Electromagnet design for ultra-low-field MRI

Abstract

Comparing with superconducting and permanent magnet magnetic resonance imaging (MRI) system, ultra-low- field MRI (uMRI) system has a much lower weight, which can be used in some special scenes, such as real-time image monitoring on bedside or in the ambulance for human brain disease (stroke). In order to make the uMRI system more compact, lighter, and smaller, in this work, we proposed a hybrid method for designing uMRI electromagnet. The method consists of integer linear programming (ILP) and nonlinear optimization, which were bridged by empirical principles. ILP was used to determine the coil distribution of uMRI electromagnet. Considering that the coil distribution, acquired with ILP, is not conducive for magnet processing, two empirical principles were applied to adjust the coil distribution with decreasing electromagnet performance as little as possible. At last, nonlinear optimization was exploited to eliminate the negative influence on magnetic field distribution which is introduced by coil adjustment and make electromagnet satisfy the design requirement of magnetic field homogeneity. This hybrid method combines mathematical models and empirical principles, which provides a way to design electromagnet efficiently. A small electromagnet with homogeneity of 10 ppm in a spherical region of interest (ROI) with diameter of 80 mm is designed and built in this paper.

Keywords

Ultra-low-field MRI electromagnet design integer linear programming nonlinear optimization

1. Introduction

Traditional MRI system includes superconducting system and permanent magnet system. Superconducting system works under superconducting state with complicate refrigeration system, permanent magnet is too heavy (usually tens of tons), that make traditional systems costly to maintaining and hard to move, thus limiting the application of MRI [1].

UMRI system is small in size, light in weight, which can extend the application of MRI, and make the portable MRI system possible [1–3]. However, because of low magnetic field, the imaging time of uMRI system is too long, and the images, obtained by uMRI system, are not as clear as those acquired by traditional high field MRI systems; hence the development of uMRI received little attention for a long time. In recent years, with the development of low noise hardware, computational MRI pulse sequences [4], and deep learning-based approaches to image formation [5], the performance of ultra-low-field MRI system has been improved. In 2015, Sarracanie et al. reported a 6.5 mT MRI scanner used for rapid brain imaging [6]; in 2019, Mäkinen et al. published work using a hybrid MEG-ULF MRI system [7].

In the presently described work, we focus on the design of an electromagnet for uMRI system with the objection of compact and lightweight. Electromagnet design plays a key role in the research of MRI system, with the development of high field MRI, superconducting electromagnet has got a lot of attention for a long time, and design methods for superconducting magnet have been well developed. The representative methods include Monte Carlo [8], simulated annealing [9], genetic algorithm [10], inverse method [11], linear programming and hybrid method [12,13]. Basically, Monte Carlo algorithm, simulated annealing algorithm, and genetic algorithm need a lot of computing resources to deal with large-scale variable optimization problems, which could be constrained by the computing hardware; inverse method was always solved with establishing and solve complicate numerical electromagnetic calculation model, and that could result in low calculating speed; hybrid method is the combination of several methods mentioned above, and usually proposed to synthesize the advantages of various methods to simplify electromagnet design.

The method, proposed in this work, is a hybrid method, consists of ILP and nonlinear optimization which were bridged by empirical principles. We simplified the electromagnet design by converting it to two optimization problems, which could be solved in a reasonable time without causing hardware pressure. Moreover, we also take electromagnet processing into account in this method, coil adjustment with two empirical principles, was applied to make sure the electromagnet could be conducive for processing with guaranteeing precision.

With proper method, we should also make the design goal clear. With considering the comfort experience of patient, the design of superconducting electromagnet developed toward to shorter cavity and larger aperture, which would decrease the efficiency of electromagnet (efficiency was defined by ratio of the weight of electromagnet and magnetic flux density in ROI, energized by unit current) [14]. For the high field electromagnet, it is acceptable to sacrifice the efficiency, however, for a low field electromagnet, it is necessary to improve the efficiency to make sure the weight and volume is acceptable in a portable system. Therefore, the design goal of uMRI electromagnet should be generating a required magnetic field with less coils and a compact structure.

