Design method of inserted gradient coil based on intrinsic basis of curved surface

Abstract

The present invention relates to the field of gradient coil manufacturing technology, and in particular to a design method for an insertable gradient coil based on an intrinsic basis of a curved surface. The method comprises: establishing an optimization model; constructing an objective function and performing sensitivity analysis and derivation; obtaining a set of linear equations related to the coefficients corresponding to the intrinsic basis through sensitivity derivation; obtaining the basis coefficients corresponding to the optimal stream function solution by solving the set of linear equations; defining the stream function through the intrinsic basis, obtaining the final stream function optimization result, and obtaining the gradient coil configuration. The advantages are: using a smooth intrinsic orthogonal basis to define the stream function; when applying this method to coil design, the resulting coil has good smoothness, avoiding the additional discussion of the coil smoothness problem in traditional methods; and having good application effects on both traditional cylindrical surfaces and complex general surfaces.

Description

Technical Field

[0001] The present invention relates to the technical field of gradient coil manufacturing, and in particular to a design method for an inserted gradient coil based on a curved surface intrinsic substrate. Background Art

[0002] Magnetic resonance imaging (MRI) is an advanced imaging technology widely used in clinical medicine, biology, polymer chemistry and other fields. The gradient coil is an important component of the MRI system. Its function is to generate a gradient magnetic field to provide spatial encoding of the imaging area. Traditional coil structures mostly use regular curved surfaces such as cylinders as current-carrying surfaces. In recent years, various coil structures with anisotropic current-carrying surfaces that fit the imaging object have been proposed. However, complex current-carrying surfaces also bring new challenges to coil design. The smoothness of the designed coil is also more difficult to ensure on non-developable surfaces. Therefore, it is necessary to develop new gradient coil design solutions to meet the increasingly complex needs of coil structures in the future.

[0003] In the existing technology, there is a problem of difficult-to-control coil smoothness caused by numerical oscillation in the traditional stream function method. Therefore, during the design process, designers need to make trade-offs between various coil parameters, resulting in poor robustness of the final coil. Summary of the Invention

[0004] In order to solve the above problems, the present invention provides a design method of an inserted gradient coil based on a curved surface intrinsic basis.

[0005] The present invention aims to provide a method for designing an insertable gradient coil based on a curved surface intrinsic basis, which specifically comprises the following steps:

[0006] S1. Determine the coil's current-carrying surface and imaging area, and establish an optimization model.

[0007] S2. Constructing an objective function based on the optimization model, the objective function includes a primary objective function and an auxiliary objective function;

[0008] S3. Assume that the sensitivity of the objective function to the design variables is zero. After discretizing and calculating the objective function, the corresponding linear equations are obtained. By solving the linear equations, the stream function result is obtained.

[0009] S4. Obtain a set of orthogonal and complete basis functions by calculating and solving the eigenvalue problem of the current-carrying surface;

[0010] S5. Perform preliminary optimization calculations with different numbers of bases, compare the coil parameters of the optimization results, and select the optimal number of bases that can achieve a stable coil structure;

[0011] S6. Obtaining an optimal auxiliary target weight using an L-curve method based on the determined optimal number of bases;

[0012] S7. Perform final optimization calculations based on the coil parameters obtained from the optimization results to obtain the optimal stream function distribution and the final configuration of the gradient coil; obtain the final gradient coil loop based on the definition of the stream function.

[0013] Preferably, the current density on the current-carrying surface is defined by the following formula:

[0014] ;

[0015] in, is the tangential gradient operator on the current-carrying surface, is the stream function defined on the current-carrying surface, n is the external normal vector on the current-carrying surface;

[0016] Stream Function The expression is as follows:

[0017]

[0018] in, Nodes on the current-carrying surface The stream function value at is the intrinsic basis number, is the number of intrinsic bases used, is the coefficient corresponding to the basis, is the basis function at the node The value at .

