Algorithm for adiabatic pulse design using the Shinnar Le-Roux transform

1. A computer implemented method for providing an adiabatic RF pulse that is an inversion or refocusing pulse for a RF pulse sequence, for a magnetic resonance imaging system, comprising:

determining a linear phase frequency profile (F lp (ω)) for the adiabatic RF pulse;

applying a quadratic phase to the linear phase frequency profile for the adiabatic RF pulse to obtain F(ω), wherein the applying the quadratic phase comprises setting F(ω)=F lp (ω) e ikω 2 , where k is a specified constant and ω is a frequency vector over which the spectral profile is defined;

setting a polynomial β=Fourier Transform (F(ω));

determining a corresponding minimum phase a polynomial for the β polynomial; setting (α,β) as inputs to an inverse Shinnar Le-Roux transform to generate an adiabatic RF waveform; and

truncating the adiabatic RF waveform to produce the adiabatic RF pulse, wherein a pulse duration and a transition width is specified, wherein k>0.03π/(effective transition width) where effective transition width is given by (ω s −ω p )/(N+1) and k<k max , where k max is a value at which the adiabatic RF pulse is truncated at 25% of a maximum RF amplitude.

2. The method, as recited in claim 1 , wherein k>0.1π/(effective transition width) and where k max is a value at which the adiabatic RF pulse is truncated at 5% of a maximum RF amplitude.

3. The method, as recited in claim 2 , wherein the determining the linear phase frequency profile comprises determining a linear phase finite impulse response filter with a magnitude response that is equal to a desired spectral profile of the pulse.

4. The method, as recited in claim 3 , wherein the linear phase finite impulse response filter minimizes a weighted, integrated squared error between an ideal piecewise linear function and a magnitude response of the filter over a set of specified frequency bands.

5. The method, as recited in claim 4 , wherein the adiabatic RF pulse is an 180° RF pulse.

6. The method, as recited in claim 5 , wherein the adiabatic RF pulse is used as a refocusing or inversion pulse in conjunction with at least one excitation pulse within a pulse sequence which is applied by a magnetic resonance imaging system to create a magnetic resonance image.

7. The method, as recited in claim 6 , wherein the magnetic resonance imaging system uses surface coils or a high magnetic field sufficient to cause B 1 inhomogeneities.

8. The method, as recited in claim 1 , wherein the adiabatic RF pulse is an 180° RF pulse.

9. The method, as recited in claim 1 , wherein the adiabatic RF pulse is used as a refocusing or inversion pulse in conjunction with at least one excitation pulse within a pulse sequence which is applied by a magnetic resonance imaging system to create a magnetic resonance image.

10. A method for magnetic resonance imaging (MRI) in an MRI system, comprising:

designing an adiabatic RF pulse that is an inversion or refocusing pulse for a RF pulse sequence, for a magnetic resonance imaging system, comprising:

determining a linear phase frequency profile (F lp (ω)) for the adiabatic RF pulse;

applying a quadratic phase to the linear phase frequency profile for the adiabatic RF pulse to obtain F(ω), wherein the applying the quadratic phase comprises setting F(ω)=F lp (ω) e ikω 2 , where k is a specified constant and ω is a frequency vector over which the spectral profile is defined;

setting a polynomial β=Fourier Transform (F(ω));

determining a corresponding minimum phase a polynomial for the β polynomial;

setting (α,β) as inputs to an inverse Shinnar Le-Roux transform to generate an adiabatic RF waveform; and

truncating the adiabatic RF waveform to produce the adiabatic RF pulse, wherein a pulse duration and a transition width is specified, wherein k>0.03π/(effective transition width) where effective transition width is given by (ω s −ω p )/(N+1) and k<k max , where k max is a value at which the adiabatic RF pulse is truncated at 25% of a maximum RF amplitude;

applying a B 0 field;

applying the a sequence of RF pulses including designed adiabatic RF pulse for excitation;

reading out a signal at an echo; and

creating an image from a Fourier transform of the read out signal.

11. The method, as recited in claim 10 , wherein k>0.1π/(effective transition width) and where k max is a value at which the adiabatic RF pulse is truncated at 5% of a maximum RF amplitude.

12. The method of claim 11 , wherein the designed adiabatic RF pulse is largely insensitive to B 1 inhomogeneities.

13. The method, as recited in claim 12 , wherein the magnetic resonance imaging system uses surface coils or a high main magnetic field resulting in B 1 inhomogeneities.

14. The method, as recited in claim 10 , wherein the determining the linear phase frequency profile comprises determining a linear phase finite impulse response filter with a magnitude response that is equal to a desired spectral profile of the pulse.

15. The method, as recited in claim 14 , wherein the linear phase finite impulse response filter minimizes a weighted, integrated squared error between an ideal piecewise linear function and a magnitude response of the filter over a set of specified frequency bands.

16. The method, as recited in claim 10 , wherein the adiabatic RF pulse is an 180° RF pulse.

17. The method, as recited in claim 10 , further comprising displaying the created image.

18. An apparatus for providing magnetic resonance imaging, comprising:

a magnetic resonance imaging excitation and detection system; and

a controller electrically connected to the magnetic resonance imaging excitation and detection system, comprising:

a display;

at least one processor; and

non-transitory computer readable media, comprising:

computer readable code for designing an adiabatic RF pulse that is an inversion or refocusing pulse for a RF pulse sequence for the magnetic resonance imaging excitation and detection system, comprising:

computer readable code for determining the a linear phase frequency profile (F lp (ω)) for the adiabatic RF pulse;

computer readable code for applying a quadratic phase to the linear phase frequency profile for the adiabatic RF pulse to obtain F(ω), wherein the applying the quadratic phase comprises setting F(ω)=F lp (ω)e ikω 2 , where k is a specified constant and ω is a frequency vector over which the spectral profile is defined;

computer readable code for setting a polynomial β=Fourier Transform (F(ω));

computer readable code for determining a corresponding minimum phase α polynomial for the β polynomial;

computer readable code for setting (α,β) as inputs to an inverse Shinnar Le-Roux transform to generate an adiabatic RF waveform; and

computer readable code for truncating the adiabatic RF waveform to produce the adiabatic RF pulse, wherein a pulse duration and a transition width is specified, wherein k>0.03π/(effective transition width) where effective transition width is given by (ω s −ω p )/(N+1) and k<k max , where k max is a value at which the adiabatic RF pulse is truncated at 25% of a maximum RF amplitude;

computer readable code for applying the a sequence of RF pulses including designed adiabatic RF pulse for excitation;

computer readable code for reading out a signal at an echo; and

computer readable code for creating an image from a Fourier transform of the read out signal.

19. The apparatus, as recited in claim 18 , further comprising computer readable code for displaying the created image on the display.

20. The apparatus, as recited in claim 18 , wherein k>0.1π/(effective transition width) and where k max is a value at which the adiabatic RF pulse is truncated at 5% of a maximum RF amplitude.