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Course on Medical Modelling at the London Taught Course Centre (LTCC) Nicholas Ovenden Department of Mathematics University College London Email: [email protected] ucl. ac. uk Lectures: Mondays (23 rd Feb, 2 nd Mar, 9 th Mar, 16 th Mar, 23 rd Mar) 10: 30 – 12: 30 in De Morgan House

Medical Modelling The demand for mathematics in the biological and medical sciences has been growing enormously in recent years. Models are constructed to gain quantitative understanding of physiological processes from microscope to macroscopic level and to enable predictions on outcome to be made. Ideally the ultimate goal of medical modelling is to assist clinicians in improving quality of life via better detection and clinical management of disease. .

Research areas • • • Cell membranes and cellular modelling Tissue engineering Angiogenesis Wound healing/tumour modelling Haemodynamics/respiration/urinary tract Modelling of entire organs (heart, liver, skin etc. ) Virtual Physiological Human (VPH) Development and psychology Epidemiology and agent interaction

Medical Modelling - techniques This depends on the physical phenomena driving the dynamics – Fluid Mechanics (Navier-Stokes equations, Poiseuille Flow) – Solid Mechanics (membrane mechanics, elastic tubes)

Medical Modelling - techniques – Reaction-diffusion equations – Lumped parameter models (e. g. Windkessel) P I r C R

Medical Modelling - techniques – Statistical or Stochastic approaches (Monte-Carlo, Markov chains) – Numerical/Asymptotic Techniques • • • Perturbation Methods of Multiple-scales Finite-difference Finite element Lattice-Boltzmann – Others (e. g. agent-based game theory, functional analysis)

Medical Modelling - Difficulties – – – Large number of parameters Parameters are often difficult/impossible to measure Large variability between subjects (patient-specific) Obtaining enough data for comparison/validation Inability to test ideas/theories (ethics/unmeasurable data) Communication of ideas between disciplines

Modelling vs Computation ". . . I want to distinguish between mathematical modelling and mere computation. If the geometry and motion of a blood vessel and its branches are prescribed, if the inflow and outflow boundary conditions are given and if a simple rheology (e. g. Newtonian) is assumed for blood, so that the governing equations (e. g. Navier. Stokes) are known, then the full details of the flow can in principle be computed. The output of the computation, in the form of distributions of wall shear stress and pressure, or predictions of the mass transport to the wall of biologically important molecules, may represent a useful contribution to biomechanical science, but that DOES NOT make it a mathematical model. Nor is it likely to be a significant contribution to fluid mechanics, unless it reveals or aids the physical understanding of a novel fluid mechanical phenomenon. The process of mathematical modelling requires the formulation of a simplifying hypothesis that permits the isolation of a particular physical phenomenon or mechanism of interest, and leads to a reduced system of equations that can be solved or at least investigated by mathematical methods (usually of course aided by computation these days). " - Timothy Pedley, Lighthill Memorial Paper, Journal of Engineering Mathematics vol 47 2003.

A Course on Medical Modelling How could this course be designed? 1. Present work on a particular topic in great depth – e. g. ultrasound imaging 2. Lecture on classical aspects – arterial blood flow, S. I. R. model of epidemics 3. Demonstrate the ‘art’ of mathematical modelling through case studies. For a postgraduate audience with varied backgrounds and projects – option (iii) seems best suited.

Study Groups in Mathematics originated in Oxford and have been running in the UK since 1968. The idea is to bring mathematics and experts from industry/medicine together to tackle applied problems. The study group usually runs for one week and the mathematicians have to “brainstorm” full-time during this week to come up with “potential” solutions. Study groups have been a very successful way of forming new collaborations and finding new research areas. Because of their success in uncovering noval applied mathematics and forging new contacts with industry/medicine the study group format has been copied around the world.

Study Groups There is an annual study group for “maths in medicine” organised in the UK. In this course – we will use previous study group problems as case studies. First problem – “Atherosclerotic Plaque Rupture” by J. E. F. Green, S. L. Waters, L. J. Cummings, J. B. van den Berg, J. H. Siggers, A. Grief, Study Group Nottingham 2002

Medical Modelling – a rough guide A clinician approaches you with a problem that may be analysable using mathematical model. How should one approach the problem? – Understand, focus and formulate – spend time asking important (even if seeming naïve) questions to make sure you understand what exactly the clinician wants to learn/gain from a model? Is there any data to emulate? Communication is essential to focus the problem into something that can be formulated. Beware of subtle differences of terminology (examples “steadily going up and down” “rigid pulsating sphere”). – Isolate the dominant mechanisms – it is better to build upon an over simplistic model than to attempt to grasp intuition from an overcomplicated one. From the identified mechanisms obtain a closed governing system of equations. – Don’t forget important initial and boundary conditions

Medical Modelling – a rough guide – Estimate parameters – Although the actual value (or statistical distribution) of parameters may not be known – estimates of each parameter’s magnitude are important to determine what terms in the equations are driving the dynamics. Sometimes these magnitudes need to be guessed or approximated using known values (e. g. diffusivity and viscosity of water). – Nondimensionalise – This is the absolute key to intuition – by using estimates of parameter values one can highlight the dominant terms in the equation that drive the dynamics in different spatial and/or temporal regions or in diseased and healthy patients. Multiple scales may be needed (e. g. short time scale for heart beat but longer timescale for development of disease) – Model dynamics – now compute solutions to the reduced model and observe the solution’s dynamics – are they realistic? Go back and review ignored mechanisms or parameter estimates. Examine asymptotics to gain further insight.