MR hardware II
Transmit/ receive RF coils (categories)
• The RF coils can be divided into three general categories: 1) transmit coils, 2) receive coils and 3) transceive coils (transmit and receive coils).
• The transmission and reception of RF in MRI can be compared to a broadcasting system, which uses a high power transmitting antenna and a local receiving antenna
sensitive to low strength signals.
• The coils in MR, however, operate in the ‘near-field’ region as we are interested in the fields inside the coil structures.
Transmit/ receive RF coils (a variety of RF coils)
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Transmit RF coils (B1 field)
• The radiofrequency or RF field (B1) has to be generated at the Larmor precessional frequency. It is therefore a high frequency, oscillatory electromagnetic field (MHz range).
• The component of the RF magnetic field perpendicular to B0 is able to exert a torque on magnetic moments precessing with a Larmor frequency equal to the frequency of the applied RF field.
• Consider (again) the effect of an RF field of suitable strength and duration to nutate the net magnetization 90o from its equilibrium.
Transmit RF coils (transmit/receive coils)
• The devices that generate RF magnetic fields at the Larmor frequency are termed transmit RF coils.
• In MRI, the RF coils are designed to generate fields that are directed transverse to the applied static field B0.
Transmit RF coils (linear polarization)
• Linearly polarised RF fields are those that oscillate linearly along a particular axial direction.
• A linearly polarised RF field may be mathematically decomposed into two circularly polarised fields, each with half the amplitude and rotating in opposite directions.
• The counter-clockwise component of the RF field rotates in the opposite direction to the precessing spins, and ‘resonance-wise’ it exerts negligible effects on the spin system. So only half of the RF energy is effective in nutating the spins.
Let us see the proof
Transmit RF coils (proof)
• In the laboratory frame, linear polarisation along the x-axis is described: B1 (t) = B1 cos(ω0t)xˆ • The RF field in the rotating frame can be calculated from the RF field in the laboratory
frame by using a transformation matrix.
Bx’ cosωt −sinωt 0Bx
B ‘ = sinωt cosωt 0B yy
B’0 01B z z
• where the dashed elements represent those in the rotating frame. For an x-directed linearly polarised B1 field in the laboratory frame, the result is:
B ‘ cosωt −sinωt 0B cosωt B cos2 ωt 0.5⋅B (1+cos2ωt) x 11 1
By ‘ = sinωt cosωt 0 0 = B1 cosωt sinωt = 0.5⋅sin2ωt B’0 0100 0
z
• in which all of the terms rotating at 2ωt have no effect on the spin system as they are ‘resonance-wise’ ineffective at the Larmor frequency. In the rotating frame, this means that only half of the linearly applied RF field is useful in the excitation of spins:
B ‘= B1
x
2
Transmit RF coils (circular polarization)
• To improve the efficiency of the system, it is possible to use a quadrature coil system which generates circularly polarised RF fields. In the laboratory frame, these are described as:
B (t) = B (cos(ωt)xˆ −sin(ωt)yˆ) 11
• Using the rotation matrix, we calculate the effect of a circularly polarised field in the rotating frame:
B ‘ cosωt −sinωt 0 B cosωt B cos2 ωt+B sin2 ωt B x 1 1 1 1
B ‘=sinωt cosωt 0−B sinωt=B cosωtsinωt−B sinωtcosωt=0 y111
B’ 0 0 1 0 0 0 z
• This means, to generate a 90o pulse, a quadrature coil requires only half the amplitude a linear coil requires.
•Note that we have ignored system noise in this analysis.
A generalised quadrature coil feeding arrangement
•generate fields at the Larmor precessional frequency and therefore be selectively tuned to that frequency.
• irradiate the entire sample with the same value of RF field (and therefore produce the same nutation angle)
Tuning and matching of the resonant circuit.
• generate RF fields efficiently from the applied RF power (maximum power transfer).
Transmit RF coils (geometry)
• The geometry of the main or ‘body’ RF coil is often cylindrical and is positioned just inside the gradient coil set.
