Method of correcting inhomogeneity of the static magnetic field generated by the magnet of a MRI machine and device for carrying out such method

Abstract

The present invention relates to a method of correcting inhomogeneity of the static magnetic field generated by the magnet of a Nuclear Magnetic Resonance imaging machine, wherein the magnet is flat and the magnetic field on one side of said magnet is corrected such that a volume is defined, which is bounded by a spherical cap surface, in which volume and along which surface the magnetic field is homogeneous, i.e. has field lines having equal parallel directions and equal intensities.

Claims

1. A method of correcting inhomogeneity of a static magnetic field generated by a magnet of a Nuclear Magnetic Resonance imaging machine, which method has the steps of: a) generating a polynomial that represents the magnetic field generated by the magnet and comprises a plurality of harmonic terms, each associated with a coefficient; b) measuring the magnetic field at a plurality of points, with a predetermined distribution in space; c) determining the coefficients from the measured values; d) comparing the coefficients from the measured values with those that describe the field having the desired characteristics; e) defining a grid for positioning a plurality of correction elements and relating it to the field structure; f) calculating the position and magnitude parameters of said correction elements to obtain the desired field characteristics; wherein said magnet is flat and said polynomial is a solution of the Laplace's equation with boundary conditions on a spherical cap, such that the magnetic field on one side of said magnet in a volume bounded by a spherical cap surface, in which volume and along which surface the magnetic field is homogeneous such that the magnetic field defines field lines having equal parallel directions and equal intensities.

2. A method as claimed in claim 1, wherein said spherical harmonics belong to eight families of symmetries, and the desired field characteristics are obtained by simply minimizing the coefficients of said four symmetries.

3. A method as claimed in claim 1, wherein, for said Laplace's equation of the magnetic field to be solved over a spherical cap, an orthogonal basis of even functions and an orthogonal basis of odd functions are generated, either basis being alternately usable for spherical harmonics expansion to generate said polynomial.

4. A method as claimed in claim 1, wherein the step a) comprises the steps of: aa) identifying the following solution of the Laplace's equation of the magnetic field on a spherical cap: $B_{l}^{m} (r, ϑ, ϕ) = {(\frac{r}{r_{0}})}^{l} P_{l}^{m} (\cos ϑ) (a_{l}^{m} \cos φ + b_{l}^{m} \sin φ);$ ab) computing the values of
l.sub.k=l.sub.k(m, custom character )with k≦m εNand 0≦<π to a desired order m=M.sub.MAX such that the functions B.sub.l.sup.m form at least one basis of orthogonal functions over the spherical cap; ac) generating said polynomial that represents the magnetic field generated by the magnet by expansion into series of normalized spherical cap harmonics; wherein: B is the magnetic field in polar coordinates, wherein r is a radius r, custom character is a polar angle, and φ and Φ is an azimuthal angle; l and m are the order and degree of the harmonics; r.sub.0 is a reference radius which is needed to avoid singularities when r.sub.0 approaches 0; a.sub.l.sup.m and b.sub.l.sup.m are the coefficients defining a magnitude of the harmonics; P.sub.l.sup.m(cos custom character ) are associated Legendre functions of the first kind.

5. A method as claimed in claim 4, wherein the step ab) comprises computing the zeros of Legendre functions P.sub.l.sup.m(cos custom character ) and the derivatives of Legendre functions at , $\frac{{dP}_{l}^{m} (\cos ϑ)}{d ϑ}$ by generating two bases of orthogonal functions, which are nonorthogonal to each other, which bases may originate two independent expansions to represent the magnetic field on the spherical cap surface.

