The technical progress in sensors manufacturing during the last decades and the access to now extensive computational resources constitute a major shift of paradigm for the theory of medical imaging. The fidelity of the images produced from the measurements relies then heavily on the reconstruction algorithm and the underlying mathematical model. In this work, we focus on medical ultrasound imaging where ultrasonic waves are emitted by a transducer array in a region of interest and the echoes generated by the tissues are measured by the same transducers and numerically backpropagated to obtain the image. We aim at constructing an estimator of the effective velocity in a tissue-mimicking medium where echoes come from numerous (up to a few hundred by wavelength) unresolved scatterers randomly distributed throughout the medium.

We first derive a mathematical model for the propagation of ultrasounds in random multi-scale media using quantitative stochastic homogenization techniques [2]. We obtain an asymptotic expansion of the scattered field with respect to the size of the scatterers and estimate the error between the exact solution and our approximation. We also present numerical simulations to illustrate our results. In this setting the mean density of scatterers is constant with respect to their size. 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 perform a quantitative sensitivity analysis and confront our results with numerical simulations and experimental results.

Josselin Garnier(CMAP), Laure Giovangigli (POEMS, ENSTA Paris), Quentin Goepfert (CMAP, et POEMS, ENSTA Paris) and Pierre Millien (Institut Langevin).

References.
[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).