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.
The investigator and his colleagues worked on a theory of pattern formation, nonlinear dynamics and transport in systems with anomalous diffusion, and applied it to a number of significant problems such as pattern-forming reaction-diffusion systems and drug delivery problems. Reaction-diffusion systems are ubiquitous in many branches of science and engineering and have been attracting the attention of scientists, engineers and mathematicians for decades. Since the ground-breaking discoveries of Turing who showed that diffusion in a mixture of chemically reacting species could cause instability of a spatially uniform state leading to the formation of spatio-temporal patterns, and Belousov and Zhabotinskii who discovered oscillating chemical reactions, reaction-diffusion systems have become one of the paradigms for the formation of spatio-temporal patterns in systems far from thermodynamic equilibrium. The formation of such fascinating structures as spiral waves, spatially-regular, stationary patterns with different symmetries (hexagonal, stripe, etc.) as well as chemical turbulence have made reaction-diffusion systems the subject of numerous ongoing investigations. A characteristic feature of most R-D systems that have been studied is that diffusion is normal. However, there is experimental evidence of anomalous diffusion that can be either very fast (superdiffusion) or very slow (subdiffusion). In our work we developed theories of pattern formation and nonlinear dynamics, including theories of traveling waves, in sub- and superdiffusive media. 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. The common principle on which various drug delivery devices are based is 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 of 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 very slow, i.e., sub-diffusion, as the drug molecule has to diffuse through a very crowded environment. We studied drug transport governed by subdiffusion in order to obtain better understanding of drug delivery processes. The results of our research are applicable to numerous systems arising in engineering, physics and especially bio-medicine, and are of interest to a broad and diverse audience, including mathematicians, scientists and engineers. The results of the research were incorporated in a number of courses at Northwestern University. The research contributed to the training of a Ph.D. student and provided him with the opportunity to both use existing methods of applied mathematics and to develop new methods to meet the challenges posed by the new problems. The results of the research were broadly disseminated (six papers published and three more are at the final stage of preparation) and a number of invited talks and lectures at national and international conferences were given.
