Cylindrical Bi-Planar Gradient Coil for MRI

Abstract

Cylindrical bi-planar gradient coil assemblies for use in open magnetic resonance imaging, wherein each of the coil assemblies contains in sequential order (i) a circular primary coil set placed flat above a cylindrical planar substrate, (ii) cooling means, (iii) 0th and 2nd order shims, (iv) shield layers, and (v) 1st order shims. In use the gradient coil assemblies are disposed symmetrically to each other about a plane of symmetry parallel to each.

1 . A method for designing a plurality of coil sets comprising planar X and Y-primary coil sets and planar X and Y-shield coil sets of a bi-planar gradient coil assembly said method comprising the steps of:
 specifying a surface primary and shield current distributions as well as a plurality of magnetic fields in polar coordinates and have reciprocal space expressions related by a Fourier-Hankel transform; and   obtaining a surface transverse primary and shield spatial current distributions that generate a plurality of desired transverse gradient magnetic fields by:
 i. specifying polar surface current distributions on four coplanar planes that share a common z-axis through their respective polar centers so the z-axis is in an orthogonal relation to the planes, setting the z-axis center equidistant between two inner planes so that these planes are positioned symmetrically at z=±a, then defining the current distributions on these planes as the primary coils where a plurality of primary gradient magnetic fields are generated therebetween; 
 ii. specifying an outer plane to be positioned symmetrically at z=±b, defining the current distributions on these planes as the shield coils so that a plurality of shield gradient magnetic fields are generated in opposition to the primary gradient magnetic fields, together producing a total gradient magnetic field in the space between z<a and z>−a, and zero magnetic field in the space above z>b and below z<−b; 
 iii. using a shielding condition of producing zero magnetic field in the space above z>b and below z<−b to obtain an algebraic relation between the primary and shield current distributions in reciprocal polar space; 
 iv. using this primary and shield current distribution relation to express a total shielded magnetic field in terms of just the primary current distribution; 
 v. expressing the shielded gradient magnetic field in its Fourier-Hankel expansion so that a current density is in its reciprocal space representation; 
 vi. evaluating the magnetic field expression at specified discrete values, these points all having a same axial position z=c, where c<a, and a same polar angle φ, but different radial positions ρj, forming a term where for each j the magnetic field value so expressed is subtracted from the desired magnetic field values at that particular jth location, then summing all these N terms where each term is multiplied by a different constant, the Lagrange multipliers, and this sum added to an energy formula in its reciprocal space representation through a Fourier-Hankel expansion to finally yield an expression which is an energy functional; 
 vii. the energy functional differentiated with respect to the current distribution and set to zero, subsequently solving the resultant expression to obtain an extremized current distribution in reciprocal space which becomes a sum over the Lagrange multipliers multiplied by terms containing a plurality of corresponding radial constraint points; 
 viii. the extremized current density substituted back into the expression for the shielded gradient magnetic field that will contain the sum over the N discrete radial positions with the associated multiplicative Lagrange multipliers; 
 ix. the field calculated at each discrete constraint value and set equal to the corresponding constraint discrete magnetic field value to form a matrix equation of calculated field values at the discrete constraint radial positions times the corresponding Lagrange multipliers equaling the constraint magnetic field values; 
 x. inverting the matrix equation to finally obtain the Lagrange multipliers; 
 xi. substituting the Lagrange multiplier values in the expression for the extremized current density; 
 xii. the primary and shield magnetic fields calculated from the extremized current distribution by an inverse Fourier-Hankel transform to obtain the spatial field distributions, an efficiency and a linearity of the gradient magnetic fields so produced calculated, further an inductance also calculated from said current distribution; 
 xiii. a parameter constructed as a figure of merit equal to the efficiency squared and divided by the product of the inductance times the square-root of the linearity value; and 
 xiv. optimizing the figure of merit by a Nelder-Mead simplex optimization algorithm that has an associated cost function to limit the planar or radial extent of the current distribution whereby in each iteration of the algorithm the extremized current distribution is obtained first, subsequently the fields, linearity and inductances calculated to get the figure of merit, then the radial constraint points with the associated magnetic field values varied by the simplex algorithm to obtain a new extremized current distribution from which to calculate a new figure of merit, the algorithm terminating when no change is obtained in the figure of merit finally yielding the desired transverse primary and shield current distributions. 
   
