atlas).
• Three Switching Schemes: Three distinct implementation schemes were compared:
  • Scheme 1: Uses $\bar{E}_{\alpha}$ as the switching parameter. If $\bar{E}_{\alpha} \leq 0$, the fiber contribution is set to zero. (Common in Abaqus/FEBio).
  • Scheme 2: Uses $\bar{I}_{4\alpha}$ as the switching parameter. If $\bar{I}_{4\alpha} \leq 1$ (compression), the fiber contribution is set to zero. (Common in LS-DYNA/ANSYS user subroutines).
  • Scheme 3: Uses $\bar{I}_{4\alpha}$ as the switching parameter. If $\bar{I}_{4\alpha} \leq 1$, the fiber contribution reduces to the isotropic matrix term only ($\kappa(\bar{I}_1 - 3)$), retaining the original GOH formulation.
• Simulation Setup:
  • Model: A detailed FE head model including scalp, skull, brain, CSF, ventricles, and meninges.
  • Loading: 38 whole-head impact simulations were performed, including 2 concussive cases (one with loss of consciousness) and 36 NFL football impacts (15 concussed, 21 non-concussed).
  • Metrics: The study analyzed Maximum Principal Strain (MPS) at the element level and Maximum Fiber Strain (MFS) at the fiber level.
• Analysis: Statistical comparisons (Friedman test and Nemenyi post-hoc test) were performed on the strain peaks across all 38 impacts to determine significant differences between the schemes.

3. Key Contributions

• Theoretical Clarification: The study mathematically demonstrated that $\bar{E}_{\alpha}$ (used in Scheme 1) is an inappropriate proxy for fiber stretch. It was shown that $\bar{E}_{\alpha} > 0$ does not necessarily imply the fiber is in tension ($\bar{I}_{4\alpha} > 1$). Consequently, Scheme 1 can erroneously activate fiber stiffening even when fibers are kinematically compressed.
• Comprehensive Comparison: Unlike previous studies that focused on single aspects, this work simultaneously evaluated the effects of the switching parameter ($\bar{E}_{\alpha}$ vs. $\bar{I}_{4\alpha}$) and the treatment of compression (zero contribution vs. isotropic retention).
• Implementation Guidance: The authors provided a unified framework for implementing these schemes in LS-DYNA via user-defined materials, allowing for direct, controlled comparison within the same computational environment.

4. Results

• Significant Strain Differences: Both the choice of switching parameter and the treatment of compressed fibers led to statistically significant differences in predicted brain strains.
  • Switching Parameter: Scheme 2 (using $\bar{I}_{4\alpha}$) predicted significantly higher strains than Scheme 1 (using $\bar{E}_{\alpha}$). In a representative case, the difference in MPS was 25.7%, and in MFS, it was substantial (Scheme 2: 0.405 vs. Scheme 1: 0.302).
  • Compression Treatment: Scheme 2 (zero contribution) yielded higher strains than Scheme 3 (isotropic retention). The difference in MPS was 33.8%.
• Spatial Distribution: The spatial distribution of elements identified as having "compressed fibers" varied drastically between schemes. Scheme 1 identified fewer compressed elements in critical areas like the brainstem compared to Schemes 2 and 3, leading to different local deformation patterns.
• Statistical Significance: Group-level analysis across 38 impacts confirmed that:
  • MPS differed significantly between all pairs (1 vs. 2, 2 vs. 3).
  • MFS differed significantly between Scheme 1 and Scheme 2, but not between 2 and 3 (though trends were observed).
• Mechanism of Error: The use of $\bar{E}_{\alpha}$ (Scheme 1) results in "spurious stiffening" because it fails to correctly identify compressed fibers, artificially increasing the tissue's resistance to deformation and altering injury predictions.

5. Significance and Conclusions

• Recommendation for Modeling: The study strongly advocates for the adoption of $\bar{I}_{4\alpha}$ (fiber stretch) as the switching parameter, as originally proposed by Gasser et al., rather than the strain-like $\bar{E}_{\alpha}$. This ensures that fibers are only activated when physically stretched.
• Impact on Injury Prediction: Since fiber strain is a primary metric for predicting Diffuse Axonal Injury (DAI), the choice of switch formulation directly biases injury risk assessment. Using an inappropriate switch (like Scheme 1) may lead to underestimation or overestimation of injury thresholds depending on the specific loading scenario.
• Call for Experimental Validation: The authors highlight a critical knowledge gap: there is a lack of direct experimental evidence confirming that axonal fibers buckle immediately under compression in human brain tissue. They call for targeted experiments to verify the tension-compression switch hypothesis at the microstructural level.
• Practical Implication: Researchers using FE head models for TBI must carefully review their constitutive model implementations. The widespread use of Scheme 1 in commercial software (e.g., Abaqus) and literature may compromise the accuracy of anisotropic brain modeling unless corrected.

In summary, this paper provides a crucial correction to the field of computational brain biomechanics, demonstrating that the mathematical formulation of tension-compression switches is not a minor implementation detail but a fundamental factor that significantly alters predicted brain mechanics and injury outcomes.