CN101996273A - Finite difference design method of magnatic resonance imaging (MRI) system gradient coil - Google Patents

Abstract

The invention discloses a finite difference design method of a magnatic resonance imaging (MRI) system gradient coil, which comprises the following steps: firstly, dividing finite difference grids in the coil space, and establishing finite difference relation between a node stream function and grid current density on each grid node; subsequently, selecting a constraint field point in the magnetic field related region of the MRI system gradient coil, and computing the magnetic induction intensity on the constraint field point according to the design requirements; then, establishing a system of linear equations between the node stream function and the magnetic induction intensity of the constraint field point according to the Biot-Savart law; simultaneously, establishing a penalty function of the system of linear equations according to the actual engineering requirements; and finally, solving the system of linear equations by a method of regularization to acquire a node stream function value, and determining the current style of the coil by a constant stream function line. The invention is simple and effective and can be suitable for designing gradient coil systems in any structure.

Description

The finite difference method for designing of MRI system gradient coil

Technical field

The present invention relates to the finite difference method for designing of gradient coil in a kind of magnetic resonance imaging (being called for short MRI) system.

Background technology

In the magnetic resonance imaging field,,, carried out a large amount of work for obtaining uniform gradient magnetic about gradient coil.Mainly contain two kinds of methods for designing.A kind of method that is based on the discrete lead space, such as simulated annealing (S.Crozier and D.M.Doddrell, " Gradient-coil design by simulatedannealing, " J.Magn.Reson.A, vol.103, pp.354-357,1993.); Another kind is based on the method for designing of continuous current density space, such as target field method (R.Turner, " A target field approach tooptimal coil design; " Journal of Physics D:Applied Physics, vol.19, pp.L147-L151,1986.); More than two kinds of methods relative merits are respectively arranged.

Based on the method in discrete lead space, at coil space supposition current-carrying element (annular or arc), utilize the random optimization method to adjust the current-carrying positions of elements then earlier to reach designing requirement.As simulated annealing, Crozier etc. are mapped to deformation space with circle, form the egg deltoid, are used for representing current carrying conductor, adjust the position of lead then with simulated annealing, electric current, loop shape etc.This method relatively directly and is easily implemented, but optimizing process is more consuming time.

Based on the method for designing of continuous current density space, be a kind of more effective method for designing, need utilize a discretize process to come the approximate current Density Distribution.Loop construction for regular shape, express current density on the coil plane with analytic expression or series expansion, as the target field method, represent current density on the coil plane with the Fourier analytic expression, find the solution current density according to the target field distribution of hope with Fourier transform then, utilize FFT can realize this process easily very rapidly; For non-regular loop construction, coil space is carried out grid dividing, utilize finite element or boundary element method (M.Poole and R.Bowtell then, " Novel Gradient Coils Designed Using a Boundary Element Method; " Concepts inMagnetic Resonance Part B (Magnetic Resonance Engineering), vol.31, pp.162-175,2007.) come the approximate current Density Distribution.Utilize the stream function method that current density is mapped to arrangement of conductors at last.

Summary of the invention

The finite difference method for designing that the purpose of this invention is to provide a kind of MRI system gradient coil.

For achieving the above object, the technical solution adopted in the present invention is: the finite difference method for designing of its MRI system gradient coil comprises following each step:

(1) coil space to MRI system gradient coil carries out the finite difference grid dividing;

(2) each grid that division obtains to step (1) is set up the finite difference relation between node stream function and the mesh current density;

(3) the target magnetic induction density value on selected confining field point on the relevant range, magnetic field of MRI system gradient coil and setting confining field point, according to node stream function and the relation of the finite difference between the mesh current density that coordinate, target magnetic induction density value and the step (2) of described each confining field point are set up, utilize Biot-Sa to cut down theorem and set up about the target magnetic induction density of each confining field point and the system of linear equations between all node stream functions;

(4) engineering of the corresponding requirements of MRI system gradient coil being set up about all node stream functions according to actual engineering retrains penalty function, uses the described system of linear equations of regularization method solution procedure (3), obtains each node value of stream function;

(5) according to of the size of current requirement of actual engineering to MRI system gradient coil, all node value of stream function that step (4) obtains are divided line of constant stream function, obtain MRI system gradient coil the distribution of current pattern.

Further, MRI of the present invention system gradient coil comprises main coil and potted coil, and the described coil space of step (1) comprises main coil and potted coil space, and the described relevant range of step (3) comprises the imaging region interested and the shielding area of MRI system.

