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Radial basis function-generated finite difference scheme for simulating the brain cancer growth model under radiotherapy in various types of computational domains.
Computer Methods and Programs in Biomedicine ( IF 6.1 ) Pub Date : 2020-07-10 , DOI: 10.1016/j.cmpb.2020.105641
Mehdi Dehghan 1 , Niusha Narimani 1
Affiliation  

Background and Objectives

We extend the original mathematical model, i.e., Swanson’s reaction-diffusion equation to the surfaces with no boundary, and we find a new numerical method based on a meshless approach for solving numerically Swanson’s reaction-diffusion model in the square and on the sphere.

Methods

To solve numerically the Swanson’s reaction-diffusion model and its extension version, a collocation meshless technique, namely radial basis function-generated finite difference (RBF-FD) scheme is employed for approximating the spatial variables in the square domain and on the sphere, respectively. Also, to approximate the time variable of the studied models, a first-order semi-implicit backward Euler scheme is used. The resulting fully discrete scheme is a linear system of algebraic equations per time step that is solved via the biconjugate gradient stabilized (BiCGSTAB) iterative algorithm with a zero-fill incomplete lower-upper (ILU) preconditioner.

Results

The numerical simulations show the growth of untreated and treated brain tumors with radiotherapy using estimated and clinical data (given from magnetic resonance imaging (MRI) scans of patients). Moreover, the results reported here can be used for improving the treatment strategies of the invasive brain tumor.

Conclusions

Using the developed numerical scheme in this paper, we can simulate the behavior of the invasive form of brain tumor response to radiotherapy. Also, we can see the effects of radiation response on the brain tumor cell concentration of individual patients. The proposed meshless technique, which is applied for solving numerically the studied model, does not depend on any background mesh or triangulation for approximation in comparison with mesh-dependent methods. Moreover, we apply this technique to the sphere via any set of distributed points easily.



中文翻译:

径向基函数生成的有限差分方案,用于在各种类型的计算域中模拟放射治疗下的脑癌生长模型。

背景和目标

我们将原始的数学模型(即斯旺森反应扩散方程)扩展到无边界的表面,并找到了一种基于无网格方法的新数值方法,用于求解正方形和球面上的斯旺森反应扩散模型。

方法

为了对Swanson的反应扩散模型及其扩展版本进行数值求解,采用搭配无网格技术,即径向基函数生成的有限差分(RBF-FD)方案分别逼近平方域和球面上的空间变量。 。同样,为了近似研究模型的时间变量,使用了一阶半隐式后向欧拉方法。产生的完全离散方案是每个时间步长的线性代数方程组系统,可通过双共轭梯度稳定(BiCGSTAB)迭代算法和零填充不完全下上部(ILU)前置条件求解。

结果

数值模拟显示了使用估计和临床数据(来自患者的磁共振成像(MRI)扫描)进行放疗后未治疗和已治疗的脑肿瘤的生长情况。此外,此处报道的结果可用于改善浸润性脑肿瘤的治疗策略。

结论

使用本文中开发的数值方案,我们可以模拟侵袭性形式的脑肿瘤对放疗的反应。此外,我们可以看到辐射反应对个别患者脑肿瘤细胞浓度的影响。所提出的无网格技术用于数值求解所研究的模型,与依赖于网格的方法相比,它不依赖于任何背景网格或三角剖分进行近似。此外,我们可以通过任意一组分布点将这种技术轻松地应用于球体。

更新日期:2020-07-10
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