• The requirements of an RF probe are to:
Optimal current density distribution Typical body coil ‘bird-cage’ structure B1 uniformity for generating a pulse transverse field (power ratings generally ~15-20kW)
Transmit RF chain (description)
• The B1 field transmission chain starts with the frequency synthesizer which adjusts the carrier frequency to the Larmor precessional frequency of the spins to be excited.
• The carrier is then fed to the amplitude modulator which in most pulse sequences modulates the carrier into an apodized RF pulse (typically sinc-pulse). The modulated signal is passed through a power amplifier which then delivers the energy to the transmit RF coil.
Transmit RF chain (description)
• The transmit coil should ideally have a large FOV and a very homogeneous RF magnetic field distribution, which is desirable for uniform slice/volume excitation. Frequency modulation schemes are also possible.
• The transmitted energy is in form of an electromagnetic wave that excites the spin ensemble of the imaging object. As the spins relax back to the low energy state electromotive force is induced in the receive RF coil, which is described next.
•Further improvement in both SNR and resolution via reduced FOV may be obtained by using local receiver coils. These are small coils placed very close to the region of the body to be imaged.
• Receive coil units may consist of a single coil unit or multiple coils (typically phased array).
• Multiple receiver channels (amplifiers, demodulators etc) are normally used in phased array systems.
• Coils are often overlapped to reduce (inductive) mutual coupling.
(a) Single surface coil
(b) Petal coil
(c) Phased array
Receive RF coils (reciprocity)
• Transmit coils may also be used for receiving the MR signal and, by reciprocity,
their efficiency in generating RF fields is closely related receiving MR signals.
to their sensitivity in
Receive RF coils (dedicated receivers)
• Receive coils act to restrict the areas from which MR signal is received and therefore enable specific anatomical locations to be selectively imaged at higher resolution or faster imaging times than would be possible with a body transmit/receive coil
(a) Using a body-coil receiver; (b) Using surface coil as receiver
Receive RF coils (isolation)
• Both transmit and receive coils are often tuned to the same frequency. So what stops the receive coil absorbing the energy radiated by the transmit coil ???
– Nothing, unless we act to prevent this.
• The effect of this interaction would be that the receive coil would re-radiate the absorbed energy from the transmit coil and the patient could conceivably suffer local heating or burns as a result!
• Receive and transmit coils contain circuitry that enables them to be detuned at the appropriate time. For example, during a transmission pulse the receiver is open circuited and during MR signal reception the transmitter is open circuited.
The circuit for a simple diplexor.
MR signal processing (demodulation)
• Just as AM (Amplitude Modulation) and FM (Frequency Modulation) radio broadcasts are made at many MHz, so is the received MR signal. Clearly in AM/FM radio, we can only hear signals up to ~20kHz and the information we want is embedded in the carrier (or high frequency) signal. We must extract this information from the high frequency signal, and this is also the case in MR.
• The information we are interested in is the variation of the carrier or Larmor signal, ω0, so that we can consider the total signal frequency to be ω0+−δω, and we are most interested in δω.
• As it is not generally practical for the electronics to directly digitise the MR signal, it is necessary to convert the signal to a lower frequency before we can convert it to a digital representation. The signal must be in a digital form so that it can be processed by a computer.
MR signal processing (demodulation)
• We learned that spins either side of the iso-centre could be discriminated by the application of field gradients. When applied, the gradients ensure that spins on each side have higher or lower frequencies than spins at the iso-centre. Relative to the central frequency (ω0) these are either differentially positive or negative frequencies. The receiver system must be able to decipher this information.
• The first step is to multiply the signal by the frequency standard signal generated by the frequency synthesizer, which supplies both transmission and reception chains. This results in an intermediate frequency (IF) signal.
• After demodulation and filtering:
MR signal processing (demodulation)
• The next stage in the system is to demodulate the signal prior analog-to-digital conversion (ADC). It is here that we must discriminate between the positive and negative frequencies.
• Consider two magnetic moments at equal distances either side of the iso-centre along the x- axis with the frequency encoding gradient imposed. The two frequencies are:
ω1 =ω0 +γGx∆x ω2 =ω0 −γGx∆x
S1(t)∝S0 cos(+γGx∆x) S2(t)∝S0 cos(−γGx∆x)
• But since cos(θ) = cos(-θ), these signals are indistinguishable. A similar problem is encountered in spectroscopy, even without gradients.
