Simulation Study of Transient Eddy Current Effect in 3T Animal Gradient Coil

In the process of high-field magnetic resonance imaging (MRI), the eddy current modeling and simulation of the cold screen and metal parts are not highly accurate. To solve this problem, this paper uses the target field method to design a 3T animal gradient coil model. To get more realistic simulation results, a complex model including a gradient system, cryogenic vessel, cold screen, epoxy resin, and temperature holes was constructed. The model was refined using the finite element method, and the effects of eddy currents induced by the gradient coil on the cylindrical cryogenic container and cold screen in the superconducting magnetic resonance instrument under the state of multi-physical field coupling were simulated. The model effectively reflects the transient eddy current distributions on the cold screen and cryogenic vessel in the MRI system and their time-dependent temperature rise effects.


Introduction
MRI is in the presence of a gradient magnetic field and a main magnetic field, which ultimately generates a cross-sectional image of human tissues by interfering with the spin of protons and extracting the signal of their recovery process.MRI demonstrates better performance in multi-angle imaging and cancer detection compared to CT and is also safer.During the imaging process, the gradient magnetic field serves to change the imaging area.The gradient magnetic field is generated by a gradient coil, which is divided into three groups: X, Y, and Z.By controlling the size and direction of the current, the encoded information of different spatial sections can be obtained.The switching frequency and field strength of the gradient magnetic field determines the quality of imaging, but it also generates eddy current effects that adversely affect the entire magnetic resonance device.
The gradient coils are surrounded by various kinds of conductor structures, i.e., superconducting magnet skeletons, room-temperature uniform field coils, vacuum containers, liquid helium containers, and radiation cold screens.During an MRI scan, a high-frequency, high-voltage current is passed through the gradient system to accomplish spatial localization of the NMR signal, and by Faraday's Law of Electromagnetic Induction [1], the varying magnetic field produces induced currents, or eddy currents, in the surrounding conductors.These eddy currents usually have a more complex spatiotemporal distribution [2].From the corrugator's law, the induced eddy currents oppositely change with time to the currents in a gradient coil.The cold screen is a good conductor, and the low temperature is affected by the gradient coil significantly; the eddy current effect produced by its surface is much larger than other metal skeleton parts, and the larger eddy current is bound to produce a secondary magnetic field and a temperature rise, interfere with the accuracy of the imaging, accelerate the evaporation of liquid helium, and even cause the coil to lose superconducting properties.Compensation of eddy currents in practical applications requires the construction of complex models.Due to the high manufacturing cost of MRI instruments, controlled experiments with multiple schemes are uneconomical.Meanwhile, due to the complexity of the whole process, when the new optimization scheme contains local corrections to the structure or adjustments to the position of some of the coils, it is not possible to split the modifications again in a short period.This prompts higher demand for large models with high simulation accuracy and coupled multi-physics fields.Referencing a large body of related literature, I have tried to propose and construct a higher precision, coupled multi-physics field model of eddy current secondary effects in gradient system, which is different from the traditional eddy current simulation using pure numerical computation or single-variable simulation, and constructs a three-dimensional complex MRI instrument model containing gradient coils, cryogenic vessel, cold screen, metal skeleton, temperature holes, and other multi-structures.

Design of gradient coils for 3T animals
In the gradient coil design scheme, there are two classical design methods, based on the discrete current design method and based on the distributed current density design method [3].The discrete current-based design approach is essentially a positive design scheme.The design process of this method is simple.However, it incorporates insufficient parameters for consideration, does not provide sufficient constraints on the coil, and the designed coil is often far from ideal.The design methods based on distributed current density are inverse methods, which do not pre-constrain the coil structure, among which the target field method is a more mature method [4][5][6], which incorporates some steps of the analytical method on top of the inverse design.