Compared with prior art, the invention has the beneficial effects as follows: can carry out the division of finite difference grid to arbitrary surfaces, be applicable to the gradient coil system of arbitrary structures, very big advantage be arranged for the coil system that develops non-traditional structure; For the coil requirement of engineering reality, such as inductance minimize, the maximization of minimum power, coil-span etc., all can realize by setting up corresponding penalty function; The present invention uses the Tikhonov regularization method to find the solution inverse problem, is suitable for ill Solving Linear.The inventive method does not need to carry out the analytic expression or the series expansion of current density, can be applied to the gradient coil system design of arbitrary structures, and is more simple than using finite element or Element BEM simultaneously yet; Considering simultaneously now to have developed adopts method of difference to carry out the research of gradient coil electro-magnetic biological effect, the electro-magnetic biological effect that the application of method of difference makes the design of coil can be fused to gradient coil in the inventive method goes in studying, and can further investigate the problems such as security of gradient coil.

Description of drawings

Fig. 1 is the configuration structure and the region of interest of plane type X gradient coil.

Fig. 2 is the grid dividing in plane type X gradient main coil space.

Fig. 3 is the sampling of plane type X gradient coil design confining field point.

Fig. 4 is the right half part of disk on the X gradient main coil.

Fig. 5 is the right half part of disk on the X gradient potted coil.

Fig. 6 is the gradient intensity distribution situation of imaging region interested at the xz central plane

Fig. 7 is the partly shielding effect zone magnetic induction situation distribution situation on the xz central plane of the potted coil outside.

Embodiment

Below be example explanation the inventive method with the plane type X gradient coil design in the open MRI systems, the present invention is applicable to the gradient coil system design of arbitrary structures.

Figure 1 shows that the configuration structure of plane type X gradient coil, coil distribution is being on central shaft and the disk about the z=0 plane symmetry with the z axle, and it is on the center ball of the centre of sphere that imaging region interested is distributed in the initial point.In the present embodiment, the goal gradient field strength of imaging region interested requires to be G _xIt is 0.38m that=6.25mT/m, sphere diameter DSV (diameter of spherical volume) require, and two main coil disks are positioned at z=a and z=-a, and spacing is 2a=0.5m, and it is R that coil size is limited to radius _aIn the circle of=0.43m, two potted coil disks are positioned at z=b and z=-b, and spacing is 2b=0.7m, and coil radius is R _b=0.5m.Shielding area is a potted coil along the zone in the z direction outside, requires as far as possible for a short time at the magnetic induction density of shielding area, generally requires its absolute value less than 5 Gausses in the actual engineering.

At first the finite difference grid dividing is carried out in plane type X gradient coil space.As shown in Figure 2, with the example that is divided on main coil, in the circular cylindrical coordinate system, evenly divide on angle and radial direction respectively on each coil disk, radial direction is divided into the M=56 five equilibrium in this example, and angle direction is divided into the N=56 five equilibrium, (i j) represents, wherein i=1 grid node with two-dimensional coordinate on each disk, 2 ..., M+1, j=1,2 ..., N.According to the notion of stream function, for the surface current J that is distributed on the disk, can defining scalar stream function Ψ, it concerns as shown in Equation (1)

$\stackrel{\→}{J}={\stackrel{\→}{e}}_r{J}_r+{\stackrel{\→}{e}}_{\mathrm{\θ}}{J}_{\mathrm{\θ}}$

(1)

$={\stackrel{\→}{e}}_r\frac{\∂\mathrm{\Ψ}}{r\∂\mathrm{\θ}}+{\stackrel{\→}{e}}_{\mathrm{\θ}}(-\frac{\∂\mathrm{\Ψ}}{\∂r})$

In the formula (1), Represent the unit vector of radial direction,

Represent the unit vector of angle direction, r is the radial direction coordinate, and θ is the angle direction coordinate, J _rBe the current density of radial direction on the coil disk, J _θIt is the current density of coil disk upper angle direction.

Utilize the difference approximation method, can get finite difference relation between node stream function and the mesh current density shown in formula (2) and (3) by formula (1),

$J_r(i,j)\≈\frac{1}{r}\frac{\mathrm{\Ψ}(i,j+1)-\mathrm{\Ψ}(i,j)}{\mathrm{\Δ\θ}}---\left(2\right)$

$J_{\mathrm{\θ}}(i,j)\≈-\frac{\mathrm{\Ψ}(i+1,j)-\mathrm{\Ψ}(i,j)}{\mathrm{\Δr}}---\left(3\right)$

In formula (2) and (3), the nodal pitch of Δ r representative on radial direction, Δ θ represents nodal pitch on the angle direction, and (i j) is node (i, stream function j), J to Ψ _r(i j) is (i, radial direction current density j), the J of grid on the coil disk _θ(i j) is (i, angle direction current density j) of grid on the coil disk.

