The present mathematical models of microtumours consider, in general, volumetric growth and spherical tumour invasion shapes. Nevertheless in many cases, such as in gliomas, a need for more accurate delineation of tumour infiltration areas in a patient-specific manner has arisen. The objective of this study was to build a mathematical model able to describe in a case-specific way as well as to predict in a probabilistic way the growth and the real invasion pattern of multicellular tumour spheroids (in vitro model of an avascular microtumour) immersed in a collagen matrix. The two-dimensional theoretical model was represented by a reaction-convection-diffusion equation that considers logistic proliferation, volumetric growth, a rim with proliferative cells at the tumour surface and invasion with diffusive and convective components. Population parameter values of the model were extracted from the experimental dataset and a shape function that describes the invasion area was derived from each experimental case by image processing. New possible and aleatory shape functions were generated by data mining and Monte Carlo tools by means of a satellite EGARCH model, which were fed with all the shape functions of the dataset. Then the main model is used in two different ways: to reproduce the growth and invasion of a given experimental tumour in a case-specific manner when fed with the corresponding shape function (descriptive simulations) or to generate new possible tumour cases that respond to the general population pattern when fed with an aleatory-generated shape function (predictive simulations). Both types of simulations are in good agreement with empirical data, as it was revealed by area quantification and Bland-Altman analysis. This kind of experimental-numerical interaction has wide application potential in designing new strategies able to predict as much as possible the invasive behaviour of a tumour based on its particular characteristics and microenvironment.

中文翻译：

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基于体外实验的微肿瘤浸润的数学模型。

当前的微肿瘤数学模型通常考虑体积生长和球形肿瘤浸润形状。然而，在许多情况下，例如在神经胶质瘤中，已经出现了以患者特异性方式更精确地描绘肿瘤浸润区域的需求。这项研究的目的是建立一个数学模型，该模型能够以案例特定的方式描述并以概率方式预测浸没的多细胞肿瘤球体（无血管微肿瘤的体外模型）的生长和实际侵袭模式在胶原蛋白基质中。二维理论模型由反应-对流-扩散方程表示，该方程考虑逻辑扩散，体积增长，肿瘤表面有增殖细胞的边缘以及扩散和对流成分的侵入。从实验数据集中提取模型的种群参数值，并通过图像处理从每个实验案例中得出描述入侵区域的形状函数。数据挖掘和蒙特卡洛工具借助卫星EGARCH模型生成了新的可能的和偶然的形状函数，这些函数被提供了数据集的所有形状函数。然后以两种不同的方式使用主要模型：在喂入相应的形状函数后，以案例特定的方式重现给定实验性肿瘤的生长和侵袭（描述性模拟），或生成新的可能的肿瘤案例以应对特定的肿瘤。用偶然生成的形状函数喂食时的总体种群模式（预测模拟）。两种模拟都与经验数据非常吻合，正如通过面积定量和Bland-Altman分析所揭示的那样。这种实验-数字相互作用在设计新策略方面具有广泛的应用潜力，这些新策略能够根据肿瘤的特定特征和微环境来尽可能地预测肿瘤的侵袭行为。