MRI gradient waveform design using convex optimization

• US 20050077895A1
• Filed: 10/08/2003
• Published: 04/14/2005
• Est. Priority Date: 10/08/2003
• Status: Active Grant

• Current Assignee: Board of Trustees of the Leland Stanford Junior University (Stanford Management Co.)
• Original Assignee: Board of Trustees of the Leland Stanford Junior University (Stanford Management Co.)
• Inventors: Conolly, Steven M., Hargreaves, Brian A.
• Granted Patent: US 7,301,341 B2
• Time in Patent Office: Days
• US Class Current: 324/307
• CPC Class Codes: G01R 33/4824 using a non-Cartesian trajectory
• CPC Class Codes: G01R 33/561 by reduction of the scanning time, i.e. fast acquiring systems, e.g. using echo-planar pulse sequences

Abstract

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).

Claims

1. A method of magnetic gradient waveform design for rapid magnetic resonance imaging using constrained optimization 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 the solution.

2. The method as defined by claim 1 wherein step e) utilizes linear programming to identify the solution.

3. The method as defined by claim 2 wherein the linear programming finds the vector x that minimizes a cost function f^Tx subject to the constraint Ax=<

b, where matrix A and vector b are formed by combining all of the linear constraint equations for the amplifier and pulse sequence constraints.

4. The method as defined by claim 1 wherein steps c), d), and e) utilize L1-norm formulation.

5. The method as defined by claim 4 wherein the gradient is one dimensional.

6. The method as defined by claim 4 wherein the gradient is two dimensional.

7. The method as defined by claim 4 wherein the gradient is three dimensional.

8. The method as defined by claim 4 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):

$<br><br>G<br><br>\left(t\right)<br><br>\le <br><br>\mathrm{\eta <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>I_{\max }<br><br>\text{\hspace{1em}<br><br>}<br><br>\mathrm{and}<br><br>\text{\hspace{1em}<br><br>}<br><br><br><br>L<br><br>\frac{d<br><br>}{d<br><br>t}<br><br>G<br><br>\left(t\right)+\mathrm{RG}<br><br>\left(t\right)<br><br>\le <br><br>\mathrm{\eta <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>V_{\max }$

where L, R, and η

are the gradient coil inductance, resistance, and efficiency.

9. The method as defined by claim 1 wherein steps c), d), and e) include adding slack variables to the optimization.

10. The method as defined by claim 9 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 |α

Gx(n)+β

Gx(n+1)|.

11. The method as defined by claim 10 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]+α

G_x[n]+β

G_x[n+1]≦

0
−

S_x[n]−

α

G_x[n]−

β

G_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;

$J<br><br>\left(S_x<br><br>\left[n\right],H_x<br><br>\left[n\right]\right)=\underset{n=1}{\overset{N}{\sum <br><br>}}<br><br>\left(H_x<br><br>\left[n\right]+S_x<br><br>\left[n\right]\right).$

12. The method as defined by claim 1 wherein step e) utilizes second-order cone programming (SOCP) to find a solution x that minimizes a linear pulse function f^Tx subject to a second order cone constraint

∥

Ax+b∥

₂≦

Cx+d

which is a superset of the linear constraints in linear programming where matrix A and vector b are formed by combining all linear constraint equations for amplifier and pulse sequence constraints.

13. The method as defined by claim 1 wherein steps c), d) and e) use a binary-search to minimize time in identifying the minimum value of n which provides a solution.

14. The method as defined by claim 13 wherein the gradient is one dimensional.

15. The method as defined by claim 13 wherein the gradient is two dimensional.

16. The method as defined by claim 13 wherein the gradient is three dimensional.

17. The method as defined by claim 13 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):

$<br><br>G<br><br>\left(t\right)<br><br>\le <br><br>\mathrm{\eta <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>I_{\max }<br><br>\text{\hspace{1em}<br><br>}<br><br>\mathrm{and}<br><br>\text{\hspace{1em}<br><br>}<br><br><br><br>L<br><br>\frac{d<br><br>}{d<br><br>t}<br><br>G<br><br>\left(t\right)+\mathrm{RG}<br><br>\left(t\right)<br><br>\le <br><br>\mathrm{\eta <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>V_{\max }$

where L, R, and η

are the gradient coil inductance, resistance, and efficiency.

18. The method as defined by claim 1 wherein the gradient is one dimensional.

19. The method as defined by claim 1 wherein the gradient is two dimensional.

20. The method as defined by claim 1 wherein the gradient is three dimensional.

21. The method as defined by claim 1 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):

$<br><br>G<br><br>\left(t\right)<br><br>\le <br><br>\mathrm{\eta <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>I_{\max }<br><br>\text{\hspace{1em}<br><br>}<br><br>\mathrm{and}<br><br>\text{\hspace{1em}<br><br>}<br><br><br><br>L<br><br>\frac{d<br><br>}{d<br><br>t}<br><br>G<br><br>\left(t\right)+\mathrm{RG}<br><br>\left(t\right)<br><br>\le <br><br>\mathrm{\eta <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>V_{\max }$

where L, R, and η

are the gradient coil inductance, resistance, and efficiency.

22. The method as defined by claim 1 and further including:

f) establishing physical constraints of a gradient amplifier and a gradient coil.

23. The method as defined by claim 22 and further including:

g) establishing beginning and end gradient boundary constraints and fixed area for one or more gradient moments.

24. The method as defined by claim 1 and further including:

f) establishing beginning and end gradient boundary constraints and fixed area for one or more gradient moments.

25. The method as defined by claim 1 wherein step b) includes gradient heating constraints.

26. The method as defined by claim 25 wherein step b) includes gradient magnetostimulation constraints.

27. The method as defined by claim 1 wherein step b) includes gradient magnetostimulation constraints.