组织-细胞多尺度耦合的实体肿瘤生长数学建模与算法研究

Even though there are a number of advances in cancer research to clarify the cause of cancer and develop effective treatments in last few decades, we still have a long way to study the cancer, especially for the research of mathematical analysis, mathematical modeling, and computational method. Mathematical modeling and computational simulation are essential for understanding the dynamics of the complex cell self-control system and mechanism of tumor growth and cancer metastases. They also can help to develop cancer drugs and understand how circumvention of that control leads to cancer. The purpose of this project is to develop the multiscale mathematical modeling for solid tumor growth by considering both micro and macro scale tumor biology, tumor angiogenic factors, tumor microenvironment, tumor vessel and so on. The project will prove the existence of solution in analysis and try to describe the dynamics behavior of the proposed model. We will study the efficient and stable multiscale finite difference method and prove the accuracy, convergence, and stability of the proposed numerical scheme. We also will develop the adaptive mesh refinement and parallel computing method for proposed solid tumor modeling. Various numerical experiments will be performed to study the mechanism of solid tumor growth, to study the relationship between tumor factors, and to try to find the way of controlling tumor growth and cancer metastasis suppression. The project is a frontier project in computational mathematics and biological sciences, which will be of great significance for studying mechanism of solid tumor growth and cancer metastases and for developing the advanced high-performance computer simulation methods in biological sciences.

近几十年来，在致癌因素的查找及癌症的早期治疗等方面已取得巨大进展，但对癌症的研究仍任重道远，特别是从理论研究、数学建模及计算模拟方面。有效的数学建模与计算模拟，有助于研究复杂的细胞自我控制机理，理解肿瘤生长与癌细胞转移机制，有助于加速抗癌药物的研发和癌症抑制途径的探索。本项目将在实体肿瘤组织尺度和细胞尺度下，考虑致癌因子、肿瘤周围微环境、肿瘤血管等因素，进行实体肿瘤生长机理的多尺度耦合建模，揭示模型的动力学行为；建立模型的多尺度高效稳定算法，研究算法的稳定性、收敛性等基本数值性质，发展符合实体肿瘤生长模型的自适应网格算法和并行算法等高性能计算方法；并与医院合作开展实验验证；项目研究试图揭示实体肿瘤的生长新现象、探索抑制/控制肿瘤生长的途径，为医学上的肿瘤生长研究提供支持。该项目属数学与生命科学交叉的前沿研究，对肿瘤生长机理的研究以及探索符合肿瘤生长的高性能计算模拟有重要意义。

本项目综合运用了数学、生命科学与复杂物理场耦合的知识，在相场模型的框架下，考虑肿瘤周围微环境、肿瘤血管等因素，进行实体肿瘤生长机理的多尺度耦合建模；在数值方法方面，研究了适应复杂区域、复杂曲面的相场模型，研究算法的稳定性、收敛性等数值性质，发展自适应网格算法和并行算法等高性能计算方法；具体研究内容包括：建立组织-细胞两尺度耦合的肿瘤生长模型，分析了模型的动力学行为，发展了相对应的计算方法；开展了经典相场模型的动力学行为研究，提出了达到四阶精度的高效稳定算法，证明了算法的稳定性和理想收敛阶，发展与之相适应的多重网格算法和自适应网格算法；开展了曲面上的物质聚合和分离的建模和计算模拟，提出了达到时间和空间的二阶精度的稳定算法，证明了算法的无条件稳定性；设计了流场和压力场吻合的边界计算模型，并提出了简单而有效的多相流体计算方法，发展了相应的求解器；受本基金资助，已发表16篇SCI论文，协助培养了5名学生。相关研究成果在理论方面能够丰富实体肿瘤生长研究和相场模型的研究，在应用方面能够指导实体肿瘤治疗的研究，并且推动了数学、生命科学与复杂物理场耦合的交叉发展。

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