CT Imaging

[wikipedia]

A Computed Tomography (CT) machine acquires xray images from different angles around the head-to-toe axis of the patient.

The machine provides a stack of 2D axial xray "slices" of the patient, giving a 3D view of the internals.

Pre-operative applications are in diagnosis, modelling, and planning.

Intraoperative CT (iCT) can be used for tool guidance (e.g. brachytherapy and biopsy), but out-of-plane tool tracking is difficult as only a slice of the tool appears.

Some facts:

• Slice pixels can be as small as 0.625mm x 0.625mm.
• Spacing between slices should be maximized to reduce dose.

The CT is an xray machine that spins around the patient.

Below, $T$ is the xray tube, $X$ are the xrays, $D$ is the detector, and $R$ is the direction of rotation.

[radiopaedia.org]

CT machines have gone through several generations of development:

[radiologykey.com]

• 1st: pencil beam scans linearly, then rotates and repeats.
• 2nd: fan beam scans linearly, then rotates and repeats.
• 3rd: fan beam rotates with detectors
• 4th: fan beam rotates, detectors fixed.
• 5th: all fixed sources and detectors.
• 6th: helical scan

The raw output for one CT slice is a 2D "sinogram", $g(\theta,\rho)$:

• Each sinogram row corresponds to a particular angle, $\theta$, of the xray apparatus.
• Each sinogram column corresponds to a position, $\rho$, on the xray detector.

Angles range in only [0,180] degrees because the rays at angles $\theta$ and $\theta + 180$ have almost the same attenuation. (Why not the same?)

The value in the sinogram at $(\theta,\rho)$ is the total xray attenuation along the corresponding ray, or

$\int \mu(t) \; dt$ ,

where the integral is taken betweeen the xray source and detector.

Recall the Beer-Lambert law:

$I_\textrm{out} = I_\textrm{in} \; e^{- \int \mu(t) \; dt }$

Then the total xray attenuation stored in the sinogram is

$\int \mu(t) \; dt \; = \; \ln I_\textrm{in} - \ln I_\textrm{out}$

$I_\textrm{in}$ is the known energy of the emitted xray.

$I_\textrm{out}$ is the detected energy at the detector element.

Question: What does the sinogram of an isolated fiducial look like?