     
     
         2 . A method for designing the Z-primary coil and the Z-shield coil of the gradient coil assembly of  claim 1 , said method further comprising the steps of:
 expanding an axial magnetic field in terms of a Fourier-Hankel expansion, inverting this expression in reciprocal polar space so that a current density can be specified as a function of the magnetic field;   specifying two regions of constant axial magnetic fields as a function of radial position on an interior planar surface coaxial and parallel to a plurality of planar axial primary current distributions;   further specifying an outer axial magnetic field in opposite relation to the previous two axial magnetic fields to limit a plurality of radial extents of the planar axial primary and shield current distributions;   Fourier-Hankel transforming the field distributions and inputting them in said primary current distribution in reciprocal polar space;   obtaining an associated planar axial shield current distribution from said primary current distribution and calculating an interior axial shielded magnetic field from said Fourier-Hankel magnetic field expression so an efficiency and a linearity of the magnetic fields is found;   further calculating a corresponding inductance from said planar axial primary and shield current distributions;   forming a figure of merit by squaring the efficiency and dividing it by a product of the inductance and a square root of the linearity;   optimizing the figure of merit by a Nelder-Mead simplex optimization algorithm where the simplex algorithm varies a range and a plurality of amplitudes of the three target field distributions to obtain a new current distribution from which a new figure of merit is obtained by calculating the corresponding axial shielded magnetic fields, and thus the efficiency and linearity, along with the inductance over each iteration, the algorithm terminating when no change in the figure of merit is reached to yield an optimized planar axial primary and shield current distribution.   
     
     
         3 . A method to define a planar shim current distribution for a gradient coil assembly to generate spatial magnetic field distributions corresponding to terms in a spherical harmonic expansion of a plurality of main magnetic fields, said method comprising the steps of:
 specifying polar, surface shim current distributions placed on two parallel planes sharing a common z-axis to produce a shim magnetic field there between;   expressing the shim magnetic field in a Fourier-Hankel expansion, transforming the shim magnetic fields to its reciprocal space representation, inverting the expression so that the planar shim current distribution can be specified as a function of the shim magnetic field in reciprocal space;   each term in the spherical harmonic expansion of the main magnetic field converted to a plurality of cylindrical coordinates and multiplied by a radial envelope function to constrain radial extent, then Fourier-Hankel transformed to reciprocal space, the function entered as the shim magnetic field in reciprocal field for the shim current distribution in reciprocal space; and   the shim current distribution inverse Fourier-Hankel transformed to obtain a continuous radial and angular distribution in cylindrical coordinates.   
     
     
         4 . The planar current distributions solved for in  claim 1  for the transverse, axial and shim currents being continuous surface distributions are discretized to form conductor placement positions wherein:
 if conductors of a same cross-sectional area are to be used, a continuous central conductor path can be obtained by constructing a plurality of corresponding stream functions and generating a plurality of contours of constant intervals of an integrated current; 
 if the continuous conductor paths are to be etched from a plurality of solid conductors the etch paths are formed in between said equal, constant intervals of integrated currents from the corresponding stream functions. 
 
     
     
         5 . A method of manufacturing the coils of  claim 1 , comprising the steps of:
 cutting a metal plate in the shape of a circular coil creating a coil and a curf portion;   removing the curf position from the metal plate creating a curf void around the coil;   placing a first insulation layer into the former;   placing the coil into the former over the first insulation layer;   placing a second insulation layer over the coil in the former; and   compressing the insulating layers and the coil with a control weight.   
     
     
         6 . A method of manufacturing the coils of  claim 1 , comprising the steps of:
 cutting a metal plate creating a noncontiguous curf portion and a coil with a plurality of tabs interconnecting sections of the coil;   removing the curf portion from the metal plate creating a curf void around the coil;   placing an epoxy into the curf void around coil; and   removing the plurality of tabs from the coil.   
     
     
         7 . The method of  claim 6 , further comprising the steps of:
 fabricating a former having an open top and a closed planar bottom;   placing a first insulating layer inside the former and onto the former planar bottom;   adhering the coil and epoxy onto the first insulating layer;   adhering a second insulating layer on top of the coil and epoxy; and   compressing the first and second insulating layers, the coil and the epoxy with a control weight.   
     
     
         8 . A method of manufacturing the coils of  claim 1 , comprising the steps of:
 cutting a metal plate in the shape of a circular coil creating a coil portion and a curf portion;   removing the curf portion from the metal place creating a curf void round the coil;   forming a formed insulation layer having curf tracks; and   placing the coil on the formed insulation between the curf tracks.   
     
     
         9 . The method of  claim 8 , further comprising the steps of:
 fabricating a former having an open top and a closed planar bottom;   placing the coil and the formed insulation layer into the former;   adhering a second insulating layer on top of the coil and the formed insulation layer; and compressing the second insulating layer, the coil and the formed insulation layer with a control weight.   
     
