Abstract Cell migration is fundamental in a wide variety of physiological and pathological phenomena, being exploited in biomedical engineering as well. In this respect, we here present a hybrid non-local integro-differential model where a representative cell, reproduced by a point particle with an orientation, moves on a planar domain upon signals coming from environmental variables. From a numerical point of view, non-locality implies the need to evaluate integral terms which may present non-regular integrand functions because of heterogeneities in the environmental conditions and/or in cell sensing region. Having in mind multicellular applications, we here propose a robust computational method able to handle such non-regularities. The procedure is based on low order Runge–Kutta methods and on an ad hoc application of the Gauss–Legendre quadrature rule. The accuracy and efficiency of the resulting computational method is then tested by selected benchmark settings. In this context, the ad hoc application of the quadrature rule reveals to be crucial to obtain a high accuracy with a remarkably low number of quadrature nodes with respect to the standard Gauss–Legendre quadrature formula, and which thus results in a reduced overall computational cost. Finally, the proposed method is further coupled with the cubic spline interpolation scheme which allows to deal also with possible poor (i.e., point-wise defined) molecular spatial information. The performed simulations (which accounts also for different scenarios) show how the interpolation of the molecular variables affects the efficiency of the overall method and further justify the proposed procedure.

中文翻译：

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细胞迁移的积分微分非局部模型及其有效数值解

摘要 细胞迁移是各种生理和病理现象的基础，也被用于生物医学工程。在这方面，我们在这里提出了一个混合非局部积分微分模型，其中由具有方向的点粒子复制的代表性细胞根据来自环境变量的信号在平面域上移动。从数值的角度来看，非局部性意味着需要评估积分项，由于环境条件和/或细胞传感区域的异质性，这些积分项可能会呈现非规则的被积函数。考虑到多细胞应用，我们在这里提出了一种能够处理这种不规则性的强大计算方法。该过程基于低阶 Runge-Kutta 方法和 Gauss-Legendre 求积法则的特殊应用。然后通过选定的基准设置测试所得计算方法的准确性和效率。在这种情况下，正交规则的临时应用表明对于以相对于标准 Gauss-Legendre 正交公式而言非常少的正交节点数量获得高精度至关重要，从而降低整体计算成本. 最后，所提出的方法进一步与三次样条插值方案相结合，该方案还允许处理可能的不良（即逐点定义）分子空间信息。