5.1.1.1Transmission Computed Tomography

Transmission computed tomography (TCT) is the most popular use of CT. Since x-rays are often used in TCT, TCT is also called XCT. The radiation emitted from the emission source passes through the object and arrives on the receiver. When the radiations pass through the object, part of them is absorbed by the object, but the rest is received by the receiver. Since different parts of objects absorb different quantities of radiation, the radiation intensity received by the receiver shows the degree of radiation absorption at different parts of an object.

Let I0 denote the intensity of the source, k(x) denote the linear attenuation factor, L denote the radiation line, and I denote the intensity passing through an object. These parameters have the relation given by

$I=I_0\exp \left\{-{\displaystyle {\int }_{L}k\left(s\right)\text{d}s}\right\}\left(5.1\right)$

If the object is uniform/homogeneous,

$I=I_0\exp \left\{-kL\right\}\left(5.2\right)$

where I represents the intensity after passing through the object, I0 represents the intensity without the object, L is the length of radial inside the object, and k is the linear attenuation factor of the object.

In recent years, the scan time needed for CT has decreased dramatically, and the spatial resolution of CT has increased continuously. In 1972, the scan time required by CT was about 5 minutes and the spatial resolution was only 80 × 80. At present, the scan time for obtaining a 1024 × 1024 image is less than 1 second.

An ultrasound CT has the same working principles as TCT, except that the emission is ultrasound. Since ultrasound is not transmitted precisely along a line, some nonlinear models are required.

5.1.1.2Emission Computed Tomography

The history of emission computed tomography (ECT) can be traced back to the 1950s. The source of the ECT systems is placed inside the object that is to be examined. A common procedure is to inject ions with radioactivity into the object and detect the emission outside of the object. In this way, the distribution and movement of the ions can be detected, so the physiological states of objects can be investigated. Two kinds of ECT can be distinguished: positron emission tomography (PET) and single positron emission CT (SPECT). Both provide the space distribution information of radioisotopes. They differ in the methods used to define the ray direction.

PET has great sensitivity to detect details in nanomolar scale. Figure 5.1 illustrates the structure of a PET system. It uses ions with radioactivity, which can emit positrons in attenuation. The emitted (positive) positrons quickly collide with the (negative) electrons and are annihilated; thus, a pair of photons is produced and emitted in opposite directions. Two detectors installed face-to-face can receive these two photons and determine the radial. In PET systems, all detectors are spread clockwise and any two detectors facing each other form a pair of detectors used to detect the photon pair.

If two photons have been recorded simultaneously by a pair of detectors, then annihilation must happen in the line connecting these two detectors. A recording from such a projection event can be written as

$P=\exp \left(-{\displaystyle \int k\left(s\right)\text{d}s}\right){\displaystyle \int f\left(s\right)}\text{d}s\left(5.3\right)$

where P denotes the projection data, k(s) is the attenuation coefficient, and f(s) is the distribution function of radioisotopes.

SPECT is a technique that combines radiology and tomography. In SPECT, any radioisotope that emits decay γ-ray may be used as the basis for imaging. In contrast to the annihilation radiation, these γ-rays are emitted as single individual photons. It thus allows a much broader range of isotopes to be used, which are easily available. In contrast, because of the short half-lives of the isotopes involved in PET, a very expensive cyclotron is required.

The structure of a SPECT system is shown in Figure 5.2. To determine the ray directions, materials that are opaque to the γ-rays are used to make spatial collimators. These collimators eliminate most of the γ-rays that would otherwise strike the detector and allow only those that are incident in the prescribed direction to pass. The γ-rays arrived at the scintillation crystal are transformed to low energy photons and further transformed to electrical signals by a photomultiplier.

The sensitivity of SPECT is much less than that of PET, because only 0.1% of the γ-rays can pass the collimators. The sensitivity of ECT is

$S\propto \frac{\mathrm{A}{e}^{n}k}{4\pi r^2}\left(5.4\right)$

where A denotes the area of the detector, e is the efficiency of the detector (1 for SPECT and 2 for PET), k is the attenuation coefficient (around 0.2–0.6), and r is the radius of the object. The sensitivity ratio of PET to SPECT is about 150 divided by the resolution. When the resolution is 7.5 mm, this ratio is about 20.

5.1.1.3Reflection Computed Tomography

Reflection computed tomography (RCT) also works based on the principle of reconstruction from projections. A typical example is the radar system, in which the radar map is produced by the reflection from objects.

The structure of the synthetic aperture radar (SAR) system is shown in Figure 5.3. In SAR imaging, the radar moves while objects are assumed to be static (their relative movement is used to improve the transverse resolution). Suppose that v is the moving speed of the radar along the Y-axis, T is the effective cumulative time, and λ is the wavelength. Two point-objects are located along the moving direction of the radar, object A is located at the beam-line (X-axis), and the offset between object A and object B is d.