[0019] Preferably, the objective function is expressed as:

[0020] ;

[0021] Where, f 0 and f 1 represents the main objective function and the auxiliary objective function respectively;

[0022] The expression of the main goal is as follows:

[0023] ;

[0024] Where, represents the number of sampling points in the imaging area, Indicates the discrete node number; Represents the target gradient field, through the gradient intensity Multiply by the position along the gradient direction 、 or calculate; It is about the basis coefficient Function, representing the node at Towards the actual magnetic field;

[0025] The expression of the auxiliary objective is as follows:

[0026] ;

[0027] Where, is the material conductivity, is the thickness of the conductive layer, represents the integral over the coil design surface.

[0028] Preferably, Calculated by Biot-Savart law:

[0029] ;

[0030] Where, 、 、 represents the position along the gradient direction, c represents the basis coefficient, n x Representation surface The x-component of the normal vector, n y Representation surface The y-component of the normal vector, n z Representation surface The z-component of the normal vector.

[0031] Preferably, the linear equations in step S3 are expressed as follows:

[0032] ;

[0033] Where, The sensitivity matrix of the magnetic field at the target position to the coefficients to be solved , express The transposed matrix of represents the energy consumption matrix of coil optimization, Represents a matrix composed of basis functions arranged in rows , Represents a column vector consisting of the target magnetic field .

[0034] Preferably, in step S4, the stream function on the current-carrying surface is expressed as follows using basis functions:

[0035] ;

[0036] in, is the stream function scalar field on the current-carrying surface, is the intrinsic basis number, is the number of intrinsic bases used, is the coefficient corresponding to the basis, are basis functions defined on the current-carrying surface.

[0037] Preferably, step S6 specifically includes: selecting a series of different auxiliary target weights for calculation, constructing a relationship curve using the main target value and auxiliary target value of the corresponding calculation results, and selecting the auxiliary target weight at the inflection point from the relationship curve for calculation.

[0038] Preferably, the number of substrates is greater than 20 and does not exceed 100.

[0039] Compared with the prior art, the present invention can achieve the following beneficial effects:

[0040] The present invention provides a gradient coil design method that defines the stream function using a smooth intrinsic orthogonal basis. Based on this definition, an objective function is constructed and sensitivity analysis is performed. The sensitivity derivation yields a system of linear equations associated with the coefficients corresponding to the intrinsic basis. The basis coefficients corresponding to the optimal stream function solution are obtained by solving this system of linear equations. The stream function is defined using the intrinsic basis to determine the final stream function optimization result. This method demonstrates excellent application results on both traditional cylindrical surfaces and complex general surfaces. Furthermore, since the basis function itself is smooth, the resulting coils exhibit excellent smoothness. This avoids the additional considerations required for coil smoothness in traditional methods, including numerical oscillations that can lead to uncontrollable coil smoothness. It also eliminates the trade-offs between coil parameters that designers often face, enriching coil configurations and improving coil robustness. The method has been applied to coil design on cylindrical and human head surfaces, and computational results and simulations demonstrate its practicality and effectiveness. BRIEF DESCRIPTION OF THE DRAWINGS

[0041] Figure 1 3 is a flow chart of a method for designing an inserted gradient coil based on a curved surface intrinsic basis according to an embodiment of the present invention.

[0042] Figure 2 Schematic diagram of the gradient coil optimization model provided according to Example 1 of the present invention.

[0043] Figure 3 Schematic diagram of a gradient coil optimization model provided according to Example 2 of the present invention.

[0044] Figure 4 Schematic diagram of the definition of a convection function provided according to an embodiment of the present invention.

[0045] Figure 5 2 is a schematic diagram of a 40-order intrinsic basis of a cylindrical surface provided by an embodiment of the present invention.

[0046] Figure 6 40-order intrinsic basis diagram of the human head curve provided by an embodiment of the present invention.

[0047] Figure 7 1 is a coil parameter curve under different substrate numbers provided by an embodiment of the present invention.

[0048] Figure 8 Schematic diagram of an L-curve of a primary target with respect to an auxiliary target according to an embodiment of the present invention.