6. A method as claimed in claim 5, wherein the step ab) comprises the steps of: aba) providing a general formulation of
P.sub.l.sup.m(cos custom character ) in $P_{t}^{m} (t) = \frac{1}{2^{m} m!} \frac{Γ (l + m + 1)}{Γ (l - m + 1)} {(1 - t^{2})}^{m / 2} F (m - l, m + l + 1; m + 1; \frac{1 - t}{2}), where$ $t = \cos ϑ with - 1 < t \leq 1, Γ (z) = \int_{0}^{\infty} e^{- t} t^{z - 1} dt$ $and$ $F (a, b; c; z) = {.Math.}_{k = 0}^{\infty} \frac{{(a)}_{k} {(b)}_{k}}{k! {(c)}_{k}} z^{k}$ with |z|<1 and (a).sub.k, (b).sub.k and (c).sub.k being shifted factorials; abb) computing the zeros of $P_{l}^{m} (t_{0}) = 0$ $and$ $\frac{{dP}_{l}^{m} (t_{0})}{dt} = 0$ resulting in the following conditions to be separately fulfilled ${\begin{matrix} F (l, m, t_{o}) = 0 \\ {lt}_{0} F (l, m, t_{o}) - (l - m) F (l - 1, m, t_{o}) = 0 \\ t_{0} \neq - 1 \end{matrix}$ and wherein the condition F(l,m,t.sub.0)=0 generates an orthogonal basis of odd functions and the condition lt.sub.0F(l,m,t.sub.0)−(l−m)F(l−1,m,t.sub.0)=0 generates an orthogonal basis of even functions.

7. A method as claimed in claim 6, wherein the step ac) comprises the steps of: aca) defining the normalized spherical cap harmonic (l,m) as: $(\begin{matrix} {\overline{R}}_{l}^{m} (ϑ, φ) \\ {\overline{S}}_{l}^{m} (ϑ, φ) \end{matrix}) = {\overline{P}}_{l}^{m} (\cos ϑ) (\begin{matrix} \cos (m φ) \\ \sin (m φ) \end{matrix})$ wherein P.sub.l.sup.m(cos custom character ) are said even functions or said odd functions and wherein normalization results in: $\frac{1}{2 π (1 - \cos ϑ_{0})} \underset{Cap}{\int \int} {({\overline{R}}_{l}^{m})}^{2} d σ = \frac{1}{2 π (1 - \cos ϑ_{0})} \underset{Cap}{\int \int} {({\overline{S}}_{l}^{m})}^{2} d σ = 1$ acb) generating said polynomial that represents the magnetic field generated by the magnet as an expansion into a series of spherical cap harmonics in the following form: $B (r, ϑ, φ) = {.Math.}_{n = 0}^{N_{m ax}} {.Math.}_{m = 0}^{n} {(\frac{r}{r_{0}})}^{l_{n} (m)} [a_{l_{n}}^{m} {\overline{R}}_{l_{n}}^{m} (ϑ, φ) + b_{l_{n}}^{m} {\overline{S}}_{l_{n}}^{m} (ϑ, φ)]$ wherein: R.sub.l.sup.m ( custom character ,φ) represents even spherical cap harmonics; and S.sub.l.sup.m (,φ) represents odd spherical cap harmonics.

8. A method as claimed in claim 7, wherein the coefficients of said expansions are computed using the equation: $(\begin{matrix} a_{l_{n}}^{m} \\ b_{l_{n}}^{m} \end{matrix}) = \frac{1}{2 π (1 - \cos ϑ_{0})} \underset{Cap}{\int \int} B (r_{0}, ϑ, φ) (\begin{matrix} {\overline{R}}_{l_{n}}^{m} (ϑ, φ) \\ {\overline{S}}_{l_{n}}^{m} (ϑ, φ) \end{matrix}) d σ$ are calculated with the Gauss quadrature method: $\int_{a}^{b} f (x) d x ≅ {.Math.}_{i = 1}^{N_{ma x}} w_{i} f (x_{i}),$ Gauss abscissas x.sub.i being used to determine a sampling grid comprising said plurality of points and depending on the selected orthogonal basis of functions.

9. A method as claimed in claim 8, wherein the step b) comprises measuring the magnetic field in a first sampling grid, if said expansion is generated by the orthogonal set of odd functions, or measuring the derivative of the magnetic field at custom character in a second sampling grid, if said expansion is generated by the orthogonal set of even functions.

10. A Nuclear Magnetic Resonance Imaging device, comprising a magnet for generating a static magnetic field, means for generating magnetic field gradients, means for emitting excitation pulses, means for receiving the magnetic resonance signals emitted by the target body and means for correcting inhomogeneity of said static magnetic field, which inhomogeneity correcting means include one or more correction elements for obtaining the desired field characteristics and processing means for calculating the position and magnitude parameters of said one or more correction elements, wherein said magnet is flat and means are provided for correcting magnetic field inhomogeneity, such that the magnetic field on one side of said magnet is corrected in an imaging volume bounded by a spherical cap surface, in which volume and along which surface the magnetic field is homogeneous such that the magnetic field has field lines having equal parallel directions and equal intensities.

