Operation of computed tomography

Numbers (1)–(7) shown in Fig. 3 (see the numbers within the parentheses) respectively indicate: (1) = an object; (2) = the parallel beam light source; (3) = the screen; (4) = transmission beam; (5) = the datum circle (a datum feature); (6) = the origin (a datum feature); and (7) = a fluoroscopic image (a one-dimensional image; p (s, θ)).

Two datum coordinate systems xy and ts are imagined in order to explain the positional relations and movements of features (0)–(7) in the figure. The xy and ts coordinate systems share the origin (6) and they are positioned on the same plane. That is, the xy plane and the ts plane are the same plane. Henceforth, this virtual plane will be called “the datum plane”. In addition, a virtual circle centered at the abovementioned origin (6) is set on the datum plane (it will be called “the datum circle” henceforth). This datum circle (5) will be represents the orbit of the parallel beam irradiation optical system. Naturally, the origin (6), the datum circle (5), and the datum coordinate systems are virtual features which are imagined for mathematical purposes.

The μ(x,y) is absorption coefficient of the object (3) at each (x,y), p(s,θ) (7) is the collection of fluoroscopic images.

The parallel beam irradiation optical system is the key component of a CT scanner. It consists of a parallel beam X-ray source (2) and the screen (3). They are positioned so that they face each other in parallel with the origin (6) in between, both being in contact with the datum circle (6).

These two features ((2) and (3)) can rotate counterclockwise [Notes 1] around the origin (6) together with the ts coordinate system while maintaining the relative positional relations between themselves and with the ts coordinate system (so, these two features ((2) and (3)) are always opposed each other). The ts plane is positioned so that the direction from a collimated X-ray source (2) to the screen (3) matches the positive direction of the t-axis while the s-axis parallels these two features. Henceforth, the angle between the x- and the s-axes will be indicated as θ. That is, parallel beam irradiation optical system where the angle between the object and the transmission beam equals θ. This datum circle (6) will be represents the orbit of the parallel beam irradiation optical system.

On the other hand, the object (1) will be scanned by CT scanner is fixed to xy coordination system. Hence, object (1) will not be moved while the parallel beam irradiation optical system are rotated around the object (1). The object (1) must be smaller than datum circle.

The distance the table moves for every 360° rotation of the X-ray generator is called the increment or table feed for axial scan modes. For helical scan modes, it is called table speed.[3] Setting an increment that is smaller than the slice thickness results in overlap between the slices. A beneficial effect of this is a smoother transition between images when scrolling through the stack.[4]

During the above-mentioned motion (that is pivoting around the object(1)) of parallel beam irradiation optical system, the collimated X-ray source (2) emits transmission beam (4) which are effectively “parallel rays” in a geometrical optical sense. The traveling direction of each ray of the transmission beam (4) is parallel to the t-axis. The transmission beam (4), emitted by the X-ray source (2), penetrates the object and reaches the screen (3) after attenuation due to absorption by the object.

Optical transmission can be presumed to occur ideally. That is, transmission beam penetrates without diffraction, diffusion, or reflection although it is absorbed by the object and its attenuation is assumed to occur in accordance with the Beer-Lambert law.

Consequently, a fluoroscopic image (7) is recorded on the screen as a one-dimensional image (one image is recorded for every θ corresponding to all s values). When the angle between the object and transmission beam is θ and if the intensity of transmission beam (4) having reached each "s" point on the screen is expressed as p(s, θ), it expresses a fluoroscopic image (7) corresponding to each θ.

In this section, the basic principle of tomography in the case that especially uses tomography utilizing the parallel beam irradiation optical system will be explained.

Tomography is a technology that uses a tomographic optical system to obtain virtual 'slices' (a tomographic image) of specific cross section of a scanned object, allowing the user to see inside the object without cutting. There are several types of tomographic optical system including the parallel beam irradiation optical system. Parallel beam irradiation optical system may be the easiest and most practical example of a tomographic optical system therefore, in this article, explanation of "How to obtain the Tomographic image" will be based on "the parallel beam irradiation optical system". The resolution in tomography is typically described by the Crowther criterion.

Fig. 3: Considering a parallel beam irradiation optical system where the angle between the object and all transmission lights equals θ. Here, the numbers in the figure (see the numbers within the parentheses) respectively indicate: (1) = an object; (2) = the parallel beam light source; (3) = the screen; (4) = transmission beam; (5) = the datum circle; (6) = the origin; and (7) = a fluoroscopic image (a one-dimensional image; p_θ(s)). Two datum coordinate systems xy and ts are also imagined in order to explain the positional relations and movements of features (0)–(7) in the figure. In addition, a virtual circle centered at the above-mentioned origin (6) is set on the datum plane (it will be called “the datum circle” henceforth). This datum circle (6) represents the orbit of the parallel beam irradiation optical system. In the figure above, X-Y plane rotates around the point of origin in the plane in such a way “to keep mutual positional relationship between the light source (2) and screen (7) passing through the trajectory (5).” Rotation angle of this case is defined as θ. In the figure set out above, absorption coefficient at a cross-sectional coordinate (x, y) of the subject is modeled as μ(x, y).

Fig. 3 is intended to illustrate the mathematical model and to illustrate the principle of tomography. In Fig.3, absorption coefficient at a cross-sectional coordinate (x, y) of the subject is modeled as μ(x, y). Consideration based on the above assumptions may clarify the following items. Therefore, in this section, the explanation is advanced according to the order as follows:

• (1)Results of measurement, i.e. a series of images obtained by transmitted light are expressed (modeled) as a function p (s,θ) obtained by performing radon transform to μ(x, y), and
• (2)μ(x, y) is restored by performing inverse radon transform to measurement results.