[0049] Reference numerals:

[0050] 101. Cylindrical current-carrying surface;

[0051] 102. Current-carrying surface of human head;

[0052] 2. Imaging area. DETAILED DESCRIPTION

[0053] Hereinafter, embodiments of the present invention will be described with reference to the accompanying drawings. In the following description, identical modules are denoted by identical reference numerals. In the case of identical reference numerals, their names and functions are also identical. Therefore, their detailed description will not be repeated.

[0054] In order to make the purpose, technical solutions and advantages of the present invention more clearly understood, the present invention will be further described in detail below in conjunction with the accompanying drawings and specific embodiments. It should be understood that the specific embodiments described herein are only used to explain the present invention and do not constitute a limitation of the present invention.

[0055] The present invention provides a design method for an inserted gradient coil based on a curved surface intrinsic basis (see flowchart). Figure 1 ), the gradient coil designed by this method can generate a magnetic field with a gradient distribution along a certain direction after being energized; specifically comprising the following steps:

[0056] S1. Determine the coil's current-carrying surface and imaging area and establish an optimization model; the current-carrying surface is used to carry the current density, and the imaging area is used to calculate the magnetic field;

[0057] The current density on a current-carrying surface is defined by:

[0058] ;

[0059] in, is the tangential gradient operator on the current-carrying surface, is the stream function defined on the current-carrying surface,n is the external normal vector on the current-carrying surface;

[0060] Stream Function By defining the intrinsic basis, the expression is as follows:

[0061]

[0062] in, Nodes on the current-carrying surface The stream function value at is the intrinsic basis number, is the number of intrinsic bases used, is the coefficient corresponding to the basis, is the basis function at the node The value at .

[0063] S2. Construct an objective function based on the optimization model. The objective function includes a primary objective function and an auxiliary objective function. The primary objective is to control the gradient linearity of the overall coil structure, while the auxiliary objective is to control the energy consumption of the coil. (The primary objective function is usually the linearity of the magnetic field generated by the coil, and the auxiliary objective is usually the energy consumption of the coil.)

[0064] The expression of the objective function is:

[0065] ;

[0066] Where, f 0 and f 1 represents the main objective function and the auxiliary objective function respectively;

[0067] The main target expression is as follows:

[0068] ;

[0069] Where, represents the number of sampling points in the imaging area, Indicates the discrete node number; Represents the target gradient field, which can be expressed by the gradient intensity Multiply by the position along the gradient direction (or , )calculate; is a coefficient about the basis Function, representing the node at The actual magnetic field can be calculated using the Biot-Savart law:

[0070] ;

[0071] Where, 、 、 represents the position along the gradient direction, c represents the basis coefficient, n x Representation surface The x-component of the normal vector, n y Representation surface The y-component of the normal vector, n z Representation surface The z-component of the normal vector;

[0072] The expression of the auxiliary objective is as follows:

[0073] ;

[0074] Where, is the material conductivity, is the thickness of the conductive layer, represents the integral over the coil design surface.

[0075] S3. Calculate the sensitivities of the primary and auxiliary objectives with respect to the design variables, respectively; set the sum of the sensitivities of the primary and auxiliary objectives with respect to the design variables to zero, discretize and calculate the linear equations corresponding to the optimization problem; obtain the stream function result by solving the linear equations;

[0076] The system of linear equations is expressed as follows:

[0077] ;

[0078] Where, express , that is, the sensitivity matrix of the magnetic field at the target position to the coefficients to be solved; express The transposed matrix of represents the energy consumption matrix of coil optimization; express , that is, the matrix composed of basis functions arranged in rows; express , which is a column vector consisting of the target magnetic field.

[0079] S4. By calculating and solving the eigenvalue problem on the current-carrying surface, a set of orthogonal and complete basis functions is obtained. These basis functions can linearly represent any second-order smooth function on the current-carrying surface. Therefore, the stream function on the current-carrying surface can be linearly represented as:

[0080] ;

[0081] in, is the stream function scalar field on the current-carrying surface, is the intrinsic basis number, is the number of intrinsic bases used, is the coefficient corresponding to the basis, are basis functions defined on the current-carrying surface.

[0082] S5. Perform preliminary optimization calculations with different numbers of substrates, compare the coil parameters of the optimization results, and select the optimal number of substrates that can obtain a stable coil structure; usually the number of substrates should be greater than 20 and not more than 100.

