Roniotis Alexandros, Marias Kostas, Sakkalis Vangelis, Tsibidis George D, Zervakis Michalis
Institute of Computer Science, Foundation for Research and Technology (FORTH), Heraklion 71110, Greece.
Annu Int Conf IEEE Eng Med Biol Soc. 2009;2009:2807-10. doi: 10.1109/IEMBS.2009.5333776.
Glioma is the most aggressive type of brain cancer. Several mathematical models have been developed towards identifying the mechanism of tumor growth. The most successful models have used variations of the diffusion-reaction equation, with the recent ones taking into account brain tissue heterogeneity and anisotropy. However, to the best of our knowledge, there hasn't been any work studying in detail the mathematical solution and implementation of the 3D diffusion model, addressing related heterogeneity and anisotropy issues. To this end, this paper introduces a complete mathematical framework on how to derive the solution of the equation using different numerical approximation of finite differences. It indicates how different proliferation rate schemes can be incorporated in this solution and presents a comparative study of different numerical approaches.
神经胶质瘤是最具侵袭性的脑癌类型。已经开发了几种数学模型来确定肿瘤生长的机制。最成功的模型使用了扩散反应方程的变体，最近的模型考虑了脑组织的异质性和各向异性。然而，据我们所知，尚未有任何工作详细研究三维扩散模型的数学解和实现，以及解决相关的异质性和各向异性问题。为此，本文介绍了一个完整的数学框架，说明如何使用不同的有限差分数值近似来推导方程的解。它指出了不同的增殖率方案如何能够纳入该解，并对不同的数值方法进行了比较研究。
