CN107831461A - Longitudinal gradient coil design method based on 01 integer programmings - Google Patents

Abstract

The invention discloses a longitudinal gradient coil design method based on 0-1 integer programming. Aiming at the problems existing in the separate winding method, the invention proposes a 0-1 integer programming method for longitudinal gradient coil design. This method divides the area where the coil is located into several one-dimensional grids, and takes the center of the grid as the position of the current ring to be calculated. Given the current I of the ring coil, if the current ring somewhere contributes to the magnetic field, the variable is 1, the current is 1*I, and if there is no contribution, the variable is 0, and the current is 0. The calculation of this method is simple and direct, and the minimum inductance of the coil or the minimum material can be used as the goal. It is easy to achieve a good linearity of the gradient magnetic field, and it is convenient to impose constraints on the spacing of the coils.

Description

Design Method of Longitudinal Gradient Coil Based on 0-1 Integer Programming

technical field

The invention relates to the technical field of magnetic resonance imaging, in particular to a design method for longitudinal gradient coils based on 0-1 integer programming.

Background technique

Magnetic resonance imaging is an imaging technique based on the phenomenon of nuclear magnetic resonance. Gradient coils are an important part of the magnetic resonance imaging system. They are used to generate linearly changing magnetic fields along three orthogonal directions in the imaging area, which are respectively applied to slice selection, phase encoding and frequency encoding, so as to provide positioning basis for image reconstruction. . The structure of the gradient coil mainly includes a closed cylindrical structure and an open planar structure.

At present, most permanent magnetic resonance imaging systems adopt an open planar disk structure, and superconducting magnetic resonance imaging systems mostly adopt a closed cylindrical structure. With the increase of the cost of permanent magnetic resonance imaging system and the improvement of people's requirements for imaging quality, superconducting magnetic resonance imaging system will replace the permanent magnetic resonance imaging system is the future development trend.

The indicators to measure the performance of gradient coils are: gradient strength G, gradient nonlinearity E, eddy current size and coil inductance. Generally speaking, the higher the gradient strength, the smaller the nonlinearity and the smaller the inductance of the gradient coil, the better the performance of the coil.

In the magnetic resonance imaging system, the direction of the main magnetic field is defined as the z direction. The coils along the direction of the main magnetic field are called longitudinal gradient coils (z-direction gradient coils), and the coils perpendicular to the main magnetic field direction are called transverse gradient coils (x-direction gradient coils and y-direction gradient coils).

The longitudinal gradient coil generally adopts the form of "Maxwell coil pair", which is a pair of circular coils with radius r, symmetrical about the origin, and the current direction is opposite. In order to obtain better linearity and gradient strength, actual gradient coils often use multiple pairs of coils.

The basic design methods of gradient coils can be divided into two categories: one is the regular separation winding method, which selects a predetermined regular coil geometry, and then optimizes the coil structure according to the principle of obtaining the best linear gradient of the magnetic field; the other One is the distributed winding method (also known as the current density method). According to the Maxwell equation, according to the required gradient field distribution, an ideal continuous surface current density in a certain space is obtained through an optimization algorithm, and then This current density distribution is simulated with distributed wire or conductive copper plates. The distributed winding method can achieve the current density distribution of the field according to the requirements of the target field, but there are errors when the current density is discrete, which leads to the degradation of the coil performance. This issue is especially important in the design of self-shielded gradient coils. Also, the distributed winding approach does not easily impose coil spacing constraints. The advantage of the separate winding method is that it is simple and direct, and it is convenient for engineering calculation and realization. However, the performance index of the coil has a great relationship with the pre-determined coil shape, and it is difficult to find the global optimal solution of the coil performance parameters. On the whole, it is more likely to obtain a technical solution to the above-mentioned problems by improving the separate winding method.

Contents of the invention

In view of this, the object of the present invention is to provide a longitudinal gradient coil design method based on 0-1 integer programming that is convenient for finding the global optimal solution of the coil performance parameters, so as to solve the existing problems in the prior art that are not easy to impose constraints, It is difficult to obtain the technical problem of the global optimal solution of the coil performance parameters.

