Slice Selection
How does frequency-encoding let you stimulate a given slice?
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Two steps are required to uniquely excite a slice in 2D MR imaging: (1) a slice-select gradient is imposed along an axis perpendicular to the plane of the desired slice, resulting in a linear variation of potential resonance frequencies in that direction, and (2) a specially tailored RF-pulse is simultaneously applied, whose frequency components match the narrow range of frequencies contained in the desired slice. The combination of steps (1) and (2) insures that only protons within the chosen slice are excited.
Magnetic field gradient causes the center frequency (Fc) of each slice to vary by position. The range of frequencies (ΔF) contained in a slice depends on slice thickness (ΔF) and the strength of the gradient (Gss).
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Each slice has a different center frequency Fc determined by its position (z) along the slice-select gradient (Gss), given by
Fc = γ(Bo + z•Gss) = fo + γ•z•Gss
where Bo is the main magnetic field strength and fo is its corresponding Larmor frequency.
Each slice has a finite width (Δz) and so it contains a range of frequencies (ΔF) centered around Fc. These quantities are related by the equation
ΔF = γ • Gss • Δz
In common practice ΔF is held constant (on the order of 1000-2000 Hz) and slice thickness Δz is varied by adjusting Gss. Stronger gradients produce thinner slices, and vice versa.
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The frequency-encoding gradient thus uniquely identifies a slice using center frequency (Fc) to determine slice position and frequency span (ΔF) to determine slice thickness (Δz). The next step is to stimulate the slice with an RF-pulse that will excite uniformly that particular band of frequencies and no others.
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Mathematically it can be shown that the ideal RF-excitation to achieve this frequency profile is a so-called sinc pulse. The sinc pulse is an amplitude-modulated sine wave with base frequency Fc whose equation is given below
Here the range of frequencies (ΔF) subtending a slice equals the transmit bandwidth of the pulse. Transmitter bandwidth should be distinguished from receiver bandwidth (discussed in a prior Q&A). Receiver BW is function of the digitization rate of the recorded MR signal and has nothing to do with slice selection.
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Idealized multi-lobe sinc pulse
Truncated sinc with two side lobes obtained by author from his own scanner. Only the envelope of the pulse is seen as the 64 MHz carrier wave cannot be resolved at this setting of the oscilloscope.
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A sinc pulse would require an infinite number of side lobes (and hence infinite transmission time) to uniformly and exclusively excite a discrete band of frequencies. In practice the sinc pulse is modified by a process known as apodization (literally cutting off the "feet", limiting the number of side lobes and changing their shape by digital filtering). Other pulse shapes, including those with a Gaussian envelope, are also used.
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Today most RF-pulses cannot be described by mathematically simple waveforms, having eclectic shapes based on an computer-based polynomial design process known as the Shinnar-Le Roux (SLR) algorithm. SLR-generated pulses provide more careful control of RF phase properties, allowing the designer to trade off analytically several parameters describing pulse performance including frequency profile, ripple, duration, and energy deposition.
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Examples of two 90° SLR pulses. In red, a linear phase SLR pulse resembling the classic sinc pulse. In blue, an SLR pulse that minimizes phase dispersions due to flow.
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References
Pauly J, Le Roux P, Nishimura D, Macovski A. Parameter relations for the Shinnar-Le Roux selective excitation pulse design algorithm. IEEE Transact Med Imaging 1991; 10:53-65.
Balchandani P, Pauly J, Spielman D. Designing adiabatic radiofrequency pulses using the Shinnar-Le Roux algorithm. Magn Reson Med 2010; 54:843-851.
Pauly J, Le Roux P, Nishimura D, Macovski A. Parameter relations for the Shinnar-Le Roux selective excitation pulse design algorithm. IEEE Transact Med Imaging 1991; 10:53-65.
Balchandani P, Pauly J, Spielman D. Designing adiabatic radiofrequency pulses using the Shinnar-Le Roux algorithm. Magn Reson Med 2010; 54:843-851.