Abstract: Elastography is a noninvasive medical imaging technique that aims to visualize the elastic properties of soft tissue. The underlying idea is that elastic behavior can distinguish healthy from unhealthy tissue in specific situations. For instance, cancerous tumors and scarred liver tissue will often be stiffer than the surrounding healthy tissue. Elastography based devices emit elastic waves that interact with the tissue under study and record the response received. Using the recorded data, adequate mathematical algorithms provide information on stiffness variations in the explored region. Such devices are already being used for the diagnosis of liver and prostate diseases. However, there is a need for improved methods and algorithms, allowing to resolve for tiny tumors or for little contrast regions with quantified uncertainty. Here, we propose a full wave form inversion scheme for localized inhomogeneities that proceeds as follows. First, we identify the most prominent anomalous regions in the tissue by visualizing topological fields associated to functionals comparing the true recorded data with the data that would be obtained varying the stiffness fields. Then, we improve this information by optimization strategies. Finally, we quantify uncertainty in the outcome resorting to a Bayesian inversion framework. We illustrate the method using data from prostate and liver studies.

Keywords: Medical elastography, optimization, topological sensitivity, uncertainty quantification, cancer

Acknowledgements: Research partially supported by Spanish FEDER/MICINN-AEI grants MTM2017-84446-C2-1-R and PID2020-112796RB-C21.

Biography: A. Carpio is a Professor of Applied Mathematics at the Universidad Complutense de Madrid. She received her Ph.D. degree at Laboratoire Jacques Louis Lions (Université, Paris VI, now Paris Sorbonne) and conducted postdoctoral research at the Oxford Centre for Industrial and Applied Mathematics (University of Oxford). She has been visiting scholar at Stanford University, Harvard University and New York University. Her research interests include inverse problems and variational methods for image analysis, shape reconstruction and parameter identification. She has pioneered the development of topological sensitivity based methods for inverse scattering problems, with applications to electrical impedance tomography, to photothermal and microwave imaging, and to the acoustic sounding of bodies. More recently, she has developed inversion techniques for holographic data in noninvasive light imaging of 3D biological samples, combining topological methods, optimization strategies and Bayesian approaches, which are now being extended to medical elastography.