Considering that the efficiency of cylindrical electromagnet is better, comparing to biplanar electromagnet and saddle electromagnet, in this work, a cylindrical electromagnet was designed and built with the hybrid method.

2. Method

The hybrid method, used for electromagnet design, consists of ILP and nonlinear optimization. It is implemented in 3 steps. (1) establishing integer linear programming model, and obtaining the preliminary coil distribution of the electromagnet; (2) adjusting the shape of coil bundles according to the principle of equal area and maximum overlapped area, that would make electromagnet easy to be manufactured. (3) adjusting the location and radius of every coil bundle, which was implemented to eliminate the negative influence, introduced in second step, on magnetic field homogeneity in ROI.

2.1. Integer linear programming (ILP)

Linear programming is a technique for the optimization of a linear objective function, subject to linear equality and linear inequality constraints, so the linear programming problem is fast and efficient to solve. There are many classic algorithms to solve this problem. In MATLAB (The MathWorks, Inc.), this problem could be solved by linprog function. Linear programming for electromagnet design was first proposed in [15], in this work, we used ILP to further simplify electromagnetic design, where the results, obtained by solving linear programming, are integer values.

In order to convert this design problem to an ILP problem, the feasible coil space is densely partitioned by an array of candidate coils. Figure 1 shows the cross section of an electromagnet and the ROI. Small rectangles in Fig. 1 denotes feasible coils, and the circular region denotes ROI.

Fig. 1.

The feasible coils space.

The magnetic field in ROI could be regarded as the result of the superposition of the magnetic field generated by candidate coils, which are energized. The purpose of ILP is to find an optimal combination of candidate coils in Fig. 1; with this candidate coil energized, the superimposed magnetic field can meet the design requirements, such as magnetic flux density, magnetic field homogeneity and the size of ROI.

To build a mathematical model of ILP, the magnetic field in ROI, generated by every candidate coil with unit current, need to be calculated. According to Helmholtz theorem, the magnetic field distribution of electromagnet in ROI is determined by that distributing on the boundary of ROI [15], and because of the symmetry structure of electromagnet, the magnetic field in ROI can be evaluated by that distribution on the arc in first quadrant in Fig. 1. Therefore, we calculate the magnetic field at target points (the target points are indicated with black cross) on the arc; where, the target points are uniformly chosen on the arc.

The feasible space for coil is partitioned into identical rectangles, whose index is n, and the index of target points in ROI is m; we calculated the magnetic flux density at each target point generated by candidate coil at the position of each partitioned rectangle, excited by unit current; the calculation results make up the sensitivity matrix A, where A _nm is the magnetic flux density at the mth target point generated by unit current in nth candidate coil. In this ILP problem, the variable to be optimized is the currents i _n in each feasible coil. The magnetic flux density at target points is calculated as Eq. (1), where I is the column vector of current value in each coil; and b is the column vector which denotes magnetic flux density at each target point. $\begin{eqnarray}\displaystyle A\cdot I=b. & & \displaystyle\end{eqnarray}$ (1)

The objective function, applied in this work, is power dissipation. With the objection of low power dissipation, we could obtain an electromagnet with low resistance, which means the electromagnet could be wound with relative short wire; thus, the structure of electromagnet can be compact. The power dissipation of the magnet could be regarded as the sum of power dissipation of each coil loop and was calculated as Eq. ((2)). $\begin{eqnarray}\displaystyle P=\mathop{\sum }_{n=1}^{N}|i_{n}|^{2}\left(\frac{2{\pi}r_{n}}{{\sigma}_{c}S_{n}}\right) & & \displaystyle\end{eqnarray}$ (2) Where, σ_c is the electrical conductivity of wire, winding electromagnet, S _n and r _n are the cross-section area and radius of the nth coil, respectively. Practically, every coil loop of electromagnet was in series, and has the same current, so the ratio |i _n|∕S _n is a constant and described by J, then we can rewrite the power dissipation equation as: $\begin{eqnarray}\displaystyle P=\frac{2{\pi}J}{{\sigma}_{c}}\mathop{\sum }_{n=1}^{N}|i_{n}|r_{n}. & & \displaystyle\end{eqnarray}$ (3)

