多项分数阶模型的高效数值算法及其在核磁共振成像上的应用

The magnetic resonance imagining (MRI) has been established as a routine but indispensable tool in the imaging procedures. Recently, to meet the growing imaging needs, many researchers favor the high-field MRI and the high b-value diffusion-weighted imaging. Although effectively improving the imaging quality, these technologies lead to deviations in MRI signal relaxation and diffusion, which cannot be appropriately described by the classical mono-exponential models..Recently, the application of fractional models to account for these anomalies has achieved some preliminary success. Given that the varied values of fractional orders are related to the tissue structure and composition, we proposed the multi-term time fractional Bloch and Bloch-Torrey equations. However, owing to the complexity of the fractional-order definition, it is difficult to find the solutions with a high efficiency to ensure the ensuing applications. To investigate the anomalous decay of MRI signal relaxation and diffusion, this project centers on solving the multi-term time fractional Bloch and Bloch-Torrey equations with less requirement of storage and computation by using novel fast calculation and parallel computation, which is surely of great theoretical and pratical significance.

核磁共振成像技术是医学影像技术中不可或缺的常规成像手段之一。近年来，为满足更高的成像需求，研究人员开始青睐高场强核磁共振成像和高b值弥散加权成像。二者虽然能有效改善成像质量，但是会对应产生传统指数模型所无法描述的反常衰减和反常扩散现象。.最近，分数阶模型在描述核磁共振信号的反常现象上取得了初步进展。考虑到组织的结构和成分与分数阶导数密切相关，项目申请人提出了多项分数阶Bloch以及多项分数阶Bloch-Torrey方程。然而，由于分数阶定义本身的复杂性，对其的数值求解相对困难且效率低，难以开展模型的进一步应用。本项目将围绕核磁共振的多项分数阶模型，提出计算量小、存储量少的新快速算法和并行算法以及对应的数值分析，从而研究高场强和高b值情况下核磁共振信号的反常衰减和反常扩散行为，这在理论和实际应用上都具有重要意义。

现代科学和工程中的许多问题，都可以用基于微分方程的数学模型来进行描述。由于模型的复杂性，方程的精确解析解通常难以获得，需要寻求近似的数值解。本课题围绕新建立的数学模型，包括分数阶和整数阶偏微分方程，提出了相应的高效并行算法和快速算法对其求解。通过结合真实人体医学影像数据（包括MRI和CT），验证模型及算法的有效性，旨在提取具有临床价值的重要信息来指导相关疾病的诊断与治疗。主要研究内容包括：（1）基于核磁共振的分数阶模型研究MRI信号的反常衰减现象，着重介绍模型和算法的可靠性；（2）基于CT影像提取的人体血管几何模型进行大规模个性化血流数值模拟，重点强调算法的高可扩展性。项目开展三年以来，共发表高水平学术论文6篇，其中SCI收录5篇，EI收录1篇，多次参加学术交流会议，并邀请同行专家进行学术交流指导。