Crossover Frequency as a Model-Independent Viscoelastic Constant for Soft Tissue Biomechanics

Abstract

Magnetic resonance elastography (MRE) and related elastography techniques are emerging as quantitative diagnostic tools for assessing tissue microstructure and pathology. To determine descriptive parameters of the tissues' properties, a frequency-dependent viscoelastic material model is required, which is calibrated to the measured response in a parameter identification process. However, the selection of this model and the fitting strategy is challenging, since it may influence the identified viscoelastic parameters notably. Here, we address this limitation by proposing the crossover frequency (fc, defined as the frequency at which storage and loss moduli intersect G'(fc) = G''(fc)) as a model-independent viscoelastic constant for soft tissues. Fresh porcine specimens of the corona radiata, the putamen, the thalamus, and the liver were investigated using tabletop MRE and the frequency-dependent viscoelasticity was characterized with a fractional Kelvin-Voigt model. By validating the crossover frequency against the viscoelastic parameters, we demonstrated that the crossover frequency accurately reflects the viscoelastic behavior, independent of the material model or the fitting strategy. Across all samples, fc distinguished brain regions and separated brain from liver tissue by median frequencies of 85Hz (95% CI: 69-269Hz) in the corona radiata, 423Hz (95% CI: 316-575Hz) in the putamen, 426Hz (95% CI: 302-601Hz) in the thalamus and 1174Hz (95% CI: 1074-1300Hz) in the liver (p<0.001). These results suggest that crossover frequencies capture distinct viscoelastic fingerprints without requiring viscoelastic model selection. The crossover frequency may therefore serve as a practical, model-independent biomaterial constant to improve comparability of viscoelastic measurements across elastography studies.

Figures (6)

• Figure 1: Exemplary tabletop MRE samples of a) brain and b) liver tissues. The black bar demonstrates the position of the image slice, the arrows indicates the vibration direction and the white bars serve as a length scale with a length of 10mm,
• Figure 2: Measured storage (G') and loss moduli (G") (mean $\pm$ standard deviation) for a) the corona radiata (n = 9), b) the putamen (n = 5), c) the thalamus (n = 5) and d) the liver (n = 5).
• Figure 3: Comparison of the viscoelastic parameters for the corona radiata (CR), the putamen (P), the thalamus (T) and the liver (L). A fractional Kelvin-Voigt model with two fractional elements (shear modulus $\mu_i$ and powerlaw exponent $\alpha_i$) in parallel to an elastic spring (elastic shear modulus $\mu_e$) was calibrated based on the measured dynamic moduli. The box donated the median and the first and third interquartile range, the whiskers label the minimum and maximum data points and the outliers are given in red. Statistically significant differences between the groups are marked with ’*’ for p < 0.05 and ’**’ for p < 0.01.
• Figure 4: Crossover frequency $f_c$ of a representative liver sample. The crossover frequency is the frequency point, at which the storage and loss moduli intersect (G'($f_c$) = G"($f_c$)).
• Figure 5: Comparison of the crossover frequency $f_c$ at which the loss modulus exceeds the storage modulus for the corona radiata (CR), the putamen (P),the thalamus (T) and the liver (L). The box donated the median and the first and third interquartile range, the whiskers label the minimum and maximum data points and the outliers are given in red. Statistically significant differences between the groups are marked with ’*’ for p < 0.05 and ’**’ for p < 0.01.