Then at the selected confining field point in the relevant range, magnetic field of MRI system gradient coil, as shown in Figure 3, stain is represented the confining field point of choosing among Fig. 3.Symmetry based on the X gradient coil, distribution of current unanimity on the coil on the lower disc, therefore, only investigating upward, disk (is positioned at the main coil of z=a, be positioned at the potted coil of z=b) get final product, grey block is represented the main coil and the potted coil of actual investigation among the figure, and lower disc (promptly being arranged in the main coil and the potted coil that is positioned at z=-b of z=-a) does not directly relate in actual design, go but be mapped in the disk, represent with the light gray color lump among the figure; Corresponding confining field point is also only chosen z＞0 part, represents with the black round dot among the figure, and light grey round dot is corresponding z＜0 part.As shown in Figure 3, the relevant range comprises imaging region interested and shielding area, for imaging region interested, evenly chooses the confining field point on its sphere; For shielding area, on the disk at potted coil outside certain position place, carry out evenly choosing on angle and the radial direction, to choose in this example and be positioned at z=0.5m, radius is that the disk of 0.5m is chosen.For the point of the confining field on the imaging region interested (x _f, y _f, z _f) the z component of magnetic induction density be B _z=G _h* h _f, G wherein _hBe the gradient intensity that requires, h can be x, and y or z represent X respectively, the gradient coil of Y or Z direction, and in this example, h is x; Magnetic induction density for shielding area confining field point can be taken as 0T.If do not require the design potted coil in actual design, then coil space only need consider to be positioned at the main coil of z=a, and z＞0 part of imaging region interested only need be considered in the relevant range, magnetic field of MRI system gradient coil.

For any point confining field point (x _f, y _f, z _f), utilize Biot-Sa to cut down theorem, the governing equation on the X gradient coil between the current density peace treaty bundle field point magnetic induction density z component as shown in Equation (4),

Wherein subscript ± a represents lower disc on the main coil, and ± b represents lower disc on the potted coil, S, and Q formula intermediate variable, as shown in Equation (5):

S＝-x _fsinθ+y _fcosθ，Q＝x _fcosθ+y _fsinθ-r (5)

With formula (2) and (3) substitution formula (4), the discrete form that can get the governing equation between node stream function and the confining field point magnetic induction density z component as shown in Equation (6)

(6)

Utilize the symmetric relation of the distribution of current unanimity of lower disc on the X gradient coil, as shown in Equation (7):

Ψ ^+a＝Ψ ^-a，Ψ ^+b＝Ψ ^-b (7)

Following formula can be reduced to the situation of only considering to go up disk.The total grid node number of last disk coil space is V=5192, gets the confining field point and is total up to U=636, then according to (6) and (7) formula, can set up system of linear equations

$\left[\begin{array}{cccc}{A}_{11}& A_{12}& ...& A_{1V}\\ A_{21}& A_{22}& ...& A_{2V}\\ .& .& .& .\\ .& .& .& .\\ .& .& .& .\\ A_{U1}& A_{U2}& ...& A_{\mathrm{UV}}\end{array}\right]\·\left[\begin{array}{c}{\mathrm{\Ψ}}_1\\ {\mathrm{\Ψ}}_2\\ .\\ .\\ .\\ {\mathrm{\Ψ}}_V\end{array}\right]=\left[\begin{array}{c}{B}_{z1}\\ B_{z2}\\ .\\ .\\ .\\ B_{\mathrm{zU}}\end{array}\right]---\left(8\right)$

Above-mentioned system of linear equations (formula (8)) can be write a Chinese character in simplified form into A Ψ=B, and wherein A is the matrix of coefficients of stream function, and B is the array of the magnetic induction density z component of each confining field point.Formula (8) is an ill-condition equation group, need set up penalty function and find the solution, and the penalty function F that sets up in this example is the variable quantity quadratic sum minimum of each component of coil current density on orthogonal directions, as shown in Equation (9)

$F=\underset{0}{\overset{R_a}{\∫}}\underset{0}{\overset{2\mathrm{\π}}{\∫}}({\left(\frac{{\∂J}_r}{r\∂\mathrm{\θ}}\right)}^2+{\left(\frac{1}{r}\frac{\∂\left({\mathrm{rJ}}_{\mathrm{\θ}}\right)}{\∂r}\right)}^2)\mathrm{rd\θdr}+\underset{0}{\overset{R_b}{\∫}}\underset{0}{\overset{2\mathrm{\π}}{\∫}}({\left(\frac{{\∂J}_r}{r\∂\mathrm{\θ}}\right)}^2+{\left(\frac{1}{r}\frac{\∂\left({\mathrm{rJ}}_{\mathrm{\θ}}\right)}{\∂r}\right)}^2)\mathrm{rd\θdr}---\left(9\right)$

Minimizing of formula (9) can make the accumulation of coil localized heat minimum, thereby thereby avoids because the too small actual engineering that is unfavorable for that causes of the distribution of current distance between centers of tracks that obtains is made.