The technical solution of the present invention is to provide a method for designing longitudinal gradient coils based on 0-1 integer programming, comprising the following steps:

Assume that the main coil and shielding coil of the gradient coil are respectively distributed in areas with lengths L _p and L _s , radii R _p and R _s respectively, and the energized current is I; the coil area is uniformly divided into M _p along the z axis by a grid Take the center of the grid as the coil position; the main coil and the shielding coil adopt the same grid spacing, and adjust the length L _p and L _s of the coil according to the needs, so that the number of grids is just an integer _;

In the spherical imaging area DSV, N ₁ target field points are selected, and N ₂ target field points are selected in the shielding area, then they are located at z=z′ _j (j=1,...,M _p +M _s ), and the radius is r =r′ _j (j=1,...,M _p +M _s ) the magnetic field generated by the i-th (i=1,...,N ₁ +N ₂ ) field point (r _i , z _j ) of the current ring The z and r components are:

in,

K(k) and E(k) are respectively the first kind of elliptic integral and the second kind of elliptic integral; μ ₀ is the vacuum magnetic permeability;

The currents of the main coil and the shielding coil are equal in size and opposite in direction, so the magnetic field generated by all current-carrying grids at the i-th field point is

Where e _j = 0, it means that the grid current has no contribution to the magnetic field, and e _j = 1 means that the coil has a contribution to the magnetic field; in DSV, only the z component of the magnetic field is considered, and B _z and B _r are considered in the shielded area, written in matrix form for

B _zdsv =IA ₁ e

B _zshield = IA ₂ e

B _rshield = IA ₃ e

Wherein, the dimension of the coefficient matrix A ₁ is N ₁ ×(M _p +M _s ), and A ₂ and A ₃ are coefficient matrices of N ₂ ×(M _p +M _s ).

The model is established with the goal of minimizing the amount of coil material, then

Objective function:

Restrictions:

|IA ₁ eB′ _zdsv |≤ε ₁ B′ _zdsv

|IA ₂ e|≤ε ₂

|IA ₃ e|≤ε ₃

e _j =0 or e _j =1; B' _zdsv =G _z *z _j , B' _zdsv is the z component of the target magnetic field, and G _z is the target gradient field strength;

Among them, ε ₁ takes 0.05, ε ₂ and ε ₂ take 10 ^-7 ;

The linear programming model is solved to obtain the number of turns of the main coil and the shielding coil and the position of the coil distribution.

Optionally, define the minimum spacing constraint of the coil, assuming that the grid spacing is a mm, and the minimum distance between the centers of two turns of the coil is b mm, constraints can be imposed, and h is the largest integer smaller than b/a; then:

e _j +e _j+1 +e _j+2 ...+e _j+h ≤ 1 (j=1,...,M _p -h, M _p +1,...,M _p +M _s -h)

Restrictions:

|IA ₁ eB′ _zdsv |≤ε ₁ B′ _zdsv

|IA ₂ e|≤ε ₂

|IA ₃ e|≤ε ₃

Ce≤1

e _j =0 or e _j =1

Wherein, the dimension of the matrix C is (M _p +M _s −2h)×(M _p +M _s ).

Optionally, the point on the 1/4 arc is taken as the target field point in the DSV, and the positive half-axis or negative half-axis is taken on the coil structure for grid division.

Optionally, the unit of the gradient strength G _z is T/m/A, and the current I=1A.

Optionally, when the unit of the gradient strength G _z is T/m, that is, when T/m is taken, the current I is equal to a specific value first, and then the current is gradually increased, and it will be found that the amount of materials is decreasing as the current increases , until the difference between the material usage at the current current value and the material usage at the previous current value is less than the corresponding threshold value, confirm that the current value is the optimal current value to achieve the minimum material usage.

Compared with the prior art, the method of the present invention has the following advantages: The present invention aims at the problems existing in the separate winding method, and proposes a 0-1 integer programming method for longitudinal gradient coil design. This method divides the area where the coil is located into several one-dimensional grids, and takes the center of the grid as the position of the current ring to be calculated. Given the current I of the ring coil, if the current ring somewhere contributes to the magnetic field, the variable is 1, the current is 1*I, and if there is no contribution, the variable is 0, and the current is 0. The calculation of this method is simple and direct, and the minimum inductance of the coil or the minimum material can be used as the goal. It is easy to achieve a good linearity of the gradient magnetic field, and it is convenient to impose constraints on the spacing of the coils.

Description of drawings

Fig. 1 is the schematic diagram of longitudinal gradient coil under the present invention;

Fig. 2 is a schematic diagram of the winding of the main coil;

Figure 3 is a schematic diagram of the winding of the shielding coil.

Detailed ways

Preferred embodiments of the present invention will be described in detail below with reference to the accompanying drawings, but the present invention is not limited to these embodiments. The present invention covers any alternatives, modifications, equivalent methods and schemes made within the spirit and scope of the present invention.

In order to provide the public with a thorough understanding of the present invention, specific details are set forth in the following preferred embodiments of the present invention, but those skilled in the art can fully understand the present invention without the description of these details.

In the following paragraphs the invention is described more specifically by way of example with reference to the accompanying drawings. It should be noted that all the drawings are in simplified form and use inaccurate scales, which are only used to facilitate and clearly assist the purpose of illustrating the embodiments of the present invention.

