MRI gradient waveform design using convex optimization
View Patent ↗A time-optimal MRI gradient design method utilizes constrained optimization to design minimum-time gradient waveforms that satisfy gradient amplitude and slew-rate limitations. Constraints are expressed as linear equations which are solved using linear programming, L1-norm formulation, or second-order cone programming (SOCP).
1. A method of rapid magnetic resonance imaging using constrained optimization comprising the steps of:
A) applying a magnetic gradient to an object, comprising the steps of:
a) selecting a number N of discrete-time waveforms with a sampling period, τ, and time T=Nτ,
b) expressing constraints as equations,
c) identifying if a solution exists for T, and if not, increasing T until a solution exists,
d) decreasing T to find a shortest solution, and
e) solving the equations to identify said shortest solution, utilizing linear programming to identify said shortest solution, wherein the linear programming finds the vector x that minimizes a cost function f T x subject to the constraint Ax=<b, where matrix A and vector b are formed by combining all of the linear constraint equations for an amplifier and pulse sequence constraints, wherein said shortest solution is used for generation of said magnetic gradient;
B) detecting imaging signals; and
C) providing an image based on the detected image signals.
2. The method as defined by claim 1 wherein steps c), d), and e) utilize L1-norm formulation.
3. The method as defined by claim 2 wherein the gradient is one dimensional.
4. The method as defined by claim 2 wherein the gradient is two dimensional.
5. The method as defined by claim 2 wherein the gradient is three dimensional.
6. The method as defined by claim 2 wherein in step a) the gradient amplifier has current and voltage limits (I max and V max ) that result in the following limits on G(p):
G
(
t
)
≤
η
I
max
and
L
ⅆ
ⅆ
t
G
(
t
)
+
RG
(
t
)
≤
η
V
max
where L, R, and 1 are the gradient coil inductance, resistance, and efficiency.
7. A method of rapid magnetic resonance imaging using constrained optimization comprising the steps of:
A) applying a magnetic gradient to an object, comprising the steps of:
a) selecting a number N of discrete-time waveforms with a sampling period , τ, and time T=Nτ,
b) expressing constraints as equations,
c) identifying if a solution exists for T, and if not, increasing T until a solution exists,
d) decreasing T to find a shortest solution, and
e) solving the equations to identify said shortest solution, wherein steps c), d), and e) include adding slack variables to the optimization, wherein said shortest solution is used for generation of said magnetic gradient;
B) detecting imaging signals; and
C) providing an image based on the detected image signals.
8. The method as defined by claim 7 wherein step e) solves for gradient Gx(n) and a set of variables H x (n) that converge to |Gx(n)| and a set of variables S x (n) that converge to |αGx(n)+βGx(n+1)|, wherein a sequence number n=1 . . . N,
α
=
R
-
L
τ
,
and
β
=
L
τ
,
where R is a gradient coil resistance and L is a gradient coil inductance.
9. The method as defined by claim 8 wherein the slack variables are forced to converge by adding linear gradient constraints:
− H x [n]+G x [n]≦ 0
− H x [n]−G x [n]≦ 0
and slew-rates constraints:
S x [n]+αG x [n]+βG x [n+ 1]≦0
S x [n]−αG x [n]−βG x [n+ 1]≦0
whereby the constraints force the Hx(n) and Sx(n) variables to approach an appropriate absolute value when combined with minimization of the following pulse function:
J
(
S
x
[
n
]
,
H
x
[
n
]
)
=
∑
n
=
1
N
(
H
x
[
n
]
+
S
x
[
n
]
)
.
10. The method as defined by claim 7 wherein steps c), d) and e) use a binary-search to minimize time in identifying the minimum value of n which provides a solution.
11. The method as defined by claim 10 wherein the gradient is one dimensional.
12. The method as defined by claim 10 wherein the gradient is two dimensional.
13. The method as defined by claim 10 wherein the gradient is three dimensional.
14. The method as defined by claim 10 wherein in step a) the gradient amplifier has current and voltage limits (I max and V max ) that result in the following limits on G(p):
G
(
t
)
≤
η
I
max
and
L
ⅆ
ⅆ
t
G
(
t
)
+
RG
(
t
)
≤
η
V
max
where L, R, and η are the gradient coil inductance, resistance, and efficiency.
15. The method as defined by claim 7 wherein the gradient is one dimensional.
16. The method as defined by claim 7 wherein the gradient is two dimensional.
17. The method as defined by claim 7 wherein the gradient is three dimensional.
18. The method as defined by claim 7 wherein in step a) the gradient amplifier has current and voltage limits (I max and V max ) that result in the following limits on G(p):
G
(
t
)
≤
η
I
max
and
L
ⅆ
ⅆ
t
G
(
t
)
+
RG
(
t
)
≤
η
V
max
where L, R, and η are the gradient coil inductance, resistance, and efficiency.
19. The method as defined by claim 7 and further including:
f) establishing physical constraints of a gradient amplifier and a gradient coil.
20. The method as defined by claim 19 and further including:
g) establishing beginning and end gradient boundary constraints and fixed area for one or more gradient moments.
21. The method as defined by claim 7 and further including:
f) establishing beginning and end gradient boundary constraints and fixed area for one or more gradient moments.
22. The method as defined by claim 7 wherein step b) includes gradient heating constraints.
23. The method as defined by claim 22 wherein step b) includes gradient magnetostimulation constraints.
24. The method as defined by claim 7 wherein step b) includes gradient magnetostimulation constraints.
25. A method of rapid magnetic resonance imaging using constrained optimization comprising the steps of:
A) applying a magnetic gradient to an object, comprising the steps of:
a) selecting a number N of discrete-time waveforms with a sampling period , τ, and time T=Nτ,
b) expressing constraints as equations,
c) identifying if a solution exists for T, and if not, increasing T until a solution exists,
d) decreasing T to find a shortest solution, and
e) solving the equations to identify said shortest solution, utilizing a second-order cone programming (SOCP) to find said shortest solution x that minimizes a linear pulse function f T X subject to a second order cone constraint
∥Ax+b∥ 2 ≦Cx+d
which is a superset of the linear constraints in linear programming where matrices A and B and vectors b and d are formed by combining all linear constraint equations for amplifier and pulse sequence constraints, wherein said shortest solution is used for generation of said magnetic gradient;
B) detecting imaging signals; and
C) providing an image based on the detected image signals.