11. A device as claimed in claim 10, wherein said processing means at least partially implement a method of correcting inhomogeneity of the static magnetic field generated by the magnet of a Nuclear Magnetic Resonance imaging machine, which method has the steps of: a) generating a polynomial that represents the magnetic field generated by the magnet and comprises a plurality of harmonic terms, each associated with a coefficient; b) measuring the magnetic field at a plurality of points, with a predetermined distribution in space; c) determining the coefficients from the field sampling values; d) comparing the measured coefficients with those that describe the field having the desired characteristics; e) defining a grid for positioning a plurality of correction elements and relating it to the field structure; and f) calculating the position and magnitude parameters of said correction elements to obtain the desired field characteristics; wherein said magnet is flat and said polynomial is a solution of the Laplace's equation with boundary conditions on a spherical cap, such that the magnetic field on one side of said magnet in a volume bounded by a spherical cap surface in which volume and along which surface the magnetic field is homogeneous such that the magnetic field defines field lines having equal parallel directions and equal intensities.

12. A device as claimed in claim 10, wherein said means for generating gradients include fixedly mounted gradient coils, gradient coils being provided which extend in a direction perpendicular to the surface of the magnet, and which have tracks arranged on a flat wall parallel to the surface of the magnet and on two parallel vertical walls, which are in opposed and spaced relationship, and are joined along the corresponding peripheral edges of said flat wall parallel to the flat surface of the magnet, on the side in which the magnetic field is corrected, and further gradient coils being provided, which extend in two further directions parallel to the surface of the magnet and are respectively arranged on two walls in opposed relationship, each overlapping said vertical walls, said coils being in such arrangement as to at least partially contain said imaging volume.

13. A device as claimed in claim 10, wherein said means for generating gradients include gradient coils mounted in at least partially removable fashion, the gradient coils extending perpendicular to the surface of the magnet consisting of first and second sets of substantially circular concentric tracks, and the gradient coils extending in second and third directions parallel to the surface of the magnet consisting each of first and second sets of tracks, each set being composed of two subsets of substantially semicircular concentric tracks separated by a diametrical axis, the diametral axis of the gradient coils extending in said second direction being oriented perpendicular to the diametral axis of the gradient coils extending in said third direction, and said first sets being arranged in mutually overlapping relationship on the surface of said magnet and said second sets being arranged in mutually overlapping relationship on a wall opposed to the surface of the magnet of a support structure, such that said coils at least partially contain said imaging volume.

14. A device as claimed in claim 13, wherein said support structure comprises a receiving coil.

15. A device as claimed in claim 10, wherein the inhomogeneity correcting means includes means for generating a polynomial that represents the magnetic field generated by the magnet and comprises a plurality of harmonic terms, each associated with a coefficient, and said polynomial is a solution of the Laplace's equation with boundary conditions on a spherical cap, such that the magnetic field on one side of said magnet in a volume bounded by a spherical cap surface, in which volume and along which surface the magnetic field is homogeneous such that the magnetic field defines field lines having equal parallel directions and equal intensities.

16. A Nuclear Magnetic Resonance Imaging device, comprising a magnet for generating a static magnetic field, means for generating magnetic field gradients, means for emitting excitation pulses, means for receiving the magnetic resonance signals emitted by the target body and means for correcting inhomogeneity of said static magnetic field, which inhomogeneity correcting means include one or more correction elements for obtaining the desired field characteristics and processing means for calculating the position and magnitude parameters of said one or more correction elements, wherein said magnet has two pole pieces in spaced relationship on opposite sides of a patient receiving volume, means being provided for correcting the magnetic field in said receiving volume such that the magnetic field is corrected in an imaging volume bounded by a spherical cap surface, in which volume and along which surface the magnetic field is homogeneous such that the magnetic field has field lines having equal parallel directions and equal intensities.

17. A device as claimed in claim 16, wherein the inhomogeneity correcting means includes means for generating a polynomial that represents the magnetic field generated by the magnet and comprises a plurality of harmonic terms, each associated with a coefficient, and said polynomial is a solution of the Laplace's equation with boundary conditions on a spherical cap, such that the magnetic field on one side of said magnet in a volume bounded by a spherical cap surface, in which volume and along which surface the magnetic field is homogeneous such that the magnetic field defines field lines having equal parallel directions and equal intensities.

