Taking Maths Further Podcast - Episode 18: Medical imaging and Fourier analysis

Episode 18: Medical imaging and Fourier analysis

04/24/15 • -1 min

This week the topic was Fourier analysis. We interviewed Heather Williams, who’s a medical physicist and works with Positron Emission Tomography (PET) scanners, as well as other medical scanning devices. We talked about her work and how maths is important in converting data from the scanner into images that can be used to diagnose patients.

Interesting links:

PET scanners on the NHS website
Being a Medical Physicist on the NHS careers website
Central Manchester University Hospitals, Heather's employer
Shape of the sine and cosine graphs at BBC Bitesize
An interactive guide to the Fourier transform at BetterExplained.com
XKCD comic 'Fourier'

Puzzle: If a function is made by adding sin(x) + cos(x), what’s the maximum value attained by this function?

Solution:

This is a periodic function, which repeats every 180 degrees (or π radians). Its maximum value is the square root of two, or √2 = 1.414213..., which it first reaches at a value of 45 degrees, or π/4. The function varies between √2 and -√2, and it looks like a sin curve. The function can also be written as √ 2 + sin(θ + π/4). For a graph of the function, and more detail, input it into Wolfram Alpha. Show/Hide

Interesting links:

PET scanners on the NHS website
Being a Medical Physicist on the NHS careers website
Central Manchester University Hospitals, Heather's employer
Shape of the sine and cosine graphs at BBC Bitesize
An interactive guide to the Fourier transform at BetterExplained.com
XKCD comic 'Fourier'

Puzzle: If a function is made by adding sin(x) + cos(x), what’s the maximum value attained by this function?

Solution:

Previous Episode

Episode 17: Nuclear reactor modelling

This week the topic was mathematical modelling and linear programming. We interviewed Rick Crawford from AMEC, who’s a mathematician studying decommissioning of nuclear reactors, and using mathematical models to determine whether it’s safe to continue using a particular reactor given that it may have degraded over time, but without actually building a physical model of it.

Interesting links:

Nuclear Power Plant at HowStuffWorks
Mathematical model on Wikipedia
Small angle approximation at John Cook's blog The Endeavour
Linear programming at Purple Math

Puzzle: A rod sits inside a cylindrical tube of the same height. The tube is 193mm tall, and 50mm in diameter. We assume the rod has zero thickness. What’s the maximum angle away from vertical that the rod can make (to the nearest degree)?

Solution:

You can imagine the rod as being a line inside a rectangle, since the cylinder is the same all the way round. Then, you need to calculate the angle made by the rod when it’s touching one bottom corner of the rectangle and resting against the opposite side. This will be a triangle whose base is 50mm and hypotenuse is the length of the rod. The angle from the vertical will be the top corner, and the sin of this angle will be the opposite (base of the triangle) over the hypotenuse. So the angle will be sin(50mm/193mm), which is 15 degrees to the nearest degree. Show/Hide

Next Episode

Episode 19: Computer games and mechanics

This week the topic was mechanics and friction. We interviewed Dan Hett, who works for CBBC writing computer games for their website. We talked about his work and how he uses a lot of mathematics in modelling how characters move, and making sure that’s done in a realistic way.

Interesting links:

CBBC games website
CBeebies story app (with pop-up book!)
Game physics on Wikipedia
A-level Mechanics topics at MathsRevision.net
Friction and Coefficients of Friction at Engineering Toolbox (with some example values)
Coefficient of friction on Wikipedia

Puzzle: Susan the Hedgehog runs at 20cm/s across the screen while the run button is held down. Once the run button is released, she slows down with constant deceleration of 8.5cm/s2. Will she stop within 32cm more of screen?

Solution:

The time taken to stop can be calculated by knowing that every second travelled, 8.5cm/s of speed is lost, so after 20/8.5=2.35 seconds, speed will be zero. We can approximate this deceleration by imagining Susan is travelling at 20cm/s for 1 second, 11.5cm/s for 1 second and 3cm/s for the remaining 0.35 seconds until she stops. This will cover more distance than the actual motion does (as your speed is lower than this for most of the time), but will cause you to travel only 31.6cm - so you will definitely stop within 32cm. (In actual fact, the distance taken to stop will be 23.53cm, because your speed continues to decrease at a constant rate for the whole time. In order to work this out, you need to use a little calculus!) Show/Hide