Physics of Medical Scans
Computed Tomography - Advanced
Detectors
X-ray source
Gantry
1D Projection
Figure 1: A simple display of the principle of CT scan operation as the X-ray tube and detector move in tandem to create one-dimensional projections at different angles of the patient.
Figure 2: A thoracic-abdominal CT scan composed of multiple images along the patient's longitudinal axis. Bright areas indicate instances of large X-ray attenuation and therefore high density tissue (eg. bone) while darker areas depict low density tissues or cavities like the lungs. (Source: University of Maryland School of Medicine)
X-ray source
X-ray after attenuation
Voxel
Figure 3: An iterative reconstruction lattice. Each voxel has its own attenuation coefficient a and the inset describes the coefficients in the highlighted voxels. The grey oval area represents the shape of the 2D slice.
Figure 4: (a) X outputs for a 4-voxel object. (b) Attenuation measurements are taken and divided between the voxels in the path to make an initial estimate. (c) More measurements done in another angle again dividing the value between the voxels. (d),(e),(f) The estimate is adjusted repeatedly to match each new measurement, stopping when the estimates match all measurements to within some tolerance.
X-ray Emission and Detection
Background
Computed Tomography (CT) is a largely popular computerised method of collecting cross-sectional slices of a part of the body to reveal the distributions of tissue located there. Its wide range of uses including patient diagnosis, treatment planning, and monitoring means that it is a versatile scanning technique. The demand for CT scans have increased massively ever since its commercial inception in 1967 by Godfrey Hounsfield, with on average over 70 million scans being performed annually in the US [1]. This section helps to give a introduction on the physics behind CT scans.
The Physics of CT scanning
The fundamental principle behind CT scanning is that a two-dimensional cross-sectional structure of an object, a ‘slice’, can be reconstructed from a series of one-dimensional projections through the object; the projections in this case are the X-ray attenuation data when a beam of X-rays is shone through an object at different angles (most commonly around the axial plane), followed by corresponding image reconstructions. The attenuation of monochromatic X-rays passing through a homogenous medium is governed by the Lambert law of absorption
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where I is X-ray intensity after passing through the medium, I0 is the intensity initially, x is the distance travelled in the medium and a is the linear attenuation coefficient. For inhomogeneous mediums like human bodies, (3.1) is expressed as
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From (3.2), it is clear to see that by mapping solutions to (ie. the different tissues) to their spatial positions, an image of a patient’s tissue distribution can be resolved; this is discussed below in the CT Image Reconstruction section. The attenuation coefficient of tissues are usually expressed through their Hounsfield unit (HU), or their "CT number", which is defined by
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where is the linear attenuation coefficient of a medium i . X-ray energies found in CT scans, are usually in the range of 70-80keV given a typical scanner operating at ~140 kVp (peak voltage at 140kV) with an X-ray tube current between 70-320 mA [2]. A collimator on the X-ray emission tube usually sets the wide X-ray beam to an angular sweep of ~45-60Ëš, and a second collimator restricts the beam thickness (perpendicular to the plane of the slice) to a few millimetres or smaller determining the scan’s spatial resolution.
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Situated opposite the X-ray source is an array of detectors and together they rotate about the gantry (the ring in which the patient lies) collecting unique cross-sectional data of the patient’s body. For one axial slice, the X-ray tubes and detectors are rotated through one complete revolution about the patient and data is continuously collected. Then the patient table moves incrementally along the head-to-feet (longitudinal) axis and the system rotates again to collect another slice. Together, the collection of slices stacked in the longitudinal direction describe a three-dimensional ‘map’ of tissues within the chosen field-of-view (FOV) of the patient (see Figure 2 to the right as an example). This method of incremental table movement is called conventional CT. While conventional CT can reliably produce images, it has since been superseded by helical CT scanning as the latter is faster and less sensitive to errors due to movement and breathing [3].
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(3.1)
(3.2)
(3.3)
Image Reconstruction
Iterative Method
Adapted by Godfrey Hounsfield, a way in which to reconstruct a two-dimensional image using a set of one-dimensional projections is to use an initial guess of the image and refine it through iterative operations. Imagine the area of a slice divided up into rectangular boxes called voxels with each voxel approximately containing one type of tissue (and therefore having a homogenous attenuation coefficient within it); this can be visualised in Figure 3 [4]. For a row or column of voxels in which X-rays pass, a sum X of the attenuation can be calculated and this is equal to the total attenuation experienced by one ray through the lattice. By incorporating (3.2), we can express this sum as
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where a_i is the i-th voxel’s coefficient, w_i is the i-th voxel’s width, and Ai is the product of the two. The value of X can be known for any given ray path as it only requires the I0 intensity from the source and the I value measured from the detectors and so by collecting values of X from different projections about the patient, each voxel’s attenuation coefficient can be solved using simultaneous equations. An example of this iterative process can be seen in Figure 4 where a 2 x 2 lattice of voxels is solved based on arbitrary attenuation values for simplicity. The computational approach to this process is called the algebraic reconstruction technique (ART) where ART consists of iterative algorithms to solve lattices such as in Figure 3 and 4 but in higher dimensions.
The problem with ART however is that the attenuation measurements X include random errors that can come from noise. In the context of CT imaging, noise appears due to the low number of incident photons at the detector sites based on appropriately low X-ray dosages. The variations associated with this noise can be described by Poisson statistics; that is, for a mean concentration of X-ray photons N on a detector, the standard deviation of N across all detectors is srt(N). This means that even at high N values there will still be noise present and therefore the estimates created by ART will never perfectly match up with real measurements. Consequently, some tolerance is required in an ART algorithm for it to finish iterating.
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(3.5)
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Back-Projection Method
Today, contemporary CT scanners have more advanced iterative processes that increase efficiency and decrease noise. Adaptive statistical iterative reconstruction (ASIR) starts the iterative process after firstly collecting an image through filtered back-projection (another method of image reconstruction); this takes less computing time than doing pure iteration and it substantially reduces overall image noise [5] [6].
CT - Standard
References:
[1] Brenner D. J. (2010), "Slowing the Increase in the Population Dose Resulting from CT Scans", Radiation Research 174(6b).
[2] Smith N. B., Webb A. (2010), "Introduction to Medical Imaging", Cambridge University Press Textbooks.
[3] Garvey C. J., Hanlon R. (2002), “Computed tomography in clinical practice”, The BMJ (Clinical research ed.) vol. 324, no. 7345,1077-80.
[4] Ranallo F. N., Szczykutowicz T. (2015), "The Correct Selection of Pitch for Optimal CT Scanning: Avoiding Common Misconceptions", Journal of the American College of Radiology Vol. 12 No. 4.
[5] Goldman L. (2007), "Principles of CT and CT Technology", Journal of Nuclear Medicine Technology Vol. 35 No. 3 pp. 115-128.
[6] Padole A., Ali Khawaja R.D., Kalra M., Singh S. (2015), CT Radiation Dose and Iterative Reconstruction Techniques, American Journal of Roentgenology Vol. 204, No. 4.