Inverse Problems Arising in Novel Modalities of Biomedical Imaging

计算机断层扫描在生物医学成像中起着核心作用。自20世纪60年代计算机辅助断层扫描发明以来，许多成像方式已经被引入，并成为生物学和医学诊断仪器中不可或缺的工具。最近，对更敏感和更可靠的技术的追求导致了诸如热声和光声层析成像（TAT和PAT）和磁声电层析成像（MAET）等有前途的耦合物理模式。这些成像方法结合了超声的高分辨率和电磁波对组织的光吸收和电导率的敏感性。后一种物理参数的急剧异常是乳腺癌、血栓形成、缺血和其他医疗状况的良好标志。因此，这些新技术克服了传统断层扫描的局限性，并以更低的成本和对患者的伤害更小的方式提供了原本无法获得的、可能挽救生命的诊断信息。这些模式中的图像是通过复杂的数学程序获得的，而不是通过直接获取。这些方法的数学原理大多还处于非常早期的发展阶段。研究者和他的合作者致力于解决核心理论问题，并为PAT、TAT、MAET和其他混合技术开发有效的数值技术。一名研究生和一名博士后在该项目中发挥着重要作用，他们有机会接触到精确科学、医学和生物学交叉的令人兴奋的领域。研究结果将通过在高质量的研究期刊上发表的出版物、在国家和国际会议上的发言以及在各主要场所举行的一系列讲座来传播。PAT、TAT、MAET等模式和基于康普顿散射的几种新技术的数学基础和实现包含许多具有挑战性的开放问题，从理论和应用的角度来看都很重要。研究者和他的合作者的目标是获得一个理论的理解，并开发这些模式的算法基础。特别是，他们致力于(1)从时间和空间上简化的数据集中推导出TAT/PAT重建问题的精确反演公式；(2)针对不同的MAET数据采集方案，包括低频损失MAET和具有各向异性电导率目标的MAET，开发了高效的重建算法；(3)设计有效的数值技术来聚集强超定康普顿数据集，并使用先前为单光子发射计算机断层扫描开发的衰减补偿技术来处理它们。此外，新开发的理论和算法工具将用于处理PI在与亚利桑那大学医学影像学研究人员共同进行的MAET实验研究中获得的真实MAET数据。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Computerized tomography plays a central role in biomedical imaging. Since the invention of computer-aided tomography in the 1960s, numerous imaging modalities have been introduced and became indispensable for diagnostic instruments in biology and medicine. Recently, the quest for more sensitive and more reliable techniques has led to such promising coupled-physics modalities as Thermoacoustic and Photoacoustic Tomography (TAT and PAT), and Magnetoacoustoelectric Tomography (MAET). These imaging methods combine high resolution of ultrasound with the sensitivity of electromagnetic waves to optical absorption and conductivity of the tissues. Sharp abnormalities in the latter physical parameters are good markers of breast cancer, thrombosis, ischemia, and other medical conditions. Thus, these new techniques overcome limitations of classical tomography, and deliver otherwise unavailable, potentially life-saving diagnostic information - at a lesser cost and with less harm to a patient. The images in these modalities are obtained by complex mathematical procedures, rather than through direct acquisition. The mathematics of these methods is, mostly, at very early stages of development. The investigator and his collaborators work to resolve the central theoretical problems and to develop efficient numerical techniques for PAT, TAT, MAET and other hybrid techniques. A graduate student and a postdoc are playing a significant role in the project, gaining exposure to the exciting area at the junction of exact sciences, medicine, and biology. The results will be disseminated through publications in high quality research journals, presentations at national and international conferences, and series of lectures at various major venues.The mathematics underlying and enabling such modalities as PAT, TAT, MAET and several novel techniques based on Compton scattering, contains a number of challenging open problems, important from both the theoretical and applied points of view. The investigator and his collaborators aim to gain a theoretical understanding and to develop algorithmic foundations for these modalities. In particular, they work on (1) deriving exact inversion formulas for the problem of TAT/PAT reconstruction from the sets of data reduced both in time and in space; (2) developing efficient reconstruction algorithms for different MAET data acquisition schemes, including MAET with loss of low frequencies, and MAET of objects with anisotropic conductivity; (3) devising efficient numerical techniques for congregating strongly over-determined Compton data sets, and processing them using attenuation compensation techniques previously developed for Single Photon Emission Computed Tomography. In addition, the newly developed theoretical and algorithmic tools will be used to process real MAET data obtained by the PI in experimental MAET research done jointly with the researchers from the Medical Imaging Department at the University of Arizona.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.