Ultra-low field movable brain magnetic resonance hybrid gradient coil system and design method

Abstract

The application discloses a kind of ultra-low-field movable brain magnetic resonance hybrid gradient coil systems and design methods, including two coaxial sleeve of cylindrical coil and a set of oppositely arranged planar coil, wherein two cylindrical coils are used to generate z direction and y direction first-order gradient magnetic field respectively, planar coil is used to generate x direction first-order gradient magnetic field, establish three dielectric models of air, eddy current resistance plate and polar plate, and the influence of dielectric interface on gradient magnetic field is equivalent to the superposition contribution of multiple order mirror current by mirror current method;The target field magnetic field model of cylindrical coil and planar coil is established respectively, the Fourier expansion coefficient is solved using Tikhonov regularization, and the discrete coil distribution is obtained based on stream function contour line;When linear error or power loss does not satisfy preset requirement, the coil design parameter is optimized by fast genetic algorithm.The application can consider gradient field linearity, power loss and eddy current suppression requirement in the limited space of movable brain MRI equipment.

Description

Technical Field

[0001] This invention relates to the field of ultra-low field magnetic resonance imaging (MRI) system design, specifically to an ultra-low field portable cranial magnetic resonance hybrid gradient coil system and its design method. Background Technology

[0002] In ultra-low field MRI systems, gradient coils are used to generate the gradient magnetic field required for spatial encoding. Portable cranial MRI devices typically employ permanent magnets, pole plates, anti-eddy current plates, and compact support structures. The gradient coils must perform multi-directional encoding within a limited space while minimizing the coupling between gradient switching and conductive or highly permeable structures.

[0003] If the gradient coil is designed solely based on the free-space model, the target field response matrix cannot reflect the changes in magnetic field boundary conditions caused by the anti-eddy current plates and poles, leading to deviations between the linearity of the gradient field and the expected design after actual installation. This problem cannot be solved by the basic magnetic field formula of a single cylindrical coil; rather, it stems from the boundary coupling of the hybrid coil within the overall system's dielectric environment.

[0004] On the other hand, the target field area, patient access passage, and magnet window of portable cranial MRI equipment collectively limit the space for coil arrangement. A single cylindrical coil is not easy to simultaneously achieve linearity and low coupling characteristics of gradient fields in three directions; a single planar coil is also easily affected by the boundaries of the pole plates and anti-eddy current plates. Therefore, it is necessary to select a combination configuration of cylindrical and planar coils according to different gradient directions, and to correct for the radial current component and dielectric boundary effects of the planar coil.

[0005] Therefore, it is necessary to propose a hybrid gradient coil system and design method suitable for ultra-low field mobile cranial MRI systems. Summary of the Invention

[0006] The purpose of this invention is to provide an ultra-low field portable cranial MRI hybrid gradient coil system and its design method. This invention addresses the problems of existing free-space coil designs failing to accurately reflect the overall installation environment, single coil configurations failing to simultaneously satisfy multi-directional gradient encoding, and insufficient correction of radial current contributions from planar coils. This invention combines cylindrical and planar coils according to their directional roles, and before solving the target field, it incorporates the overall boundary influence into the response matrix using a three-dielectric boundary model and a mirror current model. Then, it performs regularized solution, stream function discretization, and fast genetic algorithm optimization. This allows the system to balance gradient field linearity, power loss, inductance, and eddy current suppression requirements under limited space conditions, making it suitable for portable ultra-low field cranial MRI systems.

[0007] To achieve the above objectives, the present invention adopts the following technical solution: The design method of an ultra-low field movable cranial magnetic resonance hybrid gradient coil system includes the following steps: Step 1: Based on the target field area, overall spatial constraints, main magnetic field direction, and the positions of the magnet pole plates and anti-eddy current plates of the portable cranial magnetic resonance system, determine the hybrid gradient coil configuration. The hybrid gradient coil includes two coaxially nested cylindrical coils with different diameters and a set of planar coils arranged opposite each other. The two cylindrical coils are used to generate first-order gradient magnetic fields in the z and y directions, respectively, and the planar coils are used to generate first-order gradient magnetic fields in the x direction. Step 2: Place the smaller diameter cylindrical coil inside the larger diameter cylindrical coil so that the target field region is located in the center region of the cylindrical coil; arrange a set of planar coils on both sides of the cylindrical coil, and make the axis of the planar coils perpendicular to the axis of the cylindrical coil to form a hybrid gradient coil system; Step 3: Establish a three-dielectric boundary model including air, anti-eddy current plate and magnet pole plate, use the image current model to calculate the correction contribution of the surrounding medium to the magnetic field response of cylindrical gradient coil and planar gradient coil, and write the multi-order image current contribution into the target field response matrix. Step 4: Based on the target field response matrix, perform regularized target field solution, stream function discretization, and fast genetic algorithm optimization to obtain a hybrid gradient coil system that meets the requirements of target field linearity, power loss, inductance, and eddy current suppression under the whole machine medium environment.

[0008] Furthermore, the design method of the cylindrical coil includes: Step 11: Based on the Biot-Savart law and spherical harmonic series expansion, establish the magnetic field distribution expression of the cylindrical coil in the target field region for the target gradient direction of the cylindrical coil; Step 12: Based on the three-dielectric boundary model, the magnetic field distribution expression of the cylindrical coil in the target field region is corrected using the mirror current model; Step 13: Constrain and regularize the modified cylindrical coil target magnetic field using the target field method, and solve for the Fourier expansion coefficients of the cylindrical coil. ; Step 14: Calculate the Fourier expansion coefficients of the cylindrical coil. Substituting the expression for the current function of the cylindrical coil into the expression for the current function of the cylindrical coil, we obtain the current function distribution of the cylindrical coil. Step 15: Take contour lines for the flow function distribution of the cylindrical coil to obtain the discrete coil distribution of the cylindrical coil, and calculate the magnetic field nonlinearity error and the power loss of the cylindrical gradient coil in the target field region; if the magnetic field nonlinearity error or power loss does not meet the preset requirements, then execute step 16; otherwise, output the discrete coil distribution of the cylindrical coil and execute step 17. Step 16: Use a fast genetic algorithm to iteratively optimize the cylindrical coil design parameters. Repeat steps 11 to 15 for each candidate design parameter, and use magnetic field nonlinearity error, gradient coil power loss and inductance as optimization objectives to obtain the optimized discrete coil distribution of the cylindrical coil. Step 17: Prepare a cylindrical coil based on the optimized discrete coil distribution of the cylindrical coil.

