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Final year Projects 2002/3

Possible Potential PhD projects

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Project: Modelling growth of soft tissues


As a result of hypertension arterial wall tissue and heart tissue remodel to accommodate the increase in wall stress by increasing the mass and thickness of the tissues. This is a process that involves the interaction between mechanical stimulus and biological response and a key ingredient in the process is the initial residual stresses in the tissue. Residual stresses change as a result of the remodelling and an important aim of this project is to develop a mathematical model, based on the continuum theory of nonlinear elasticity and a novel theory of mass growth, that describes the change in residual stresses due to the geometrical changes in the tissues and the associated remodelling (i.e. changes in material properties). This will in part involve a general theory of residual stress in materials capable of large elastic deformations and a general theory of growth that has been developed in Glasgow [1, 2]. This will be informed by developments at the cellular level that explain how changes in mechanical stress influence the biological adaptation that lead to cell proliferation and growth. [1] A. Guillou and R.W. Ogden, Growth in soft biological tissue and residual stress development. Proceedings of the IUTAM Symposium on Mechanics of Biological Tissue (G.A. Holzapfel and R.W. Ogden, eds), Graz, Austria, June/July 2004, pp. 47-62. Springer (2006). [2] M. Shams, M. Destrade and R.W. Ogden, Initial stresses in elastic solids: constitutive laws and acoustoelasticity. Wave Motion 48 (2011), 552-567.

This project is in collaboration with Prof. R. W. Ogden, FRS

Project: Finite element-immersed boundary method and its application to mitral valves


Mathematical modelling can help us to understand mitral valve (MV) diseases and their relationship with left ventricle (LV) functions. This project will study MRI based dynamic MV models, which will include important features such as fluid-structure interaction and nonlinear soft tissue modelling. The work will be carried out through an interdisciplinary collaboration with Prof. Boyce Griffith from NYU, Dr. Berry at the Cardiovascular Research Centre, and Prof. R. W. Ogden. The PhD student will work closely with a post-doc RA working on the the same project, and develop and apply the structure-based constitutive models for human MV and LV. The computations will be carried out using the object oriented C++ finite element version of immersed boundary method with adaptive mesh refinement (IBAMR: The computational models will be validated using MRI DENSE measurements of 3D temporal displacement and strain vector field of the human MV/LV in vivo. This will tie the computational simulations with clinical applications together and allow us to identify key elements and parameters in our models. The research in this proposal will address important questions about MV mechanics. Deeper understanding of the basic mechanisms of heart valve function could result in improved clinical therapies and therefore has clear social benefit. Ultimately, the project will contribute towards delay or prevent progression of valvular disease, for example, by modulating transvalvular blood flow or engaging pharmacological approaches to modify cardiac output and valve elasticity.

This project is in collaboration with Prof. B. E. Griffith at NYU.

Project: Multi-scale modelling of the heart


Heart disease is the biggest killer in the world. To understand how to best maintain a healthy heart, it is essential for us to understand the mechanical functions of a normal heart. There is growing interest in the research on cardiac tissues in the heart, as reflected in the recent workshop on the ``Cardiac Physiome: Multi-scale and Multi-physics Mathematical Modelling Applied to the Heart'', at the Isaac Newton Institute in July 2009. However, to date, the biggest challenge and the weakest link in our understanding remains the mechanics of large-deformation, nonlinear fluid-structure interactions in the heart. It is for this purpose that we are developing EPSRC and BHF proposals, in conjunction with Prof. G. Smith at the Faculty of Biomedical & Life Sciences, and Dr. Colin Berry at the BHF Glasgow Cardiovascular Research Centre, with the aim of developing a mathematical model of the mechanics of the heart. The models will be built using the most-advanced structure-based nonlinear material model, as well the most-advanced computational immersed boundary methods IBAMR. These novel approaches will enable us to simulate the fluid-structure interactions inside the heart and will be world-leading.

This project is in collaboration with Profs. Colin Berry, Godfrey Smith, and Francis Burton at MVLS, Glasgow.

