Method for joint sparsity-enforced k-space trajectory and radiofrequency pulse design
A system and method is provided for simultaneously designing a radiofrequency (“RF”) pulse waveform and a magnetic field gradient waveform in a magnetic resonance imaging (“MRI”) system. The method includes determining a desired pattern of RF excitation and determining, from the desired pattern of RF excitation, a plurality of k-space locations indicative of the magnetic field gradient waveform and a plurality of complex weighting factors indicative of RF energy deposited at each k-space location. The method also includes calculating, from the determined k-space locations, the magnetic field gradient waveform and calculating, from the complex weighting factors, the RF pulse waveform that will produce the desired pattern of RF excitation when produced with the calculated magnetic field gradient.
1. A method for producing a radiofrequency (RF) excitation field with an RF coil that forms a part of a magnetic resonance imaging (MRI) system, the steps comprising:
a) determining a desired pattern of RF excitation;
b) calculating, using the desired pattern of RF excitation, a plurality of weighting factors related to an RF waveform and a plurality of k-space locations related to a magnetic field gradient waveform, wherein the calculation enforces joint sparsity in the plurality of k-space locations and the plurality of weighting factors;
c) calculating, from the calculated plurality of k-space locations, the magnetic field gradient waveform;
d) calculating, using the calculated plurality of weighting factors, an RF pulse waveform indicative of the desired pattern of RF excitation;
e) producing, with the MRI system, a magnetic field gradient using the magnetic field gradient waveform calculated in step c); and
f) producing, with the MRI system, an RF pulse using the RF pulse waveform calculated in step d) such that an RF excitation field is produced.
2. The method as recited in claim 1 in which the plurality of k-space locations and the plurality of weighting factors are calculated contemporaneously.
3. The method as recited in claim 1 in which the MRI system includes a parallel transmission RF coil.
4. The method as recited in claim 3 in which step d) includes determining an RF waveform for each channel in the parallel transmission RF coil.
5. The method as recited in claim 4 in which step f) includes producing each RF waveform calculated in step d) with the corresponding channel in the parallel transmission RF coil.
6. The method as recited in claim 1 in which steps e) and f) are performed substantially contemporaneously so that an RF excitation field indicative of the desired pattern of RF excitation is produced.
7. The method as recited in claim 1 in which step b) is performed by minimizing an objective function having the form:
f
1
(
d
-
S
F
tot
g
tot
)
+
f
2
(
g
(
m
)
)
;
wherein:
f 1 and f 2 are non-decreasing functions;
∥ . . . ∥ is a norm operation;
d is the desired pattern RF excitation;
S is a sensitivity profile of a RF coil that makes a part of the MRI system;
F tot is a matrix that relates the energy deposited at a k-space location to a corresponding spatial location on the RF coil;
g tot is a matrix containing the plurality of weighting factors indicative of the sparse set of k-space locations; and
g (m) is a vector having values equal to the L m -norms of the rows of g tot .
8. The method as recited in claim 1 in which step b) is performed by minimizing an objective function having the form:
(
1
-
λ
)
d
-
S
F
tot
g
tot
2
2
+
λ
g
(
l2
)
1
;
wherein:
λ is a control parameter;
∥ . . . ∥ is a norm operation;
d is the desired pattern RF excitation;
S is a sensitivity profile of a RF coil that makes a part of the MRI system;
F tot is a matrix that relates the energy deposited at a k-space location to a corresponding spatial location on the RF coil;
g tot is a matrix containing the plurality of weighting factors indicative of the sparse set of k-space locations; and
g (l2) is a vector having values equal to the L 2 -norms of the rows of g tot .
9. The method as recited in claim 1 , further including:
g) acquiring, with the MRI system, image data from a subject in response to the produced RF excitation field; and
h) reconstructing an image of the subject using the acquired image data.
