Past Junior Applied Mathematics Seminar

29 November 2011
13:15
Abstract

Turbidity currents are fast-moving streams of sediment in the ocean 
which have the power to erode the sea floor and damage man-made
infrastructure anchored to the bed. They can travel for hundreds of
kilometres from the continental shelf to the deep ocean, but they are
unpredictable and can occur randomly without much warning making them
hard to observe and measure. Our main aim is to determine the distance
downstream at which the current will become extinct. We consider the
fluid model of Parker et al. [1986] and derive a simple shallow-water
description of the current which we examine numerically and analytically
to identify supercritical and subcritical flow regimes. We then focus on
the solution of the complete model and provide a new description of the
turbulent kinetic energy. This extension of the model involves switching
from a turbulent to laminar flow regime and provides an improved
description of the extinction process. 

  • Junior Applied Mathematics Seminar
18 November 2011
15:30
Abstract

 A common way to replace body tissue is via donors, but as the world population is ageing at an unprecedented rate there will be an even smaller supply to demand ratio for replacement parts than currently exists. Tissue engineering is a process in which damaged body tissue is repaired or replaced via the engineering of artificial tissues. We consider one type of this; a two-phase flow through a rotating high-aspect ratio vessel (HARV) bioreactor that contains a porous tissue construct. We extend the work of Cummings and Waters [2007], who considered a solid tissue construct, by considering flow through the porous construct described by a rotating form of Darcy's equations. By simplifying the equations and changing to bipolar variables, we can produce analytic results for the fluid flow through the system for a given construct trajectory. It is possible to calculate the trajectory numerically and couple this with the fluid flow to produce a full description of the flow behaviour. Finally, coupling with the numerical result for the tissue trajectory, we can also analytically calculate the particle paths for the flow which will lead to being able to calculate the spatial and temporal nutrient density.

  • Junior Applied Mathematics Seminar
1 November 2011
13:15
Abstract
Cell motility is a crucial part of many biological processes including wound healing, immunity and embryonic development. The interplay between mechanical forces and biochemical control mechanisms make understanding cell motility a rich and exciting challenge for mathematical modelling. We consider the two-phase, poroviscous, reactive flow framework used in the literature to describe crawling cells and present a stripped down version. Linear stability analysis and numerical simulations provide insight into the onset of polarization of a stationary cell and reveal qualitatively distinct families of travelling wave solutions. The numerical solutions also capture the experimentally observed behaviour that cells crawl fastest when the surface they crawl over is neither too sticky nor too slippy.
  • Junior Applied Mathematics Seminar
18 October 2011
13:15
Abstract

Motivated by the study of micro-vascular disease, we have been investigating the relationship between the structure of capillary networks and the resulting blood perfusion through the muscular walls of the heart. In order to derive equations describing effective fluid transport, we employ an averaging technique called homogenisation, based on a separation of length scales. We find that the tissue-scale flow is governed by Darcy's Law, whose coefficients we are able to explicitly calculate by averaging the solution of the microscopic capillary-scale equations. By sampling from available data acquired via high-resolution imaging of the coronary capillaries, we automatically construct physiologically-realistic vessel networks on which we then numerically solve our capillary-scale equations. By validating against the explicit solution of Poiseuille flow in a discrete network of vessels, we show that our homogenisation method is indeed able to efficiently capture the averaged flow properties.

  • Junior Applied Mathematics Seminar
21 June 2011
13:15
Abstract

Bacteria are ubiquitous on Earth and perform many vital roles in addition to being responsible for a variety of diseases. Locomotion allows the bacterium to explore the environment to find nutrient-rich locations and is also crucial in the formation of large colonies, known as biofilms, on solid surfaces immersed in the fluid. Many bacteria swim by turning corkscrew-shaped flagella. This can be studied computationally by considering hydrodynamic forces acting on the bacterium as the flagellum rotates. Using a boundary element method to solve the Stokes flow equations, it is found that details of the shape of the cell and flagellum affect both swimming efficiency and attraction of the swimmer towards flat no-slip surfaces. For example, simulations show that relatively small changes in cell elongation or flagellum length could make the difference between an affinity for swimming near surfaces and a repulsion. A new model is introduced for considering elastic behaviour in the bacterial hook that links the flagellum to the motor in the cell body. This model, based on Kirchhoff rod theory, predicts upper and lower bounds on the hook stiffness for effective swimming.

