In this paper we introduce an efficient numerical method in order to solve Volterra integral equations (VIE) of the second type. We are motivated by the fact that the coupled PDE-ODE model, used to describe the metastatic tumor growth, can be reformulated in terms of VIE, whose unknowns are biological observables, such as the cumulative number of metastases and the total metastatic mass. Here in particular we focused our attention on the 2D non autonomous case, where also the treatment is considered. After reformulating the model as a VIE and introducing and studying the numerical method, we first compare it with a method previously introduced by the authors for the 1D case, and extended to the 2D case only for the sake of comparison, in term of efficiency in the run time execution. Secondly, we present numerical results on the effectiveness of different treatment protocols on the total cumulative number of metastases and the total metastatic mass.

中文翻译：

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治疗转移性肿瘤生长模型的数值解


在本文中，我们引入了一种有效的数值方法来求解第二类 Volterra 积分方程（VIE）。我们的动机是，用于描述转移性肿瘤生长的耦合 PDE-ODE 模型可以根据 VIE 重新表述，其未知数是生物可观测值，例如转移灶的累积数量和总转移质量。在这里，我们特别关注二维非自主情况，其中也考虑了治疗。在将模型重新表述为 VIE 并介绍和研究数值方法之后，我们首先将其与作者之前针对 1D 情况介绍的方法进行比较，并且仅出于比较的目的而扩展到 2D 情况，就效率而言运行时执行。其次，我们提出了不同治疗方案对总累积转移数和总转移质量的有效性的数值结果。