3.1 Gradient Coil Linearity
From the perspective of the imaging scientist, one source of error that is important to appreciate is the geometric distortion that is introduced in the image of the sample by the scanning device. There are several mechanisms by which magnetic resonance images can be spatially distorted. The principal causes are poor magnetic field homogeneity, which is dealt with in Section 3.2, and imperfect gradient coil design, which is addressed here.
As outlined previously, the ideal gradient coil consists of a cylindrical former inside the magnet, on which are wound various current paths designed to produce the pure terms Gx = dBz/dx, Gy = dBz/dy, and Gz = dBz/dz. Additionally, the perfect field gradient coil would have high gradient strength, fast switching times, and low acoustic noise. In practice it is very difficult to achieve these parameters simultaneously. Often, the linearity specification of the coil is compromised in order to achieve high gradient strength and fast switching times. This affects all MRI sequences, whether conventional or ultrafast. Romeo and Hoult  have published a formalism for expressing gradient nonlinearity in terms of spherical harmonic terms. The coefficients to these terms should be obtainable from the manufacturer if they are critical.
The practical effect that nonlinear field gradients have is to distort the shape of the image and to cause the selection of slices to occur over a slightly curved surface, rather than over rectilinear slices. Most human gradient coils have specifications of better than 2% linearity over a 40 cm sphere (meaning that the gradient error is within 2% of its nominal value over this volume). It should be noted, however, that a 2% gradient error can translate into a much more significant positional error at the extremes of this volume, since the positional error is equal to f A Gxdx. In the case of head insert gradient coils, the distortions may be significantly higher over the same volume, given the inherently lower linearity specifications of these coils. Correction of in-plane distortion can be achieved by applying the precalibrated or theoretical spherical harmonic terms to remap the distorted reference frame (x', y', z') back into a true Cartesian frame (x, y, z). Indeed, many manufacturers perform this operation in two dimensions during image reconstruction. Correction of slice selection over a curvilinear surface is, however, rarely made.
A second related problem that can occur is if the gradient strength is not properly calibrated. Generally, systems are calibrated to a test phantom of known size. Over time, though, these calibrations can drift slightly, or may be inaccurately performed. For both of these reasons, great care should be taken when making absolute distance measurements from MRI data. Manufacturers strongly discourage the use of MRI scanners in stereotactic measurement because it is very difficult to eliminate all forms of geometric distortion from magnetic resonance images. Similarly, if accurate repeat measurements of tissue volume are to be made (e.g., to follow brain atrophy), then similar placement of the subject in the magnet should be encouraged so that the local gradient-induced anatomical distortions are similar for each study. If this is not the case, then a rigid-body registration between data sets collected in different studies is unlikely to match exactly even when no atrophy has occurred.
The theoretical framework described in Section 2 assumed that the only terms contributing to the magnetic field at position (x, y, z) were the main static magnetic field (assumed to be perfectly homogeneous in magnitude at all points in the sample, and demodulated out during signal detection) and the applied linear field gradients Gx, Gy and Gz. As was shown in Section 3.1, the linear field gradient terms may in fact contain nonlinear contributions. An additional source of geometric distortion is caused by the static magnetic field itself being spatially dependent. This may result from imperfections in the design of the magnet or from geometric or magnetic susceptibility properties of the sample. Practically, offset currents can be applied to "shim" coils wound on the gradient former which seek to optimize the homogeneity of the magnetic field within the volume of interest. Shim coils are rarely supplied beyond second-order spatial polynomial terms, however. This leaves a high-order spatially varying field profile across the sample that is dominated both by the magnetic susceptibility differences between the sample and the surrounding air and the magnetic susceptibility differences between the various component tissues. An example of this is shown in Fig. 2, which shows a 2D field map through the brain of a normal volunteer. The low order static magnetic field variations have been minimized using the available shim coils, but anatomically related high spatial frequency magnetic field variations remain. These are particularly prominent around the frontal sinuses (air-tissue boundary) and the petrous bone.
The result of the residual magnetic field inhomogeneities is that Eq. (3) is modified to give
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