[0083] S6. Based on the determined optimal number of bases, the optimal auxiliary target weight is obtained through the L-curve method. The specific method is as follows: a series of different auxiliary target weights are selected for calculation, and a relationship curve (usually L-shaped) is constructed using the main target value and the auxiliary target value of the corresponding calculation results. The auxiliary target weight at the inflection point of the relationship curve is selected for calculation.

[0084] S7. Perform final optimization calculations based on the coil parameters obtained from the optimization results to obtain the optimal stream function distribution and the final configuration of the gradient coil; obtain the final gradient coil loop based on the definition of the stream function.

[0085] Example 1

[0086] This embodiment provides a method for designing an inserted gradient coil based on a curved surface intrinsic basis, which specifically includes the following steps:

[0087] S1. Establish an optimization model: Determine the cylindrical current-carrying surface 101 and imaging area 2 of the coil and establish an optimization model; the cylindrical current-carrying surface 101 is used to carry the current density, and the imaging area 2 is used to calculate the magnetic field (see Figure 2 );

[0088] The current density on the cylindrical current-carrying surface 101 is defined by the following formula:

[0089] ;

[0090] in, is the tangential gradient operator on the current-carrying surface, is the stream function defined on the current-carrying surface, n is the external normal vector on the current-carrying surface;

[0091] Different from the existing technology, the stream function By defining the intrinsic basis, the expression is as follows:

[0092]

[0093] in, Nodes on the current-carrying surface The stream function value at is the intrinsic basis number, is the number of intrinsic bases used, is the coefficient corresponding to the basis, is the basis function at the node The value at (schematic diagram as shown Figure 4 ).

[0094] S2. Construct objective function: Construct objective function based on the optimization model. The objective function includes main objective function and auxiliary objective function. The main objective is to control the gradient linearity of the coil structure as a whole, and the auxiliary objective is to control the energy consumption of the coil (the main objective function is usually selected as the linearity of the magnetic field generated by the coil, and the auxiliary objective function is usually selected as the energy consumption of the coil).

[0095] S3. Sensitivity Analysis: Calculate the sensitivity of the primary and auxiliary objectives with respect to the design variables. Set the sum of the sensitivities of the primary and auxiliary objectives with respect to the design variables to zero, discretize them, and calculate the linear equations corresponding to solving the optimization problem. Obtain the stream function result by solving the linear equations.

[0096] S4. By calculating and solving the eigenvalue problem of the cylindrical current-carrying surface 101, a set of orthogonal and complete basis functions is obtained. The basis functions can linearly represent any second-order smooth function on the current-carrying surface. The first 40-order intrinsic basis of the cylindrical current-carrying surface 101 is as follows: Figure 5 shown.

[0097] S5. Perform preliminary optimization calculations with different numbers of substrates, compare the various parameters of the coils obtained from the optimization results, and select the optimal number of substrates that can achieve a stable coil structure. The coil parameter curves under different numbers of substrates are shown in Figure 2. Figure 6 shown.

[0098] S6. Based on the determined optimal number of bases, the optimal auxiliary target weight is obtained using the L-curve method. The specific method is as follows: a series of different auxiliary target weights are selected for calculation, and a relationship curve is constructed using the primary target value and the auxiliary target value of the corresponding calculation results ( Figure 7 ), the auxiliary target weight at the inflection point is selected from the relationship curve for calculation.

[0099] S7. Perform the final optimization calculation based on the coil parameters of the optimization results to obtain the optimal stream function distribution and the final configuration of the gradient coil; obtain the final gradient coil loop according to the definition of the stream function. The coil optimization results are as follows: Figure 8 shown.

