Asymmetric tesseral shim coils for magnetic resonance
AssigneeNMR Holdings No. 2 Pty Limited
InventorsLawrence Kennedy Forbes, Stuart Crozier
Life of this patent
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Abstract
Asymmetric tesseral coils and sets for magnetic resonance applications are provided. Also provided are methods for designing such coils, as well as coils generating arbitrary field profiles. In certain embodiments, the coils are rigorously constrained to have finite dimensions.
Claims43
A tesseral coil having (i) a longitudinal axis and a radius r describing a primary cylindrical surface and (ii) a predetermined volume in which at least one predetermined tesseral harmonic is generated (the tesseral harmonic volume) and comprising a plurality of current-carrying windings associated with the primary cylindrical surface, the tesseral coil producing a magnetic field, the longitudinal component of which is given by:
- B ( r , θ , φ ) = ∑ n = 0 ∞ ∑ m = 0 m = n r n [ A n m cos ( m φ ) + B n m sin ( m φ ) ] P n m ( cos θ )
- where Anm and Bnm are the amplitudes of spherical harmonics, Pnm(cos θ) are associated Legendre polynomials, n is the order and m the degree of the polynomial, and r ,θ and φ are polar co-ordinates;
- wherein:
- (i) the at least one predetermined tesseral harmonic has a degree m′ and an order n′ which satisfy the relationships:
- m′>0, and
- n′≧2;
- (ii) the primary cylindrical surface has first and second ends which define a length 2L therebetween;
- and
- (iii) the tesseral harmonic volume extends along the longitudinal axis from z=pL to z=qL, where
- (a) −1<p<q<1;
- (b) |P|≠|q|;
- and
- (c) z=0 is midway between the first and second ends of the primary cylindrical surface.
A method for designing a coil, where the coil has a longitudinal axis which lies along the z-axis of a three dimensional coordinate system, said method comprising the steps of:
- (a) selecting one or more cylindrical surfaces Si (i≧1) for calculating current densities for the coil, each surface (1) having a radius a1, (2) surrounding the longitudinal axis, and (3) extending from −Li to +Li;
- (b) selecting a set of desired values for the longitudinal component of the magnetic field Bz (or HT) to be produced by the coil at target locations on one or more cylindrical surfaces Tj(j≧1) which extend along the longitudinal axis from z=pLi to z=qLi where i=1 and −1<p<q<1;
- and
- (c) determining current density distributions ji(θ,z) at the Si surfaces for the coil by:
- (1) expressing Bz (or HT) in terms of ji(θ,z);
- (2) expressing the z-component of each of the ji(θ,z) (jzi(θ,z)) in terms of a basis function expansion and constraining jzi(θ,z) to equal 0 at −Li and +Li, said expansion including a set of coefficients;
- (3) expressing Bz (or HT) in terms of said basis function expansions;
- and
- (4) for each of jzi(θ,z), determining values for said set of coefficients of said basis function expansion by:
- i. selecting an error function which comprises a measure of the difference between the desired Bz (or HT) values at the target locations and the Bz values calculated at those locations using the basis function expansions of the jzi(θ,z);
- ii. regularizing said error function;
- and
- iii. determining values for the sets of coefficients using the regularized error function;
- and
- (5) determining the azimuthal-component of ji(θ,z) (jθi(θ,z)) using the values for the sets of coefficients determined in step (c)(4) and the continuity equation.
A method for designing a coil comprising:
- (a) selecting one or more surfaces Si (i≧1) for calculating current densities for the coil;
- (b) selecting a set of desired values for one or more components of the magnetic field B (or H) to be produced by the coil at target locations on one or more surfaces Tj (j≧1);
- and
- (c) determining current density distributions ji at the Si surfaces for the coil by:
- (1) expressing B (or H) in terms of ji;
- (2) expressing each of the ji in terms of a basis function expansion having a set of coefficients;
- (3) expressing B (or H) in terms of said basis function expansions;
- and
- (4) for each ji determining values for said set of coefficients of said basis function expansion by:
- i. selecting an error function which comprises a measure of the difference between the desired B (or H) values at the target locations and B (or H) values calculated at those locations using the basis function expansions of the ji;
- ii. regularizing said error function using one or more regularization parameters, one parameter for each ji, said parameter being the curvature of a streamfunction for ji;
- and
- iii. determining values for the sets of coefficients using the regularized error function.
Dependencies are inferred from the claim text.