[0009] Furthermore, the design method of the planar coil includes: Step 21: Based on the Biot-Savart law, establish the magnetic field distribution expression of the planar coil in the target field region for the target gradient direction of the planar coil; Step 22: Based on the three-dielectric boundary model, the magnetic field distribution expression of the planar coil in the target field region is corrected using the mirror current model; Step 23: Constrain and regularize the modified planar coil target field magnetic field using the target field method, and solve for the Fourier expansion coefficients of the planar coil. ; Step 24: Expand the Fourier coefficients of the planar coil Substituting the expression for the planar coil current function, we obtain the distribution of the planar coil current function; Step 25: Take contour lines for the flow function distribution of the planar coil to obtain the discrete coil distribution of the planar coil, and calculate the magnetic field nonlinearity error and the power loss of the planar gradient coil in the target field region; if the magnetic field nonlinearity error or power loss does not meet the preset requirements, then proceed to step 26; otherwise, output the discrete coil distribution of the planar coil and proceed to step 27. Step 26: Use a fast genetic algorithm to iteratively optimize the design parameters of the planar coil. Repeat steps 21 to 25 for each candidate design parameter, and use the magnetic field nonlinearity error, planar gradient coil power loss and inductance as optimization objectives to obtain the optimized discrete coil distribution of the planar coil. Step 27: Prepare a planar coil based on the optimized planar coil discrete coil distribution.

[0010] Furthermore, in step 11, the expression for the magnetic field distribution of the cylindrical coil within the target field region is established, specifically as follows: A first-order transverse gradient magnetic field is constructed using a cylindrical coil as a carrier. This first-order transverse gradient magnetic field corresponds to the first-order non-axisymmetric mode in a spherical harmonic expansion. For a radius of... , length is The cylindrical coil will have an axial current density and circumferential current density Represented in Fourier trigonometric series form: (1) In the formula, and represent the order and degree in the spherical harmonic series expansion and the current density function expansion, respectively; Q Let be the order of the Fourier expansion of the current density function; This represents the Fourier expansion coefficient vector in the trigonometric series expansion of the current density of a cylindrical coil. For the first Fourier coefficients of the current density; The axial coordinates of the source point on the cylindrical surface. The upper limit of the order of a spherical harmonic expansion is determined by the truncation order of the expansion. Sure; ; ; ; , It is used to select sine or cosine basis functions in axial expansion based on modal parity symmetry; The main magnetic field of the magnet is along the y-axis. According to Biot-Savart's law, the y-direction component of the magnetic field generated by the source point is... Represented as:

[0011] In the formula, Indicates the polar angle and azimuth angle of the field point. Indicates the polar angle and azimuth angle of the source point. for The first-order associated Legendre function is used to characterize the angular distribution in spherical harmonic expansions. , , and The intermediate coefficients are those whose subscripts correspond to the order of the spherical harmonic expansion. Consistent, the specific expression is as follows:

[0012] Further, the y-direction component of the magnetic field generated by the cylindrical coil Except Other integral terms are denoted as :

[0013] The above formula can be simplified to: .

[0014] Furthermore, in step 21, the expression for the magnetic field distribution of the planar coil in the target field region is established based on Biot-Savart's law, specifically as follows: Trigonometric functions are introduced as basis functions for the current density of the planar coil during Fourier preprocessing; for the planar coil used to generate the gradient magnetic field in the x-direction, its current density is expanded into the following components:

[0015] In the formula, Indicates the current density in the radial direction The components on; Indicates current density in azimuth angle Components in direction; This is the distance from a point on the coil in the plane to the central axis; The azimuth angle of a point on the coil; Indicates the center position of the current density expansion; Let be the order of the Fourier expansion of the current density function. The order number in the Fourier expansion; Here are the Fourier expansion coefficients of the planar coil. For the first The expansion coefficients corresponding to the four-order Fourier basis functions; The spatial frequency scaling factor determines the rate of radial oscillation of the cosine function; Based on the Biot-Savart law, for a gradient coil consisting of two sets of planar coils arranged with the y-direction as the normal direction, the field point... place This represents the linear superposition of the contributions of each basis function:

[0016] In the formula, and The two distance quantities corresponding to the upper and lower planar coils represent the distances from the field point to the surface elements of the upper and lower planar coils, respectively; kernel function symbol. Indicates the first The Fourier basis function contributes to the magnetic field at the target field point; Further, the y-direction component of the magnetic field generated by the planar coil Except Other integral terms are denoted as : Then the y-direction component of the magnetic field generated by the planar coil will be... Simplified to: .

[0017] Furthermore, in step 12, the magnetic field distribution expression of the cylindrical coil in the target field region is corrected using the mirror current model based on the established three-dielectric boundary model; and in step 22, the magnetic field distribution expression of the planar coil in the target field region is corrected using the mirror current model based on the established three-dielectric boundary model. Specifically: A three-medium boundary model is established, consisting of air, an anti-vortex plate, and an electrode. The interface between air and the anti-vortex plate, and the interface between the anti-vortex plate and the electrode, are respectively considered as mirror boundaries. The reflection coefficient at each interface is expressed as:

[0018] Based on image theory and dielectric boundary conditions, the boundary influence of the original coil current in the air domain is equivalent to multiple order image currents; the amplitude expressions for each order of equivalent image current are as follows:

[0019] In the formula, The primary coil current, This refers to the mirror current that is not reflected between the anti-eddy current plate and the electrode. To generate between the anti-eddy current plate and the electrode plate The mirror current formed by the secondary reflection; The cylindrical coil and planar coil use the same three-dielectric boundary model and image current model in steps 12 and 22, and these models are used in subsequent formulas. Unified representation and ,use Unified representation and ,use Unified representation and ; Corrected total magnetic field Magnetic field generated by the primary coil current and the magnetic fields generated by the image currents of each order. The result of superposition is represented as:

[0020] For the first The magnetic field generated by the mirror current; In the revised magnetic field calculation expression, except for Other integral terms; substitute into the magnetic field calculation formula for cylindrical or planar coils according to the spatial location of each mirror current. and The integral terms corresponding to the image currents of each order are obtained. and And correct the integral term in the target field method based on the magnitude of the mirror current:

[0021] In the formula, Unified representation and ; For the first The ratio of the maximum value of the current function corresponding to the first-order mirror current to the maximum value of the current function corresponding to the primary coil current is used to characterize the intensity contribution of different-order mirror currents relative to the primary coil current. In the revised magnetic field calculation expression, except for Other integral terms.

[0022] Furthermore, in steps 13 and 23, the following steps are adopted: Unified representation and The specific process includes: Uniform sampling of the target field area Based on magnetic field sampling information of the target field region at discrete points, the corrected response terms of the cylindrical coil and the planar coil are synthesized into a unified response matrix, and the hybrid gradient coil design problem is then transformed into a linear equation. :

[0023] In the formula, For the first The target magnetic field at each sampling point ; Each row in the matrix Each element is In the subscript That is to represent the first One sampling point, Represents the order number in the Fourier expansion; In the coil design, both power loss constraints and field linearity constraints are introduced, and the solution is obtained by combining the Tikhonov regularization strategy. The objective function is:

[0024] In the formula, , , The power loss of the gradient coils are respectively ,inductance Stream function curvature Regularization weights, and respectively through The calculated magnetic field distribution and the target magnetic field distribution, and Represents passing The calculated first The magnetic field distribution at discrete points and the target magnetic field distribution; The above power loss ,inductance Stream function curvature The constraints are further projected into the modal coefficient space and transformed into Fourier coefficients in the trigonometric series expansion of the current density. The quadratic form:

[0025] In the formula, , and These are the characteristic matrices corresponding to power loss, inductance, and flow function curvature, respectively. Based on the first-order optimality condition, the objective function is related to the Fourier coefficients. Differentiate and set the gradient to zero to determine the Fourier coefficients in the trigonometric series expansion of the current density. The optimal solution: .