Project: Mathematical modelling of cerebral aneurysm inception


Cerebral aneurysms appear as sac-like out-pouchings of the arterial wall inflated by the pressure of the blood. Prevalance rates in populations without comorbidity are estimated to be around 5%. Most remain asymptomatic; however, there is a small but inherent risk of rupture: 0.1% to 1% of detected aneurysms rupture every year. Subarachnoid haemorrhage (SAH) due to aneurysm rupture is associated with a 50% chance of fatality and of those that survive, nearly half have long term physical and mental sequelae. Pre-emptive treatment may prevent aneurysm SAH and thus reduce the associated (large) financial burden, e.g. the total annual economic cost of aneurysm SAH is £510M in the UK. However, management of unruptured aneurysms by interventional procedures, i.e. minimally invasive endovascular approaches or surgical-clipping, is highly controversial and not without risk. Given the very low risk of rupture, there is both a clinical and an economic need to identify those aneurysms which are actually in need of intervention. It is envisaged that mathematical models will guide understanding of the aetiology of the disease and ultimately assist in diagnostic decisions. The early stages of aneurysm formation are characterised by a loss of elastin and apoptosis of smooth muscles cells. As the wall weakens the geometry enlarges to maintain force balance. However, the functional role of the vascular smooth muscles is to contract or relax to control the diameter of the artery to maintain the wall shear stress that acts on the luminal layer of the artery to normotensive levels. Hence two competing mechanisms are at play: (i) the loss of load bearing due to apoptosis of smooth muscle cells leading to increased arterial diameter; (ii) change in basal tone of smooth muscle cells to maintain arterial diameter. There is a real need for a novel mathematical model to guide our understanding of these competing physiological processes. In this project, a novel mathematical model of aneurysm inception will be developed. The aneurysm will be modelled as a nonlinear elastic cylindrical membrane comprised of elastin and collagen [1]; growth and remodelling of the material constituents gives rise to enlargement. This mathematical model will be extended to explicitly represent the active response of vascular smooth muscle [2]. The influence of the (above) competing mechanisms on aneurysm evolution will be explored. The research student will work in close consultation with Dr Paul Watton (University of Sheffield), an expert in aneurysm modelling. Opportunities to visit Sheffield for a short duration to pursue the research are possible. The research will expose the student to state of the art approaches in mathematical and computational modelling of vascular mechanobiology. It is anticipated it will lead to a publication. [1] Watton PN, Ventikos P, Holzapfel GA, (2009) Modelling Growth and Stabilisation of Cerebral Aneurysms, Mathematical Medicine and Biology, 26:133-164. [2] Rachev A., Hayashi K.,(1999) Theoretical Study of the Effects of Vascular Smooth muscle Contraction on Strain and Stress Distributions in Arteries, Annals of Biomedical Engineering, Vol. 27, pp. 459-468.

This project is in collaboration with Dr. Paul Watton at the University of Sheffield.

Project: Flows in flexible vessels


One of the principal questions of interest in flow in collapsible tubes is the mechanism of the self-excited oscillations. There are numerous physiological applications which are related to flow in collapsible tubes: arteries compressed by a sphygmomanometer cuff, intra-myocardial coronary blood vessels during systole, pulmonary blood vessels in the lung, the urethra during micturition, and the glottis during phonation. Many experiments with model systems of collapsible tubes in the laboratory have revealed a rich variety of self-excited oscillations. This has stimulated numerous theoretical and numerical studies. Most of the studies, however, have consisted of linear or nonlinear instability theories for flow in a long, parallel-sided channel, so in the basic state the steady flow is unidirectional and the elastic walls are planar.

Various numerical simulations and analytical tools have been developed to model these systems. However in most of the existing models, the elastic wall has been over-simplified, hence the mechanism of the instability identified may only be of limited value when applied to real vessels. In this project, a fibre-reinforced soft tissue wall model will be developed and the instability of the fluid-structure interaction will be studied. It is expected that the research will shed new light on the mechanism of self-excited oscillations of flow in realistic vessels.

This project is in collaboration with with Dr. Peter Stewart at GU and Prof. T. J. Pedley at DAMTP, Cambridge

1. X.Y. Luo, Z.X. Cai, W.G. LI, T.J. Pedley, The cascade structure of linear stabilities of flow in collapsible channels, J. Fluid Mechanics, 600,45-76, 2008

2. Z. X. Cai, & X.Y. Luo, A fluid-beam model for flow in collapsible channel. J. of Fluids and Structures, 17 (1), 123-144, 2003.

3. X.Y. Luo & T.J. Pedley, Flow limitation and multiple solutions in 2-D collapsible channel flow. J. of Fluid Mechanics, 420, 301-324, 2000.

4. X.Y. Luo & T.J. Pedley, The effects of the wall inertia on the 2-D collapsible channel flow. J. of Fluid Mechanics, 363, 253-280, 1998.

5. T.J. Pedley & X.Y. Luo, Modelling flow and oscillations in collapsible tubes. J. of Theoret. Comp. Fluid Dynamics, 10, 277-294, 1998.

6. X.Y. Luo & T.J. Pedley, A numerical simulation of unsteady flow in a 2-D collapsible channel. J. of Fluid Mechanics. 314, 191-225, 1996.

7. X.Y. Luo & T.J. Pedley, A numerical simulation of steady flow in a 2-D collapsible channel. J. of Fluids & Structures, 9, 149-174, 1995.

Project: Large Mechanical modelling of sleep apnoea

Obstructive sleep apnoea is becoming a major health care topic. It affects 4 percentage of the adult population, and has many consequences such as excessive daytime sleepiness or hypertension. Obstructive sleep apnoea consists periodic episodes of soft tissue collapse within the upper airway during sleep. From a fluid mechanical point of view, the partial or the total collapse of the upper airway, as observed during obstructive sleep apnoea, can be understood as a spectacular example of fluid-walls interaction. While the most important parameters influencing this effect in-vivo are well known, this phenomenon is still difficult to model and thus to predict. In this PhD project, the aim is to develop and to validate a mechanical model for the flow induced collapse inside an elastic walled conduit. The project will focus two crucial aspects. One is the description of the flow, and in particular to the movement of the point of flow separation associated with the deformation of the conduit [1]. Several theoretical descriptions (Boundary-layer method, RNSP) have been considered and tested against experimental data obtained on an in vitro replica of the human airways. However, these models suffer from some over-simplifications. Another aspect is related with the description of the deformable tissues. It is quite clear that, for clinical applications, distributed or lumped models are unable to reproduce accurately the behaviour of the human tissues. This is arguably one of the major limitations of all existing models [2-4]. In the current project, a nonlinear elastic solid mechanical model will be developed to explore the key impact of the soft tissue effect. .

This project is in collaboration with Dr. Annemie VAN HIRTUM GIPSA-Lab, Grenoble, France .




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