[0100] Example 2

[0101] This embodiment provides a method for designing an inserted gradient coil based on a curved surface intrinsic basis, which specifically includes the following steps:

[0102] S1. Establish an optimization model: Determine the coil's head current-carrying surface 102 and imaging area 2, and establish an optimization model; the head current-carrying surface 102 is used to carry current density, and the imaging area 2 is used to calculate the magnetic field (see Figure 3 );

[0103] The current density on the cylindrical current-carrying surface 1 is defined by the following formula:

[0104] ;

[0105] in, is the tangential gradient operator on the current-carrying surface, is the stream function defined on the current-carrying surface, n is the external normal vector on the current-carrying surface;

[0106] Different from the existing technology, the stream function By defining the intrinsic basis, the expression is as follows:

[0107]

[0108] in, Nodes on the current-carrying surface The stream function value at is the intrinsic basis number, is the number of intrinsic bases used, is the coefficient corresponding to the basis, is the basis function at the node The value at (schematic diagram as shown Figure 4 ).

[0109] S2. Construct objective function: Construct objective function based on the optimization model. The objective function includes main objective function and auxiliary objective function. The main objective is to control the gradient linearity of the coil structure as a whole, and the auxiliary objective is to control the energy consumption of the coil (the main objective function is usually selected as the linearity of the magnetic field generated by the coil, and the auxiliary objective function is usually selected as the energy consumption of the coil).

[0110] S3. Sensitivity Analysis: Calculate the sensitivity of the primary and auxiliary objectives with respect to the design variables. Set the sum of the sensitivities of the primary and auxiliary objectives with respect to the design variables to zero, discretize them, and calculate the linear equations corresponding to solving the optimization problem. Obtain the stream function result by solving the linear equations.

[0111] S4. By calculating and solving the eigenvalue problem of the head current-carrying surface 102, a set of orthogonal and complete basis functions is obtained. The basis functions can linearly represent any second-order smooth function on the current-carrying surface. The first 40-order intrinsic basis of the cylindrical current-carrying surface 1 is as follows: Figure 6 shown.

[0112] S5. Perform preliminary optimization calculations with different numbers of bases, compare the coil parameters of the optimization results, and select the optimal number of bases that can obtain a stable coil structure.

[0113] S6. Based on the determined optimal number of bases, the optimal auxiliary target weight is obtained through the L-curve method; the specific method is as follows: a series of different auxiliary target weights are selected for calculation, a relationship curve is constructed using the main target value and the auxiliary target value of the corresponding calculation results, and the auxiliary target weight at the inflection point of the relationship curve is selected for calculation.

[0114] S7. Perform the final optimization calculation based on the coil parameters of the optimization results to obtain the optimal stream function distribution and the final configuration of the gradient coil; obtain the final gradient coil loop according to the definition of the stream function. The coil optimization results are as follows: Figure 8 shown.

[0115] Brief description of the present invention: The optimization model includes the current-carrying surface and imaging area of ​​the coil design. The objective function should include the main objective function and the auxiliary objective function, wherein the main objective function is usually selected as the linearity of the magnetic field generated by the coil, and the auxiliary objective function is usually selected as the energy consumption of the coil. Sensitivity analysis requires calculating the sensitivity of the main objective and the auxiliary objective relative to the design variables respectively. Then, by setting the sum of their sensitivities to zero, a set of linear equations for solving the optimization problem is obtained. Obtaining the intrinsic basis of the surface requires calculating the eigenvalue problem of the surface and obtaining a series of linearly independent and complete basis functions. Determining the number of bases is to obtain the minimum number of bases required to ensure the convergence of the coil structure by calculation. Determining the auxiliary objective weight is to obtain the optimal auxiliary objective weight that can balance the two design objectives through numerical methods. Finally, based on the various parameters obtained in the above steps, the optimal stream function distribution is calculated.

[0116] It should be understood that the various forms of the processes shown above can be used to reorder, add, or delete steps. For example, the steps described in the present disclosure can be performed in parallel, sequentially, or in a different order, as long as the desired results of the technical solutions disclosed in the present disclosure can be achieved. This is not limited herein.

[0117] The above specific embodiments do not limit the scope of protection of the present invention. Those skilled in the art will appreciate that various modifications, combinations, sub-combinations, and substitutions may be made based on design requirements and other factors. Any modifications, equivalent substitutions, and improvements made within the spirit and principles of the present invention are intended to be included within the scope of protection of the present invention.

