课程题目：Mathematics of Imaging Modalities Using Resonant Contrast Agents

授课 人：Mourad Sini

所在单位：Austrian Academy of Sciences

授课时间：2025年10月14日-21日

授课地点：正新楼 209教室

课程介绍:  

In this lecture series, we delve into the mathematical modeling, analysis, and practical applications of imaging modalities that leverage resonant contrast agents—an area that has garnered significant attention and rapid development within the applied engineering community. Our focus lies in presenting a systematic, rigorous approach to formulating and solving the family of inverse problems inherent to these advanced imaging techniques, while bridging theoretical insights with real-world imaging scenarios.

We begin with foundational discussions on classical inverse problems tied to interfaces, starting with the establishment of mathematical models and the critical concept of wellposedness—laying the groundwork for understanding how imaging data relates to underlying medium properties. We then explore key aspects of these classical problems, including proving uniqueness (ensuring a single, definitive solution exists for given data) and deriving stability estimates (quantifying how small perturbations in data affect solution accuracy), as well as advancing reconstruction methods that utilize multi-frequency scattering data to enhance the precision of interface-related imaging.

A core segment of the course is dedicated to wave propagation in resonant media, where we first motivate the use of sub-wavelength resonators—explaining why these miniature structures are transformative for imaging, particularly in enhancing contrast and resolving fine-scale features that elude traditional techniques. We then deepen into the mathematics of resonances, focusing on their connection to resolvent estimates: analyzing how resonances influence the time-harmonic re-solvent (a key operator in frequency-domain wave analysis) and exploring the large-time behavior of resonant waves. This analysis reveals how resonant phenomena shape wave propagation over extended periods, a critical factor for time-dependent imaging applications.

Building on this theoretical base, we transition to two prominent classes of resonant contrast agents and their associated imaging modalities. First, we examine Minnaert resonances and their role in acoustic imaging using bubbles. Here, we split our exploration into inverse problems in the time-harmonic regime—where we leverage frequency-specific bubble resonances to extract information about the surrounding acoustic environment—and the time-domain regime, where we analyze transient wave responses to recover dynamic properties of the medium. Second, we turn to plasmonic resonances and their application in nanoscale imaging, covering both optical imaging (utilizing plasmonic nanoparticles to amplify optical signals for high-resolution visualization) and photo-acoustic imaging (combining plasmonic light absorption with acoustic wave detection to merge optical contrast with deep-tissue penetration).

We further expand into elastography, a vital technique for mapping mechanical properties of biological tissues, by investigating elastic resonant inclusions. A key focus here is developing methods to recover the mass density of elastic media using pressure waves, demonstrating how resonant inclusions can enhance the sensitivity of elastographic measurements to subtle density variations—critical for detecting abnormalities like tumors.

Throughout the course, we emphasize the advantages of resonant contrast agents in overcoming longstanding challenges in traditional imaging: they mitigate the non-linearity of inverse problems and improve stability in remote measurements, addressing two major limitations of contrast-agent-free techniques. Finally, we conclude with a discussion of open problems and future directions, inviting participants to engage with unresolved questions—such as refining reconstruction algorithms for complex clustered contrast agents or extending resonant imaging to new modalities—and explore the broader potential of this framework to revolutionize emerging imaging technologies.

授课人简介：M. Sini got his PhD degree from University of Provence, France, in October 2002. From September 2006, he joined the Radon Institute, of the Austrian Academy of Sciences, where he was tenured since 2011 as a senior fellow (equivalent to university professor) after securing the Habilitation degree from J. Kepler University in 2009. M. Sini was invited to deliver the IAAM Award Lecture (Sweden, August 2019) and an invited plenary speaker in the Applied Inverse Problems (AIP) conference (Grenoble, France, July 2019). He attracted around 2 Mi euro as third-party funding. M. Sini is interested in the analysis of partial differential equations applied to inverse problems, mathematical imaging, mathematical therapy and material theory. By now, he published nearly 100 papers, most of them in top journals, several book chapters and co-edited a book.  He is member of the editorial board of several journals including Mathematical Methods in Applied Sciences (Wiley) and Communication on Analysis and Computation (AIMS).

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