August 27th and 28th 2015

Tower Building, Baxter Suite 1.36, University of Dundee, Dundee, Scotland


Abstracts


Uri AscherData completion in applied inverse problems
Data completion in applied inverse problems

We consider inverse problems whose forward operator involves PDEs that depend on some material property that forms a surface. The inverse problem is to estimate that material property, requiring a given functional of the PDE solution to match given noisy measured data. Often data is available only at a restricted set of locations, while theory, or other considerations, demand that a fuller set (e.g., ``data everywhere') be given. It is then tempting to ``complete the data', e.g. by interpolation, before starting the inverse problem solution process. Such data completion, however, has its well-known perils as well. This talk describes our experience in handling (or avoidance) of data completion in the context of practical applications including electromagnetic in geophysical exploration; plant motion tracking and calibration in computer graphics; Monte Carlo methods for problems involving many data sets; and local volatility surface calibration for financial commodity markets.

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Nicola BellomoTheory and Applications of the Kinetic Theory for Active Particles
Theory and Applications of the Kinetic Theory for Active Particles

This lecture is presented in two parts. The first part presents the mathematical theory of active particles as a tool to model large living, hence complex systems. This method borrows from the classical kinetic theory the representation of the overall system by a probability distribution function and the balance of particles in the elementary volume of the space of microscopic states, while interactions are modeled by evolutionary stochastic games. The second part shows how this general framework can be applied to model various systems in life sciences. More in detail, the following applications are considered: Behavioural crowd and social dynamics, asymptotic methods to derive macroscopic tissue models from the underlying description at the cellular scale. Simulations are obtained by suitable development of Monte Carlo methods.

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Thomas CarraroA new sensitivity based method to approximate nonlinear confidence regions
A new sensitivity based method to approximate nonlinear confidence regions

Biological systems are characterized by inherent stochastic behaviour representing an essential feature. Therefore, mathematical model calibration and parameter estimation of biological processes should consider the variability of the experimental data. Modelling approaches taking into account the data variability are typically computationally expensive. Examples of these are the well known methods based on Monte Carlo techniques (e.g. Markov Chain Monte Carlo). The advantage of these methods is that the convergence rate is independent of the number of parameters. Their disadvantage is the slow convergence rate. In fact, the underlying system has to be solved typically several thousand times to perform parameter estimation. An additional advantage of the Monte Carlo type methods is that they can also estimate the confidence regions of the parameters.

To overcome the drawback of high computational costs of the Monte Carlo type methods it is desirable to use sensitivity based approaches. These methods are designated to approximate linearized confidence regions and therefore do not perform well with high nonlinearities. Nevertheless, nonlinearities play an important role especially with large variability of the data. This aspect is important for biological applications because they are strongly nonlinear and have to deal with large experimental variability.

Here, we present a new method called "Successive Approximation of Nonlinear Confidence Regions" (SANCR) to estimate nonlinear confidence regions based on consecutive applications of a sensitivity approach. Interestingly, the presented method shows good performance for prototypical problems. These results and the application to complex biological processes will be discussed.

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Mark ChaplainPerspectives on the mathematical modelling of cancer invasion enabling data inversion
Perspectives on the mathematical modelling of cancer invasion enabling data inversion

The invasion of surrounding tissue by a cancer has long been noted as a key feature of the disease, and more recently, it has been identified as one of the “hallmarks of cancer”. As a process, invasion is a complex, multiscale phenomenon involving many inter-related genetic, biochemical, cellular and tissue processes at different spatial and temporal scales. However, a key aspect of invasion is the ability of cancer cells to degrade and modify the extracellular matrix. This fact, along with the loss of cell-cell adhesion and alterations to cell-matrix adhesion, leads to the local spread of the cancer cells into the surrounding tissue and is the first step in the metastatic cascade. In this talk we will examine several models of cancer invasion (both individual-based and continuum) and highlight the opportunities for such models to enable data inversion.