Design of gradient coil by target field method
The target field method is a method of inversion solution using the analytical method [7].It is assumed that the target coils are distributed on a hollow cylinder with a length of 0.6 m and a diameter of 0.25 m.
Assuming the distribution of its gradient magnetic field, and using the clear correspondence between the magnetic induction intensity and the current density, the current density distribution of the target area is obtained using the multi-level Fourier transform.We discretize the cylinder current density, i.e. separate the wires to simulate.Usually, the more turns of the wire, the better the linearization.Here we set the number of turns of the main coil of the transverse coil to 22, and that of the turns of the shielding coil to 12.The number of turns of the longitudinal coil main coil wire is 22, and the number of turns of the shielding coil is 14.The current density is approximated by a coil with finite root windings.The Fourier transform of J is expressed as the following equations: where z j and j  are the transverse and longitudinal current densities in the spatial coordinate system, which can be found from the current continuity equation, combined with 0 J    and Equations ( 1) and ( 2), as shown below.
A Fourier transform of the above equation is performed to solve for its relationship in the frequency domain, as expressed below: where a represents the radius of the column surface.The relation between the magnetic fields is obtained using Bessel's formula for the above equation: where and are the first and second types of deformed Bessel functions, respectively.
The target magnetic field is set as , and when c r  , the magnetic field the frequency domain is calculated using Equation ( 7) below.
The current density in the frequency domain can be calculated from the above equation.Also, it should be ensured that the gradient system does not produce significant self-interference under high field strength and strong current excitation conditions.We further optimize the spatial current density according to the inductance equation and constraint equation.
Finally, the optimized spatial current density is obtained, and the final gradient coils are then obtained by separating the wires via discretization [8][9].However, the gradient coils obtained by this method have unsmooth and cross boundaries, which need to be further optimized before they can be used for 3D simulation.

High-precision MRI system modeling
The overall system modeling consists of two parts: 3D modeling and boundary condition constraints.The 3D modeling includes gradient coil modeling and overall MRI system modeling except gradient coil modeling.Boundary condition constraints include mesh delineation and physical field boundary constraints.4

Gradient coil modeling
For the coil cross-linking problem, the quadrant division method is mainly used to categorize the data in different quadrants, using Python software to constrain the classification of coordinate points, such as the Y coil group, whose transverse coordinates will be categorized into positive and negative parts; then the longitudinal coordinates are classified, which will result in the data of the internal main coil and the external shielding coil, and the two coils are then categorized to further obtain an independent coil for each group.The problems of local sharpening and jaggedness are handled by removing the non-compliant points by Python programming.The improved gradient coils satisfy the needs of 3D modeling in terms of shape and smoothness.It should be noted that the distance between the X and Y coils in the main coil and the shielding coil is around 0.005 m.The gradient coils obtained by the sharpening process and spatial coordinate separation meet the requirements of model parameters in the physical field.However, the distance between each group of coils is only 0.005 m.Attention should be drawn to the distance between the coils during the 3D modeling process to avoid entanglement interference [10].

Overall MRI system modeling
To simulate the eddy current effect in reality, the model structure should be set up to meet realistic conditions, and the liquid helium cold screen, temperature holes, cryogenic container, and epoxy resin domain are outsourced in the model of the MRI system in addition to the construction of gradient coils.The appearance of the overall model is a hollow cylinder with a length of 0.6 m, an outer radius of 0.17 m, and an inner radius of 0.84 m.During the operation of MRI equipment, the liquid helium cold screen is most affected by eddy currents, with a surface temperature of about 50 k and a thickness of about 0.003 m, which isolates the internal and external temperatures and reduces the effect of eddy currents in the gradient system on other metal parts.

SECMP-2023 Journal of Physics: Conference Series 2624 (2023) 012017
The outer thermostatic container and the cold screen are made of aluminum, the three-dimensional structure of the transverse gradient coil is rectangular, and the coil is covered with epoxy resin.The constant temperature hole is set outside, and the hexahedral air domain is set to wrap up the overall model.

Model Mesh Segmentation
The setting of boundary conditions is the core of FEA and determines whether the model converges and achieves the target accuracy [11][12].
For the cryogenic vessel and cold screen, the sweeping form of division is used to increase the accuracy of the surface calculation and reduce the amount of calculation of the invisible part, and the surface of each cell is set to be a small rectangular surface of 0.02 m.The surface of each cell is set to be a small rectangular surface.The gradient coil is not regular in shape, so it should be further refined, using free tetrahedral division and setting its maximum cell size to 0.02 m, and the minimum cell size to 0.005 m; and the smallest cell is set to avoid overly dense networks in the surface location.The narrow region resolution is set to 2. This part of the setup should be adjusted with the difference in geometry, the larger the difference the larger the setup parameter should be, and should be taken to 1 under perfect conditions.