[0026] Furthermore, in step 14, the Fourier expansion coefficients of the cylindrical coil are... Substituting the expression for the current function of the cylindrical coil into the equation, we obtain the distribution of the current function of the cylindrical coil, as follows:

[0027] In the formula, The current function distribution is for a cylindrical coil. In step 24, the Fourier expansion coefficients of the planar coil are... Substituting the expression for the flow function of a planar coil, we obtain the flow function distribution of a cylindrical coil, specifically:

[0028] In the formula, The distribution of the flow function for a planar coil.

[0029] Furthermore, using Unified representation of the flow function of a cylindrical coil and planar coil current function Steps 15 and 25 specifically include: Based on the difference between the equipotential lines of the flow function corresponding to the current flux in the coil wire, the coil current... Represented as:

[0030] In the formula, This refers to the number of coil turns. Based on the distribution results of the discrete coils, the nonlinear error of the magnetic field in the target region, the power loss of the gradient coil, and the inductance of the gradient coil are further calculated as follows:

[0031]

[0032]

[0033] In the formula, This indicates the linearity error of the magnetic field within the target field region. and They represent the target field regions respectively. Magnetic field calculation results and target magnetic field value at each sampling point; This represents the total number of sampling points in the target field area. and These represent the power loss and inductance value of the cylindrical gradient coil, respectively. and These represent the power loss and inductance of the planar gradient coil, respectively. Indicates the resistivity of the coil material. Indicates the diameter of the coil conductor; and These represent the expressions for the circumferential and axial current densities of the cylindrical coil, respectively. and These represent the expressions for the current density in the azimuth and radial directions of the planar coil, respectively. and These represent the vectors pointing from the center point to the field point and the source point, respectively. Steps 16 and 26 specifically include: The various design parameters of the coil are selected as decision variables in the optimization process. For a planar coil, the decision variable vector is: For cylindrical coils, the design vector expansion is as follows: ;in, For the order of the Fourier expansion, Let the order of the image current expansion be denoted as . The number of coil turns. These are the regularization parameters for coil power loss, coil inductance, and flow function curvature, respectively. The order of the spherical harmonic maximum expansion is given; an initial population is randomly generated based on the value range of each decision variable; for each candidate individual in the initial population, the corresponding magnetic field nonlinearity error in the target field region is calculated. Gradient coil power loss and inductor Subsequently, based on the nonlinear error reference value, power loss reference value, and inductance reference value, the nonlinear error, power loss, and inductance are dimensionless, and combined with pre-set weighting coefficients, the comprehensive fitness function is constructed as follows:

[0034] In the formula, using A unified representation of power loss in cylindrical coils and planar coil power loss ,use A unified representation of cylindrical coil inductance and planar coil inductance ; This is the overall fitness function, and the fitness result of each individual is calculated based on this function; , and These are the nonlinear error reference value, the coil power loss threshold, and the inductance threshold, respectively. , and These are the weighting coefficients for linearity error, power loss, and inductance, respectively. , and Determined by system design specifications and engineering constraints; , and It depends on the focus of the optimization objective.

[0035] The ultra-low field movable cranial magnetic resonance hybrid gradient coil system was obtained using the above design method.

[0036] Compared with the prior art, the present invention has at least the following beneficial technical effects: This invention employs a hybrid coil configuration, allowing different gradient directions to be matched with coil forms more suitable for the overall installation space. The cylindrical coil reduces electromagnetic coupling caused by adjacent conductive structures, while the planar coil improves the magnetic field coverage of the target area within a confined space. This ensures assembly adaptability while improving the space utilization and magnetic field efficiency of the gradient coils. Simultaneously, this invention uses a mirror current model based on a three-dielectric boundary model to correct the target field response matrix, ensuring that the target field calculation process simultaneously considers the influence of anti-eddy current plates, poles, and their boundaries on the magnetic field distribution. This allows for pre-compensation for magnetic field deviations caused by conductive structures and boundary conditions during the design phase, reducing the difference between the theoretical design results in free space and the actual magnetic field after installation, thus improving the feasibility and accuracy of the coil design. Furthermore, this invention utilizes a fast genetic algorithm to jointly optimize configuration parameters, mirror order, Fourier basis function order, and regularization weights. This enables the coil design to achieve a comprehensive balance among multiple indicators such as magnetic field linearity, power loss, inductance, and eddy current suppression, avoiding local optima or indicator imbalances caused by manual adjustment of a single parameter, thereby improving the overall system performance and design efficiency. Attached Figure Description

[0037] The accompanying drawings are provided to further understand the invention and constitute a part of this invention. The illustrative embodiments of the invention and their descriptions are used to explain the invention and do not constitute an improper limitation of the invention.

[0038] Figure 1 This diagram shows the magnet distribution and its positional relationship with the gradient coil of a portable ultra-low field cranial MRI prototype.

[0039] Figure 2 This is a side view showing the spatial distribution of planar and cylindrical coils.

[0040] Figure 3 A front view of the hybrid coil design.

[0041] Figure 4 This is a three-dielectric boundary model; the orange area represents the anti-eddy current plate region, and the blue area represents the electrode plate region. The distance from the coil plane to the upper surface of the anti-eddy current plate is... The thickness of the anti-vortex plate is The magnetic permeabilities of the three media—air, anti-eddy current plate, and electrode plate—are respectively... , and .

[0042] Figure 5 This is a mirror current model.

[0043] Figure 6The Z-gradient coil and target field distribution are shown in the side view. The gradient coil radius is 147.15 mm, the target field radius is 115 mm, the coil has 28 turns, the coil current is 59.11 A, the linearity is 3.1%, and the gradient efficiency is 0.423 mT·m. - ¹·A - ¹.

[0044] Figure 7 The Z-gradient coil and the target field distribution are shown in the front view.

[0045] Figure 8 The Y-gradient coil and target field distribution are shown in the side view. The gradient coil radius is 141.45 mm, the target field radius is 115 mm, the coil has 32 turns, the coil current is 49.3 A, the linearity is 3.9%, and the gradient efficiency is 0.507 mT·m. - ¹·A - ¹.

[0046] Figure 9 The Y-gradient coil and target field distribution are shown in the front view.

[0047] Figure 10 The X-gradient coil and target field distribution are shown, with a spacing of 303.7 mm between the upper and lower coils and a coil radius of 285 mm. The upper and lower planar coils are completely identical, and the single-planar coil also exhibits a symmetrical structure. Each upper and lower planar coil has 32 turns, a coil current of 174.87 A, and a linearity of 3.3%.

[0048] Figure 11 The distribution diagram of the X gradient coil current function.

[0049] Figure 12 It is a finite element model that includes gradient coils, magnet structures, and an external air domain.

[0050] Figure 13 This is a graph showing the difference between the magnetic field results obtained from the mirror current model and the finite element simulation. Detailed Implementation

[0051] To enable those skilled in the art to better understand the present invention, the technical solutions of the present invention will be clearly and completely described below with reference to the accompanying drawings of the embodiments of the present invention. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort should fall within the scope of protection of the present invention.