  • Junior Applied Mathematics Seminar
7 June 2011
13:15
Abstract

Human T-lymphotropic virus type I (HTLV-I) is a persistent human retrovirus characterised by a high proviral load and risk of developing ATL, an aggressive blood cancer, or HAM/TSP, a progressive neurological and inflammatory disease. Infected individuals typically mount a large, chronically activated HTLV-I-specific CTL response, yet the virus has developed complex mechanisms to evade host immunity and avoid viral clearance. Moreover, identification of determinants to the development of disease has thus far been elusive.

 This model is based on a recent experimental hypothesis for the persistence of HTLV-I infection and is a direct extension of the model studied by Li and Lim (2011). A four-dimensional system of ordinary differential equations is constructed that describes the dynamic interactions among viral expression, infected target cell activation, and the human immune response. Focussing on the particular roles of viral expression and host immunity in chronic HTLV-I infection offers important insights to viral persistence and pathogenesis.

  • Junior Applied Mathematics Seminar
8 March 2011
13:15
Sophie Kershaw
Abstract

How best to use the cellular Potts model? This is a boundary dynamic method for computational cell-based modelling, in which evolution of the domain is achieved through a process of free energy minimisation. Historically its roots lie in statistical mechanics, yet in modern day it has been implemented in the study of metallic grain growth, foam coarsening and most recently, biological cells. I shall present examples of its successful application to the Steinberg cell sorting experiments of the early 1960s, before examining the specific case of the colorectal crypt. This scenario highlights the somewhat problematic nuances of the CPM, and provides useful insights into the process of selecting a cell-based framework that is suited to the complex biological tissue of interest.

  • Junior Applied Mathematics Seminar
22 February 2011
13:15
Yi Ming Lai
Abstract
&nbsp;We examine several aspects of introducing stochasticity into dynamical systems, with specific applications to modelling<br />populations of neurons. In particular, we use the example of a interacting<br />populations of excitatory and inhibitory neurons (E-I networks). As each<br />network consists of a large but finite number of neurons that fire<br />stochastically, we can study the effect of this intrinsic noise using a master<br />equation formulation. In the parameter regime where each E-I network acts as a<br />limit cycle oscillator, we combine phase reduction and averaging to study the<br />stationary distribution of phase differences in an ensemble of uncoupled E-I<br />oscillators, and explore how the intrinsic noise disrupts synchronization due<br />to a common external noise source.<pre> </pre>
  • Junior Applied Mathematics Seminar
25 January 2011
13:15
Hermes Gadelha
Abstract

Abstract: Flagella and cilia are ubiquitous in biology as a means of motility and critical for male gametes migration in reproduction, to mucociliary clearance in the lung, to the virulence of devastating parasitic pathogens such as the Trypanosomatids, to the filter feeding of the choanoflagellates, which are constitute a critical link in the global food chain. Despite this ubiquity and importance, the details of how the ciliary or flagellar waveform emerges from the underlying mechanics and how the cell, or the environs, may control the beating pattern by regulating the axoneme is far from fully understood. We demonstrate in this talk that mechanics and modelling can be utilised to interpret observations of axonemal dynamics, swimming trajectories and beat patterns for flagellated motility impacts on the science underlying numerous areas of reproductive health, disease and marine ecology. It also highlights that this is a fertile and challenging area of inter-disciplinary research for applied mathematicians and demonstrates the importance of future observational and theoretical studies in understanding the underlying mechanics of these motile cell appendages.

  • Junior Applied Mathematics Seminar
30 November 2010
13:15
Almut Eisentrager
Abstract
<p>In a healthy human brain, cerebrospinal fluid (CSF), a water-like liquid, fills a system of cavities, known as ventricles, inside the brain and also surrounds the brain and spinal cord. Abnormalities in CSF dynamics, such as hydrocephalus, are not uncommon and can be fatal for the patient. We will consider two types of models for the so-called infusion test, during which additional fluid is injected into the CSF space at a constant rate, while measuring the pressure continuously, to get an insight into the CSF dynamics of that patient.</p> <p>&nbsp;</p> <p>In compartment type models, all fluids are lumped into compartments, whose pressure and volume interactions can be modelled with compliances and resistances, equivalent to electric circuits. Since these models have no spatial variation, thus cannot give information such as stresses in the brain tissue, we also consider a model based on the theory of poroelasticity, but including strain-dependent permeability and arterial blood as a second fluid interacting with the CSF only through the porous elastic solid.</p>
  • Junior Applied Mathematics Seminar

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