• Implementing quadrature detection, also called phase sensitive detection, solves this problem.
• Since the references are quadrature signals it turns out that the demodulated outputs are also in quadrature and the output channels are often called the real and imaginary channels because of Fourier Transform synergies.
• Quadrature signals, by their nature, allow the direction of rotation to be determined.
• The system noise is generally ‘white’ meaning that it covers the entire spectrum equally. A wider bandwidth introduces more noise.
The essential equipment for quadrature detection.
MR signal processing (quadrature detection)
• A quadrature demodulator (or mixer) uses two Double-Balances Mixers (DBMs) and a 90o phase shifter to demodulate the signal and generate two output signals.
• This is equivalent to multiplying the signal by cosine and sine versions of an intermediate frequency (IF) reference signal.
MR signal processing (quadrature detection)
• Consider the signal of a magnetic moment at +∆x. The two reference signals multiplied by the IF MR signal result in:
Sr (t) ∝ S0 (cos(ωif t + γGx ∆xt) ⋅ cos(ωif t)) Si (t) ∝ S0 (cos(ωif t + γGx ∆xt) ⋅ sin(ωif t))
Which are equivalent to:
Sr (t) ∝ 0.5 ⋅ S0 (cos(2ωif t + γGx ∆xt) + sin(γGx ∆xt))
Si (t) ∝ 0.5 ⋅ S0 (sin(2ωif t + γGx ∆xt) − cos(γGx ∆xt))
After low-pass filtering, the real/imaginary components of the time-varying signal result in:
Real: Imaginary:
For +∆x: +0.5⋅S0 cos(γGx∆xt)
For -∆x: +0.5⋅S0 cos(γGx∆xt)
−0.5⋅S0 sin(γGx∆xt) These signals indicate unique rotation direction.
+0.5⋅S0 sin(γGx∆xt)
• Modern receiver systems employ digital filtering.
MR signal processing (sampling the signal)
• Sampling of the continuous-time signals is often performed in order to manipulate a signal with a computer.
• In the simplest form of sampling, the continuous signal is sampled at regular intervals. The rate of sampling is controlled by the scanner and is limited by the receiver hardware. The sampling interval often varies from sequence to sequence.
A snapshot of two continuous x1(t) and x2(t) signals are sampled by the same impulse functions of sampling period τ.
• For example, a sinusoidal waveform with 1ms periodicity must be sampled at least every 0.5ms to have taken truly representative samples. For signals with more complex shapes and temporal behaviour, some knowledge of the frequency components in the signal is needed to choose an appropriate sampling rate.
• Fourier analysis can greatly assist in this regard. If a signal is sampled too slowly, it cannot be correctly reconstructed from the samples. The error of reconstruction in this case is often called aliasing and is also important in spatial sampling of MR images.
MR signal processing (Nyquist criterion in MR)
• The Nyquist sampling theorem states that, in order to correctly represent a signal by a series of samples, the rate at which the samples are taken must be at least twice the frequency of the highest spectral component of the original signal.
Normal and aliased MR image
MR signal processing (Bandwidth)
• In the analog-to-digital (ADC) converter, the value of the signal is often held for a short period to allow the ADC to perform conversion – this is accomplished by a sample and hold circuit.
An example of finite time sampling of a continuous signal m(t), where T0 is the ‘hold time’ and τ is the sampling period.
MR signal processing (ADC)
• In the ADC, quantised levels represent the analog signal. The larger the number of bits in the ADC, the greater the dynamic range of the receiver. This means that a larger variation in signal strength may be accounted for or that finer variations in the signal may be detected.
• The range of the ADC is given by 2N, where N is the number of bits. 16-bit ADCs are common giving a range of 216 or 65,536. For an input signal that fills the entire input range of the ADC (say -1 to 1 Volts), the quantisation error will be 2/65,536 or about 30μV. This figure is important in terms of errors/noise introduced by the receiver.
Digitised sine wave with 3-bit resolution.