10. A computer readable storage medium having stored thereon a computer program that when executed by a computer processor causes the processor to:
a) determine a desired pattern of RF excitation;
b) calculate, using the desired pattern of RF excitation:
i) a plurality of weighting factors related to an RF waveform; and
ii) a plurality of k-space locations related to a magnetic field gradient waveform;
c) calculate, using the plurality of calculated k-space locations the magnetic field gradient waveform; and
d) calculate, using the plurality of calculated weighting factors, the RF pulse waveform that will produce the desired pattern of RF excitation when produced, by a magnetic resonance imaging (MRI) system, substantially contemporaneously with the calculated magnetic field gradient.
11. The computer readable storage medium as recited in claim 10 in which the plurality of k-space locations and the plurality of weighting factors are jointly calculated.
12. The computer readable storage medium as recited in claim 11 in which step b) is performed by minimizing an objective function having the form:
f
1
(
d
-
S
F
tot
g
tot
)
+
f
2
(
g
(
m
)
)
;
wherein:
f 1 and f 2 are non-decreasing functions;
∥ . . . ∥ is a norm operation;
d is the desired pattern RF excitation;
S is a sensitivity profile of a RF coil that makes a part of the MRI system;
F tot is a matrix that relates the energy deposited at a k-space location to a corresponding spatial location on the RF coil;
g tot is a matrix containing the plurality of weighting factors indicative of the sparse set of k-space locations; and
g (m) is a vector having values equal to the L m -norms of the rows of g tot .
13. The computer readable storage medium as recited in claim 11 in which step b) is performed by minimizing an objective function having the form:
(
1
-
λ
)
d
-
S
F
tot
g
tot
2
2
+
λ
g
(
l2
)
1
;
wherein:
λ is a control parameter;
∥ . . . ∥ is a norm operation;
d is the desired pattern RF excitation;
S is a sensitivity profile of a RF coil that makes a part of the MRI system;
F tot is a matrix that relates the energy deposited at a k-space location to a corresponding spatial location on the RF coil;
g tot is a matrix containing the plurality of weighting factors indicative of the sparse set of k-space locations; and
g (l2) is a vector having values equal to the L 2 -norms of the rows of g tot .
14. The computer readable storage medium as recited in claim 10 in which the computer program further causes the processor, when performing step b), to enforce joint sparsity in the plurality of k-space locations and the plurality of weighting factors.
15. The computer readable storage medium as recited in claim 10 in which the computer program further causes the processor to:
e) produce, with the MRI system, a magnetic field gradient using the calculated magnetic field gradient waveform; and
f) produce, with the MRI system, an RF pulse using the calculated RF pulse waveform.
16. The computer readable storage medium as recited in claim 15 in which steps e) and f) are performed substantially contemporaneously so that an RF excitation field indicative of the desired pattern of RF excitation is produced.
17. The computer readable storage medium as recited in claim 10 in which the MRI system includes a parallel transmission RF coil.
18. The computer readable storage medium as recited in claim 17 in which step d) includes calculating an RF waveform for each channel in the parallel transmission RF coil.
19. A method for producing a radiofrequency (RF) excitation field with an RF coil that forms a part of a magnetic resonance imaging (MRI) system, the steps comprising:
a) determining a desired pattern of RF excitation;
b) determining a k-space trajectory from a set of candidate k-space trajectory segments by enforcing sparsity on the set of candidate segments;
c) calculating, using the desired pattern of RF excitation, an RF pulse waveform capable of forming a high-fidelity version of the desired RF excitation pattern, wherein the RF pulse waveform is calculated substantially concurrently with the determination of the k-space trajectory in step b); and
d) producing, with the MRI system, an RF excitation pulse by transmitting the calculated RF pulse waveform along the determined k-space trajectory such that an RF excitation field is produced by the RF coil.
20. The method as recited in claim 19 in which step b) includes minimizing an objective function that substantially concurrently determines:
a plurality of k-space locations at which RF energy is to be deposited;
a plurality of weighting factors indicative of how much RF energy is to be deposited at the plurality of k-space locations; and
wherein the plurality of k-space locations are determined using a sparsity enforcement algorithm.