     
         10 . A method of manufacturing a gradient coil assembly, comprising the steps of:
 fabricating a former having an open top and a closed planar bottom;   placing an insulation layer inside the former and onto the former planar bottom;   fabricating and X-primary coil with a coil portion and a curf void portion;   placing the X-primary coil into the former;   forming an insulation layer in the X-primary coil curf void portion and on top of the X-primary coil portion;   applying control weight while the insulation layer cures;   fabricating a Y-primary coil with a coil portion and a curf void portion;   placing the Y-primary coil into the former;   forming an insulation layer in the Y-primary coil curf void portion and on top of the Y-primary coil portion;   applying control weight while the insulation layer cures;   fabricating a Z-primary coil with a coil portion and a curf void portion;   placing the Z-primary coil into the former;   forming an insulation layer in the Z-primary coil curf void portion and on top of the Z-primary coil portion;   applying control weight while the insulation layer cures;   placing one or more cooling tubes on top of Z-primary coil portion and the insulation layer;   pouring epoxy on top of and between cooling tubes forming an insulation layer;   applying control weight while the insulation layer cures;   placing a 0 th  and a 2 nd  order shim on top of cooling tubes and the insulation layer;   pouring epoxy onto shims forming an insulation layer;   applying control weight while the insulation layer cures;   fabricating a Z-shield coil with a coil portion and a curf void portion;   placing the Z-shield coil into the former;   forming an insulation layer in the Z-shield coil curf void portion and on top of the Z-shield coil portion;   applying control weight while the insulation layer cures;   fabricating an X-shield coil with a coil portion and a curf void portion;   placing the X-shield coil into the former;   forming an insulation layer in the X-shield coil curf void portion and on top of the X-shield coil portion;   applying control weight while the insulation layer cures;   fabricating a Y-shield coil with a coil portion and a curf void portion;   placing the Y-shield coil into the former;   forming an insulation layer in the Y-shield coil curf void portion and on top of the Y-shield coil portion;   applying control weight while the insulation layer cures;   placing 1 st  order shims on top of Y-shield coil portion and insulation layer;   pouring epoxy onto shim forming an insulation layer;   applying control weight while the insulation layer cures; and   sealing the top of former.   
     
     
         11 . The method of  claim 10 , wherein the shims comprise self-insulated coils. 
     
     
         12 . The method of  claim 10 , wherein the former, the coils, the shims and the insulation layers have alignment holes to align the former, the coils, the shims and the insulation layers relative to each other. 
     
     
         13 . The method of  claim 10 , wherein the insulation layers consist of a fiber cloth substrate and epoxy. 
     
     
         14 . A gradient coil manufacturing method comprising in sequential order the steps of:
 i. fabricating a pair of hollow cylindrical formers with a plurality of mounting holes placed around an edge of the formers and a plurality of additional mounting holes strategically placed in a central planar surface of the formers;   ii. etching a X-primary coil from a solid copper plate by a water-jet cutting machine and placing the X-primary coil first on the former using a plurality of pins through the central mounting holes that extend vertically through all the layers to properly align the placement of the plate on the former;   iii. placing a single layer of thin fiber cloth on the X-primary coil and pouring an epoxy resin and a hardener mixture over it, then using a heavy, heated flat plate placed on top to help the epoxy resin settle forming an insulated layer and a flat controlled surface;   iv. etching a Y-primary coil from a solid copper plate as in step ii and placing it in an orthogonal relation to the X-primary coil on top of the insulating layer using the aligning pins as in step ii;   v. applying step iii to the Y-primary coil;   vi. etching a Z-primary coil from a solid copper plate and placing it on top of the insulating layer above the Y-primary coil using the aligning pins as in step ii;   vii. applying step iii to the Z-primary coil;   viii. placing a plurality of non-conducting cooling conduits on the insulating layer over the Z-primary coil then applying step iii;   ix. subsequently, placing a preassembled set of shim layers comprising the zeroth and second order shims on top of the insulating layer above the cooling layer then applying step iii to the shim layers;   x. a layer made of thick fiber cloth and a filler material mixed with the epoxy resin and the hardener mixture over the cooling and insulating layer forms an insulative support layer when step iii is applied to it to fill the gap between the shim and the shield layers;   xi. an X-shield coil etched from a solid copper plate is placed on the support layer as in step ii, co-aligned with the X-primary coil, then step iii is applied to it afterwards;   xii. a Y-shield coil etched from a solid copper plate and placed in an orthogonal relation to the X-shield coil using step ii and subsequently applying step iii to it forming an insulation layer;   xiii. the Z-shield coil etched from a solid copper plate is placed on the insulation layer above the Y-shield coil using step ii and subsequently step iii is applied to it to form an insulating layer;   xiv. a preassembled set of shim layers comprising first, third and higher order shim coils is placed on the insulating layer using step ii; and   xv. applying step iii to the shim layer with additional fiber cloth layers forms a final insulative layer that caps the entire layer.   
     
     
         15 . The gradient coil manufacturing method of  claim 14 , further comprising the step of placing the central mounting holes strategically away from the dense parts of a plurality of current distributions throughout the entire shielded gradient coils to minimize disturbing the performance of the shielded coils and limit a gradient field leakage adjacent the pole faces. 
     
     
         16 . The gradient coil manufacturing method of  claim 14 , further comprising the step of placing a plurality of center fed busbars for the X and Y-primary and shield coils in the channels between the two halves of each respective coil to save vertical stack up space. 
     
     
         17 . The gradient coil manufacturing method of  claim 14 , further comprising the step cutting a plurality of slits into a plurality of conductor tracks of the etched solid copper plates for the gradient coils.