Page 252 - Biomedical Engineering and Design Handbook Volume 2, Applications

DESIGN OF MAGNETIC RESONANCE SYSTEMS 231

The static magnetic field is sampled at numerous points on a spherical or cylindrical surface. Then field
errors are expanded in terms of, for example, spherical harmonics. Shim coils typically are designed 9
to produce fields that approximate the desired harmonic (or other expansion). The current appropriate
for correcting each term is then set for each coil shim. Alternatively, the correct size and position of
each shim is calculated. The lowest-order shims can usually be achieved by placing constant currents
on the three gradient coil axes.

8.2.3 Forces
Forces on ferrous objects near magnets may be of concern. The acceleration a (normalized to that of
gravity g) of objects experiencing magnetic forces depends on the permeability of free space μ , sus-
0
10
ceptibility χ, density ρ, and on the magnetic field B and its spatial gradient :
χ ∂ B
a = B (8.7)
μρ g z ∂
0
From Eq. (8.7) it is clear that the greatest forces on ferromagnetic objects are where the product of
field strength and spatial gradient is the largest. For superconducting magnets, this position is nor-
mally close to the coil windings.

8.3 GRADIENT CHARACTERISTICS

A computer commonly generates digital waveforms for the three gradient axes and for the radiofre-
quency coils. These waveforms (which for gradients may include corrections for eddy currents) are
converted into analog signals, amplified, and sent to the appropriate coils. Received signals are con-
4
verted into digital signals and reconstructed into images using Fourier transforms. The reconstructed
images are then electronically displayed. The computer system may also monitor MR scanner sub-
systems, including those associated with patient safety.
In MR, the static magnetic field is usually taken as the z direction. Linear variations of the static
magnetic field (i.e., ∂Bz/∂x, ∂Bz/∂y, and ∂Bz/∂z) are produced by separate gradient coil sets for the
three (x, y, and z) coordinates. Only gradient field components in the z direction matter for MR imag-
ing physics. However, magnetic fields form closed loops. So, other non-z components are produced
as well. These other components may produce unwanted imaging artifacts and may influence patient
physiological responses to switched gradients.
Considerations in gradient coil design include gradient linearity, gradient slew rate (i.e., how
quickly the gradient amplitude can change), gradient power dissipation, eddy currents, and
gradient-induced nerve stimulation. For simplicity in discussing these issues, consider the
11
Maxwell pair (see Fig. 8.2). If two filamentary, circular loops carry current I in opposing direc-
tions, have radii a, and are spaced a distance 2d apart, then the magnetic induction B may be
expressed as

μ Ia 2 μ Ia 2
B = 0 − 0
2
2 3 2 /
2 3 2 /
2
+
−
2( a + ( z d) ) 2( a + ( z d) )
(8.8)
2
4
4
2 2
2
2
2
⎛ 3μIa d ⎞ ⎛ 5μIa d( 3a 2 − 4d )⎞ ⎛ 21 Ia d( 5 − 20a d + 8d )⎞
μ
a
5
≈ ≈ ⎜ ⎟ z − ⎜ ⎟ z 3 + ⎜ ⎟ z + higher orders
2 5 2 /
2
2 13 2
2 9 2
a
⎝ a( 2 + d ) ⎠ ⎝ a ( 2 2 + d ) / ⎠ ⎝ 8(a + d ) / ⎠