Anomalous reaction-transport equations: applications to the theory of cancer spreading and subdiffusion in dendrites

具有异常输运的反应输运系统具有重要的实际意义，因为它们为复杂介质如无序固体，随机多孔介质，活组织等提供了现实的模型。所提出的研究将导致对具有异常输运的复杂介质的基本性质的理解增加。应用包括树突中的异常离子传输，流行病和癌细胞的传播，固体氧化物燃料电池中的电化学过程，人类或动物群体的分散，复杂的化学反应和污染物传输。拟议的研究具有潜在的经济影响，化学和核工业的传统方法的基础上使用的反应扩散模型。该项目的目的是建立应用数学家，化学和核工业工程师，神经生理学家，计算神经科学家和细胞生物学家之间的对话。为了确保他们能够受益，并将我们的发现传达给广大受众，我们打算在相关期刊上发表论文。我们希望这个项目将允许应用数学家和神经生物学和细胞生物学的研究人员合作，从而能够在生物系统内的异常运输领域取得重大进展。与神经生物学家和细胞生物学专家（项目合作伙伴）的合作在提高生活质量和健康方面具有潜在的社会影响。与美国德克萨斯大学圣安东尼奥神经科学研究所的Santamaria博士合作，旨在了解棘状树突的亚扩散如何调节强调学习和记忆的突触可塑性。我们将提供一个新的异常运输理论和项目合作伙伴将有助于测量浦肯野和海马锥体细胞亚扩散的实验。与我们的项目合作伙伴，美国新墨西哥州大学病理学系的Chauviere博士合作，旨在提供一种新的理论工具，可用于设计控制癌细胞侵袭的新疗法。临床干预旨在通过应用增加细胞死亡率或降低细胞运动性的化疗药物来延缓恶性侵袭。我们的数学模型可以提供重要的见解死亡率和异常运动之间的关系。该研究项目将通过贡献科学知识和新的想法在异常运输反应系统的多学科领域，并扩展英国的研究专业知识到这个新的数学领域产生现实的影响。我们打算开发分数阶偏微分方程的创新方法，并提高人们对这些方程在工业和学术界的重要性的认识。该项目将促进与美国和西班牙的国际研究合作。

Reaction-transport systems with anomalous transport are of great practical importance because they provide realistic models for complex media such as disordered solids, random porous media, living tissues, etc. The proposed research will lead to an increased understanding of the fundamental properties of complex media with anomalous transport. Applications include anomalous ion transport in dendrites, spread of epidemics and cancer cells, electrochemical processes in solid oxide fuel cells, dispersion of human or animal groups, complex chemical reactions and contaminant transport. The proposed research has a potential economic impact for chemical and nuclear industries where traditional approaches based on reaction-diffusion models are used. The aim of this project is to establish dialogues between applied mathematicians, engineers from chemical and nuclear industries, neurophysiologists, computational neuroscientists and cell biologists. In order to ensure that they can benefit and to communicate our findings to a wide audience we intend to publish papers in relevant journals. We expect that this project will allow applied mathematicians and researchers in neurobiology and cell biology to collaborate and thus to be able to make significant advances in the areas of anomalous transport within biological systems. Collaboration with neurobiologists and experts in cell biology (Project Partners) has a potential social impact in enhancing quality of life and health. Collaboration with Dr. Santamaria from Neuroscience Institute at The University of Texas at San Antonio, USA will aim at understanding how subdiffusion in spiny dendrites regulates synaptic plasticity that underlines learning and memory. We will provide a new anomalous transport theory and the Project Partner will contribute experiments in measuring subdiffusion in Purkinje and hippocampal pyramidal cells. Collaboration with our Project Partner Dr. Chauviere from Department of Pathology, University of New Mexico, USA will aim at providing a new theoretical tool that can be potentially used in designing new therapies to control cancer cell invasion. Clinical interventions aim at retarding malignant invasion by applying chemotherapeutic drugs that increase the death rate of the cells or reduce cell motility. Our mathematical models can provide important insights into the relationship between the death rate and anomalous motility. The research project will have a realistic impact by contributing scientific knowledge and new ideas in multidisciplinary area of anomalous transport-reaction systems and extending UK research expertise into this new area of mathematics. We intend to develop innovative methodologies for fractional partial differential equations and raise awareness of the importance of these equations in industries and academia. This project will foster international research collaborations with the USA and Spain.