In this study, we model avascular tumour growth in epithelial tissue. This can help us to understand that how an avascular tumour interacts with its microenvironment and what type of physical changes can be observed within the tumour spheroid before angiogenesis. This understanding is likely to assist in the development of better diagnostics, improved therapies, and prognostics. In biological systems, most of the diffusive processes are through cellular membranes which are porous in nature. Due to its porous nature, diffusion in biological systems are heterogeneous. The fractional diffusion equation is well suited to model heterogeneous biological systems, though most of the early studies did not use this fact. They described tumour growth with simple diffusion-based model. We have developed a spherical model based on simple diffusion initially, and then the model is upgraded with fractional diffusion equations to express the anomalous nature of biological system. In this study, two types of fractional models are developed: one of fixed order and the other of variable order. The memory formalism technique is also included in these anomalous diffusion models. These three models are investigated from phenomenological point view by measuring some parameters for characterizing avascular tumour growth over time. Tumour microenvironment is very complex in nature due to several concurrent molecular mechanisms. Diffusion with memory (fixed as well as variable) formation may be an oversimplified technique, and does not reflect the detailed view of the tumour microenvironment. However, it is found that all the models offer realistic and insightful information of the tumour microenvironment at the macroscopic level, and approximate well the physical phenomena. Also, it is observed that the anomalous diffusion based models offer a closer description to clinical facts than the simple model. As the simulation parameters get modified due to different biochemical and biophysical processes, the robustness of the model is determined. It is found that the anomalous diffusion models are moderately sensitive to the parameters.

中文翻译：

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基于异常扩散的无血管肿瘤生长模型

在这项研究中，我们对上皮组织中的无血管肿瘤生长进行建模。这可以帮助我们了解无血管肿瘤如何与其微环境相互作用，以及在血管生成之前可以在肿瘤球体内观察到哪些类型的物理变化。这种理解可能有助于开发更好的诊断方法、改进的治疗方法和预后。在生物系统中，大多数扩散过程是通过本质上多孔的细胞膜进行的。由于其多孔性质，生物系统中的扩散是异质的。分数扩散方程非常适合模拟异质生物系统，尽管大多数早期研究没有使用这一事实。他们用简单的基于扩散的模型描述了肿瘤的生长。我们最初开发了基于简单扩散的球形模型，然后用分数扩散方程对模型进行升级，以表达生物系统的异常性质。在这项研究中，开发了两种类型的分数模型：一种是固定阶数，另一种是可变阶数。记忆形式主义技术也包含在这些异常扩散模型中。通过测量一些参数来表征无血管肿瘤随时间的生长，从现象学的角度对这三个模型进行了研究。由于几种并发的分子机制，肿瘤微环境在本质上非常复杂。扩散与记忆（固定以及可变）形成可能是一种过于简单化的技术，并不能反映肿瘤微环境的详细视图。然而，发现所有模型都在宏观层面提供了肿瘤微环境的真实和有洞察力的信息，并且很好地近似了物理现象。此外，据观察，与简单模型相比，基于异常扩散的模型提供了对临床事实的更接近的描述。由于模拟参数因不同的生化和生物物理过程而被修改，因此确定了模型的稳健性。发现异常扩散模型对参数具有中等敏感性。确定模型的稳健性。发现异常扩散模型对参数具有中等敏感性。确定模型的稳健性。发现异常扩散模型对参数具有中等敏感性。