The constraints of electromagnet design problem include location constraint of coils and magnetic field constraint in ROI. The location constraint is determined by the required size of the electromagnet, and considered when the sensitivity matrix (A) is calculated. The magnetic field constraint consists of size requirement of ROI, flux density and homogeneity requirement in ROI. The size requirement of ROI was also considered in sensitive matrix calculation, and the magnetic flux density and homogeneity requirements were constrained by (4): $\begin{eqnarray}\displaystyle \frac{|b-B_{0}|}{B_{0}}\leq {\varepsilon} & & \displaystyle\end{eqnarray}$ (4) where, b, calculated in Eq. (1), is the magnetic flux density at target points, generated by current in energized coils; B ₀ is the required magnetic field in design, which is a constant; ϵ is the maximum relative error between magnetic field generated by designed electromagnet and B ₀. ϵ was used to evaluate the homogeneity of magnetic field; the smaller the ϵ, the higher the homogeneity.

Above all, the electromagnetic design problem could be converted to an optimization problem, shown as ((5)). $\begin{eqnarray}\displaystyle \begin{array}{@{}l@{}}\text{Minimize:}∼\displaystyle \sum r_{n}|i_{n}|\\[5.0pt] \text{Subject to:}∼B_{0}(1-{\varepsilon})\leq A\cdot I\leq B_{0}(1+{\varepsilon}).\end{array} & & \displaystyle\end{eqnarray}$ (5) Considering that the value of i _n could only be 0 or the maximum current energizing electromagnet. Therefore, this problem was transformed to an ILP, shown as ((6)), where i _max is the maximum current, energizing electromagnet. $\begin{eqnarray} \displaystyle \begin{array}{@{} ll@{}}\text{Minimize:} & r_{n}\cdot i_{n}\\ \text{Subject to:} & B_{0}(1-{\varepsilon})\leq A\cdot i_{n}\leq B_{0}(1+{\varepsilon})\\ & i_{n}\leq i_{\max } \quad i_{n}\in \{0,i_{\max }\}. \end{array} & & \displaystyle \end{eqnarray}$ (6)

The optimization problem above is a L1-norm problem. A critical merit of L1-norm problems is the sparseness of its solution, and that result in the sparseness of the coil distribution. Figure 2 shows an example of integer linear programming result; this coil distribution area corresponds to the first quadrant in Fig. 1. The purple district represents energized coils. These coils gather in four regions and form four bundles, and the superimposed magnetic field generated by these four bundles could meet the design requirements.

Fig. 2.

Result of linear programming.

2.2. Coil bundle adjustment

We could obtain an optimal-structure electromagnet efficiently, using ILP. However, the optimal-structure electromagnet is not conducive for manufacturing; as is shown in Fig. 2, the cross section of each coil bundle is irregular. In order to reduce the difficulty of manufacturing, we adjust the irregular shape of coil bundle cross section to rectangle. Inevitably, the adjustment would change the optimal distribution of the energized coils, which would deteriorate the performance of electromagnet obtained with ILP, and make electromagnet performance fall below the design requirement.

In order to minimize the negative influence of coil bundles adjustment, the principle of equal area and the principle of maximum overlapped area are proposed based on the experience of simulation and calculation. The principle of equal area is used to ensure that the cross-section area of the coil bundle is unchanged when it is adjusted from irregular to rectangle. As is shown in Fig. 3, the width and length of rectangle are adjusted to match the irregular cross section of coil bundle, while the area of the rectangle was kept as a constant. With the principle of equal area, the total number of energized coils of each bundle is unchanged in adjustment. With the principle of maximum overlapped area, we adjust the position of rectangle while adjusting the shape of rectangle; when the overlapped area between irregular and rectangle cross section reach the maximum, the bundle adjustment ends.

Fig. 3.

Principle of equal area.

Fig. 4.

Principle of maximum overlapped area.

