基于医学影像的病灶识别的数学方法及理论

The project takes aim at the growth mechanism of the vascular plaques and the lesions in tissue nodules, and the features in medical images including morphology, anatomy, function, etc.: 1) In this project, the feature extraction and mathematical representations of lesions will be derived. The mathematical models on lesion recognition, classifications and segmentation are proposed. And variational partial differential equation models and geometric models of registration, fusion, tracking and evaluationof medical images will be built; 2) The well-posedness of the proposed mathematical models will be studied. Furthermore, positivity of solutions, propagation of level sets and support sets, measure estimates of nodal sets and singular sets of solutions will be studied; 3) We will establish the efficient and fast algorithms of proposed models and further to investigate the algorithms’ convergence, error estimation, convergence speed, stability, etc.; 4) We will analyzethe cases and data of vascular plaques and tissue nodules in clinic and do simulations by applying the proposed models, theories and algorithms, to improve the precision and accuracy of lesion recognition in clinic. The analytical and geometric method of lesion recognition is rare and the corresponding mathematical theories and theories of algorithms are vacant. Since the proposed mathematical models are accompanied with the items of feature, degeneration, non-convexity and so on, it has important significances on mathematical theories by establishing the behavior of solutions, theory of algorithms, etc., which can promote the in-depth understanding of images and will be helpful for the accurate diagnosis and treatment of lesions based on medical images.

本项目针对血管斑块和组织结节等病灶生长机制和医学影像的病灶的形态、解剖、功能等特征，1）研究给出病灶特征的提取和数学表示，建立病灶识别、分类与分割的数学模型，以及病灶医学影像的配准、融合、跟踪与评价的变分偏微分方程和几何模型；2）研究所建立的数学模型的适定性，进一步研究解的正性、水平集和支撑集的传播、零点集和奇异集的测度估计等；3）建立所提出模型的高效快速算法，进一步研究算法的收敛性、误差估计、收敛速度、稳定性等； 4）应用所建立的模型、理论和算法对临床血管斑块和组织结节的病例和数据进行分析和模拟，提高临床病灶识别的精确性和准确性。目前针对病灶识别的分析和几何方法非常少见，相应的数学理论和算法理论几乎是空白。 由于出现的数学模型带特征项、退化、非凸等，建立解的性态、算法理论等具有重要的数学理论意义，能促进对图像的深入理解，又能为基于医学影像的病灶的精确诊断治疗提供帮助。

本项目自实施以来，紧紧围绕着研究目标开展研究，以肝癌和斑块两类病灶的影像为切入点，应用几何先验、测度论和变分等结合深度学习，探索提取人体血管斑块、肿瘤等病变（灶）的影像特征及其数学表示；应用变分学、微分几何、偏微分方程和数学形态学等的理论和思想, 分别建立器官、血管斑块和组织结节等病灶（变）的医学图像的分类、分割、配准和融合的精准高效模型。建立了相应的偏微分方程、微分几何模型解的适定性、正则性、零点集、临界点集和奇异集的测度估计和几何结构等。构建适用于临床精度和速度的高效快速算法，进一步研究算法的收敛性、稳定性等。应用所建立的模型、理论和算法对临床病灶影像数据开展模拟和三维重建，提高了对肝癌、血管及斑块的识别和定位的精度和速度。完成了项目的研究目标。发表研究论文40篇，获得国家授权发明专利3项，11名研究生获得博士学位，23名研究生获得硕士学位。