Given a desired magnetization profile, determining the RF pulse that produces it is generally nonlinear, due to the non-linearity of the Bloch equations. At low tip angles, the RF excitation waveform can be approximated by the inverse Fourier Transform of the desired frequency profile, using the excitation kspace analysis.[8][9] The small tip angle approximation continues to hold well for tip angles on the order of 90 degree.[8] However, for tip angles greater than 90 degree, a different approach must be used.[1]
A direct solution to the pulse design problem was independently proposed by Shinnar [2][3][4][5] and Le Roux [6] based on a discrete approximation to the spin domain version of the Bloch equations.
Theory
The SLR algorithm simplifies the solution of the Bloch equations to the design of two polynomials, which can be solved using well-known digital filter design algorithms.[1]
Where N is the number of bins, or hard pulse divisions that you wish to approximate with, and φ(t) is the phase of the B1(t) waveform at a given time t.
The mapping of the RF pulse into two complex polynomials will be denoted as the Forward SLR Transform. Given two polynomials the SLR transform can be inverted to calculate the RF pulse that produces these polynomials. The order of the polynomials is . A minimum phase results in a minimum energy RF pulse.
References
^ abcdPauly, J; P Le Roux; D Nishimura; A Macovski (1991). "Parameter relations for the Shinnar-Le Roux selective excitation pulse design algorithm". IEEE Transactions on Medical Imaging. 10 (1): 53–65. doi:10.1109/42.75611. PMID18222800.
^ abM. Shinnar, L. Bolinger, and J. S. Leigh, “Use of finite impulse response filters in pulse design,” in Proc. 7th SMRM, Aug. 1988, p. 695.
^ abM. Shinnar, L. Bolinger, and J. S. Leigh, “Synthesis of soft pulses with specified frequency responses,” in Proc. 7th SMRM, Aug. 1988, p. 1040.
^ abM. Shinnar, S. Eleff, H. Subramanian, and J. S. Leigh, “The synthesis of pulse sequences yielding arbitrary magnetization vectors,” Magnet. Resonance Med., vol. 12, pp. 74-80, Oct. 1989.
^ abM. Shinnar, L. Bolinger, and J. S. Leigh, “The use of finite impulse response filters in pulse design,” Magnetic Resonance Med., vol. 12, pp. 75-87, Oct. 1989.
^ abP. Le Roux, “Exact synthesis of radio frequency waveforms,” in Proc. 7th SMRM, Aug. 1988, p. 1049.
![{\displaystyle [B_{1}(t),\varphi (t)]\Longleftarrow SLR\Longrightarrow [A_{N}(z),B_{N}(z)]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/548f6c825fcd8704d7662505dadd4375f8bd146c)
![{\displaystyle [A_{N}(z),B_{N}(z)]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/73ab70a8fa01762885f5ab51b114225b5a1e5c87)