Consider the mathematical model where the absorption coefficient of the object at each point ${\displaystyle (x,y)}$ is represented by the function ${\displaystyle \mu (x,y)}$ and suppose that the transmission beam penetrates without diffraction, diffusion, or reflection. Also assume the beam is absorbed by the object and its attenuation occurs in accordance with the Beer-Lambert law. What we want to know then is the values of the function ${\displaystyle \mu }$. What we can measure will be the values of the function ${\displaystyle p(s,\theta )}$.

When the attenuation conforms to the Beer-Lambert law, the relation between ${\displaystyle {I}_{0}}$ and ${\displaystyle I}$ is given by equation (1) and the absorbance ${\displaystyle p_{l}}$ along the light beam path ${\displaystyle l(t)}$ is given by equation (2). Here ${\displaystyle {I}_{0}}$ is the intensity of the light beam before transmission, while ${\displaystyle I}$ is the beam intensity after transmission.

${\displaystyle {\begin{aligned}I=I_{0}\exp \left({-\int \mu (x,y)\,dl}\right)=I_{0}\exp \left({-{\int }_{-\infty }^{\infty }\mu (l(t))\,|{\dot {l}}(t)|dt}\right)\end{aligned}}}$

(1)

${\displaystyle {\begin{aligned}p_{l}=\ln(I/I_{0})=-\int \mu (x,y)\,dl=-{\int }_{-\infty }^{\infty }\mu (l(t))\,|{\dot {l}}(t)|dt\end{aligned}}}$

(2)

Here, the direction from the light source toward the screen is defined as the ${\displaystyle t}$ direction and that perpendicular to the ${\displaystyle t}$ direction and parallel with the screen is defined as the ${\displaystyle s}$ direction. (Both the ${\displaystyle (t,s)}$ and ${\displaystyle (x,y)}$ coordinate systems are chosen such that they are reflections of each other without mirror-reflective transformation.)

By using a parallel beam irradiation optical system, one can experimentally obtain the series of fluoroscopic images (these are one-dimensional images ${\displaystyle p_{\theta }(s)}$ of a specific cross section of a scanned object) for each angle ${\displaystyle \theta }$ between the object and the transmitted light beam. In Fig.3, the ${\displaystyle (x,y)}$ plane rotates counter clockwise. [Notes 1] around the point of origin in the plane in such a way “to keep mutual positional relationship between the light source (2) and screen (7) passing through the trajectory (5).” Rotation angle of this case is same as above-mentioned θ.

The beam having an angle ${\displaystyle \theta }$ is the collection of lines ${\displaystyle {l}_{\theta ,s}(t)}$, represented by equation (3) below.

${\displaystyle {l}_{\theta ,s}(t)=t{\begin{bmatrix}-\sin \theta \\\cos \theta \\\end{bmatrix}}+{\begin{bmatrix}s\cos \theta \\s\sin \theta \\\end{bmatrix}}}$

(3)

The function ${\displaystyle p_{\theta }(s)}$ is defined by equation (4). That ${\displaystyle p_{\theta }(s)}$ is equal to the line integral of μ(x,y) along ${\displaystyle {l}_{[\theta ,s]}(t)}$ of (eq. 3) as the same manner of (eq.2). This means that, ${\displaystyle p(s,\theta )}$ of following (eq. 5) is a resultant of Radon transformation of μ(x,y).

${\displaystyle p_{\theta }(s)=-{\int }_{-\infty }^{\infty }\mu (s\cos \theta -t\sin \theta ,s\sin \theta +t\cos \theta )\,dt}$

(4)

One can define following function of two variables (5). In this article, ${\displaystyle p(s,\theta )}$ is the collection of fluoroscopic images.

${\displaystyle p(s,\theta )=p_{\theta }(s)}$

(5)

“What we want to know (μ(x,y))” can be reconstructed from “What we measured ( p(s,θ))” by using inverse radon transformation . In the above-mentioned descriptions, “What we measured” is p(s,θ) . On the other hand, “What we want to know ” is μ(x,y). So, the next will be "How to reconstruct μ(x,y) from p(s,θ)".

Since its invention by Kalender in the 1980s, helical scan CT machines have steadily increased the number of rows of detectors (slices) they deploy. The prototype 16 multi-slice scanner was introduced in 2001 and in 2004, 64 multislice scanners are on the market. These can produce an image in less than a second and thus can obtain images of the heart and its blood vessels (coronary vessels) as if frozen in time.

In order to illuminate multiple rows of detector elements in a multi-slice scanner, the x-ray source must emit a beam which is divergent along the axial direction (i.e. a cone beam instead of a fan beam).

The field of view (FOV) multiplied by scan range creates a volume of voxels (abdominal CT pictured).

A helical CT beam trajectory is characterized by its pitch, which is equal to the table feed distance along the scan range over one gantry rotation divided by the section collimation.[9] When pitch is greater than 1, the radiation dose for a given axial field of view (FOV) is decreased compared to conventional CT. At high pitches there is, however, a trade-off in terms of noise and longitudinal resolution.[10]

In cone-beam computed tomography (commonly abbreviated CBCT), the X-ray beam is conical.[11]

Helical (or spiral) cone beam computed tomography is a type of three-dimensional computed tomography (CT) in which the source (usually of X-rays) describes a helical trajectory relative to the object while a two-dimensional array of detectors measures the transmitted radiation on part of a cone of rays emanating from the source.

In practical helical cone beam X-ray CT machines, the source and array of detectors are mounted on a rotating gantry while the patient is moved axially at a uniform rate. Earlier X-ray CT scanners imaged one slice at a time by rotating source and one-dimensional array of detectors while the patient remained static. The helical scan method reduces the X-ray dose to the patient required for a given resolution while scanning more quickly. This is, however, at the cost of greater mathematical complexity in the reconstruction of the image from the measurements.