4J-3 A New Rheological Model Based on Fractional Derivatives for Biological Tissues

The intuitive use of viscoelastic properties is routinely done by physicians via the palpation. However, such an examination relies on the physician experience and is not quantitative nor reproducible. Thus, elastography has been developed to complement the palpation by reliable and reproducible measurements. The principle of elastography is to image elastic waves propagation in a medium. To evaluate the shear moduli, Helmholtz transforms are applied to the displacement images. Then the waves movie is interpreted using a rheological model. For instance, the Voigt model explains the observed frequency dependence of the measured shear wave speed. It has been shown to be a reliable model for mimicking tissue phantoms such as gelatin based phantoms. However, we show here that neither this model nor the Maxwell model are applicable to biological tissues using in vivo data (breast) and ex vivo data (liver). Although these simple models are widely used in the elastography community, they have been substituted by more relevant models in the micro-rheology community in order to reveal the solid-liquid duality of tissues. Based on these experimental observations, we introduce a new rheological model relying on fractional derivatives closer to the viscoelastic properties of in vivo data. Indeed, we observed that the dynamic modulus (G_d) and the loss modulus (G_l) have the same frequency behavior: a non-integer frequency power law smaller than 1. This frequency behavior is contradictory with the use of the Voigt model or any kind of simple arrangements of dash pots and springs. In order to explain this frequency behavior the concept of spring pot was introduced. Moreover, we observed that the ratio G_l /G_d is constant and not linked to the non-integer power observed, not predicted by the spring pot model. Thus we build a network of spring pots where the basic element is responsible of the frequency power and the network is responsible o- f the ratio G_l/G_d . The experiments were conducted on fresh liver samples and phantoms between 50 Hz and 100 Hz. The frequency behavior was analyzed by plotting the real and imaginary parts of the complex shear modulus using MR-elastography as well as 3D ultrasound based elastography. By applying the fractional derivatives model to these data sets, we observed that the frequency power law in the liver was equal to 0.75 (a liquid-like behavior), while the ratio parameter of the network was equal to 0.15 (a solid-like behavior). The dispersion curves of G_d and G_l obtained through this model correlates much better with the experimental observations. The model parameters values seem to emphasize the necessity to take into account the solid-liquid duality of tissues in the rheological model choice for elastography reconstructions