In this work, we present a phase-field model for tumour growth, where a diffuse interface separates a tumour from the surrounding host tissue. In our model, we consider transport processes by an internal, non-solenoidal velocity field. We include viscoelastic effects with the help of a general Oldroyd-B type description with relaxation and possible stress generation by growth. The elastic energy density is coupled to the phase-field variable which allows to model invasive growth towards areas with less mechanical resistance. The main analytical result is the existence of weak solutions in two and three space dimensions in the case of additional stress diffusion. The idea behind the proof is to use a numerical approximation with a fully-practical, stable and (subsequence) converging finite element scheme. The physical properties of the model are preserved with the help of a regularization technique, uniform estimates and a limit passage on the fully-discrete level. Finally, we illustrate the practicability of the discrete scheme with the help of numerical simulations in two and three dimensions.

翻译：本文提出了一种用于肿瘤生长的相场模型，其中弥散界面将肿瘤与周围宿主组织分隔开来。在我们的模型中，我们考虑了由内部非无散速度场驱动的输运过程。借助包含松弛效应及生长可能产生应力的广义Oldroyd-B型描述，我们纳入了粘弹性效应。弹性能密度与相场变量相耦合，从而能够模拟向机械阻力较小区域的侵袭性生长。主要分析结果是在附加应力扩散情况下，二维和三维空间中弱解的存在性。该证明思路是利用一种完全实用、稳定且（子序列）收敛的有限元格式进行数值逼近。通过正则化技术、一致估计以及在完全离散层面上的极限过程，保留了模型的物理特性。最后，借助二维和三维数值模拟，我们展示了该离散格式的实用性。