Page 256 - Biomedical Engineering and Design Handbook Volume 2, Applications

DESIGN OF MAGNETIC RESONANCE SYSTEMS 235

There is also a force on the coil that is independent of the static magnetic field. Consider the poten-
49
tial energy U of a gradient coil with inductance L :
b
2
U = 1 / 2 LI = ∫ F dq (8.14)
c
b
F is the force on the coil due to a displacement and q is an arbitrary coordinate. Note that the force
c
on the coil will be in a direction that increases inductance. Hence, the force will try to increase the
radius or compress the turns axially. From the derivative of the above equation, an expression for the
49
force may be obtained :
I
F = dL ⎞
c
2 ⎛ ⎜ ⎝ dq⎠ ⎟ (8.15)
The acoustic noise produced by forces on gradient coils must be limited to appropriate levels to
avoid hearing loss. 18–20,50–55 Various schemes to reduce gradient-produced acoustic noise have been
reported. 56–58

8.4 RADIO-FREQUENCY MAGNETIC FIELD AND COILS

Resonant radio-frequency (RF) magnetic fields, orthogonal to the static magnetic field, are used in
magnetic resonance (MR) to interrogate (excite) a region of interest for imaging or for spectroscopy. 3,4
The patient may absorb some portion of the transmitted RF energy. 59–63 Heating is the potential safety
64
concern with absorbed RF energy. It is essential for patient safety to limit whole-body and localized
heating to appropriate levels. 18–20,59–84
Resonant frequency scales with static field strength and nuclei of interest. For protons the reso-
3
nant RF frequency is 42.57 MHz/T . Adjusting tip angle maximizes received signals in MR. Tip
angles are proportional to area under the envelope of RF waveforms. For a given waveform, RF
energy is proportional to the square of tip angle. Only the magnetic component of the RF field is use-
ful in MR. Efforts are made by manufacturers to reduce electric field coupling to patients. The dis-
tribution of RF power deposition in MR tends to be peripheral due to magnetic induction. 59–61 Note
that plane wave exposures (in non-MR applications) may lead to greater heating at depth. 63,64
RF pules are typically transmitted by resonant RF coils. Transmit RF coils may be whole-body
coils or local coils. Safety concerns with whole-body RF transmit coils are primarily to limit whole-
body temperature elevation to appropriate levels. As shall be explored later, elevation of core body
temperatures to sufficiently high levels may be life-threatening. 59–84 With local transmit coils, the
primary safety concern is to limit local heating to prevent localized burns. 85–87
Average RF power is proportional to the number of images per unit time. Patient geometry,
RF waveform, tip angle, and whether the system is quadrature during transmit determine peak
power. Quadrature excitation lowers RF power requirements by a factor of 2 and stirs any field
inhomogeneities. 60,63 Both mechanisms lower the local specific absorption rate (SAR).

8.4.1 Transmit Birdcage Coils
One means of achieving rather homogeneous radiofrequency magnetic fields in MR is through

the use of birdcage transmit coils 88 (see Fig. 8.4). Birdcage coils ideally would produce uniform
B 1 fields. Let A be the magnetic vector potential. Components of A (and thus the electric field

as well) must be parallel to the current density on the conductors that produced them. Perfectly
uniform B 1 requires an infinitely long birdcage coil (or a spherical current density). The B 1 field
is related to magnetic vector potential:
⎡ ∂ A ∂ A ⎤ ⎡ ∂ A ∂ A ⎤ ⎡ ∂ A ∂ A ⎤
B =∇× = ˆ a x ⎢ z − y ⎥ + a ˆ y ⎢ x − z ⎥ + a ˆ z ⎢ y − x ⎥ (8.16)
A
1
⎣ y ∂ z ∂ ⎦ ⎣ z ∂ ∂x ⎦ ⎣ x ∂ y ∂ ⎦
x