Inverse Problems Arising in Novel Modalities of Biomedical Imaging

计算机断层成像在生物医学成像中发挥着核心作用。自20世纪60年代计算机辅助层析成像技术问世以来，各种成像方法相继问世，成为生物和医学中不可缺少的诊断仪器。最近，对更灵敏和更可靠技术的追求导致了一些很有前途的耦合物理模式，如热声和光声层析成像(TAT和PAT)以及磁声电层析成像(MAET)。这些成像方法结合了超声波的高分辨率和电磁波对组织的光吸收和传导性的敏感性。后一项生理参数的急剧异常是乳腺癌、血栓形成、缺血和其他医疗条件的良好标志。因此，这些新技术克服了经典断层扫描的局限性，以更低的成本提供了原本无法获得的、可能挽救生命的诊断信息，对患者的伤害也更小。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。这些方法的数学计算大多处于非常早期的发展阶段。研究人员和他的合作者致力于解决核心的理论问题，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这个项目中扮演着重要的角色，他们接触到了精确科学、医学和生物学交界处这一令人兴奋的领域。这些结果将通过高质量研究期刊上的出版物、在国内和国际会议上的演讲以及在不同主要场所的一系列讲座来传播。作为PAT、TAT、MAET和基于康普顿散射的几种新技术的基础和实现方式的数学，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是重要的。研究人员和他的合作者旨在获得理论上的理解，并为这些模式开发算法基础。特别是，他们的工作包括：(1)从时间和空间上减少的数据集推导TAT/PAT重建问题的精确反演公式；(2)为不同的MAET数据采集方案开发高效的重建算法，包括低频损失的MAET和具有各向异性导电性物体的MAET；(3)设计高效的数值技术来收集强过确定的康普顿数据集，并使用先前为单光子发射计算机层析成像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成像系的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NSF的法定使命，并通过使用基金会的智力优势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.