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Mathematical Modelling and Analysis of Cerebrospinal Mechanics: an Investigation Into the Pathogenesis of Syringomyelia

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Mathematical Modelling and Analysis of Cerebrospinal Mechanics: an Investigation Into the Pathogenesis of Syringomyelia University of Warwick institutional repository: http://go.warwick.ac.uk/wrap A Thesis Submitted for the Degree of PhD at the University of Warwick http://go.warwick.ac.uk/wrap/2751 This thesis is made available online and is protected by original copyright. Please scroll down to view the document itself. Please refer to the repository record for this item for information to help you to cite it. Our policy information is available from the repository home page. Mathematical Modelling and Analysis of Cerebrospinal Mechanics: An Investigation Into the Pathogenesis of Syringomyelia Novak Samuel Jon Elliott BSc(Hons) BE(Hons) MBiomedE This report is submitted as partial fulfilment of the requirements for the PhD Programme of the School of Engineering University of Warwick August 2009 AUTHOR: Novak Samuel Jon Elliott DEGREE: PhD TITLE: Mathematical Modelling and Analysis of Cerebrospinal Me- chanics: An Investigation Into the Pathogenesis of Syringomyelia DATE OF DEPOSIT: ............................. I agree that this thesis shall be available in accordance with the regulations gov- erning the University of Warwick theses. I agree that the summary of this thesis may be submitted for publication. I agree that the thesis may be photocopied (single copies for study purposes only). Theses with no restriction on photocopying will also be made available to the British Library for microfilming. The British Library may supply copies to individuals or li- braries. subject to a statement from them that the copy is supplied for non-publishing purposes. All copies supplied by the British Library will carry the following statement: \Attention is drawn to the fact that the copyright of this thesis rests with its author. This copy of the thesis has been supplied on the condition that anyone who consults it is understood to recognise that its copyright rests with its author and that no quotation from the thesis and no in- formation derived from it may be published without the author's written consent." AUTHOR'S SIGNATURE: ........................................................ USER'S DECLARATION 1. I undertake not to quote or make use of any information from this thesis without making acknowledgement to the author. 2. I further undertake to allow no-one else to use this thesis while it is in my care. DATE SIGNATURE ADDRESS ............................................................................. ............................................................................. ............................................................................. ............................................................................. ............................................................................. ii www.warwick.ac.uk This thesis is dedicated to the memory of Professor Peter Carpenter who taught me to challenge accepted wisdom in the pursuit of scientific rigour. iii Abstract Syringomyelia is a disease in which fluid-filled cavities, called syrinxes, form in the spinal cord causing progressive loss of sensory and motor functions. Inva- sive monitoring of pressure waves in the spinal subarachnoid space implicates a hydrodynamic origin. Poor treatment outcomes have led to myriad hypotheses for its pathogenesis, which unfortunately are often based on small numbers of patients due to the relative rarity of the disease. However, only recently have models begun to appear based on the principles of mechanics. One such model is the mathematically rigorous work of Carpenter and colleagues. They suggest that a pressure wave due to a cough or sneeze could form a shock-like elastic jump, which when incident at a stenosis, such as a hindbrain tonsil, would gen- erate a transient region of high pressure within the spinal cord and lead to fluid accumulation. The salient physiological parameters of this model are reviewed from the literature and the assumptions and predictions re-evaluated from a me- chanical standpoint. It is found that, while the spinal geometry does allow for elastic jumps to occur, their effects are likely to be weak and subsumed by the small amounts of damping that have been measured in the subarachnoid space. The analysis presented here does not support the elastic-jump hypothesis for sy- rinx formation. Furthermore, the site of maximum transpial pressure differential due to a cough-induced pulse is most likely to be at the site of pulse origin|not, iv as supposed, at a distant reflection site. This suggests that there must be some other localising factor more critical to providing the necessary conditions for sy- rinx formation. Two coaxial tube models are developed that incorporate