Shown in Fig. 1 with reference to, a kind of optimal design method of magnetic resonance imaging longitudinal gradient coil, described method is realized by following steps successively:

Taking the design of superconducting longitudinal gradient coils as an example to illustrate the design method of longitudinal gradient coils:

Firstly, since the shape of the gradient coil is circular, the coils are distributed on the roz plane. The main coil has a length L _p and a radius R _p , and the shielding coil has a length L _s and a radius R _s . The main coil and the shielding coil are evenly divided into M _p and M _s equal parts, and the center of the grid is taken as the ring coil position. The main coil and the shielding coil adopt the same grid spacing, and the lengths L _p and L _s of the coils are adjusted as required so that the number of grids is just an integer.

According to the Biot-Savart law, the magnetic field component generated at any point (r, z) in space by the energized circular coil at z′, current I, and radius r′ is

in,

K(k) and E(k) are the elliptic integrals of the first kind and the elliptic integrals of the second kind respectively.

Select the target field points of the longitudinal gradient coil optimization problem in the imaging area and the shielding area respectively:

On the arc of the 1/2 spherical area in the imaging area, select N ₁ target points, and consider the symmetry of the gradient coil, only the positive semi-axis of the z-axis of the main coil and the shielding coil can be meshed. Select the point on the 1/4 arc of the target field point in the imaging area, and select the positive semi-axis of the z-axis in the shielded area.

Among them, r _dsv is the radius of the spherical imaging area, G _z is the given gradient strength value, and B' _zdsv is the ideal z component value of the magnetic field.

In the shielded area, select N ₂ points on the side of the cylindrical surface with radius R _stray and length L _s as target field points,

Since the energizing currents of the main coil and the shielding coil are equal in size and opposite in direction, the energizing current of the main coil is I, and the energizing current of the shielding coil is -I. In the case of given operating current, spherical imaging area radius r _dsv and gradient strength Gz, a 0-1 integer programming model is established with the amount of coil material as the objective function _f :

Wherein, e _j is the optimization variable (e _j =0 or e _j =1), M _p and M _s are respectively the number of divisions of the main coil and the shielding coil, and

In the imaging area, only the z component of the magnetic field is considered, and B _z and B _r are considered in the shielding area, so the constraints are:

|IA ₁ eB′ _zdsv |≤ε ₁ B′ _zdsv

|IA ₂ e|≤ε ₂

|IA ₃ e|≤ε ₃

e _j =0 or e _j =1

Here, ε ₁ is 0.05, and ε ₂ and ε ₂ are 10 ^-7 . The dimension of coefficient matrix A ₁ is N ₁ ×(M _p +M _s ), and A ₂ and A ₃ are coefficient matrices of N ₂ ×(M _p +M _s ).

By solving the linear programming model, the number of turns of the coil and the position of the coil distribution can be obtained, and the optimization result sometimes shows that the coils are concentrated. Considering the actual size of the coils and the spacing between the coils, a minimum distance constraint between the coils must be imposed at design time. According to the size of the divided mesh, we can define the minimum spacing constraints of the coils. Assuming that the grid spacing is 4mm and the minimum distance between the centers of two coils is 10mm, the coil spacing constraints can be imposed:

e _j +e _j+1 +e _j+2 ≤1 (j=1,..., M _p -2, M _p +1,..., M _p +M _s -2)

At this point the constraints are:

|IA ₁ eB′ _zdsv |≤ε ₁ B′ _zdsv

|IA ₂ e|≤ε ₂

|IA ₃ e|≤ε ₃

Ce≤1

e _j =0 or e _j =1

Wherein, the dimension of the matrix C is (M _p +M _s −4)×(M _p +M _s ).

If the grid spacing is 3mm, the coil constraints are

e _j +e _j+1 +e _j+2 +e _j+3 ≤1(j=1,...,M _p -3, M _p +1,...,M _p +M _s -3), in the constraints The dimension of the matrix C becomes (M _p +M _s −6)×(M _p +M _s ).

In the above-mentioned design of the longitudinal gradient coil, considering the symmetry of the gradient coil, only the positive semi-axis of the z-axis of the main coil and the shielding coil can be meshed. Select the point on the 1/4 arc of the target field point in the imaging area, and select the positive semi-axis of the z-axis in the shielded area.

Keeping the constraints of the magnetic field and coil spacing constant, and changing the objective function f, different linear programming or nonlinear programming models can also be obtained.

The unit of the gradient strength G _z can be T/m/A or T/m. When T/m/A is taken, the current I=1A is taken. When T/m is taken, the current I=100A can be taken first. Then gradually increase the current, and you will find that as the current increases, the amount of material used decreases. When a current value is reached, as the current continues to increase, the amount of material used changes little.

