Gradients and Shims

GRADIENTS WITH INTEGRATED SHIMS - BFG-S SERIES

GRADIENT LINEARITY DEFINITIONS

Gradient linearity, or more precisely gradient non-linearity, is defined as the normalized difference between the actual value of some measure of the field gradient and the ideal value of the same measure of the field gradient.

The measure of a field gradient may involve either fields or gradients, and the differences may be between actual and ideal values at points in space or between actual and ideal root mean square values evaluated over surfaces or throughout volumes.

The differences are normalized to the magnitude of the ideal value at some fixed point in space when dealing with values at individual points and to the root mean square ideal value when dealing with r.m.s. values evaluated over surfaces or volumes. The resulting fraction is typically expressed as a percentage.

No matter what definition of measure is applicable there is a region of interest over which the actual value of the measure of field gradient must not deviate by more than a specified amount from the ideal value.

This region is known as the linearity volume, or the field of view, (FOV), or sometimes as the homogeneity volume if the field gradient is associated with the region of interest of a uniform background magnetic field.

The linearity volume, typically, is spherical in applications involving magnetic resonance imaging, (MRI), and cylindrical in high resolution nuclear magnetic resonance applications. However, the gradient linearity as defined above is general and can be applied to volumes of arbitrary shape.

Depending on the application for which the gradient is intended there will be some definition of the measure that is relevant and appropriate and there will be some definition of the maximum deviation of that measure from the ideal over the linearity volume that is acceptable.

A researcher designing an experiment involving the use of field gradients that ideally would be perfectly linear will doubtless be aware of the sort of deviation from the ideal field profile that would be relevant and the maximum magnitude of the deviation that would be acceptable. However, it is important that he should be able to unambiguously communicate his requirements to whomever would be designing the gradient coils that would generate the gradients.

Specifications based on measures involving maximum deviations of gradients at a point are more difficult to achieve than those based on measures involving maximum deviations of fields at a point. In other words, a specification based on local gradient deviation would result in a design fulfilling a much tighter local field deviation specification.

Therefore, specifying according to gradient deviation when field deviation should be specified would result in a more linear but more expensive gradient coil. Conversely, specifying according to field deviation when gradient deviation should be specified would result in a less expensive gradient coil but one which would fail to meet the linearity requirements necessary for a successful experimental outcome.

When the measure of gradient linearity involves differences between actual and ideal values at points in space the gradient linearity is a continuously varying function of position in space; at some points it will be a positive quantity and at other points it will be negative. In other words the actual value of the desired measure of gradient will somewhere be greater than the ideal value and elsewhere be smaller.

The extremes, positive and negative, of the differences can be determined by performing a search throughout the linearity volume and in general it will be found that these extremes occur on the surface of the volume of interest. The above discussion can be illustrated by some specific examples.

Firstly, suppose that the measure relevant to the needs of an experimenter involve differences between actual and ideal fields at points throughout a certain volume.

Let the gradient coil be energized so that it generates the desired value, e.g., Gz for a Z gradient coil or Gx for an X gradient coil. The ideal values of field at a given point in space, (x, y, z), will then be respectively or , i.e., the product of the desired gradient strength and the distance from the origin in the direction in which the field is supposed to change; this is the ideal linear variation of field with respect to distance.

If a search is performed throughout this volume the actual fields, , at corresponding points can be determined and the difference between the actual fields and the ideal fields can be calculated from . The differential linearity based on a measure of field is then obtained by normalizing this to the maximum value of the ideal field in the region of interest, i.e., . is found from or .

If the desired linearity volume is spherical then both and are equal to the radius of the linearity volume. If the desired linearity volume is cylindrical and coaxial with the z axis of the system then is the half length of the cylinder and is the radius of the cylinder.

The search will result in a maximum positive and maximum negative differential field non-linearity and these will occur at points on the surface of the volume of interest. If we select the largest of the absolute values of the positive and negative excursions and call it then we can define a maximum differential field non-linearity as, .

If such a maximum field deviation is the important parameter then the experimenter must determine the maximum excursion that is acceptable, specify its value to the designer or supplier of the gradient coils and clearly state that non-linearity based on maximum field deviation is the relevant criterion.

Another measure of differential field non-linearity could be the normalized maximum peak to peak variation, i.e., the difference between maximum positive and negative excursions with respect to the ideal within the volume.

If peak to peak deviation is the important parameter then this should be so specified. Note however, that usually the excursions positive and negative with respect to the ideal are not of equal magnitude and therefore that the maximum deviation from the ideal at the worst point will be greater than half of the peak to peak value.