Physical field boundary condition setting
In the eddy current simulation, the three basic physical fields of the coupled electric field, magnetic field, and heat transfer are used, and the coil current is set to 1000 HZ , 50 A .The maximum number of iterative refinement mesh refinement is set to 15 times, and the error factor is taken to be 0.1.A transient solver is used in PARDISO, the maximum number of iterative refinement mesh refinement is set to 15 times, and the error factor is taken to be 0.1.The number of mesh refinement is set to 15 times, and the error factor is taken to be 0.1.and the error factor is taken as 0.1.The frequency domain-transient constraint equations are set according to Ampere's law and relative permeability.
In the B H   is the relative magnetic permeability of the material itself.The electrical conduction model is set to , where  is the conductivity of the material.The potential shift is set as , where r  is the relative dielectric constant of the material.A current of 100 A at a frequency of 1000 HZ is passed through the coil, and the number of iterations is set to 20 to simulate and test the local model.
When building the cold screen model, the number of units on the cold screen should be reduced as far as possible because the number of units already divided is too large and the liquid helium cold screen is relatively regular.Surface modeling and rectangular fac-division are also used.This can reduce the calculation amount of the invisible part and increase the calculation accuracy of the surface vortex effect.The constraint equation for heat transfer in solids is expressed as the following: where the q k T    discretization uses quadratic Lagrangian cells.Thermal insulation conforms to 0 n q    , ensuring that thermal changes come from eddy current effects generated by the coil on the cold screen.Below are the electromagnetic thermal coupling constraint equations.

 
The electric and magnetic fields should be added to the A-field norm fixing 0 A    due to the uncertainty of the solution, such that the electromagnetic field conforms to the Coulomb norm.

Convergence problem
The step size should be set as small as possible to ensure that the matrix operation within each time step is convergent, due to the large number of model cells, reaching more than 3 million grid cells.The amount of operation and the accuracy of the operation are measured, and the value of a 0.02-s time step should be taken, as well as the overall model simulation of ten seconds of transient eddy current distribution and temperature rise transformation.

Simulation of eddy current effects in cold screens
The simulation part of the study focuses on the eddy current effects generated by coils with two magnetic field directions on a cold screen, including the distribution of eddy currents and the temperature rise phenomenon over time.

Eddy currents generated by gradient coils on cold screens
The following two figures show the eddy current simulations for the transverse gradient coil and the longitudinal gradient coil.It can be seen that the eddy currents generated by the transverse gradient coil are more irregular, with a larger eddy current effect; and on the liquid helium cold screen, the current density image consistent with the general structure of the coil is presented, and the local highest current density reaches

The temperature of cryogenic vessels and cold screens as a function of time
The temperature rise of the cold screen and metal container induced by the gradient coils can adversely affect the MRI system and also varies greatly from one magnetic field direction to another.In the case of a refrigerant-free scenario, the transverse gradient coil in the cold screen within ten seconds produced a 10-40 k temperature rise, and the peak temperature rose to 90 k, by the isolation of the cold screen of its shell temperature rising parameter to be smaller than the cold screen temperature rising of 20-30 k.Unlike the transverse gradient coils, the Z-coil produces a very lowtemperature rise effect in the shell, with a modest rise in values within 1 k over ten seconds.

Conclusions
In this paper, a 3T animal gradient coil group is designed using the reverse design method, and a 3D loop model is constructed.Based on the simple loop model, a complex 3D transient eddy current model coupling multiple physical fields is then constructed.Next, this model is used to compare the vortex effects of different directions and obtain the time-varying vortex distribution and temperature rise image of different coil groups, thus providing technical support for the modeling analysis of highfield MRI imaging.

Figure 5 .Figure 6 .
Figure 5. Vortex distribution of X-gradient coil radiation on a cold screen Figure 6.Vortex distribution of Z-gradient coil radiation on a cold screen

.
The longitudinal gradient coil produces greater current density on both sides of the winding, but with a peak value of only 7 2 9 10 / A m  , indicating that its eddy currents are much smaller than those of the longitudinal gradient coil.

Figure 7 .
Figure 7. Time-dependent temperature rise on the cold screen and housing

Table 1 .
Material parameter setting