Figure 2 and Figure 3 are schematic diagrams of the designed superconducting longitudinal gradient coil winding. r _dsv =0.225m, L _p =1.2m, L _s =1.4m, R _p =0.36m, R _s =0.39m, G _z =55*(1e-6)T/m/A, R _stray =R _s +0.15.

Although the above embodiments are described and illustrated separately, some common technologies are involved, and in the eyes of those of ordinary skill in the art, the embodiments can be replaced and integrated, and the content that is not clearly recorded in one of the embodiments can be Reference is made to another example documented.

The implementation methods described above do not constitute a limitation to the scope of protection of the technical solution. Any modifications, equivalent replacements and improvements made within the spirit and principles of the above implementation methods shall be included in the protection scope of the technical solution.

Claims (5)

1. A longitudinal gradient coil design method based on 0-1 integer programming comprises the following steps:

the main gradient coil and the shield coil are assumed to be distributed in a length L_pAnd L_sRadius is R respectively_pAnd R_sThe area (2) has an energization current of I; uniformly dividing the coil area into M along the z-axis by a grid_pAnd M_sEqually dividing, and taking the center of a grid as a coil position;

within the spherical imaging region DSV, N is selected₁Selecting N for each target field point and shielding area₂The target site is located at z ═ z'_j(j＝1，…，M_p+M_s) Where the radius is r ═ r'_j(j＝1，…，M_p+M_s) In the ith (i-1, …, N) current ring₁+N₂) Individual field point (r)_i，z_i) The z-component and r-component of the generated magnetic field are:

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k (k) and e (k) are elliptical integrals of a first kind and elliptical integrals of a second kind, respectively; mu.s₀Is a vacuum magnetic conductivity;

the currents of the main coil and the shielding coil are equal in magnitude and opposite in direction, so that the magnetic field generated by all current-carrying grids at the ith field point is

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Wherein e_j0, indicating that the grid current does not contribute to the magnetic field, e_j1 illustrates that the coil contributes to the magnetic field; in a DSV, only the z-component of the magnetic field is considered, and B is considered in the shielded region_zAnd B_rWritten in matrix form as

B_zdsv＝IA₁e

B_zshield＝IA₂e

B_rshield＝IA₃e

Wherein, the coefficient matrix A₁Has a dimension of N₁×(M_p+M_s)，A₂And A₃Is N₂×(M_p+M_s) The coefficient matrix of (2).

The least amount of coil material is used as a target for establishing a model, then

An objective function:

constraint conditions are as follows:

|IA₁e-B′_zdsv|≤ε₁B′_zdsv

|IA₂e|≤ε₂

|IA₃e|≤ε₃

e_j0 or e_j＝1；B′_zdsv＝G_z*z_j，B′_zdsvFor the z-component of the target magnetic field, G_zIs the target gradient field strength;

wherein epsilon₁Take 0.05, ε₂And ε₂Get 10^-7；

And solving the linear programming model to obtain the number of turns of the main coil and the shielding coil and the distribution position of the coils.

2. The method of claim 1, wherein the method comprises: defining minimum spacing constraint of the coils, assuming that the grid spacing is a mm, the minimum distance between the centers of two turns of coils is b mm, applying constraint conditions, and taking h as the maximum integer smaller than b/a; then there are:

e_j+e_j+1+e_j+2...+e_j+h≤1(j＝1，…，M_p-h，M_p+1，…，M_p+M_s-h)

optimization constraints are rewritten as

Constraint conditions are as follows:

|IA₁e-B′_zdsv|≤ε₁B′_zdsv

|IA₂e|≤ε₂

|IA₃e|≤ε₃

Ce≤1

e_j0 or e_j＝1

Wherein the dimension of the matrix C is (M)_p+M_s-2h)×(M_p+M_s)。

3. A method of designing a longitudinal gradient coil based on an integer program of 0-1 according to claim 1 or 2, characterized in that: taking a point on an 1/4 arc line in the DSV as a target field point, and taking a positive half shaft part or a negative half shaft part on the coil structure for meshing division.

4. The method of claim 1, wherein the method comprises: gradient intensity G_zThe unit of (1) is T/m/A, and the current I is 1A.

5. The method of claim 3, wherein the method comprises: gradient intensity G_zWhen the unit of (1) is T/m, namely when T/m is taken, the current I is taken to be equal to a specific value firstly, then the current is increased step by step, and the material usage is found to be reduced along with the increase of the current until the difference between the material usage at the current value and the material usage at the previous valueAnd when the current value is smaller than the corresponding threshold value, the current value is determined to be the optimal current value for realizing the minimum material consumption.