[0052] It should be noted that the terms "first," "second," etc., in the specification, claims, and accompanying drawings of this invention are used to distinguish similar objects and are not necessarily used to describe a specific order or sequence. It should be understood that such data can be interchanged where appropriate so that the embodiments of the invention described herein can be implemented in orders other than those illustrated or described herein. Furthermore, the terms "comprising" and "having," and any variations thereof, are intended to cover a non-exclusive inclusion; for example, a process, method, system, product, or apparatus that comprises a series of steps or units is not necessarily limited to those steps or units explicitly listed, but may include other steps or units not explicitly listed or inherent to such processes, methods, products, or apparatus.

[0053] Example 1 This invention provides a design method for an ultra-low field movable cranial magnetic resonance hybrid gradient coil system. This gradient coil system employs a composite structure combining planar coils and cylindrical coils. Specifically, the z-direction gradient coil and the y-direction gradient coil utilize cylindrical coil structures, such as... Figure 2 As shown, to reduce the eddy current effect caused during gradient switching, the x-direction gradient coil adopts a planar coil structure to meet the linearity requirements within the target imaging area under the constraints of coil size and target field range. Although the x-direction gradient coil adopts a planar structure, it is not arranged in close proximity to the permanent magnet, but rather a certain installation distance is maintained between it and the permanent magnet, such as... Figure 1 As shown, this significantly reduces electromagnetic coupling and lowers eddy current and remanent magnetization effects. Through the above composite structure design, this invention can balance gradient field performance, linearity, and low eddy current characteristics in an ultra-low field MRI system. Based on the above structure, the spatial arrangement and design principles of each gradient coil are further explained as follows: like Figure 2 As shown, the X, Y, and Z gradient coils generate first-order gradient magnetic fields in the x, y, and z directions, respectively. The Z and Y gradient coils are arranged around the target imaging region and coaxially nested along the system's central axis; the X gradient coils consist of two sets of planar coils symmetrically arranged along the main magnetic field direction, located on opposite sides of the target imaging region's axial direction. The cylindrical coils are... The coils are wound with 2.5mm enameled copper wire, and the planar coils are wound with 2mm×5mm rectangular coils. They are mounted on a non-conductive support structure and arranged in layers according to the spatial dimensions.

[0054] A first-order transverse gradient magnetic field is constructed using a cylindrical coil as a carrier. For a radius of... , length is For a cylindrical coil, the axial and circumferential current densities can be expressed in Fourier trigonometric series form: (1) In the formula, and represent the order and degree in the spherical harmonic series expansion and the current density function expansion, respectively; Let be the order of the Fourier expansion of the current density function; The Fourier expansion coefficients in the trigonometric series expansion of the current density of a cylindrical coil are represented. For the first Fourier coefficients of the current density; The axial coordinates of the source point on the cylindrical surface. The upper limit of the order of a spherical harmonic expansion is determined by the truncation order of the expansion. Sure; ; ; ; , It is used to select the sine basis function or cosine basis function in the axial expansion based on the parity of the modal.

[0055] According to Biot-Savart's law, the infinitesimal magnetic field strength generated by the infinitesimal current density element at the source point on the cylindrical surface at the field point inside the coil is... for: (2) Green's function Expand the series using spherical coordinates: (3) In the formula, The vacuum permeability; Represents the distance from the source point to the origin; Neumann factor ; and These represent the polar angle and azimuth angle of the field point and the source point in the coordinate system, respectively.

[0056] When the main magnetic field of the magnet is along the y-axis, the y-direction component of the magnetic field generated by the source point at the field point is expressed as: (4) In the formula, for The first-order associated Legendre function is used to characterize the angular distribution in spherical harmonic expansion; , , and The intermediate coefficients are those whose subscripts correspond to the order of the spherical harmonic expansion. Consistent, the specific expression is as follows: (5) Further, the y-direction component of the magnetic field generated by the cylindrical coil Except Other integral terms are denoted as : (6) Therefore, the expression for the magnetic field of a cylindrical coil can be derived. Simplified to: .

[0057] For a planar coil used to generate a gradient magnetic field in the x-direction, trigonometric functions are introduced as basis functions for the planar current density, and the planar coil current density is expanded as follows: (7) In the formula, Indicates the current density in the radial direction The components on; Indicates current density in azimuth angle Components in direction; This is the distance from a point on the coil in the plane to the central axis; The azimuth angle of a point on the coil; Indicates the center position of the current density expansion; Let be the order of the Fourier expansion of the current density function. The order number in the Fourier expansion; Here are the Fourier expansion coefficients of the planar coil. For the first The expansion coefficients corresponding to the first-order basis functions; This is the spatial frequency scaling factor, which determines the rate of radial oscillation of the cosine function.

[0058] Based on the Biot-Savart law, the magnetic field components of a gradient coil, consisting of two sets of planar coils arranged with the y-direction as the normal direction, can be represented as a linear superposition of the contributions of each basis function: (8) In the formula, and The two distance quantities corresponding to the upper and lower planar coils represent the distances from the field point to the surface elements of the upper and lower planar coils, respectively; kernel function symbol. Indicates the first The contribution of the basis functions to the magnetic field at the target field point.

[0059] Further, the y-direction component of the magnetic field generated by the planar coil Except Other integral terms are denoted as : The y-direction component of the magnetic field generated by the planar coil can be... Simplified to: .

[0060] Considering that the gradient coil is located near the magnet pole and the anti-eddy current plate, the dielectric boundaries of the pole and the anti-eddy current plate will change the magnetic field distribution within the target field region. This invention establishes a three-dielectric boundary model of air, the anti-eddy current plate, and the pole, and uses an equivalent mirror current model to represent the influence of the dielectric interfaces on the magnetic field. The reflection coefficients at each interface are expressed as: (9) In the formula, , and represents the permeability of air, anti-eddy current plate and electrode plate respectively; the reflection coefficient in equation (9) is determined by the difference in permeability between adjacent media.

[0061] According to the mirror theory and dielectric boundary conditions, the primary coil current forms multiple equivalent mirror currents in the air domain, and their amplitude expressions are as follows: (10) In the formula, The current in the primary coil is denoted as , and the current symbol with the image order subscript in equation (10) is the equivalent image current of the corresponding order; the image order ranges from 0 to the highest order involved in the superposition calculation.

[0062] The three-dielectric boundary model and the image current model are the same for cylindrical and planar coils. The following formulas will use... Unified representation and ,use Unified representation and ,use Unified representation and ; This represents the magnetic field generated by the current in the primary coil. This represents the magnetic field generated by the corresponding e-th order mirror current; and Let y-direction components of the magnetic fields generated by the primary coil current and the e-th order mirror current be represented respectively. Except The integral term corresponding to the integral term other than the integral term.

[0063] After determining the spatial location of each order of the image current, substitute it into the magnetic field expression of the cylindrical coil or the planar coil to obtain the integral term corresponding to each order of the image current. And correct the integral term in the target field method based on the magnitude of the mirror current: (11) In the formula, For the first The ratio of the maximum value of the current function corresponding to the first-order mirror current to the maximum value of the current function corresponding to the primary coil current is used to characterize the intensity contribution of different-order mirror currents relative to the primary coil current.

