In the recent years the research in ultrasound imaging has been focusing on refining the reconstruction algorithm and the underlying mathematical model. In soft tissues, the measured echoes come from numerous weakly contrasted unresolved scatterers. In this work we first aim at providing a mathematical framework for wave propagation in tissue-mimicking random multi-scale media. We derive a quantitative asymptotic expansion of the measured field with respect to the size of the scatterers using stochastic homogenization [2]. Secondly we use this asymptotics of the scattered field to justify the estimators of the effective speed of sound
inside biological tissues introduced by A. Aubry [1]. By analyzing the dependence of the imaging functional with respect to the backpropagation speed, we build an estimator of the sound speed in the random multi-scale medium. We then confront our results with numerical simulations and experimental results.
[1] F. Bureau, Multi-dimensional analysis of the reflection matrix for quantitative ultrasound imaging, theses, Université Paris sciences et lettres (2023).
[2] J. Garnier, L. Giovangigli, Q. Goepfert and P. Millien, Scattered wavefield in the stochastic homogenization regime, (2023).
