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FIGURE 2 Axial slice through the brain of a well-shimmed volunteer (a). An MRI field map (b) in the same slice as (a). The field map is scaled to show the range + 75 Hz. Note that the residual field variations are caused by tissue-air (frontal sinuses) and tissue-bone (petrous) interfaces in the head.

FIGURE 2 Axial slice through the brain of a well-shimmed volunteer (a). An MRI field map (b) in the same slice as (a). The field map is scaled to show the range + 75 Hz. Note that the residual field variations are caused by tissue-air (frontal sinuses) and tissue-bone (petrous) interfaces in the head.

Thus, the readout dimension of fc-space is sampled following the spin excitation by digitizing N (typically 128 or 256) points separated by dwell time DW while simultaneously applying a steady field gradient, Gx. Conversely, the phase-encode dimension of fc-space is sampled by repeating the experiment M times and applying a brief gradient pulse of varying amplitude for a constant time. Thus, fc!T = ymAGyt.

(8), where AB(x, y) is the magnetic field inhomogeneity in units of tesla.

In a conventional 2D Fourier MRI pulse sequence (standard spin echo, standard gradient echo, etc.) a poor magnetic field inhomogeneity will result in displacement of pixels in only one of the two dimensions of the image. To understand why this is so, it is necessary to define the difference between the readout (first) dimension and the phase-encode (second) dimension in fc-space for a conventional MRI pulse sequence. In the case of the readout dimension, gradient history points are sampled by incrementing time and maintaining a constant gradient strength, i.e., where AGy is the phase-encode increment size and tpe is the duration of the phase-encode gradient. Under this scheme an N x M pixel image will result and will have isotropic field of view if Afcn = Afcy. This principle is shown in Fig. 3a, demonstrating how fc-space is filled by successive spin excitations, each with a different phase-encode gradient step.

A consequence of the difference between variable-time/ constant-gradient (readout) and variable-gradient/constant-time ( phase-encode) sampling is that the magnetic field inhomogeneities only cause pixel shifts in the readout dimension of the image. The magnitude of the 1D pixel shift at position (x, y) is given by yAB(x, y)NDW pixels. Typically yAB(x, y) is rarely greater than + 50 Hz at 1.5 tesla, and DWis in the range 25-40 ^s, resulting in up to a 0.5 pixel shift for a 256 x 256 image. At high magnetic field strengths (> 1.5 tesla) the shift will be proportionately greater.

No pixel shifts are observed in the phase-encode dimension due to the zero time increment in the gradient history of adjacent points in fcy, i.e., no phase evolution, J AB(x, y)dt, is possible in this dimension.

FIGURE 3 Schematic pulse sequences and fc-space filling patterns for a conventional gradient echo sequence (a) and an echo planar imaging sequence (b). For the conventional sequence M excitations of the spins are made to build up the M phase-encode lines. For the EPI sequence a single excitation of the spins is made. A rapidly oscillating readout gradient sweeps back and forth across fc-space enabling all phase-encode lines to be collected. Note that a given point in fcx (horizontal) is always sampled at the same time following the excitation pulse in (a) but is sampled at increasing times Atpe following the excitation pulse in (b).

FIGURE 3 Schematic pulse sequences and fc-space filling patterns for a conventional gradient echo sequence (a) and an echo planar imaging sequence (b). For the conventional sequence M excitations of the spins are made to build up the M phase-encode lines. For the EPI sequence a single excitation of the spins is made. A rapidly oscillating readout gradient sweeps back and forth across fc-space enabling all phase-encode lines to be collected. Note that a given point in fcx (horizontal) is always sampled at the same time following the excitation pulse in (a) but is sampled at increasing times Atpe following the excitation pulse in (b).

For the echo planar image (EPI) pulse sequence [29], which is commonly used in neurofunctional and cardiac MRI applications, the situation is further complicated by the fact that the phase encode lines are collected sequentially following a single excitation of the spin system. Indeed, this is the reason that EPI is such a rapid imaging sequence and why it is favored by the neurofunctional and cardiac MRI communities. The implication of collecting all fc-space lines sequentially following a single excitation of the spin system is that it is no longer the case for EPI that the time evolution between adjacent points in the second dimension of fc-space is zero. This is shown schematically in Fig. 3b, revealing that the time increment between successive phase-encode lines is approximately equal to NDW. Thus, for EPI the sensitivity to magnetic field inhomogeneities in the phase-encode dimension is significantly greater than in the readout dimension by an approximate factor N. Putting numbers to this, the DW values used in EPI are generally in the range 5-10 ^s, and N is in the range 64-128. This leads to negligible pixel shifts in the readout dimension (< 0.1 pixels), but substantial pixel shifts in the phase-encode dimension (1-8 pixels). This gross difference is a simple consequence of the long effective dwell time between the acquisition of adjacent points in the phase-encode dimension of fc-space. These effects can be corrected if a knowledge is gained of the field map distribution through the sample [22].

Intensity Distortion

Field inhomogeneities throughout the sample also lead to intensity changes in the data. The nature of the intensity changes depends on the pulse sequence being performed. Broadly, the intensity of pixel (xn, yxm) in a conventional spinecho image is modulated according to the amount of signal in the ym phase-encode line contributing to the frequency interval yGxxn ^yGxxn+1. When the magnetic field homogeneity is perfect, the pixel value should be equal to p(x,y). In the presence of field inhomogeneities, though, the pixel intensity may be artifactually increased if signal is relocated into pixel (xn, ym), or may be artifactually decreased if signal is relocated out of pixel (xn, ym). Gradient-echo images may be further modulated by intrapixel destructive phase interference. This can result in signal dropout (low signal intensity) in gradientecho images in the same locations that might result in high signal intensity in spin-echo images. Details of the difference between spin-echo and gradient-echo MRI pulse sequences may be found in the introductory texts listed in Section 2.

3.3 Radio-Frequency Coil Inhomogeneity

As described in Section 2, the NMR signal is generated by exciting the longitudinal magnetization into the transverse plane so that a signal can be detected. This is accomplished using a copper resonant coil that is used to generate an oscillating or rotating radio-frequency (B1) magnetic field transverse to the static magnetic field. Classically, the longitudinal magnetization vector may be thought to rotate about the applied B1 field by an angle 2nyB1%, where % is the duration of the radio-frequency pulse. When 2nyB1% corresponds to 90°, the longitudinal magnetization is fully tipped into the transverse plane and will subsequently precess about the static field until the transverse signal decays and the magnetization vector recovers along the longitudinal direction. Many gradient-echo pulse sequences use flip angles less than 90° so that only a small disturbance is made to the longitudinal magnetization component — preserving it for subsequent excitations. Clearly, if the B1 field is not homogeneous (because of imperfect coil design or sample interactions) then the MRI signal intensity will be affected by variations in the B1 radio-frequency magnetic field. For a gradient-echo pulse sequence the signal is modulated in proportion to B1 sin(2reyB1%). For a spin echo pulse sequence the signal is modulated in proportion to B1 sin3(2reyB1%). The linear term in B1 appears because the NMR signal is generally received using the same radio-frequency resonant coil.

Methods have been published in the literature to deal with this problem, at least to some extent [5,41,44]. It is possible either to calibrate the B1 inhomogeneities as a function of position in the sample and to apply a correction for the intensity variations, or to fit a polynomial surface to the image data from which a post-hoc "bias field" can be deduced. Again, an intensity correction can then be applied. Note that the latter method may partly be influenced by other signal intensity variations across the image, such as those caused by static field inhomogeneities or different tissue types.

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