Collaborative Research: Improved Algorithms for Multiwave Imaging in Complex Media: Theory and Computation

改进的成像技术对于指导和告知日益复杂的外科手术是必要的。例如，出生时患有先天性心脏病的婴儿需要接受心脏搭桥手术；然而，大约50%的儿童在围手术期出现脑损伤，导致先天性心脏病幸存者的破坏性并发症。需要改进的脑成像技术来评估外科手术对脑血流灌注的影响。最有前途的脑成像技术之一是多波成像方法，如光声和热声断层成像，它结合了不同波的特性，获得了传统模式无法达到的高对比度和分辨率。该项目的动机是可能应用多波成像来监测婴儿的脑血流灌注。它的目的是将头部媒体特性(大脑、血管系统、血液、脑脊液、头骨、皮肤)的复杂性纳入这些成像技术，同时使这些技术对噪声和不确定性更具健壮性。这个项目的目标既是理论上的，也是计算上的。第一个是确定一般假设，在这些假设下，光声测量可以同时恢复内部源和波速。这将通过扩展关于同时恢复声源和介质性质的数学理论来实现，这是关于光声、热声和磁声成像的主要未决问题之一。在前期工作的基础上，调查人员放宽了同时恢复的先验条件。下一步是确定哪些条件是必要的，哪些条件可以删除。第二个目标是计算，利用光声效应的内在特性，产生高对比度的压力分布。其目的是设计新的间断Galerkin方法，用于精确、低色散的高对比度压力波模拟，以用于新的成像算法。这些方法的设计目的是在存在逼真的介质和几何图形的情况下在GPU架构上实现高性能。第三个目标是将这些新的发展纳入PDE约束的优化算法中，这种算法除了近似介质的未知波速外，还将提供针对噪声和缺乏信息的稳健性。

Improved imaging techniques are necessary to guide and inform increasingly complex surgical procedures. For example, cardiac bypass surgeries are required for infants born with congenital heart defects; however, approximately 50% of children develop brain injury perioperatively, leading to the damaging complications for survivors of congenital heart disease. Improved cerebral imaging techniques are needed to evaluate the effects of surgical procedures on brain perfusion. One of the most promising techniques for cerebral imaging are multiwave imaging methods such as photo-acoustic and thermo-acoustic tomography, which combine properties of different waves to obtain high contrast and resolution unattainable by traditional modalities. This project is motivated by the possible application of multiwave imaging to monitor cerebral blood perfusion in infants. It aims to incorporate the complexity of media properties in the head (brain, vasculature, blood, cerebrospinal fluid, skull, skin) into these imaging techniques, while making these techniques more robust to noise and uncertainty. The goals of this project are both theoretical and computational. The first is to identify general assumptions under which the photo-acoustic measurements render the simultaneous recovery of the internal source and the wave speed. This will be accomplished by expanding the mathematical theory concerning the simultaneous recovery of acoustic sources and media properties, which is one of the main open questions regarding photo-, thermo- and magneto- acoustic imaging. Based on preliminary work, the investigators have relaxed a-priori conditions under which simultaneous recovery is possible. The next step is to determine which conditions are essential and which ones can be removed. The second goal is computational and uses the intrinsic property of the photo-acoustic effect, which produces high contrast pressure profiles. The aim is to design novel discontinuous Galerkin methods for accurate, low dispersion simulations of high contrast pressure waves for use in new imaging algorithms. These methods are designed to achieve high performance on GPU architectures in the presence of realistic media and geometries. The third goal is to incorporate these new developments into PDE-constrained optimization algorithms, which, in addition to approximating the unknown wave speed of the medium, will also provide robustness with respect to noise and lack of information.