Adopt the Tikhonov regularization method to find the solution system of linear equations, it is as follows that the employing target field is worth minimized regularization mode solution formula:

${\mathrm{\Ψ}}_{\mathrm{\λ}}=\arg \min \{{\left|\right|\mathrm{A\Ψ}-B\left|\right|}_2^2+{\mathrm{\λ}}^2{\left|\right|L\left(\mathrm{\Ψ}\right)\left|\right|}_2^2\}---\left(10\right)$

λ is a regularization parameter in the formula (10), L is a constraint matrix, push away by penalty function F, its relational expression as shown in Equation (11):

$F={\left|\right|L\left(\mathrm{\Ψ}\right)\left|\right|}_2^2---\left(11\right)$

By formula (10) and (11), can get stream function and find the solution expression formula as shown in Equation (12):

Ψ _λ＝(A ^TA+λ ²L ^TL) ^-1A ^TB (12)

Get λ=7.785e-10 in this example, try to achieve the value of stream function on each grid node of coil space.Theory according to stream function, equidistantly the line of constant stream function that distributes has been represented the arrangement of conductors pattern with steady current flux, therefore the line of constant stream function of dividing according to current requirements is electrical current distribution lines, is the arrangement of conductors pattern of coil, in this example on the lead size of current be made as 100A.Distribution of current for clarity sake only shows the right half part of disk on the X gradient about the y rotational symmetry on the last disk of the X gradient coil that obtains (comprising main coil and potted coil), is illustrated in figure 4 as the last disk right half part of main coil; Fig. 5 is the right half part of disk on the potted coil.Can this arrangement of conductors mode be that benchmark carries out the coil making promptly in the actual fabrication process.

Fig. 6 is the simulation calculation to the gradient intensity distribution of imaging area-of-interest, be chosen at the distribution situation signal on the xz central plane, be the zone of gradient intensity ± 5% that departs from objectives among the figure in the middle circle territory, radius of circle is 0.19m, satisfies the designing requirement that DSV is 0.38m; Fig. 7 is the simulation calculation of the field strength distribution situation of shielding area, the subregion of choosing on the xz central plane of the potted coil outside is investigated, field intensity is all less than 1 Gauss in the zone, requires in the realistic engineering in the magnetic induction density absolute value of the shielding area condition less than 5 Gausses.

So far, finished the design of biplane X gradient coil, comprised main coil and potted coil, and satisfied design conditions by the result of simulation calculation checking design.

More than be that example describes the present invention with open MRI systems biplane X gradient coil, can easily be generalized to biplane Z gradient coil in the reality, the cylindrical gradient coil in the cylindrical MR I system, and the gradient coil design of non-traditional structure.

Claims (2)

1. the finite difference method for designing of a MRI system gradient coil is characterized in that comprising following each step:

(1) coil space to MRI system gradient coil carries out the finite difference grid dividing;

(2) each grid that division obtains to step (1) is set up the finite difference relation between node stream function and the mesh current density;

(3) the target magnetic induction density value on selected confining field point on the relevant range, magnetic field of MRI system gradient coil and setting confining field point, according to node stream function and the relation of the finite difference between the mesh current density that coordinate, target magnetic induction density value and the step (2) of described each confining field point are set up, utilize Biot-Sa to cut down theorem and set up about the target magnetic induction density of each confining field point and the system of linear equations between all node stream functions;

(4) engineering of the corresponding requirements of MRI system gradient coil being set up about all node stream functions according to actual engineering retrains penalty function, uses the described system of linear equations of regularization method solution procedure (3), obtains each node value of stream function;

(5) according to of the size of current requirement of actual engineering to MRI system gradient coil, all node value of stream function that step (4) obtains are divided line of constant stream function, obtain MRI system gradient coil the distribution of current pattern.

2. the finite difference method for designing of MRI according to claim 1 system gradient coil, it is characterized in that: described MRI system gradient coil comprises main coil and potted coil, the described coil space of step (1) comprises main coil and potted coil space, and the described relevant range of step (3) comprises the imaging region interested and the shielding area of MRI system.

Images