Claims

1. A design method for an inserted gradient coil based on a surface intrinsic basis, characterized in that: The specific steps include: S1. Determine the coil's current-carrying surface and imaging area, and establish an optimization model. S2. Construct an objective function based on the optimization model. The objective function includes a primary objective function and an auxiliary objective function. The primary objective function is selected as the linearity of the magnetic field generated by the coil, and the auxiliary objective function is selected as the energy consumption of the coil. S3. Assume that the sensitivity of the objective function to the design variables is zero. After discretizing and calculating the objective function, the corresponding linear equations are obtained. By solving the linear equations, the stream function result is obtained. S4. Obtain a set of orthogonal and complete basis functions by calculating and solving the eigenvalue problem of the current-carrying surface; S5. Perform preliminary optimization calculations with different numbers of bases, compare the coil parameters of the optimization results, and select the optimal number of bases that can achieve a stable coil structure; S6. Obtaining an optimal auxiliary target weight using an L-curve method based on the determined optimal number of bases; S7. Perform final optimization calculations based on the coil parameters obtained from the optimization results to obtain the optimal stream function distribution and the final configuration of the gradient coil; obtain the final gradient coil loop based on the definition of the stream function.

2. The method for designing an inserted gradient coil based on a curved surface intrinsic basis according to claim 1, characterized in that: The current density on the current-carrying surface is defined by the following formula: ； in, is the tangential gradient operator on the current-carrying surface, is the stream function defined on the current-carrying surface, n is the external normal vector on the current-carrying surface; Stream Function The expression is as follows: in, Nodes on the current-carrying surface The stream function value at is the intrinsic basis number, is the number of intrinsic bases used, is the coefficient corresponding to the basis, is the basis function at the node The value at .

3. The method for designing an inserted gradient coil based on a curved surface intrinsic basis according to claim 1, characterized in that: The expression of the objective function is: ； Where, f 0 and f 1 represents the main objective function and the auxiliary objective function respectively; The expression of the main goal is as follows: ； Where, represents the number of sampling points in the imaging area, Indicates the discrete node number; Represents the target gradient field, through the gradient intensity Multiply by the position along the gradient direction 、 or calculate; It is about the basis coefficient Function, representing the node at Towards the actual magnetic field; The expression of the auxiliary objective is as follows: ； Where, is the material conductivity, is the thickness of the conductive layer, represents the integral over the coil design surface.

4. The method for designing an inserted gradient coil based on a curved surface intrinsic basis according to claim 3, characterized in that: Calculated by Biot-Savart law: ； Where, 、 、 represents the position along the gradient direction, c represents the basis coefficient, n x Representation surface The x-component of the normal vector, n y Representation surface The y-component of the normal vector, n z Representation surface The z-component of the normal vector.

5. The method for designing an inserted gradient coil based on a curved surface intrinsic basis according to claim 1, characterized in that: The linear equations in step S3 are expressed as follows: ； Where, The sensitivity matrix of the magnetic field at the target position to the coefficients to be solved , express The transposed matrix of represents the energy consumption matrix of coil optimization, Represents a matrix composed of basis functions arranged in rows , Represents a column vector consisting of the target magnetic field .

6. The method for designing an inserted gradient coil based on a curved surface intrinsic basis according to claim 1, characterized in that: In step S4, the stream function on the current-carrying surface is expressed as follows using basis functions: ； in, is the stream function scalar field on the current-carrying surface, is the intrinsic basis number, is the number of intrinsic bases used, is the coefficient corresponding to the basis, are basis functions defined on the current-carrying surface.

7. The method for designing an inserted gradient coil based on a curved surface intrinsic basis according to claim 1, characterized in that: The step S6 specifically includes: selecting a series of different auxiliary target weights for calculation, constructing a relationship curve using the main target value and the auxiliary target value of the corresponding calculation results, and selecting the auxiliary target weight at the inflection point from the relationship curve for calculation.

8. The method for designing an inserted gradient coil based on a curved surface intrinsic basis according to any one of claims 1 to 7, characterized in that: The number of bases is greater than 20 and does not exceed 100.

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