Course on Medical Modelling at the
London Taught Course Centre (LTCC)
Nicholas Ovenden
Department of Mathematics
University College London
Email: nicko@math.ucl.ac.uk
Lectures:
Mondays (23rd Feb, 2nd Mar, 9th Mar, 16th Mar, 23rd 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.
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Research areas
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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)
1
Incompress ible N - S : .v  0 v t  ( v.) v   p   2 v

Poiseuille Flow :
U  6Qy (1  y )
p x  12 Q
– Solid Mechanics (membrane mechanics, elastic tubes)
Arterial Tube :
A0 ( x) 
4 Eh 
1 

p(x, t) - p 0 

3 R0 ( x) 
A( x, t ) 
p( x, t )  p0 is the transmura l pressure, E is Young' s modulus, h is the wall thickness ,
A0 ( x)  R0 ( x) is the cross - sectional area at zero transmura l pressure and A( x, t ) is the
cross - sectional area along the vessel at time t.
Medical Modelling - techniques
– Reaction-diffusion equations

U  DU  R (U )
t
1 - D Fisher - Kolmogorov
D is a diagonal diffusion matrix

 2
  (1   )  2
t
x
    ( x  ct )
– Lumped parameter models (e.g. Windkessel)
P
I
r
C
R
dP
dI V
i (r  R)
r 

dt
dt RC
RC
Medical Modelling - techniques
– Statistical or Stochastic approaches (Monte-Carlo, Markov chains)
Wright - Fisher Model
2N genes at time t  0, x of these genes are type A, (0  x  2N)
Yn is the random variable for the number of A genes at time n
y
P(Yn  yn | Yn 1
n
y 
2 N!
 yn 1  
 yn 1 ) 

 1  n 1 
yn !( 2 N  yn )!  2 N  
2N 
2 N  yn
– Numerical/Asymptotic Techniques
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Perturbation Methods
Methods of Multiple-scales
Finite-difference
Finite element
Lattice-Boltzmann
– Others (e.g. agent-based game theory, functional analysis)
Medical Modelling - Difficulties
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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. NavierStokes) 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
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.