Less frequently encountered is a measure of field linearity over a certain region based on a root mean square criterion; it is, however, defined in an analogous way. If the r.m.s. field deviation on the surface of a volume is the parameter of interest then where dA is an element of surface of the total surface, (A), of the volume and where the integral is taken over the surface. If it is the whole volume that is important then the integral is taken with respect to the volume.

Note that for a given gradient coil design the magnitude of the r.m.s. field non-linearity is always smaller than the magnitude of the maximum field non-linearity. In other words, inadvertently specifying non-linearity in terms of r.m.s. magnitudes will result in a design that has a higher, (worse),magnitude of maximum non-linearity.

There is another important way of specifying a gradient non-linearity. It is based not on the deviation of actual field from ideal field at a point but upon deviation of actual gradient from ideal gradient at a point. The gradient deviation definition can be developed in a manner similar but somewhat simpler to that used in the field deviation definition.

Suppose that the measure relevant to the needs of an experimenter involve differences between actual and ideal gradients at points throughout a certain volume. Let the gradient coil be energized so that it generates the desired gradient value, e.g., Gz for a Z gradient coil or Gx for an X gradient coil.

The ideal values of gradient at all points in space, (x, y, z), will be these values, i.e., or . However, the actual values of gradient will be different.

Remembering the definition of gradient in this context, i.e., for a Z gradient coil or for an X gradient coil, the local values of the gradient will be given by or , i.e. it is the rate of change of field with respect to distance locally. If a search is performed throughout this volume the actual gradients , can be determined and the difference between the actual gradients and the ideal gradients can be calculated from . The differential linearity based on a measure of gradient is then obtained by normalizing this to the ideal gradient, i.e., .

This search will result in a maximum positive and maximum negative differential gradient non-linearity and these also will occur at points on the surface of the volume of interest. If we select the largest of the absolute values of the positive and negative excursions and call it then we can define a maximum differential gradient non-linearity as, .

If such a maximum gradient deviation is the important parameter then the experimenter must determine the maximum excursion that is acceptable, specify its value to the designer or supplier of the gradient coils and clearly state that non-linearity based on maximum gradient deviation is the relevant criterion.

A set of comments concerning peak to peak gradient non-linearity excursions similar to those made about peak to peak field non-linearity excursions are applicable. The important point is that the maximum deviation from the ideal at the worst point will generally be greater than half of the peak to peak value.

An analogous situation exists with respect to r.m.s. gradient non-linearity. An r.m.s. gradient non-linearity can

be defined by if the surface is relevant as can a similar integral with respect to volume if the gradient characteristics over the volume are important.

As in the case of the field measure described above, it should be noted that for a given gradient coil design the magnitude of the r.m.s. gradient non-linearity is always smaller than the magnitude of the maximum gradient non-linearity. In other words, inadvertently specifying non-linearity in terms of r.m.s. magnitudes will result in a design that has a higher, (worse), magnitude of maximum non-linearity.

It is extremely important for the experimenter to be very clear about whether the gradient non-linearity that is being specified is based on a measure involving field deviation or on a measure based on gradient deviation.

For a given maximum deviation over a given volume of space it is more difficult to meet the specification if it is based on gradient deviation than if it is based on field deviation. For example, for Z gradient coils, the maximum gradient deviation corresponding to a given field deviation, both having been defined as above, is typically 3 to 5 times higher and the situation is more extreme for X and Y gradient coils.

In other words, if a gradient coil is specified as having to meet a maximum non-linearity difference of 10 % based incorrectly on a maximum gradient deviation measure then its design will result in it meeting a maximum non-linearity difference of about 2 % to 3.3 % based on a field measure.

If the gradient coil should have been correctly specified according to a maximum field deviation measure then the resulting gradient coil would be over-specified. In order to meet the tighter than necessary non-linearity specifications compromises in other design areas would have had to have been made. These, for example, might result in longer pulse rise time, higher power dissipation, reduced duty cycle, the need for larger gradient amplifiers, etc.

It is also important that the volume over which the non-linearity specification is applicable be explicitly stated. For example, non-linearity specified as being effective over a distance defined as plus or minus on all axes could be interpreted as (a) the specification is to be met only on axis and nothing off axis is relevant, (b) the relevant volume is a cube of side , (c) the relevant volume is a cylinder of radius and length or (d) the relevant volume is a sphere of radius . The designs resulting from these different interpretations would be vastly different.

Therefore, when specifying a gradient non-linearity state explicitly that the specification refers to:

1. A field difference measure or a gradient difference measure

2. A maximum difference from ideal or a root mean square difference from ideal

3. A volume of defined shape and size.