[0064] The corrected total magnetic field is the magnetic field generated by the current in the original coil. and the magnetic field generated by the image currents of each order The result is obtained by superposition. In the revised magnetic field calculation expression, except for Total integral term outside: (12) Based on the magnetic field sampling information of the target field region, the design problem of cylindrical coils or planar coils is transformed into linear equations: (13) In the formula, the left side of the equation For the first The target magnetic field at each sampling point ; Each row in the matrix can be represented as Consistent with the definition in claim 6, each element is In the subscript That is to represent the first One sampling point, This represents the order number in the Fourier expansion.

[0065] In the coil design, power loss constraints, field linearity constraints, and stream function smoothing constraints are introduced simultaneously, and the solution is obtained by combining the Tikhonov regularization strategy: (14) In the formula, , , Power loss ,inductance Stream function curvature Regularization weights, and respectively through The calculated magnetic field distribution and the target magnetic field distribution, Represents passing The calculated first The magnetic field distribution at discrete points and the target magnetic field distribution; The above power loss ,inductance Stream function curvature After projection onto the modal coefficient space, it transforms into a quadratic form in terms of the modal coefficient vector: (15) In the formula, , and These are the characteristic matrices corresponding to power loss, inductance, and flow function curvature, respectively.

[0066] Based on the first-order optimality condition, the objective function with respect to... Taking the derivative and setting the gradient to zero, we obtain the Fourier expansion coefficient vector. The optimal solution: (16) The results of calculating the Fourier expansion coefficient vectors of cylindrical and planar coils. Substituting the current function expressions for the cylindrical coil and the planar coil respectively, we obtain the current function results for the cylindrical coil and the planar coil: (17) (18) In the formula, The current function distribution is for a cylindrical coil; For the distribution of the current function of the planar coil; using Unified representation of the flow function of a cylindrical coil and planar coil current function .

[0067] The difference between the equipotential lines of the stream function corresponds to the current flux in the coil conductor, and the coil operating current. Represented as: (19) In the formula, This is the coil operating current. This represents the number of turns in the coil.

[0068] Based on the discrete coil distribution results, the magnetic field linearity error and gradient coil power loss within the target region are further calculated: (20) (twenty one) (twenty two) (twenty three) (twenty four) In the formula, This indicates the linearity error of the magnetic field within the target field region. and They represent the target field regions respectively. Magnetic field calculation results and target magnetic field value at each sampling point; This represents the total number of sampling points in the target field area. and These represent the power loss and inductance value of the cylindrical gradient coil, respectively. and These represent the power loss and inductance of the planar gradient coil, respectively. Indicates the resistivity of the coil material. Indicates the diameter of the coil conductor; and These represent the expressions for the circumferential and axial current densities of the cylindrical coil, respectively. and These represent the expressions for the current density in the azimuth and radial directions of the planar coil, respectively. and These represent the vectors pointing from the center point to the field point and the source point, respectively.

[0069] In the process of coil parameter optimization, various design parameters of the coil are selected as decision variables. For a planar coil, the decision variable vector is: For cylindrical coils, the design vector expansion is as follows: .in, For the order of the Fourier expansion, Let the order of the image current expansion be denoted as . The number of coil turns. These are the regularization parameters for coil power loss, coil inductance, and flow function curvature, respectively. Let be the order of the spherical harmonic expansion. For each candidate individual in the initial population, calculate its corresponding magnetic field linearity error within the target field region. Gradient coil power loss and inductor Subsequently, based on the nonlinear error reference value, coil power loss, and inductance reference value, the nonlinear error, power loss, and inductance are dimensionless, and combined with pre-set weighting coefficients, a comprehensive fitness function is constructed as follows: (25) In the formula, using Unified representation and ,use Unified representation and ; This is the overall fitness function, and the fitness result of each individual is calculated based on this function; , and These are the nonlinear error reference value, the coil power loss threshold, and the inductance threshold, respectively. , and These are the weighting coefficients for nonlinear error, coil power loss, and inductance, respectively.

[0070] In summary, this embodiment uses the three-dielectric boundary model and the mirror current model as the core correction steps. Before solving the target field method, the boundary effects caused by air, anti-eddy current plate and electrode plate are introduced into the magnetic field response matrix, so that the coil distribution obtained by subsequent stream function discretization and fast genetic algorithm optimization is closer to the actual working environment of the whole machine.

[0071] Example 2 like Figure 1 As shown, this embodiment provides a magnet and gradient coil structure suitable for a permanent magnet type portable ultra-low field brain MRI system. The structure mainly includes magnetic poles, pole plates, an anti-eddy current plate, and a shimming ring and gradient coil disposed inside the magnet. The magnetic poles and pole plates provide a static magnetic field environment, the shimming ring improves the magnetic field uniformity within the target field region, and the anti-eddy current plate is disposed within the magnet structure to reduce the eddy current effects generated during gradient magnetic field switching.

[0072] In this embodiment, the gradient coils are arranged around the target field region. Planar or cylindrical gradient coil structures can be used depending on the requirements of the gradient field in different directions. During the design process, the influence of the pole plates and anti-eddy current plates on the magnetic field distribution is comprehensively considered to improve the adaptability and design accuracy of the gradient coils in actual working environments. This structure can effectively generate the gradient magnetic field in the target region within a limited space, while also considering field linearity and low eddy current performance requirements.

[0073] like Figure 2 As shown, this embodiment employs a hybrid gradient coil structure, with the target field positioned in the central region of the coils. Specifically, the X gradient coil utilizes two sets of planar coil structures, positioned above and below the target field, respectively; the Y and Z gradient coils employ cylindrical coil structures, surrounding the outer periphery of the target field.

[0074] Figure 3 for Figure 2 The diagram shows a front view of the hybrid coil structure. The X gradient coil is located on the upper and lower sides of the target field region, while the Y and Z gradient coils are coaxially arranged around the periphery of the target field.

[0075] like Figure 4 As shown, this embodiment establishes a three-dielectric boundary model consisting of a coil, an anti-eddy current plate, and an electrode plate. The coil is positioned above the air region, and the anti-eddy current plate and electrode plate are arranged sequentially along the axial direction, each corresponding to a medium with different permeabilities. In the coil design process, the air, anti-eddy current plate, and electrode plate are treated as different dielectric layers to characterize the influence of material boundaries on the magnetic field distribution in the actual structure, thereby making the gradient coil design more consistent with the system's real-world operating environment.

[0076] like Figure 5 As shown, based on the three-dielectric model, this embodiment uses a mirror current model to equivalently treat the magnetic field influence caused by the dielectric interface. The original coil current is used as an example. Based on this, zero-order to higher-order mirror currents are constructed along the normal direction of the dielectric layer. , , … The spatial location and amplitude of each mirror current are then substituted into the calculation of the target field's magnetic field response. The coordinate positions of each mirror current are as follows: (26) in, Let y be the position of the original coil. Let y be the 0th order image current. For the first The y-coordinate of the image current; where the thickness of the anti-eddy current plate, the distance from the coil to the anti-eddy current plate, and the image order together determine the position of each image current.

