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This workshop seeks to bring together topics that highlight multiscale
and hybrid modeling: tumor growth, bifurcation theory and pattern
formation in reaction-diffusion systems with continuously deforming shapes and surfaces.
The rapid development in scientific computing has given new
impetus to mathematical modelling and numerical computations
in Developmental Biology and Bio-medicine. One major scientific
challenge for this century is the coupling of multiscale models
through hybrid modelling: to achieve this, novel mathematical and
numerical methods are needed.
A goal is to identify challenges as well as report on successes
in these areas and thereby cultivate interdisciplinary collaborative
research.
The meeting is held at The University of Sussex,
Mantell Building, Seminar Room 2A01,
on Wed 13th December 2006 . The principal
organisers is Dr. Anotida Madzvamuse, from the University of Sussex with the support
of Dr Petr Plechac from the University of Warwick.
Schedule
Participants
- Imperial College, London
- Sophia Yarilaki
- Imperial College London
- Tomas Alarcon
- Warwick University
- Erwin George
- University of Ulster
- Kurt Saetzler
- University of Sussex
- Anotida Madzvamuse
- Mark Broom
- Zhiyue Zhang
- Vini Pereira
- Jose A. Fernandez Leon
- Vanessa Styles
- Richard Welford
- Racquel Maria Barreira
Abstracts
- Speaker:
- Anotida Madzvamuse
- Title:
- Velocity-induced numerical solutions of reaction-diffusion systems on continuously growing domains
- Adbstract:
-
Reaction-diffusion systems have been widely studied in developmental
biology, chemistry and more recently in financial mathematics. Most of these
systems comprise nonlinear reaction terms which makes it difficult to find
closed form solutions. It therefore becomes convenient to look for numerical
solutions: finite difference, finite element, finite volume and
spectral methods are typical examples of the numerical methods used. Most of
these methods are locally based schemes. We examine the implications of
mesh structure on numerically computed solutions of a well-studied
reaction-diffusion model system on two-dimensional fixed and growing domains.
The incorporation of domain growth creates an additional parameter -- the
grid-point velocity -- and this greatly influences the selection of certain symmetric solutions
for the ADI finite difference scheme when a uniform square mesh structure is used.
Domain growth coupled with grid-point
velocity on a uniform square mesh stabilises certain patterns which are however very sensitive
to any kind of perturbation in mesh structure.
We compare our results to those obtained by use of finite elements on
unstructured triangular elements.
- Speaker:
- Sophia Yaliraki
- Title:
- Multiscale modelling in biomolecular assemblies
- Adbstract:
-
Biomolecular assemblies occur at a range of time and lengthscales that
span many orders of magnitude which render a single approach unsuitable
for a complete description. We are pursuing coarse-graining approaches
based on algebraic geometry, combinatorial rigidity and non-convex
optimisation that retain the essentials of the molecular description
at different scales. Applications to lipid membranes, viral capsids and
protein aggregation will be discussed.
- Speaker:
- Erwin George
- Title:
- A meshfree method for the numerical solution of partial differential equations
- Abstract:
-
We give an overview of meshfree methods and describe the partition of unity meshfree method for solving partial differential equations. We summarise our implementation of the partition of unity method in a C++ code, and outline some intended applications of the method in biological transport problems.
- Speaker:
- Tomas Alarcon
- Title:
- A multiscale model of vascular growth
- Abstract:
-
Cancer progression induces abnormalities at all levels of biological organisation from gene expression and regulation to the structure vascular network and blood flow. Thus, these diseases are characterised by a strong multiscale character. We claim that this is one of the main obstacles to overcome for understanding cancer, as a full understanding of tumour growth involves the integration of information coming from disparate sources into one single framework, and that this is unlikely to be achieved unless mathematical modelling and computer simulation are used. In this talk, we present a mathematical model of vascular tumour growth that integrates three levels of bioloical scales (corresponding to three different physical time and length scales): cell-cycle and apoptosis regulation, competition between tumour- and normal cells, and structure of vascular networks. Recent results are presented along with current limitations and future developments
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