Microwave Detection and Imaging of Tumours

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Microwave Detection and Imaging of Tumours
Produced by P. Jaques at the Centre for Maths in
Industry, Massey University, As part of a 2008
New Zealand Sciences, Mathematics and Technology
Teacher Fellowship funded by the New Zealand
government and administered by the Royal Society
of New Zealand.

• How do you convert readings from a microwave
  receiver into information about where a tumour is
  located and how big it is?

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Principles I

• When you look at a face, your brain processes the
  image you see and enables you to recognise that
  face

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A visual image

• Have you ever looked into a pond or pool and seen
  something on the bottom but when you pick it
  up, it is deeper down than you thought?
• You see the image, but your brain has to process
  the image mathematically to realise that the
  object is deeper than it looks.

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Principles II

• The microwave receiver has to do the same thing
  it gets information on the reflected microwaves,
  just like your eye does, and has to try to make
  an image of it using mathematics and
  engineering in software

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Analogy

• Imagine you are in a dark room with a head light
  on
• You turn your head towards a fish bowl
• The light goes through the bowl, bounces off the
  fish, and returns to your eye
• Your brain enables you to see the fish

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Principles III

• When you look at something, you see light that
  has been reflected off that object
• You detect colours because the object has done
  something to the light that has reflected off it
• Your eye can see different colours because there
  is a variety of materials in the object

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A microwave light image

• Microwave receivers can already detect food that
  has not frozen properly because there is a big
  difference in the colour from frozen water and
  thawed ice
• (Microwave eyes can see dielectric constant)

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There is a big difference between normal cells
and cancer cells too
Normal cells
Malignant cells
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So this is the basic idea
1
2
3
Microwave light shines on the surface
Reflected microwaves are detected by the
microwave receiver eye
d3
d1
d2
Layers in the body
Tumour
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These are the equations needed to make an image
from what the eye has received
So to get the Z for layer 1, you need to find
the Z for layer 2 first
i is the square root of -1
The impedance at the front of layer 1 is
So to get the Z for layer 2, you need to find
the Z for layer 3 first
The impedance at the front of layer 2 is
Which simplifies to
Substituting for Z2 in the top equation gives .
i2 -1 gives the minus signs
Characteristic impedance of that layer Phase
constant of that layer Electrical impedance of
that layer
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• But what we really want is the thickness of each
  layer (d), to find how deep the tumour is so we
  turn the equation around

and solve it to find d1. Easy until you realise
that your answer will depend on d2, which depends
on d3. The solution is found by taking equations
like these for each of the ds, and solving them
all at once using Newtons iterative method in
matrix form. The process does not always converge
to a sensible solution, but within certain
ranges you can make progress.
similar to the Newton-Raphson method
Of course, a tumour is not flat a more valid
model would be a sphere. To work up to that,
first consider the tumour to be a cylinder
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Two Dimensions

• this is what it looks like from the top

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And the equations become

• Where

and the J and H functions are special functions
called Bessel and Hankel functions which are
defined as infinite series and come out of
solving a particular type of differential
equation.
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When you work backwards to solve the equations,
it takes about 10 steps to get close to the
correct result
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Three Dimensions

• As a final step, we model the tumour as a
• 3-D sphere

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And the equations become, after some
simplifications
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But they can be solved to get reasonable answers
Calculated values of depth of tumour
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This maths has been tested by experiment, and the
results are very encouraging
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This could lead to

• A more comfortable, portable and accurate way of
  screening people for tumours, especially breast
  cancer tumours. It would also be cheaper, so it
  would make early detection more affordable.
• More research to refine the method and improve
  the accuracy.

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How did this work happen?

• Keam Holdem Associates Ltd, New Zealand provided
  the industrial support and motivation for the
  research to develop their products into new
  areas.
• Technology New Zealand provided a TIF
  (Technology Innovation Fund) fellowship for PhD
  study on Microwave signal processing for breast
  cancer detection.
• Other people are actively working in this popular
  field in New Zealand and overseas, sharing their
  findings and collaborating at times

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Questions?

• Acknowledgments
• Dr Galkadowite Senaratne (Massey University)
• Rick Keam (Keam Holdem Associates)
• Dr Winston Sweatman (Massey University)
• Professor Graeme Wake (Massey University)