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Pierre DegondA hybrid agent-based continuum model of vasculogenesis
A hybrid agent-based continuum model of vasculogenesis

We present a hybrid agent-based continuum model of vasculogenesis.The model combines a porous medium description of blood flow, a convection diffusion equation for nutrient transport and an agent-based model of capilary formation/destruction.

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Ping LinA discrete homogenisation process and its computation for particle systems
A discrete homogenisation process and its computation for particle systems

Many applications are involved or partially involved with the interaction of particles, for example, atomistic systems in material sciences and cellular systems in biological processes. A major difficulty for such systems is their huge size and thus computing their solution may be extremely expensive or even impossible. In this talk we will focus on a complex-lattice atomistic system as an example and show how a discrete homogenisation process may be introduced into the system and then used to derive a coarse-grained model so as to largely reduce the computational cost. We also demonstrate that the introduced homogenisation concept can help to extend the applicable range of the model and to analyse the error between the original model and the coarse-grained model.

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John MackenzieA Computational Method for the Coupled Solution of Reaction-Diffusion Equations on Evolving Domains and Surfaces: Application to a Model of Cell Migration and Chemotaxis
A Computational Method for the Coupled Solution of Reaction-Diffusion Equations on Evolving Domains and Surfaces: Application to a Model of Cell Migration and Chemotaxis

In this talk I will present details about a moving mesh finite element method for the approximate solution of partial differential equations on an evolving bulk domain in two dimensions, coupled to the solution of partial differential equations on the evolving domain boundary. Problems of this type occur frequently in the modeling of eukaryotic cell migration and chemotaxis - for these applications the bulk domain is either the interior or exterior of the cell and the domain boundary is the cell membrane. Fundamental to the success of the method is the robust generation of bulk and surface meshes for the evolving domains. For this purpose we use a moving mesh partial differential equation (MMPDE) approach. The developed method is applied to model problems with known solutions which indicate second-order spatial and temporal accuracy. The method is then applied to a model of the two-way interaction of a migrating cell with an external chemotactic field.

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Anotida MadzvamuseMathematical and computational approaches to modelling cell motility
Mathematical and computational approaches to modelling cell motility

In this talk I will present an overview of current mathematical and computational approaches to modelling cell motility whereby we couple bulk, surface and extra-cellular dynamics in a mathematically consistent framework. This framework allows us to derive from basic physical principles such models. State-of-art efficient and robust numerical methods for approximating solutions in 2- and 3-diminsions are discussed. Computational results are validated through comparisons to real experimental data.

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Raimondo PentaEffective governing equations for poroelastic growing media
Effective governing equations for poroelastic growing media

A new mathematical model is developed for the macroscopic behaviour of a porous, linear elastic solid, saturated with a slowly flowing incompressible, viscous fluid, with surface accretion of the solid phase. The derivation uses a formal two-scale asymptotic expansion to exploit the well separated length scales of the material: the pores are small compared to the macroscale, with a spatially periodic microstructure. Surface accretion occurs at the interface between the solid and fluid phases, resulting in growth of the solid phase through mass exchange from the fluid at a prescribed rate (and vice versa). The averaging derives a new poroelastic model, which reduces to the classical result of Burridge and Keller in the limit of no growth. The new model is of relevance to a large range of biological applications, including early stage tumor growth and mineralization of the bone structure.

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Cornelis WeijerPorous media effects on cell-cell communication and cell behaviour
Porous media effects on cell-cell communication and cell behaviour

In most developing biological systems cells tend to be closely packed. Development requires effective cell-cell communication to integrate critical behaviours such as relative cell movement to make more complex structures. Cell-cell communication often involves the secretion of ligands into a restricted extracellular space, which may be packed with factors that sequester or degrade these signalling molecules. These porous media affect the kinetics of cell-cell signalling as well as resulting cell behaviours such as cell movement. I will illustrate the effects of some of these properties on the kinetics of development of the cellular slime mould Dictyostelium discoideum and early development of the chick embryo resulting in some questions relevant for modelling these systems.

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Porous Media Modelling in Biological Processes: Perspectives on Analytical and Computational Methods Enabling Data Inversion