## Review of Image Formation

It is not the purpose of this chapter to survey the theory of image formation in the MRI scanner. Excellent descriptions are given in, for example, Morris [32], Stark and Bradley [43], and Callaghan [9]. However, it is useful to summarize the basic equations involved in magnetic resonance imaging. Many artifacts in MRI can be understood simply in terms of their linear additive effect on the signal or its Fourier transformation. For this reason a formalism for the construction of the nuclear magnetic resonance signal is worth summarizing.

The fundamental equation when considering the MRI signal from an elemental volume of a sample of spin density p(x, y, z) is

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where ^(x, y, z) is the phase of the elemental volume of the sample. In most MRI studies the spin density term, p(x, y, z), is simply the density of mobile water molecules (bound water is not generally seen in conventional images). The phase of the elemental signal is dictated by the time history of the local magnetic field at position (x, y, z). Following demodulation of the background static magnetic field component to the signal (the radio-frequency carrier signal), the phase term is thus given by

with Bz (x, y, z) being the net static magnetic field in tesla at position (x, y, z), which is defined to point along the z axis, and y being the gyromagnetic ratio, which relates the magnetic field strength to the frequency of the NMR resonance. For *H nuclei this is 42.575 MHz/tesla.

In order to generate an image, the net magnetic field must be made spatially varying. This is accomplished by passing current through appropriately wound "gradient"; coils that are able to generate the terms Gx = dBz/dx, Gy = dBz/dy, and Gz = dBz/dz to modulate the magnetic field. Thus, in a perfect magnet the total signal detected from the sample at an arbitrary time t following radio-frequency excitation of the signal into the transverse detection plane is:

S(t) = J J p(x, y) exp[ — 2nyi{ J Gx(x, y) x dt (3)

The simplification of considering a two-dimensional plane has been made in Eq. (3); since the radio-frequency excitation pulse typically selects a single slice of spins rather than an entire volume.

It is convenient to make the substitutions fcx = y f Gxxdt and fcy = y f Gyydt in which fcx and fcy represent the Fourier space of the image (as well as representing the field gradient history). This can easily be seen if Eq. (3) is rewritten as

, fcy) = J J p(x, y) exp[- 2rej(fexx + fcyy)] dxdy. (4)

In most modern MRI pulse sequences, the raw signal is generated by sampling the (fcx, fcy) space in a rectilinear fashion by appropriately pulsing the currents in the gradient coils (and thus by generating a series of gradient histories that sample all points in fc-space). Once the points in fc-space have been acquired, the image is generated by inverting Eq. (4) through Fourier transformation to yield a map of spin density p(x, y).

2.1 Nuclear Relaxation Times Tl, T2

Two principal nuclear relaxation time constants affect the signal in the image. The longitudinal relaxation time, Tl, defines the time constant for acquisition of longitudinal magnetization in the sample. Although longitudinal magnetization cannot be observed directly, since it lies entirely along the z-axis and is time invariant, it is necessary to allow enough time for longitudinal magnetization to build up before any sampling of signal can occur. Moreover, when repeated excitations of the spins are made with a repeat interval TR (as would be the case for almost all MRI pulse sequences), then a reduction in the maximum possible magnetization will generally result. This manifests in the p(x, y) in Eq. (4) being scaled according to p'(x, y) =p(x, y)[l - exp( — TR/Tl)]/

where cos 6 is the flip angle for the pulse sequence. Clearly, when TR > Tl the spin density term is unaffected, and "proton density" contrast in the image can result. However, when TR < Tl the contrast in the final image can be significantly affected by the Tl value of the different tissue types in the sample, resulting in "Tl-weighted" images. For example, in the human brain at l.5 Tesla the Tl values of gray matter, white matter and cerebro-spinal fluid are, respectively, l.O s, 0.7 s, and 4 s. A short TR will thus show CSF as very dark, gray matter as mid-intensity, and white matter as brightest.

Once a component of the longitudinal magnetization is tipped by angle 6 into the transverse plane (the plane in which signal is induced in the MRI coil), it precesses about the main static field direction and the transverse component decays with a time constant T2. The T2 time is shorter than Tl, often substantially so. In human soft tissue the T2 values range from 20 to 200 ms. The effect of T2 decay is to further scale the signal in Eq. (4) by a factor exp( —T2/TE), where TE is the time between the excitation of the magnetization and the time of the echo when the signal is read out. Thus, an image collected with a long TE will be T2-weighted.

A secondary effect of the T2-related decay of signal in the transverse plane is to modulate the fc-space data by a term exp( — t/T2), where t = 0 is the time that the spins are excited into the transverse plane. This time modulation results in a Lorentzian point spread broadening (convolution) of the image profile in the dimension in which the signal modulation occurs. Because most conventional images are acquired by sampling each line of fc-space following a separate excitation of the spins, the point spread broadening is generally only in one dimension (the readout dimension). This is shown schematically in Fig. l.

The preceding mathematical formalism can be quite useful in understanding the effects of various imperfections in the hardware, or problems with the pulse sequence during acquisition. In general, however, any processes that introduce unwanted amplitude or phase imperfections to Eq. (4) will generate artifacts in the image.

FIGURE 1 Schematic diagram showing the conventional method for collecting MRI data. Longitudinal magnetization is tipped into the transverse detection plane (a). The signal decays with a T2 (spin-echo) or T2* (gradient-echo) decay constant (b). M excitations of the spin system are made to collect all phase encode lines in fc-space (c). Fourier transformation of the filled fc-space gives the image (d). The T2(*) modulation in the readout dimension of fc-space is equivalent to a Lorentzian convolution in the image domain.

FIGURE 1 Schematic diagram showing the conventional method for collecting MRI data. Longitudinal magnetization is tipped into the transverse detection plane (a). The signal decays with a T2 (spin-echo) or T2* (gradient-echo) decay constant (b). M excitations of the spin system are made to collect all phase encode lines in fc-space (c). Fourier transformation of the filled fc-space gives the image (d). The T2(*) modulation in the readout dimension of fc-space is equivalent to a Lorentzian convolution in the image domain.

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