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

Lead Research Organisation: University of Manchester
Department Name: Mathematics

Abstract

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.

Planned Impact

The proposed research will lead to an increased understanding of the fundamental properties of complex media with anomalous transport such as disordered solids, random porous media, glassy materials, living tissues, etc. The new mathematical models involving fractional differential equations to be developed in this project will be of use to not only applied mathematicians and theoretical physicists but also engineers working on fuel cells and heterogeneous catalysis in chemical industry, engineers and managers working on the underground waste storages in nuclear industry, the scientists in biophysics, biochemistry and chemical engineering, population biologists, neurophysiologists and computational neuroscientists. Applications involve a vast range of interdisciplinary fields including anomalous ion transport in dendrites, spread of epidemics and cancer cells, contamination of groundwater, dispersion of human or animal groups, complex chemical reactions and the electrochemical prosesses within a solid fuel cells. The proposed research has a potential economic impact for chemical industry where traditional reaction-diffusion models are used. The new fractional systems of partial differential equations promise to make modelling of industrial reaction-transport processes in complex structures more accurate and cost-effective. These equations can be incorporated into current reaction-diffusion codes used in chemical industry. This could lead to better products or cost reduction for consumers. It has been recognized recently that the transport of substances in random porous media is anomalous. Therefore, this project has a potential impact for nuclear industry involving underground nuclear waste storage management. Collaboration with neurobiologists and experts in cell motility (Project Partners) has a potential social impact in improving of quality of life and healthcare. The new fractional models of transport in living tissues can be used to predict clinically relevant outcomes as invasion rates of glioma. These models can be also used for testing the efficiency of strategies aiming at reducing the invasion rate of tumour. Thus, the research project will have a realistic impact by (1) contributing scientific knowledge to multidisciplinary area of fractional transport-reaction systems and extending UK research expertise; (2) development innovative methodologies for the system of fractional partial differential equations with reactions and raising awareness of the importance of these equations for industries, applied mathematics, chemistry and biology; (3) training a PDRA in a new area of mathematics of anomalous reaction-transport systems; (4) establishing dialogues between applied mathematicians, engineers from chemical and nuclear industries, neurophysiologists, computational neuroscientists and cell biologists; (5) fostering international research collaborations with the USA and Europe.

Publications

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Falconer S (2015) Nonlinear Tempering of Subdiffusion with Chemotaxis, Volume Filling, and Adhesion in Mathematical Modelling of Natural Phenomena

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Fedotov S (2016) Single integrodifferential wave equation for a Lévy walk. in Physical review. E

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Fedotov S (2013) Nonlinear subdiffusive fractional equations and the aggregation phenomenon. in Physical review. E, Statistical, nonlinear, and soft matter physics

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Fedotov S (2014) Nonlinear degradation-enhanced transport of morphogens performing subdiffusion. in Physical review. E, Statistical, nonlinear, and soft matter physics

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Fedotov S (2015) Persistent random walk of cells involving anomalous effects and random death. in Physical review. E, Statistical, nonlinear, and soft matter physics

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Fedotov S (2015) Self-organized anomalous aggregation of particles performing nonlinear and non-Markovian random walks. in Physical review. E, Statistical, nonlinear, and soft matter physics

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Fedotov S (2013) Random death process for the regularization of subdiffusive fractional equations. in Physical review. E, Statistical, nonlinear, and soft matter physics

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Fedotov S (2015) Subdiffusion in an external potential: Anomalous effects hiding behind normal behavior. in Physical review. E, Statistical, nonlinear, and soft matter physics

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Fedotov S (2013) Non-homogeneous Random Walks, Subdiffusive Migration of Cells and Anomalous Chemotaxis in Mathematical Modelling of Natural Phenomena

 
Description We have developed a new fractional Fokker-Planck equation
that describes the cell's motility and signalling molecules in dendrites.
We have implemented a random death process into a persistent random walk model which produces sub-ballistic superdiffusion (Levy walk).
We have derived mesoscopic integro-differential and fractional reaction-transport equations for
number densities when the underlying movement is anomalous. We have used a probabilistic approach that takes into account non-Markovian processes with memory effects and multiscale heterogeneous spatial structures.
We have focused on two-component models with age-dependent kinetics and non-homogeneous medium.
We have developed a stochastic two-velocity jump model of cancer cell motility for which the switching rate depends upon the time which the cell has spent moving in one direction.
This describes the anomalous persistence of cell
motility: the longer the cell moves in one direction, the smaller the switching probability to another direction
becomes.
We have derived master equations for the cell densities with the generalized switching terms involving the tempered fractional material derivatives. We show that the random death of cancer cells has an important implication for the transport process through tempering of the superdiffusive transport of cells.
Exploitation Route new mathematical tools for the analysis of anomalous transport
of cancer cells
Sectors Environment,Healthcare,Pharmaceuticals and Medical Biotechnology

URL http://www.maths.manchester.ac.uk/~sf/anomalousdiffusion/index.html