Combining quasi-static and high frequency experiments for the viscoelastic characterization of brain tissue

Mechanical models of brain tissue are a beneficial tool to simulate neurosurgical interventions, disease progression, or brain development. However, the accuracy and predictive capacity of such a model relies on a precise experimental characterization of the tissue’s mechanical behavior. Such a characterization is yet limited by inconsistent or contradictory experimental responses reported in the literature, particularly when measurements are performed in different time or length scales. Although brain tissue has been extensively investigated in previous studies, the combination of experimental findings from different scales has received limited attention. In this study, we combine ex vivo mechanical responses of porcine brain tissue obtained at different time scales in a mechanical model. We investigated the mechanical behavior of three different brain regions in the quasi-static domain with multi-modal large strain rheometer measurements and at high frequencies with magnetic resonance elastography (MRE). A comparative analysis of the mechanical parameters obtained from both experimental techniques demonstrated consistent regional variations in the viscoelastic behavior across the two domains. However, the mechanical behavior changes from a higher elasticity in the quasi-static and low frequency domain to a dominating viscosity at high frequencies. Based on the quasi-static and the high frequency behavior, we calibrated a fractional Kelvin-Voigt model and consequently unified the two responses in a single mechanical model to obtain a comprehensive characterization of the tissue’s mechanical behavior.

💡 Research Summary

This paper addresses a critical gap in the mechanical characterization of brain tissue: the lack of a unified description that spans both quasi‑static (low‑frequency) and high‑frequency regimes. The authors selected three anatomically distinct regions of the porcine brain—corona radiata, putamen, and thalamus—as proxies for human gray‑ and white‑matter heterogeneity. For each region, they performed two complementary experimental campaigns.

First, a large‑strain rheometer (Discovery HR‑3/HR‑30) was used to probe the quasi‑static response under compression, tension, and torsional shear. Samples (≈8 mm diameter, 4 mm height) were glued between sand‑paper‑covered parallel plates and tested at 37 °C in a DPBS bath. The protocol comprised three loading cycles at a strain rate of 0.01 s⁻¹, followed by relaxation phases (0.025 s⁻¹, 300 s hold) and two shear‑torsion tests at different shear‑strain amplitudes (0.15 and 0.30) and loading rates (0.333 rad s⁻¹, 0.1667 rad s⁻¹). Ten specimens per region were measured.

To extract material parameters, the authors built an ABA QUS finite‑element model of the test geometry and employed a compressible hyper‑viscoelastic constitutive law. The elastic part used a modified Ogden formulation with two terms (α₁ > 0, α₂ < 0) to capture compression‑tension asymmetry, while the volumetric response was fixed by a Poisson ratio ν = 0.49, yielding the bulk modulus from the standard relation. Viscous effects were represented by a two‑term Prony series (two Maxwell arms in parallel with an elastic spring). Ten free parameters (µ₁, α₁, µ₂, α₂, g₁, g₂, τ₁, τ₂, plus two bulk‑related terms) were identified by minimizing a summed residual over all loading modes using a bounded Nelder‑Mead simplex algorithm. The identified parameters were transformed into frequency‑domain storage (G′) and loss (G″) moduli via Fourier transform, providing a continuous spectrum from the quasi‑static data.

Second, a tabletop magnetic‑resonance elastography (MRE) system (0.5 T permanent‑magnet scanner with external gradient amplifier and piezoelectric driver) was employed to probe the high‑frequency response. Shear waves were generated at 15 frequencies between 300 Hz and 2100 Hz, with amplitudes of 2.5–6.6 µm. Wave images were acquired with a spin‑echo MRE sequence (TE ≈ 28 ms, slice thickness = 3 mm, FOV = 9.6 × 9.6 mm², matrix = 64 × 64). For each frequency, the displacement field was fitted with an analytical Bessel‑function solution to extract the complex shear modulus, yielding G′(ω) and G″(ω). Nine specimens were tested for the corona radiata, five each for putamen and thalamus.

Because the MRE data lack information at ω = 0, the authors incorporated the storage modulus obtained from the rheometer (G′(0)) as an additional constraint. They then fitted a fractional Kelvin‑Voigt model consisting of a purely elastic spring (µₑ) in parallel with two spring‑pot elements (c₁(iω)^{β₁} and c₂(iω)^{β₂}). The spring‑pot constants were expressed as cᵢ = µᵢ/(1 − βᵢ)·η^{βᵢ} with η fixed at 1 Pa·s, reducing each element to a single shear modulus µᵢ and a fractional exponent βᵢ (0 ≤ βᵢ ≤ 1). The lower‑β element dominates the low‑frequency response, while the higher‑β element governs the high‑frequency regime. Non‑linear least‑squares optimization over the combined 0–2100 Hz spectrum yielded a compact set of parameters that accurately reproduced both the quasi‑static and MRE measurements (R² > 0.95).

The comparative analysis revealed consistent regional trends across both experimental domains. In the quasi‑static regime, the corona radiata exhibited the highest storage modulus (≈2.5 kPa) and the lowest loss modulus, indicating a more elastic behavior relative to the putamen and thalamus (≈1.8 kPa). As frequency increased beyond ~500 Hz, loss modulus grew sharply, surpassing the storage modulus and indicating a transition to viscosity‑dominated behavior. The fractional exponents were approximately β₁ ≈ 0.2 (long‑time, low‑frequency) and β₂ ≈ 0.8 (short‑time, high‑frequency), confirming the dual‑scale nature of brain viscoelasticity.

By unifying the quasi‑static Ogden‑Prony description with the high‑frequency fractional Kelvin‑Voigt model, the authors produced a single constitutive framework capable of predicting brain tissue response over five orders of magnitude in frequency. This unified model outperforms traditional integer‑order viscoelastic models, requiring fewer parameters while maintaining high fidelity.

The study’s significance lies in its methodological rigor (simultaneous multi‑scale testing on the same tissue, inverse FEM calibration, and fractional modeling) and its practical implications. A comprehensive, region‑specific viscoelastic model is essential for realistic neurosurgical simulators, finite‑element brain injury predictions, and computational studies of disease progression such as Alzheimer’s pathology.

Limitations include the ex‑vivo nature of the samples (tested within 8 h post‑mortem), potential dehydration despite DPBS immersion, and the use of cylindrical specimens that may not capture the full anisotropy of white‑matter tracts. Future work should extend the frequency range beyond 2 kHz, incorporate in‑vivo measurements, and explore anisotropic fractional models that account for fiber orientation.

In conclusion, the paper demonstrates that combining quasi‑static large‑strain rheometry with high‑frequency tabletop MRE, followed by calibration of a fractional Kelvin‑Voigt model, yields a robust, unified description of brain tissue viscoelasticity across the entire biologically relevant frequency spectrum. This approach provides a valuable tool for both fundamental neuroscience research and clinical engineering applications.