Despite major scientific, medical and technological advances over the last few decades, a cure for cancer remains elusive. The disease initiation is complex, and including initiation and avascular growth, onset of hypoxia and acidosis due to accumulation of cells beyond normal physiological conditions, inducement of angiogenesis from the surrounding vasculature, tumour vascularization and further growth, and invasion of surrounding tissue and metastasis. Although the focus historically has been to study these events through experimental and clinical observations, mathematical modelling and simulation that enable analysis at multiple time and spatial scales have also complemented these efforts. Here, we provide an overview of this multiscale modelling focusing on the growth phase of tumours and bypassing the initial stage of tumourigenesis. While we briefly review discrete modelling, our focus is on the continuum approach. We limit the scope further by considering models of tumour progression that do not distinguish tumour cells by their age. We also do not consider immune system interactions nor do we describe models of therapy. We do discuss hybrid-modelling frameworks, where the tumour tissue is modelled using both discrete (cell-scale) and continuum (tumour-scale) elements, thus connecting the micrometre to the centimetre tumour scale. We review recent examples that incorporate experimental data into model parameters. We show that recent mathematical modelling predicts that transport limitations of cell nutrients, oxygen and growth factors may result in cell death that leads to morphological instability, providing a mechanism for invasion via tumour fingering and fragmentation. These conditions induce selection pressure for cell survivability, and may lead to additional genetic mutations. Mathematical modelling further shows that parameters that control the tumour mass shape also control its ability to invade. Thus, tumour morphology may serve as a predictor of invasiveness and treatment prognosis.

中文翻译：

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癌症的非线性建模：弥合细胞和肿瘤之间的差距


尽管过去几十年来科学、医学和技术取得了重大进步，但癌症的治愈方法仍然难以实现。疾病的起始是复杂的，包括起始和无血管生长、由于细胞积累超出正常生理条件而引起的缺氧和酸中毒、诱导周围脉管系统的血管生成、肿瘤血管化和进一步生长、以及周围组织的侵袭和转移。尽管历史上的重点是通过实验和临床观察来研究这些事件，但能够在多个时间和空间尺度上进行分析的数学建模和模拟也补充了这些努力。在这里，我们概述了这种多尺度模型，重点关注肿瘤的生长阶段并绕过肿瘤发生的初始阶段。虽然我们简要回顾了离散建模，但我们的重点是连续体方法。我们通过考虑不按年龄区分肿瘤细胞的肿瘤进展模型来进一步限制范围。我们也不考虑免疫系统的相互作用，也不描述治疗模型。我们确实讨论了混合建模框架，其中使用离散（细胞尺度）和连续体（肿瘤尺度）元素对肿瘤组织进行建模，从而将微米与厘米肿瘤尺度联系起来。我们回顾了最近将实验数据纳入模型参数的示例。我们表明，最近的数学模型预测细胞营养物质、氧气和生长因子的运输限制可能导致细胞死亡，从而导致形态不稳定，从而提供了通过肿瘤指状和碎片侵袭的机制。 这些条件会引起细胞生存能力的选择压力，并可能导致额外的基因突变。数学模型进一步表明，控制肿瘤块形状的参数也控制其侵袭能力。因此，肿瘤形态可以作为侵袭性和治疗预后的预测因子。