[0077] By using mirror currents, the effects of the anti-eddy current plates and poles on the magnetic field can be equivalently transformed into the superimposed contributions of multiple mirror currents to the target field region. This process ensures that the response matrix in the target field method no longer corresponds only to the free-space coil, but also includes the magnetic field contributions of both the original coil current and multiple mirror currents, thereby improving the accuracy of gradient coil magnetic field calculations and subsequent structural optimization.

[0078] like Figures 6 to 9 As shown in the figure, this embodiment illustrates the conductor distribution of the Z-gradient coil and the Y-gradient coil, and the magnetic field distribution they generate within the target field region. The Z-gradient coil is used to establish a gradient magnetic field varying along the Z direction within the target field region, and the Y-gradient coil is used to establish a gradient magnetic field varying along the Y direction within the target field region.

[0079] In this embodiment, both the Z gradient coil and the Y gradient coil adopt a cylindrical coil structure arranged around the target field region. The coil wires are arranged according to a predetermined current density distribution to form a gradient magnetic field in the corresponding direction within the target field region. Figure 6 and Figure 8 The overall spatial arrangement of the coil and the variation of the magnetic field within the target field region are shown. Figure 7 and Figure 9 The magnetic field distribution within the cross-section of the target field is shown. As can be seen from the figure, the magnetic field strength in the target field region exhibits a continuous and gradual distribution along the corresponding direction, indicating that the designed gradient coil can form good gradient characteristics within the target field range.

[0080] like Figure 10 and Figure 11As shown in the figure, this embodiment presents the target field distribution and its stream function distribution results for the X gradient coil. The X gradient coil adopts a planar coil structure with the coils positioned vertically opposite each other, used to generate a gradient magnetic field that varies along the X direction within the target field region. Figure 10 The magnetic field strength within the target field region varies continuously along the X-direction and exhibits a relatively good symmetrical distribution. For example... Figure 11 The conductor trajectory of the X gradient coil is extracted from the stream function contour lines, which can determine the winding form of the planar coil, thereby obtaining better gradient field linearity and field strength distribution in the target field region.

[0081] In summary, this embodiment completed the equivalent magnetic field analysis of the gradient coil in a real-world dielectric environment by constructing a three-dielectric boundary model and a mirror current model. Furthermore, optimization was performed using a fast genetic algorithm to design the structural forms and wire distributions of the Z, Y, and X gradient coils. As can be seen from the target field distribution results shown in the figures, the designed gradient coils can generate a continuously varying gradient magnetic field with good linearity in the corresponding directions, thus meeting the design requirements of the gradient-encoded magnetic field for ultra-low field MRI systems.

[0082] Example 3 In this embodiment, to verify the magnetic field distribution characteristics of the designed gradient coil in the magnet structure environment and the accuracy of the established analytical calculation method, a finite element model including the gradient coil, the magnet structure, and the external air domain was constructed, as follows: Figure 12 As shown in the diagram, in the finite element model, the gradient coil is positioned near the working space of the magnet structure. The magnet structure includes upper and lower magnet components and a supporting connection structure. The gradient coil is located outside the target imaging region and maintains a predetermined relative position with the main magnet structure. To reduce the impact of boundary truncation on the simulation results, an air computational domain is further set outside the gradient coil and the magnet structure to characterize the magnetic field propagation characteristics in open space. During the finite element solution process, an excitation current consistent with the design conditions is applied to the gradient coil, and the corresponding magnetic field distribution results are extracted within the target field region to obtain the actual magnetic field response considering the influence of the magnet structure.

[0083] Based on the above finite element model, the magnetic field results of the target field region obtained by finite element simulation are compared point by point with the magnetic field results calculated by the image current method to obtain the distribution of the difference between the two, such as... Figure 13 As shown. Figure 13 The image current model and the magnetic field components in the target field region of the finite element simulation are given. The difference cloud map shows that within the circular imaging region of the target, the overall difference in the magnetic field results obtained by the two methods is relatively small, with the difference distributed within 10. -5The magnitude of the magnetic field is on the order of magnitude, and in most areas it is close to zero, with only small fluctuations in local edge regions and individual locations. This indicates that after using the mirror current method to equivalently treat the boundary effects of the magnet structure, the magnetic field distribution of the gradient coil in the actual magnet environment can be predicted more accurately. The established analytical model has good consistency with the finite element simulation results, thus verifying the effectiveness and feasibility of the gradient coil design and magnetic field analysis method described in this invention.

[0084] Those skilled in the art will understand that embodiments of the present invention can be provided as methods, systems, or computer program products. Therefore, the present invention can take the form of a completely hardware embodiment, a completely software embodiment, or an embodiment combining software and hardware aspects. Furthermore, the present invention can take the form of a computer program product embodied on one or more computer-usable storage media (including, but not limited to, disk storage, CD-ROM, optical storage, etc.) containing computer-usable program code.

[0085] This invention is described with reference to flowchart illustrations and / or block diagrams of methods, apparatus (systems), and computer program products according to embodiments of the invention. It will be understood that each block of the flowchart illustrations and / or block diagrams, and combinations of blocks in the flowchart illustrations and / or block diagrams, can be implemented by computer program instructions. These computer program instructions can be provided to a processor of a general-purpose computer, special-purpose computer, embedded processor, or other programmable data processing apparatus to produce a machine, such that the instructions, which execute via the processor of the computer or other programmable data processing apparatus, generate instructions for implementing the flowchart illustrations and / or block diagrams. Figure 1 One or more processes and / or boxes Figure 1 A device that provides the functions specified in one or more boxes.

[0086] These computer program instructions may also be stored in a computer-readable storage medium that can direct a computer or other programmable data processing device to function in a particular manner, such that the instructions stored in the computer-readable storage medium produce an article of manufacture including instruction means, which are implemented in a process Figure 1 One or more processes and / or boxes Figure 1 The function specified in one or more boxes.

[0087] These computer program instructions may also be loaded onto a computer or other programmable data processing equipment to cause a series of operational steps to be performed on the computer or other programmable equipment to produce a computer-implemented process, thereby providing instructions that execute on the computer or other programmable equipment for implementing the process. Figure 1 One or more processes and / or boxes Figure 1 The steps of the function specified in one or more boxes.

[0088] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and not to limit its scope of protection. Although the present invention has been described in detail with reference to the above embodiments, those skilled in the art should understand that after reading the present invention, they can still make various changes, modifications or equivalent substitutions to the specific implementation of the invention, but these changes, modifications or equivalent substitutions are all within the scope of protection of the pending claims of the invention.

