Anomalous diffusion in pattern-forming systems, and applications

图案形成系统中的反常扩散及其应用

• 批准号: 1108624
• 负责人: Bernard Matkowsky
• 金额: $ 35.6万
• 依托单位: Northwestern University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-08-01 至 2014-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1108624&HistoricalAwards=false
• 关键词: Anomalous; diffusion; pattern; forming; systems

This project is devoted to development of a theory of pattern formation, nonlinear dynamics and transport in systems with anomalous diffusion. Applications to a number of significant problems, such as pattern-forming reaction-diffusion problems and drug delivery problems, will be investigated. Unlike regular diffusion, in anomalous diffusion the mean square displacement behaves as a power function of time. If the exponent is less than one the diffusion process is slower than normal diffusion (subdiffusion), and if the exponent is greater than one it is faster than normal (superdiffusion). Mathematical description of anomalous diffusion involves integro-differential operators which have to be derived from appropriate continuous time random walk models and leads to novel mathematical problems. Specifically, the investigators study(i) Pattern formation in growing domains, both for normal diffusion and anomalous diffusion, focusing on the singular perturbation case when some of the diffusion coefficients are asymptotically small;(ii) Turing pattern selection in reaction - anomalous diffusion systems, in particular, the stripes-spots selection;(iii) Drug delivery problems which include development of an approximate analytic theory of subdiffusive problems with moving free boundaries; study of models of bioerodible controlled drug delivery devices governed by subdiffusive transport of the chemicals, transdermal drug release in the presence of an electric field i.e. accompanied by iontophoresis and others.Controlled drug delivery has been attracting a great deal of attention in the medical community for years as an efficient way of providing treatment for a wide class of diseases. Various drug delivery devices are based on mass transfer of the given drug towards particular organs, in which either the mass transfer rate, or place, or both are prescribed according to certain medical protocols. Much progress has been achieved in the design and development of various controlled drug delivery systems, and many people routinely take medicine designed for controlled release. Mathematical modeling of drug delivery systems is very important since it can provide a better understanding and a quantitative description of the physical, chemical and biological processes governing the performance of the systems. On the basis of this description, better controlled drug delivery systems can be designed. There exists experimental evidence that drug diffusion toward the biological target is not normal but rather slow, so-called sub-diffusion, as the drug molecule has to diffuse through a very crowded environment. The investigators will study drug transport governed by sub-diffusion in order to obtain better understanding of drug delivery processes. In addition, the investigators will study problems of pattern formation in reaction-diffusion systems where subdiffusion, as well as superdiffusion are important. The latter is typical of some processes in plasmas, semiconductors, surface reactions and many others. Training of a PhD student through this research is an integral part of the project.

该项目致力于发展具有反常扩散的系统中的图案形成、非线性动力学和输运理论。应用到一些重要的问题，如图案形成反应扩散问题和药物输送问题，将进行调查。与常规扩散不同，在反常扩散中均方位移表现为时间的幂函数。如果指数小于1，则扩散过程比正常扩散慢（亚扩散），如果指数大于1，则比正常扩散快（超扩散）。异常扩散的数学描述涉及到积分微分算子，这些算子必须从适当的连续时间随机游走模型中导出，并导致新的数学问题。具体来说，研究人员研究（i）增长域中的模式形成，包括正常扩散和异常扩散，重点是当一些扩散系数渐近小时的奇异扰动情况;（ii）反应-异常扩散系统中的图灵模式选择，特别是条纹-斑点选择;（三）药物输送问题，其中包括发展一个近似分析理论的次扩散问题与移动的自由边界;研究受化学品亚扩散运输控制的生物可侵蚀的受控药物递送装置的模型，在电场存在下的经皮药物释放，即伴随离子电渗疗法等。药物控释在医学领域已经吸引了大量的关注。多年来，社区一直是为多种疾病提供治疗的有效方式。各种药物递送装置基于给定药物向特定器官的质量传递，其中质量传递速率或位置或两者根据某些医疗协议规定。在设计和开发各种受控药物递送系统方面已经取得了很大进展，许多人常规服用为受控释放设计的药物。药物传递系统的数学建模是非常重要的，因为它可以提供一个更好的理解和定量描述的物理，化学和生物过程控制的系统的性能。在此描述的基础上，可以设计更好的受控药物递送系统。有实验证据表明，药物向生物靶标的扩散不是正常的，而是相当缓慢的，所谓的亚扩散，因为药物分子必须通过非常拥挤的环境扩散。研究人员将研究受亚扩散控制的药物转运，以便更好地了解药物递送过程。此外，研究人员还将研究反应扩散系统中的模式形成问题，其中子扩散和超扩散都很重要。后者是等离子体、半导体、表面反应和许多其他过程中的典型过程。通过这项研究培训博士生是该项目的一个组成部分。