Understanding the Maths of Computed Tomography (CT) scans


Noseman is having a headache and as an old-school hypochondriac he goes to see his doctor. His doctor is quite worried and makes an appointment with a radiologist for Noseman to get a CT scan.

Modern CT scanner from Siemens

Because Noseman always wants to know how things work he asks the radiologist about the inner workings of a CT scanner.

The basic idea is that X-rays are fired from one side of the scanner to the other. Because different sorts of tissue (like bones, brain cells, cartilage, etc.) block different amounts of the X-rays the intensity measured on the other side varies accordingly.

The problem is of course that a single picture cannot give the full details of what is inside the body because it is a combination of different sorts of tissue in the way of the respective X-rays. The solution is to rotate the scanner and combine the different slices.

How, you ask? Good old linear algebra to the rescue!

We start with the initial position and fire X-rays with an intensity of 30 (just a number for illustrative purposes) through the body:

Initial position

As can be seen in the picture the upper ray goes through areas 1, 2 and 3 and let’s say that the intensity value of 12 is measured on the other side of the scanner:

    \[30-x_1-x_2-x_3=12\]

or

    \[x_1+x_2+x_3=18\]

The rest of the formula is found accordingly:

    \[\underbrace{ \begin{pmatrix}   1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\   0 & 0 & 0 & 1 & 1 & 1 & 0 & 0 & 0 \\   0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 \end{pmatrix} }_{\bold{A}_1} \underbrace{ \begin{pmatrix}   x_1 \\   x_2 \\   x_3 \\   x_4 \\   x_5 \\   x_6 \\   x_7 \\   x_8 \\   x_9 \end{pmatrix} }_{\bold{x}} = \underbrace{ \begin{pmatrix}   18 \\   21 \\   18 \end{pmatrix} }_{\bold{b}_1}\]

We then rotate the scanner for the first time…

Position after first rotation

…which gives the following formula:

    \[\underbrace{ \begin{pmatrix}   0 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 \\   0 & 0 & 0 & 0 & 1 & 1 & 1 & 0 & 0 \\   0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 \end{pmatrix} }_{\bold{A}_2} \underbrace{ \begin{pmatrix}   x_1 \\   x_2 \\   x_3 \\   x_4 \\   x_5 \\   x_6 \\   x_7 \\   x_8 \\   x_9 \end{pmatrix} }_{\bold{x}} = \underbrace{ \begin{pmatrix}   18 \\   21 \\    9 \end{pmatrix} }_{\bold{b}_2}\]

And a second rotation…

Position after second rotation

…yields the following formula:

    \[\underbrace{ \begin{pmatrix}   0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\   0 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 0 \\   0 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 0 \end{pmatrix} }_{\bold{A}_3} \underbrace{ \begin{pmatrix}   x_1 \\   x_2 \\   x_3 \\   x_4 \\   x_5 \\   x_6 \\   x_7 \\   x_8 \\   x_9 \end{pmatrix} }_{\bold{x}} = \underbrace{ \begin{pmatrix}   18 \\   14 \\   16 \end{pmatrix} }_{\bold{b}_3}\]

Now we are combining all three systems of equations:

    \[\begin{pmatrix}   \bold{A}_1 \\   \bold{A}_2 \\   \bold{A}_3 \end{pmatrix} \bold{x} = \begin{pmatrix}   \bold{b}_1 \\   \bold{b}_2 \\   \bold{b}_3 \end{pmatrix}\]

or written out in full:

    \[\underbrace{ \begin{pmatrix}   1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\   0 & 0 & 0 & 1 & 1 & 1 & 0 & 0 & 0 \\   0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 \\   0 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 \\   0 & 0 & 0 & 0 & 1 & 1 & 1 & 0 & 0 \\   0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 \\   0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\   0 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 0 \\   0 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 0 \end{pmatrix} }_{\bold{A}} \underbrace{ \begin{pmatrix}   x_1 \\   x_2 \\   x_3 \\   x_4 \\   x_5 \\   x_6 \\   x_7 \\   x_8 \\   x_9 \end{pmatrix} }_{\bold{x}} = \underbrace{ \begin{pmatrix}   18 \\   21 \\   18 \\   18 \\   21 \\    9 \\   18 \\   14 \\   16 \end{pmatrix} }_{\bold{b}}\]

Here is the data of the matrix \bold{A} for you to download: ct-scan.txt).

We now have 9 equations with 9 unknown variables… which should easily be solvable by R, which can also depict the solution as a gray-scaled image… the actual CT-scan!

A <- read.csv("data/ct-scan.txt")
b <- c(18, 21, 18, 18, 21, 9, 18, 14, 16)
v <- solve(A, b)
matrix(v, ncol = 3, byrow = TRUE)
##      [,1] [,2] [,3]
## [1,]    9    9    0
## [2,]    9    5    7
## [3,]    9    9    0
image(matrix(v, ncol = 3), col = gray(4:0 / 4))
CT of Noseman

The radiologist looks at the picture… and has good news for Noseman: everything is how it should be! Noseman is relieved and his headache is much better now…

Real CT scans make use of the same basic principles (of course with a lot of additional engineering and maths magic 😉 )

Here are real images of CT scans of a human brain…

Source: Wikimedia

… which can be combined into a 3D-animation:

Source: Wikimedia

Isn’t it fascinating how a little bit of maths can save lives!

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