Description

(1) These and other features and advantages of the present invention will appear more clearly from the following description of a few embodiments, illustrated in the annexed drawings, in which:

(2) FIG. 1 shows an exemplary embodiment of the device of the present invention;

(3) FIG. 2 shows an exemplary embodiment of the distribution of magnetic field correction elements;

(4) FIGS. 3 and 4 show a comparison between spherical harmonics and their decomposition into spherical cap harmonics;

(5) FIGS. 5 and 6 show two exemplary embodiments of the magnetic field sampling grid;

(6) FIGS. 7 to 9 show the magnetic field error as calculated with respect to the field measured at different stages of the correction process;

(7) FIGS. 10 to 13 show an example of fixed gradient coils;

(8) FIGS. 14 to 17 show an example of partially removable gradient coils.

(9) FIG. 18 shows a schematic diagram of a gantry and two pole pieces.

(10) FIG. 1 shows a possible magnetic design for a Magnetic Resonance Imaging device, in which the magnetic field generating magnet 1 is a flat magnet.

(11) The magnetic field on one side of said magnet is corrected by the above described method such that an imaging volume is defined, which is bounded by a spherical cap surface 2, in which volume and along which surface the magnetic field is homogeneous.

(12) The spherical cap surface 2 may have any orientation.

(13) The present invention also contemplates a magnet with two opposite pole pieces, not shown, with an imaging volume being defined between said pole pieces, which is bounded by the spherical cap surface 2, and shimming being carried out thereon.

(14) FIG. 2 shows an example of the distribution of correction elements to obtain the desired field characteristics, particularly magnetic dipoles 3 in a shimming simulation.

(15) It will be appreciated that a rather small number of magnetic dipoles 3 is sufficient for an effective shimming process on the spherical cap surface 2.

(16) FIGS. 3 and 4 show two comparisons of spherical harmonics and their decomposition into spherical cap harmonics, to form the l.sub.12 order summation, for the case of the hemisphere (t.sub.0=0).

(17) Particularly, FIG. 3 shows the spherical harmonic P.sub.1,sphere.sup.0(t) and its decomposition into even spherical cap harmonics, and FIG. 4 shows the spherical harmonic P.sub.2,sphere.sup.0(t) and its decomposition into odd spherical cap harmonics.

(18) As shown in the diagram, the approximation of an odd spherical harmonic with even spherical cap harmonics is quite good over the entire range 0≦t≦1 whereas wider oscillations are found in the case P.sub.2,sphere.sup.0(t).

(19) In both cases there is a larger deviations at values corresponding to points close to the edge of the cap. This behavior, that is found for all spherical harmonics, reflects the manner in which the families of spherical cap harmonics were constructed: they do not fulfill both continuity conditions (for instance, in the first case they do not fulfill the annulation condition in t.sub.0).

(20) This theoretical “error” affects the mathematical precision of the method, but this approximation is substantially negligible in both numerical and physical terms, and does not invalidate the applicability of the method.

(21) FIGS. 5 and 6 show two exemplary embodiments of the magnetic field sampling grid in the preferred embodiment in which odd harmonics are used, and the boundary conditions to be considered are Dirichlet conditions, i.e. on the value of the field B.

(22) In FIG. 5 sampling is carried out for a cap of radius r.sub.0=60 mm, aperture custom character =45° and N.sub.MAX=8, whereas in FIG. 6 sampling is carried out for a cap of radius r.sub.0=60 mm, aperture =90° and N.sub.MAX=12.

(23) FIGS. 7 to 9 show the magnetic field error as calculated with respect to the field measured at different stages of the correction process.

(24) The field was recalculated at the points of a spherical cap with parameters r.sub.0=60 mm, custom character =90°, N.sub.max=12, i.e. a hemisphere.

(25) Correct expansion is typically assessed by recalculating the starting magnetic field at the sampling points, and determining the error (in ppm) of the reconstructed field at each point.

(26) FIG. 7 shows the chart of what is observed considering an expansion at the start of the shimming process.

(27) Here, the error resulting from recalculation of the field from spherical cap harmonics expansion is very low for almost the totality of sampling points, whereas it starts to be of some importance at the points close to the edge of the cap.