Darcy's law separately in the pial membrane and the spinal cord tissue. It is shown that permeability plays opposing roles in the spinal cord and pia for wave attenua- tion; the propagation of a pressure wave is aided by a less-permeable pia but a more-permeable spinal cord. This may have implications in a syringomyelic cord. To understand the dynamic interaction of the fluid and solid components of the spinal cord tissue Biot's theory of poroelasticity is employed. It is concluded that physiological frequencies are probably too low for poroelastic dissipation to be of significance in such a soft and weak material as the spinal cord. Accumulating evidence in the last decade from animal studies implicates arterial pulsations in syrinx formation. In particular, Bilston and colleagues suggested that a phase difference between the pressure pulse in the spinal subarachnoid space and the perivascular spaces, due to a pathologically disturbed blood supply, could result in a net influx of cerebrospinal fluid into the spinal cord. A lumped-parameter model is developed of the cerebrospinal system to investigate this conjecture. It is found that although this phase-lag mechanism may operate, it requires the spinal cord to have an intrinsic storage capacity due to the collapsibility of the contained venous reservoir. If this storage requirement is met then the results presented here suggest that, on mechanical grounds, a syringo-subarachnoid shunt may be a better surgical treatment option than a subarachnoid bypass for post-traumatic syringomyelia. Keywords: syringomyelia, syrinx formation, collapsible tubes, pressure waves, poroelasticity, lumped-parameter v List of Publications Parts of this work were presented in the following publications. Articles in refereed journals Elliott, N.S.J., Lockerby, D.A. & Brodbelt, A.R. (2009) The pathogenesis of syringomyelia: a re-evaluation of the elastic jump hypothesis, Journal of Biome- chanical Engineering, 131(4): 044503-1{6. Published conference papers Elliott, N.S.J., Carpenter, P.W. & Brodbelt, A.R. (2007) The role of pial per- meability on pressure wave propagation in a model of syringomyelia, Proceedings of the International Symposium Syringomyelia 2007 Abstracts of Free Papers, Rugby, U.K., 24 Oct; published in British Journal of Neurosurgery, 21(5):430. Elliott, N.S.J., Lockerby, D.A. & Brodbelt, A.R. (Paper accepted) A lumped- parameter model of the cerebrospinal system for simulating syringomyelia, Pro- ceedings of the 2009 Micro and Nano Flows Conference, London, U.K., 1{2 Sep. vi Elliott, N.S.J., Lockerby, D.A. & Brodbelt, A.R. (Paper accepted) Arterial- pulsation driven flow in syringomyelia|A lumped-parameter model, Proceedings of the 2009 World Congress on Medical Physics and Biomedical Engineering, Munich, Germany, 7{12 Sep. Conference posters Elliott, N.S.J. & Carpenter, P.W. (2006) Pressure propagation in the spinal cord and the origins of syringomyelia, World Congress on Medical Physics & Biomed- ical Engineering, Seoul, South Korea, 27 Aug{1 Sept. Elliott, N.S.J. & Carpenter, P.W. (2007) Modelling syringomyelia using wave propagation in collapsible tubes, Fifth Physiological Flow Network Meeting, Im- perial College, University of London, 3{4 September. Carpenter, P.W. & Elliott, N.S.J. (2007) A simple biomechanical model for sy- ringomyelia, Syringomyelia International Symposium, Rugby, United Kingdom, 24{26 October. Talks / Colloquia Elliott, N.S.J. (2006) Pressure-wave propagation in the spinal cord and the ori- gins of syringomyelia, Post-graduate Student Colloquium, School of Engineering, University of Warwick, Coventry, U.K., 14 June. vii Elliott, N.S.J. & Carpenter, P.W. (2006) Pressure waves in the spinal cord and the origins of syringomyelia, Fluid Dynamics Research Group Seminar, Department of Mechanical Engineering, Curtin University, Perth, Australia, 15 December. [Invited] Elliott, N.S.J. & Carpenter, P.W. (2007) Coughing up a storm: syringomyelia and wave propagation in the spinal cord, Workshop on Computation of Biomed- ical Processes, Simula Research Laboratory, Oslo, Norway, 11 June. [Invited, expenses paid] viii Acknowledgements During my PhD project I have been fortunate enough to receive invaluable advice, encouragement and inspiration from many people, to whom I would like to say \thank you". There isn't room enough here to list the names of everyone but in particular I would like to mention (in alphabetical order): Chris Bertram, Mark Brend, Andrew Brodbelt, Peter Carpenter, Mark Evernden, Matthias Heil, Richard Howell, James Jewkes, Duncan Lockerby, Tony Lucey, Bryn Martin, Astrid Nordin, Mark Pitman, Sarah Waters and Robert Whittaker. To my family, friends and fellow PhD students, you have been an endless source of support and I thank you all for your time. I would also like to acknowledge the following institutions and people for fund- ing my research: Graduate School, University of Warwick
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