In Figure 5.3, the minimum distance between object A and the radar is R, and this moment is defined as the origin t = 0. Suppose that around t = 0 the change of distance is δR, then it has δR = (y – d)2/2R for R $\gg$ δR. The advanced phase of the reflected signal at object A for the round trip is

${\theta }_{\mathrm{A}}\left(t\right)=-\frac{4\pi y^2}{2R\lambda }=-\frac{4\pi }{\lambda }\frac{v^2{t}^2}{2R}\left(5.5\right)$

The advanced phase of the reflected signal at object B for the round trip is

${\theta }_B\left(t,d\right)=-\frac{4\pi }{\lambda }\frac{{\left(vt-d\right)}^2}{2R}\left(5.6\right)$

If the frequency of the emitted signal is high enough, the reflected signal can be considered continuous. Therefore, these signals can be integrated. In addition, if the emission is uniform during the integration period, the reflected signal at object B is

$E\left(d\right)={\displaystyle \underset{-T/2}{\overset{T/2}{\int }}\exp }\left[-\frac{j4\pi }{2R\lambda }{\left(vt-d\right)}^2\right]\text{d}t\left(5.7\right)$

5.1.1.4Magnetic Resonance Imaging

Magnetic resonance imaging (MRI) was originally called nuclear magnetic resonance (NMR). Its process is briefly described below. Materials with an odd number of protons or neutrons possess a weak but observable nuclear magnetic moment. The static nuclear moment is far too weak to be measured when it is aligned with the strong static magnetic field. Resonance techniques permit this weak moment to be measured. The main idea is to measure the moment while it oscillates in a plane perpendicular to the static field.

When protons are placed in a magnetic field, they oscillate (resonate) at a frequency that depends on the field strength and absorb energy at the oscillation frequency. This energy is reradiated as the protons return to their round state. The reradiation involves processes (relaxation of the magnetization components parallel and perpendicular to the field) with different time constants T1 and T2. The strength of the MRI signal depends on the proton concentration (essentially the water concentration in the tissue for medical imaging), but its contrast depends on T1 and T2, which are strongly influenced by the fluid viscosity or tissue rigidity. Weighting the combination of the two signals provides control over the observed image.

The observed images are reconstructed by the integration of resonance signals along different directions in a 3-D space V:

$S\left(t\right)={\displaystyle \underset{V}{\iiint }R\left(x,y,z\right)f\left(x,y,z\right)\exp [j\theta }{\displaystyle {\int }_0^{t}w\left(x,y,z,\tau \right)d\tau }]\text{d}x\text{d}y\text{d}z\left(5.8\right)$

where R(x, y, z) depends on the nucleus of the material; f(x, y, z) is a distribution function; w(x, y, z, t) is a Larmor frequency function, which is written as w(x, y, z, t) = g[B0 + B(x, y, z, t)] (B0 is the intensity of magnetic field and B(x, y, z, t) is a time-varying inhomogeneous magnetic field); and g(•) is also related to the nucleus of the material.

MRI is used to recover the intensity distribution function based on the resonance signals inspired by magnetic fields. From a mathematical point of view, MRI is an inverse problem; that is, given S(t), w(x, y, z, t), and f(x, y, z), solve eq. (5.8) for R(x, y, z).

Another MRI method is to represent the received magnetic signals as the function of the density of spins for an object. For an object with the density of spins R(r), whose spatial-frequency signal S(q) is

$S\left(q\right)={\displaystyle \underset{r_1}{\overset{r_2}{\int }}R\left(r\right)\exp \left[-\text{j}q\left(r\right)\cdot r\right]\text{d}r}\left(5.9\right)$

where the integration covers the whole object. S(q) and R(r) form a pair of Fourier transforms.

5.1.1.5Electrical Impedance Tomography

Electrical impedance tomography (EIT) is a relatively new medical imaging modality currently under development. It uses noninvasive signals to probe the object and then detects the responses on the boundary of the object in order to reconstruct an impedance distribution inside the object. It is harmless, relatively low in cost, easier to operate, and uses nonionizing radiation. EIT is quite sensitive to the change in conductance or reactance, and is the only method for acquiring the image of the conductance. By injecting low-frequency current into an object and measuring the electrical potential from the outside surface of the object, the image representing the distribution or variation of the conductance and the reactance inside the object can be reconstructed using the techniques of reconstruction from projections. Using this technique, the structure and the inner change of the object materials can also be obtained.