Claims

1. A design method for an ultra-low field movable cranial magnetic resonance hybrid gradient coil system, characterized in that, Includes the following steps: Step 1: Based on the target field area, overall spatial constraints, main magnetic field direction, and the positions of the magnet pole plates and anti-eddy current plates of the portable cranial magnetic resonance system, determine the hybrid gradient coil configuration. The hybrid gradient coil includes two coaxially nested cylindrical coils with different diameters and a set of planar coils arranged opposite each other. The two cylindrical coils are used to generate first-order gradient magnetic fields in the z and y directions, respectively, and the planar coils are used to generate first-order gradient magnetic fields in the x direction. Step 2: Place the smaller diameter cylindrical coil inside the larger diameter cylindrical coil so that the target field region is located in the center region of the cylindrical coil; arrange a set of planar coils on both sides of the cylindrical coil, and make the axis of the planar coils perpendicular to the axis of the cylindrical coil to form a hybrid gradient coil system; Step 3: Establish a three-dielectric boundary model including air, anti-eddy current plate and magnet pole plate, use the image current model to calculate the correction contribution of the surrounding medium to the magnetic field response of cylindrical gradient coil and planar gradient coil, and write the multi-order image current contribution into the target field response matrix. Step 4: Based on the target field response matrix, perform regularized target field solution, stream function discretization, and fast genetic algorithm optimization to obtain a hybrid gradient coil system that meets the requirements of target field linearity, power loss, inductance, and eddy current suppression under the whole machine medium environment.

2. The design method of the ultra-low field movable cranial magnetic resonance hybrid gradient coil system according to claim 1, characterized in that, The design method for the cylindrical coil includes: Step 11: Based on the Biot-Savart law and spherical harmonic series expansion, establish the magnetic field distribution expression of the cylindrical coil in the target field region for the target gradient direction of the cylindrical coil; Step 12: Based on the three-dielectric boundary model, the magnetic field distribution expression of the cylindrical coil in the target field region is corrected using the mirror current model; Step 13: Constrain and regularize the modified cylindrical coil target magnetic field using the target field method, and solve for the Fourier expansion coefficients of the cylindrical coil. ; Step 14: Calculate the Fourier expansion coefficients of the cylindrical coil. Substituting the expression for the current function of the cylindrical coil into the expression for the current function of the cylindrical coil, we obtain the current function distribution of the cylindrical coil. Step 15: Take contour lines for the flow function distribution of the cylindrical coil to obtain the discrete coil distribution of the cylindrical coil, and calculate the magnetic field nonlinearity error and the power loss of the cylindrical gradient coil in the target field region; if the magnetic field nonlinearity error or power loss does not meet the preset requirements, then execute step 16; otherwise, output the discrete coil distribution of the cylindrical coil and execute step 17. Step 16: Use a fast genetic algorithm to iteratively optimize the cylindrical coil design parameters. Repeat steps 11 to 15 for each candidate design parameter, and use magnetic field nonlinearity error, gradient coil power loss and inductance as optimization objectives to obtain the optimized discrete coil distribution of the cylindrical coil. Step 17: Prepare a cylindrical coil based on the optimized discrete coil distribution of the cylindrical coil.

3. The design method of the ultra-low field movable cranial magnetic resonance hybrid gradient coil system according to claim 2, characterized in that, The design method for the planar coil includes: Step 21: Based on the Biot-Savart law, establish the magnetic field distribution expression of the planar coil in the target field region for the target gradient direction of the planar coil; Step 22: Based on the three-dielectric boundary model, the magnetic field distribution expression of the planar coil in the target field region is corrected using the mirror current model; Step 23: Constrain and regularize the modified planar coil target field magnetic field using the target field method, and solve for the Fourier expansion coefficients of the planar coil. ; Step 24: Expand the Fourier coefficients of the planar coil Substituting the expression for the planar coil current function, we obtain the distribution of the planar coil current function; Step 25: Take contour lines for the flow function distribution of the planar coil to obtain the discrete coil distribution of the planar coil, and calculate the magnetic field nonlinearity error and the power loss of the planar gradient coil in the target field region; if the magnetic field nonlinearity error or power loss does not meet the preset requirements, then proceed to step 26; otherwise, output the discrete coil distribution of the planar coil and proceed to step 27. Step 26: Use a fast genetic algorithm to iteratively optimize the design parameters of the planar coil. Repeat steps 21 to 25 for each candidate design parameter, and use the magnetic field nonlinearity error, planar gradient coil power loss and inductance as optimization objectives to obtain the optimized discrete coil distribution of the planar coil. Step 27: Prepare a planar coil based on the optimized planar coil discrete coil distribution.

4. The design method of the ultra-low field movable cranial magnetic resonance hybrid gradient coil system according to claim 3, characterized in that, In step 11, the expression for the magnetic field distribution of the cylindrical coil within the target field region is established, specifically as follows: A first-order transverse gradient magnetic field is constructed using a cylindrical coil as a carrier. This first-order transverse gradient magnetic field corresponds to the first-order non-axisymmetric mode in a spherical harmonic expansion. For a radius of... , length is The cylindrical coil will have an axial current density and circumferential current density Represented in Fourier trigonometric series form: （1） In the formula, and represent the order and degree in the spherical harmonic series expansion and the current density function expansion, respectively; Q Let be the order of the Fourier expansion of the current density function; This represents the Fourier expansion coefficient vector in the trigonometric series expansion of the current density of a cylindrical coil. For the first Fourier coefficients of the current density; The axial coordinates of the source point on the cylindrical surface. The upper limit of the order of a spherical harmonic expansion is determined by the truncation order of the expansion. Sure; ; ; ; , It is used to select sine or cosine basis functions in axial expansion based on modal parity symmetry; The main magnetic field of the magnet is along the y-axis. According to Biot-Savart's law, the y-direction component of the magnetic field generated by the source point is... Represented as: In the formula, Indicates the polar angle and azimuth angle of the field point. Indicates the polar angle and azimuth angle of the source point. for The first-order associated Legendre function is used to characterize the angular distribution in spherical harmonic expansions. , , and The intermediate coefficients are those whose subscripts correspond to the order of the spherical harmonic expansion. Consistent, the specific expression is as follows: Further, the y-direction component of the magnetic field generated by the cylindrical coil Except Other integral terms are denoted as : The above formula can be simplified to: .

5. The design method of the ultra-low field movable cranial magnetic resonance hybrid gradient coil system according to claim 4, characterized in that, In step 21, the expression for the magnetic field distribution of the planar coil in the target field region is established based on Biot-Savart's law, specifically as follows: Trigonometric functions are introduced as basis functions for the current density of the planar coil during Fourier preprocessing; for the planar coil used to generate the gradient magnetic field in the x-direction, its current density is expanded into the following components: In the formula, Indicates the current density in the radial direction The components on; Indicates current density in azimuth angle Components in direction; This is the distance from a point on the coil in the plane to the central axis; The azimuth angle of a point on the coil; Indicates the center position of the current density expansion; Let be the order of the Fourier expansion of the current density function. The order number in the Fourier expansion; Here are the Fourier expansion coefficients of the planar coil. For the first The expansion coefficients corresponding to the four-order Fourier basis functions; The spatial frequency scaling factor determines the rate of radial oscillation of the cosine function; Based on the Biot-Savart law, for a gradient coil consisting of two sets of planar coils arranged with the y-direction as the normal direction, the field point... place This represents the linear superposition of the contributions of each basis function: In the formula, and The two distance quantities corresponding to the upper and lower planar coils represent the distances from the field point to the surface elements of the upper and lower planar coils, respectively; kernel function symbol. Indicates the first The Fourier basis function contributes to the magnetic field at the target field point; Further, the y-direction component of the magnetic field generated by the planar coil Except Other integral terms are denoted as : Then the y-direction component of the magnetic field generated by the planar coil will be... Simplified to: .