(28) This error shall not be intended as purely numeric (it is actually independent of the maximum expansion order), but it substantially confirms what was previously stated about the error expected due to the method of constructing the spherical cap harmonics that form the basis of orthogonal functions.

(29) On the other hand, FIGS. 8 and 9 show the error considering an expansion for a field measurement at half of a shimming process and at the end of the shimming process.

(30) The diagrams show that as field homogeneity increases the error on the recalculated field at the points close to the cap edge considerably decreases.

(31) This behavior is substantially identical to what is observed in the validated methods concerning spheres, oblates and ellipsoids, and indicates that the main error source derives from low-order harmonics, which are predominant at the start of the shimming process, and not from high-order harmonics, which are predominant at the end of the shimming process but are of lower amplitude.

(32) The data of the figures show that the odd spherical cap harmonics expansion provides a correct representation of the magnetic field on the reference surface and that the resulting error is substantially negligible in the recursive field shimming method.

(33) The device of the present invention also has gradient generating means comprising fixedly-mounted gradient coils, as shown in FIGS. 10 to 13, in such an arrangement as to at least partially contain said imaging volume.

(34) The gradient coils that extend in a first direction perpendicular to the surface of the magnet 1, i.e. the Y direction, are shown in detail in FIG. 11 ad have plates arranged on a flat wall 40 parallel to the surface of the magnet 1, and on two vertical walls 41, which are in opposed and spaced relationship, and are joined along the corresponding peripheral edges of said flat wall 40 parallel to the surface of the magnet 1.

(35) The further gradient coils extending in two further directions parallel to the surface of the magnet 1, particularly in the X and Z directions, are shown in FIG. 10 and FIG. 12 respectively.

(36) The tracks of the gradient coils that extend in the X direction are arranged on two walls 42 perpendicular to the surface of the magnet 1 and in mutually opposed relationship, and the tracks of the gradient coils extending in the Z direction are arranged on two walls 43 perpendicular to the surface of the magnet and in mutually opposed relationship.

(37) FIG. 13 shows a view of the fixed coils in a mounted state, in which the walls 42 of the gradient coils extending in the Z direction and the walls 43 of the gradient coils extending in the X direction are in mutually overlapped relationship outside the vertical walls 41 of the gradient coils extending in the Y direction.

(38) FIGS. 14 to 17 show gradient coils mounted in at least partially removable fashion.

(39) FIG. 15 show the gradient coils that extend perpendicular to the surface of the magnet 1, i.e. in the Y direction, and are composed of first and second sets 44 of substantially circular and concentric tracks.

(40) The gradient coils extending in second and third directions parallel to the surface of the magnet 1, particularly in the X and Z directions, are shown in FIG. 14 and FIG. 16 respectively.

(41) The gradient coils that extend in the X direction are composed of first and second sets of tracks 45, each set being composed of two subsets of substantially semicircular and concentric tracks separated by a diametral axis 47, and the gradient coils that extend in the Z direction are composed of first and second sets of tracks 46, each set being composed of two subsets of substantially semicircular and concentric tracks separated by a diametral axis 48.

(42) The diametral axis 47 of the gradient coils that extend in the X direction is oriented perpendicular to the diametral axis 48 of the gradient coils that extend in the Z direction.

(43) The first sets are placed in mutually overlapped relationship on the surface of the magnet 1, and the second sets are placed in mutually overlapped relationship on a wall 50, opposite to the surface of the magnet 1, of a support structure 5, such that these coils at least partially contain the imaging volume.

(44) Preferably, the support structure 5 comprises or consists of a receiving coil, particularly the first sets are integral with the magnet and the second sets are integral with the receiving coil.

Method of correcting inhomogeneity of the static magnetic field generated by the magnet of a MRI machine and device for carrying out such method

Assignee

Inventors

Cpc classification

Classification Explorer

G01R33/3808

PHYSICS

Classification Explorer

G01R33/385

PHYSICS

Classification Explorer

G01R33/3875

PHYSICS

Classification Explorer

G01R33/3873

PHYSICS

Classification Explorer

G01R33/3806

PHYSICS

Classification Explorer

G01R33/383

PHYSICS

International classification

Classification Explorer

G01V3/00

PHYSICS

Classification Explorer

G01R33/3875

PHYSICS

Classification Explorer

G01R33/3873

PHYSICS

Abstract

Claims

Description