KAHIBARO
33.3. CT Geometry
Table of Contents
X-ray source
In a CT simulation, the X-ray source is typically represented as a small focal spot that emits a fan beam or cone beam of photons. In GATE you place this source at a specific position relative to the patient and the detector, then define its direction so that the beam passes through the patient onto the detector plane.
A simple starting point is to model the source as a point source with a given energy spectrum. The spatial position might be chosen on a circular trajectory around the patient, with radius equal to the scanner source-to-isocenter distance. In Cartesian coordinates, if the isocenter is at the origin and the rotation is in the $x$-$y$ plane, a source position for an angle $\theta$ could be
$$
x_s = R \cos \theta, \quad y_s = R \sin \theta, \quad z_s = z_0,
$$
with $R$ the source radius and $z_0$ the axial coordinate of the scan plane. The source direction is usually oriented toward the isocenter, that is the line from $(x_s, y_s, z_s)$ to $(0, 0, 0)$.
The beam opening is defined by the collimation and field of view, not by a physical object in all cases. In a simulation you control the beam shape by restricting the angular spread of the source directions so that photons cover just the detector area. For a fan beam CT, this spread is mainly in the transaxial plane, while for a cone beam CT there is a significant spread in both transaxial and axial directions.
The source to isocenter distance and source to detector distance are key geometry parameters. They define the system magnification factor, which links patient and detector coordinates. For simple 2D fan beam systems, the magnification $M$ can be written as
$$
M = \frac{D_{SD}}{D_{SO}},
$$
where $D_{SD}$ is the source to detector distance and $D_{SO}$ is the source to object or source to isocenter distance. In your simulation you typically fix these distances once, then place the source and detector consistently around the rotation circle.
It is important that the chosen source position, direction, and angular spread are geometrically compatible with your detector size and patient size. If the cone angle is too small, part of the patient will not be irradiated. If it is too large, you will simulate many photons that miss the detector and waste computation time.
In CT geometry, always maintain a consistent source to isocenter and source to detector distance for all projection angles, and ensure the source direction points toward the isocenter to preserve correct fan or cone beam geometry.
Patient
In a basic CT simulation, the patient is represented by an object placed at the isocenter of the scanner, between the X-ray source and the detector. This object can be a simple homogeneous phantom, such as a water cylinder, or a realistic voxelized phantom based on CT images. The detailed setup of voxelized geometry is covered elsewhere, so here the focus is on how the patient fits into the CT geometry layout.
The scanner is usually defined in a coordinate system where the isocenter is at $(0, 0, 0)$. The patient or phantom is centered at this point so that, as the source and detector rotate, the rays pass through different parts of the object symmetrically. If you simulate a cylindrical phantom, you choose a radius and length that fit within the field of view, for example a cylinder aligned along the $z$-axis. The radius should be smaller than the distance from isocenter to detector edges, otherwise part of the phantom will extend outside the illuminated region.
The patient geometry must be fully contained inside the world volume and must not overlap with the source or the detector volumes. In CT simulations the patient is typically stationary while the source and detector rotate around it. However, if you want to simulate helical CT or moving anatomy, the patient position can also change over time, but the geometric relationship to the rotation axis must remain well defined.
For a realistic CT geometry, you pay attention to the physical location of support structures such as the patient table. The table can be modeled as an additional volume that the beam traverses before entering the patient. Its position should be consistent with the patient center and the scanner bore, so that the beam path through table and patient matches clinical conditions.
From a geometric point of view, the key relations are distance from the patient center to the source and to the detector, and the maximum radius of the patient relative to the scanner bore. These distances determine beam path lengths and angular coverage. When designing your geometry, it is useful to visualize cross sections of the world to check that, for all projection angles, the beam from source to detector passes through the intended parts of the patient and does not go outside the world or through unintended volumes.
Place the patient or phantom at the scanner isocenter and verify that, for all rotation angles, the line from source to detector crosses the patient inside the scanner bore without leaving the world volume or intersecting unintended objects.
Detector
The CT detector is modeled as an array of sensitive elements that measure the intensity of transmitted X-rays for each projection. Geometrically, it is placed opposite the source across the patient. For a simple fan beam CT in 2D, this can be a one dimensional array of pixels in a line. For 3D cone beam CT, it is usually a two dimensional flat panel detector.
The detector center is positioned on the same rotation circle as the source but on the opposite side of the isocenter. If the source is at radius $R$ at angle $\theta$, a symmetric placement for a flat detector is to put its center at radius $R_D$ at angle $\theta + \pi$, with $R_D$ equal to the source to detector distance minus the isocenter radius if you separate those distances. In practice, it is often easier to define a local coordinate system for the detector plane, then place and rotate it so that its normal points toward the source.
You define detector size and pixel size so that the entire projected field of view is covered. For a flat detector with $N_x$ pixels of width $p_x$ and $N_y$ pixels of height $p_y$, the physical size in each direction is
$$
L_x = N_x \, p_x, \quad L_y = N_y \, p_y.
$$
These dimensions must be consistent with the source to detector distance and beam divergence so that, at each angle, the useful beam footprint lies inside the detector surface.
In GATE, each detector pixel is typically part of a larger volume that represents the full detector plane. You can create a parent volume for the full panel and then replicate or parameterize pixel volumes inside it. The parent detector volume is set as the sensitive region where hits and energy deposition are recorded. The pixel indices then provide spatial information for projection data.
The detector orientation is essential. The detector plane should be perpendicular to the line joining the source and the detector center. This alignment ensures that each pixel corresponds to a well defined ray path through the patient. If the detector is rotated incorrectly around its own normal or tilted out of the rotation plane, your simulated projections will be distorted.
Because CT involves rotating source and detector together around the patient, the detector and source positions and orientations must remain geometrically linked as the rotation angle changes. The detector center stays at a fixed distance from the isocenter, and its normal vector continues to point toward the current source position. This coupled motion defines the projection geometry used in CT reconstruction.
For consistent CT projections, ensure that the detector plane is centered on the rotation axis, at a fixed distance from the isocenter, and that its normal always points toward the current X-ray source position at every projection angle.
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