6. The design method of the ultra-low field movable cranial magnetic resonance hybrid gradient coil system according to claim 3, characterized in that, In step 12, based on the established three-dielectric boundary model, the magnetic field distribution expression of the cylindrical coil in the target field region is corrected using the image current model; and in step 22, based on the established three-dielectric boundary model, the magnetic field distribution expression of the planar coil in the target field region is corrected using the image current model; specifically: A three-medium boundary model is established, consisting of air, an anti-vortex plate, and an electrode. The interface between air and the anti-vortex plate, and the interface between the anti-vortex plate and the electrode, are respectively considered as mirror boundaries. The reflection coefficient at each interface is expressed as: Based on image theory and dielectric boundary conditions, the boundary influence of the original coil current in the air domain is equivalent to multiple order image currents; the amplitude expressions for each order of equivalent image current are as follows: In the formula, The primary coil current, This refers to the mirror current that is not reflected between the anti-eddy current plate and the electrode. To generate between the anti-eddy current plate and the electrode plate The mirror current formed by the secondary reflection; The cylindrical coil and planar coil use the same three-dielectric boundary model and image current model in steps 12 and 22, and these models are used in subsequent formulas. Unified representation and ,use Unified representation and ,use Unified representation and ; Corrected total magnetic field Magnetic field generated by the primary coil current and the magnetic fields generated by the mirror currents of each order. The result of superposition is represented as: For the first The magnetic field generated by the mirror current; In the corrected magnetic field calculation expression, except for Other integral terms; substitute into the magnetic field calculation formula for cylindrical or planar coils according to the spatial location of each mirror current. and The integral terms corresponding to the image currents of each order are obtained. and And correct the integral term in the target field method based on the magnitude of the mirror current: In the formula, Unified representation and ; For the first The ratio of the maximum value of the current function corresponding to the first-order mirror current to the maximum value of the current function corresponding to the primary coil current is used to characterize the intensity contribution of different-order mirror currents relative to the primary coil current. In the corrected magnetic field calculation expression, except for Other integral terms.

7. The design method of the ultra-low field movable cranial magnetic resonance hybrid gradient coil system according to claim 6, characterized in that, In steps 13 and 23, the following methods are adopted: Unified representation and The specific process includes: Uniform sampling of the target area Based on magnetic field sampling information of the target field region at discrete points, the corrected response terms of the cylindrical coil and the planar coil are synthesized into a unified response matrix, and the hybrid gradient coil design problem is then transformed into a linear equation. : In the formula, For the first The target magnetic field at each sampling point ; Each row in the matrix Each element is In the subscript That is to represent the first One sampling point, Represents the order number in the Fourier expansion; In the coil design, both power loss constraints and field linearity constraints are introduced, and the solution is obtained by combining the Tikhonov regularization strategy. The objective function is: In the formula, , , The power loss of the gradient coils are respectively ,inductance Stream function curvature Regularization weights, and respectively through The calculated magnetic field distribution and the target magnetic field distribution, and Represents passing The calculated first The magnetic field distribution at discrete points and the target magnetic field distribution; The above power loss ,inductance Stream function curvature The constraints are further projected into the modal coefficient space and transformed into Fourier coefficients in the trigonometric series expansion of the current density. The quadratic form: In the formula, , and These are the characteristic matrices corresponding to power loss, inductance, and flow function curvature, respectively. Based on the first-order optimality condition, the objective function is related to the Fourier coefficients. Differentiate and set the gradient to zero to determine the Fourier coefficients in the trigonometric series expansion of the current density. The optimal solution: 。 8. The design method of the ultra-low field movable cranial magnetic resonance hybrid gradient coil system according to claim 5, characterized in that, In step 14, the Fourier expansion coefficients of the cylindrical coil are... Substituting the expression for the current function of the cylindrical coil into the equation, we obtain the distribution of the current function of the cylindrical coil, as follows: In the formula, The current function distribution is for a cylindrical coil; In step 24, the Fourier expansion coefficients of the planar coil are... Substituting the expression for the flow function of a planar coil, we obtain the flow function distribution of a cylindrical coil, specifically: In the formula, The distribution of the flow function for the planar coil.

9. The design method of the ultra-low field movable cranial magnetic resonance hybrid gradient coil system according to claim 8, characterized in that, use Unified representation of the flow function of a cylindrical coil and planar coil current function Steps 15 and 25 specifically include: Based on the difference between the equipotential lines of the flow function corresponding to the current flux in the coil wire, the coil current... Represented as: In the formula, This refers to the number of coil turns. Based on the distribution results of the discrete coils, the nonlinear error of the magnetic field in the target region, the power loss of the gradient coil, and the inductance of the gradient coil are further calculated as follows: In the formula, This indicates the linearity error of the magnetic field within the target field region. and They represent the target field regions respectively. Magnetic field calculation results and target magnetic field value for each sampling point; This represents the total number of sampling points in the target field area. and These represent the power loss and inductance value of the cylindrical gradient coil, respectively. and These represent the power loss and inductance of the planar gradient coil, respectively. Indicates the resistivity of the coil material. Indicates the diameter of the coil conductor; and These represent the expressions for the circumferential and axial current densities of the cylindrical coil, respectively. and These represent the expressions for the current density in the azimuth and radial directions of the planar coil, respectively. and These represent the vectors pointing from the center point to the field point and the source point, respectively. Steps 16 and 26 specifically include: The various design parameters of the coil are selected as decision variables in the optimization process. For a planar coil, the decision variable vector is: For cylindrical coils, the design vector expansion is as follows: ;in, For the order of the Fourier expansion, Let the order of the image current expansion be denoted as . The number of coil turns. These are the regularization parameters for coil power loss, coil inductance, and flow function curvature, respectively. The order of the spherical harmonic maximum expansion is given; an initial population is randomly generated based on the value range of each decision variable; for each candidate individual in the initial population, the corresponding magnetic field nonlinearity error in the target field region is calculated. Gradient coil power loss and inductor Subsequently, based on the nonlinear error reference value, power loss reference value, and inductance reference value, the nonlinear error, power loss, and inductance are dimensionless, and combined with pre-set weighting coefficients, the comprehensive fitness function is constructed as follows: In the formula, using A unified representation of power loss in cylindrical coils and planar coil power loss ,use A unified representation of cylindrical coil inductance and planar coil inductance ; This is the overall fitness function, and the fitness result of each individual is calculated based on this function; , and These are the nonlinear error reference value, the coil power loss threshold, and the inductance threshold, respectively. , and These are the weighting coefficients for linearity error, power loss, and inductance, respectively. , and Determined by system design specifications and engineering constraints; , and It depends on the focus of the optimization objective.

10. An ultra-low field movable cranial magnetic resonance hybrid gradient coil system, characterized in that, It is obtained by using the design method described in any one of claims 1-9.

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