The distribution of the electrical potential within an isotropic conducting object through which a low-frequency current is flowing is given by Barber (2000)

$\nabla \left(c\nabla p\right)=0\left(5.10\right)$

where p is the potential distribution within the object and C is the distribution of conductivity within the object. If the conductivity is uniform, it reduces to a Laplace equation. Strictly speaking, this equation is only correct for the direct current. For the frequencies of the current used in EIT (up to 1 MHz) and the sizes of objects being imaged, it can be assumed that this equation continues to describe the instantaneous distribution of the potential within the conducting object. If this equation is solved for a given conductivity distribution and a current distribution through the surface of the object, the potential distribution developed on the surface of the object may be determined. The distribution of the potential depends on several things. It depends on the pattern of the applied current and the shape of the object. It also depends on the internal conductivity of the object, which is to be determined. In theory, the current may be applied in a continuous and nonuniform pattern at every point across the surface. In practice, the current is applied to an object through electrodes attached to the surface of the object. Theoretically, the potential may be measured at every point on the surface of the object. Again, the voltage on the surface of the object is measured in practice using electrodes attached to the surface of the object. There is a relationship between an applied current pattern Pi, p and C

$p_i=R\left(P_i,\mathrm{C}\right)\left(5.11\right)$

For one current pattern Pi, the knowledge of pi is not, in general, sufficient to determine C uniquely. However, by applying a complete set of independent current patterns, it is possible to obtain sufficient information to determine C, at least in the isotropic case. This is the inverse solution.

In practice, measurements of the surface potential or voltage can only be made at a finite number of positions, corresponding to electrodes placed on the surface of the object. This also means that only a finite number of independent current patterns can be applied. For N electrodes, N – 1 independent current patterns can be defined and N(N –1)/2 independent measurements can be made. The latter number determines the limitation of the image resolution achieved with N electrodes. In practice, it may not be possible to collect all possible independent measurements. Since only a finite number of current patterns and measurements are available, eq. (5.11) can be rewritten as

$\mathbf{V}={\mathbf{A}}_{\mathbf{c}}\mathbf{C}\left(5.12\right)$

where V is now a concatenated vector of all voltage values for all current patterns, C is a vector of the conductivity values representing the conductivity distribution divided into uniform image pixels, and AC is a matrix representing the transformation of this conductivity vector into the voltage vector. Since AC depends on the conductivity distribution, this equation is nonlinear. Although formally the preceding equation can be solved for C by inverting AC, the nonlinear nature of this equation means that this cannot be done in a single step. An iterative procedure will therefore be needed to obtain C.

A basic algorithm for reconstruction is as follows. For a given set of the current patterns, a forward transform is set up to determine the voltages V produced from the conductivity distribution C, as shown in eq. (5.12). AC is dependent on C, so it is necessary to assume an initial conductivity distribution C0. This distribution is usually assumed to be uniform. Using AC, the expected voltages V0 are calculated and compared with the actual measured voltages Vm. It can be proven that an improved estimation of C is given by

$\Delta \mathrm{C}={\left({\mathbf{S}}_{\mathrm{C}}^{\text{T}}{\mathbf{S}}_{\mathrm{C}}\right)}^{-1}{\mathbf{S}}_{\mathrm{C}}^{\text{T}}\left({\mathbf{V}}_0-{\mathbf{V}}_{\text{m}}\right)\left(5.13\right)$

${\mathrm{C}}_1={\mathrm{C}}_0+\Delta \mathrm{C}\left(5.14\right)$

where SC is the differential of AC with respect to the sensitivity matrix C. The improved value of C is then used in the next iteration to compute an improved estimate of Vm (i. e., V1). This iterative process is continued until some appropriate endpoint is reached. Although the convergence is not guaranteed, in practice, the convergence to the correct C in the absence of noise can be expected, if a good initial value is chosen. A uniform conductivity seems to be a reasonable choice. In the presence of noise on the measurements, the iteration is stopped when the difference between V and Vm is within a margin set by the known noise.

Currently used EIT systems can be divided into two classes: applied current electrical impedance tomography (ACEIT) and induced current electrical impedance tomography (ICEIT). The former measures the distribution of the impedance using injection-excitation techniques. It connects the source of the signal excitation to the driving electrodes on the surface of the object, and from the magnitude and phase of the voltage or the current on the measuring electrodes, the impedance in the imaging regions can be determined. Owing to different characteristics of various materials, a single frequency of the voltage or the current is not enough to measure the impedances of different materials. One solution for this problem is to use the sweep frequency technique, which is a relatively new branch of EIT. EIT creates alternative excitation using loops that do not touch the surface of objects. The loop produces the Faradic vortex inside the object, which can be detected from the outside.

Example 5.1 EIT images obtained with ACEIT

Two images reconstructed with ACEIT are shown in Figure 5.4, in which the gray levels represent the value of the impedances.

From the mathematical point of view, EIT is similar to various CT techniques, as all of them require the use of projection data to recover the inside structure. Compared to CT and MRI, EIT has advantages of real-time techniques, costs less, and is easy to implement for both continuous monitoring and functional imaging.

In summary, if the measured data contain the integration form of some interesting physical properties of an object, the techniques of reconstruction from projections could be used to obtain the images representing those physical properties of the object (Kak and Slaney, 2001).