Rapid method of optimal gradient waveform design for MRI

Claim 1

A method of spiral gradient design for a k-space trajectory in magnetic resonance imaging, k(τ)=Aπe i ωt where τ and ω can be functions of time, and where a discrete-time gradient waveform is g n , comprising the steps of: a) determining the angle between a given gradient sample g n and the next gradient sample g n+1 , and b) determining the magnitude of |g n+1 | using permissible change based on constraints of the magnetic resonance imaging system.

Claim 2

A method of spiral gradient design for a k-space trajectory in magnetic resonance imaging, k(τ)=Aπe i ωt where τ and ω can be functions of time, and where a discrete-time gradient waveform is g n , comprising the steps of: a) determining the angle between a given gradient sample g n and the next gradient sample g n+1 , and b) determining the magnitude of |g n+1 | using permissible change based on constraints of the magnetic resonance imaging system.

Claim 3

A method of spiral gradient design for a k-space trajectory in magnetic resonance imaging, k(τ)=Aτe i ωτ where τ and ω can be functions of time and where a discrete-time gradient waveform is g n and the discrete-time k n ==Σ n i=o g i , assuming a series inductor-resistor (LR) gradient circuit model and using the gradient circuit parameters ##EQU5## the steady-state gradient amplitude with a maximum voltage V applied, and ##EQU6## the maximum step in gradient amplitude possible starting from an amplitude of zero, said method comprising the steps of: a) determining the angle,∠, between a given gradient sample g n and the next gradient sample g n+1 by evaluating the first derivative of k(τ) at the current value of τ and obtaining ##EQU7## b) determining the magnitude of |g n+1 | based on gradient constraints represented by a circle of radius, S ##EQU8## surrounding and offset from the distal end of g n , where |g n+1 | extends along ∠g n+1 to the farthest intersection thereof with said circle.

Claim 4

A method of spiral gradient design for a k-space trajectory in magnetic resonance imaging, k(τ)=Aτe i ωτ where τ and ω can be functions of time and where a discrete-time gradient waveform is g n and the discrete-time k n ==Σ n i=o g i , assuming a series inductor-resistor (LR) gradient circuit model and using the gradient circuit parameters ##EQU5## the steady-state gradient amplitude with a maximum voltage V applied, and ##EQU6## the maximum step in gradient amplitude possible starting from an amplitude of zero, said method comprising the steps of: a) determining the angle,∠, between a given gradient sample g n and the next gradient sample g n+1 by evaluating the first derivative of k(τ) at the current value of τ and obtaining ##EQU7## b) determining the magnitude of |g n+1 | based on gradient constraints represented by a circle of radius, S ##EQU8## surrounding and offset from the distal end of g n , where |g n+1 | extends along ∠g n+1 to the farthest intersection thereof with said circle.

Claim 8

A method of spiral gradient design for a k-space trajectory in magnetic resonance imaging, k(τ)=Aτe i ωτ where τ and ω can be functions of time and where a discrete-time gradient waveform is g n and the discrete-time k n =Σ n i=0 g i , assuming a gradient model with a maximum slew rate and a maximum gradient amplitude, said method comprising the steps of: a) determining the angle, ∠, between a given gradient sample g n and the next gradient sample g n+1 by evaluating the first derivative of k(τ) at the current value of τ and obtaining ##EQU10## b) determining the magnitude of |g n+1 | based on gradient constraints represented by a maximum slew rate, S, and a maximum gradient amplitude, G, where S is the radius of a circle centered at the distal end of g n .

Claim 9

A method of spiral gradient design for a k-space trajectory in magnetic resonance imaging, k(τ)=Aτe i ωτ where τ and ω can be functions of time and where a discrete-time gradient waveform is g n and the discrete-time k n =Σ n i=0 g i , assuming a gradient model with a maximum slew rate and a maximum gradient amplitude, said method comprising the steps of: a) determining the angle, ∠, between a given gradient sample g n and the next gradient sample g n+1 by evaluating the first derivative of k(τ) at the current value of τ and obtaining ##EQU10## b) determining the magnitude of |g n+1 | based on gradient constraints represented by a maximum slew rate, S, and a maximum gradient amplitude, G, where S is the radius of a circle centered at the distal end of g n .