These two principles are used to reduce the distribution difference of the energized coils before and after adjustment, thus reducing the change of magnetic field distribution.

2.3. Nonlinear optimization

After coil bundle adjustment, the irregular bundles are adjusted to rectangle bundles, shown as Fig. 5. Although the negative influence on magnetic field distribution was considered in coil bundle adjustment, the deviation of magnetic field distribution cannot be ignored. To improve the homogeneity of magnetic field generated by the electromagnet, and make it reach the design requirement again, we further optimized the electromagnet structure by adjusting the location and radius of each bundle.

Fig. 5.

This is the adjusted coil bundles, which needs further optimization.

As is shown in Fig. 5; z = z _left and z = z _right are the boundaries of a coil bundle along z axis; r = r _in and r = r _out are the boundaries along its radial direction. Considering the shape of each coil bundle has been determined after coil bundle adjustment in 2.2, we chose z _left and r _in as parameters to be optimized, and Z _right and r _out were determined according to the shape of each bundle.

Fig. 6.

Rectangle coil bundle could be regarded as a conductor.

This parameter optimization is based on the calculation of magnetic field generated by energized coil bundle. The analytic expression to calculate magnetic field, generated by an energized coil bundle, deduced based on Biot–Savart law, is shown as Eq. (7), all the variables in Eq. (7) were labeled in Fig. 6. $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle A_{{\theta}}(r,z)=\frac{{\mu}_{0}J}{2{\pi}}\int _{z^{\prime }=0}^{2a}\int _{r^{\prime }=R_{0}-b}^{R_{0}+b}M(r^{\prime },z^{\prime };r,z)r^{\prime }dr^{\prime }dz^{\prime }\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle M(r^{\prime },z^{\prime };r,z)=\int _{0}^{{\pi}}\frac{\cos {\beta}d{\beta}}{\sqrt{(r)^{2}+r^{2}-2rr^{\prime }\cos {\beta}+(z-z^{\prime })^{2}}}\end{eqnarray}$ (7) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle B={\nabla}\times A_{{\theta}}.\nonumber\end{eqnarray}$ Based on the analytic expression, we built a nonlinear optimization mathematical model, shown as (8). Where, r _max and z _max are the boundaries of feasible coil district of electromagnet, N _pairs is the number of coil bundle pairs; B _ROI is the column vector describing magnetic flux density, generated by optimal electromagnet, at target points; max(B _ROI) and min(B _ROI) denote the maximum and minimum value of magnetic flux density, respectively; and ϵ₀ is the relative error between maximum and minimum magnetic flux density in ROI, used to evaluate magnetic field homogeneity. In this optimization problem, the objective function is total coil bundles volume, with this objection, we are supposed to obtained an electromagnet with a compact structure. The constraints include homogeneity of magnetic field at target points and the size of electromagnet, described in (8). $\begin{eqnarray}\displaystyle \begin{array}{@{}ll@{}}\text{Minimize:} & {\pi}\cdot \displaystyle \mathop{\sum }_{i=1}^{N_{pairs}}[r_{in}^{2}-r_{out}^{2}]\times [z_{right}-z_{left}]\\ \text{Subject to:} & [\max (B_{ROI})-\min (B_{ROI})]/B_{0}\leq {\varepsilon}_{0}\\ & r_{in}(i)\leq r_{out}(i);z_{left}\leq z_{right}\quad i=1,...,N_{pairs}\\ & r_{in}(1)\geq 0;z_{left}(1)\geq 0\\ & r_{out}(N_{pairs})\leq r_{\max };z_{right}(N_{pairs})\leq z_{\max }.\end{array} & & \displaystyle\end{eqnarray}$ (8)

In this paper, the nonlinear optimization problem was solved by nonlinear programming. It is a classic method depends on the initial value. With proper initial value, the optimal result could be found efficiently, otherwise, the calculation may not converge. For nonlinear optimization of electromagnet in this paper, the initial value was provided after coil bundle adjustment, and that make the optimization problem solved efficiently. The linear programming was implemented by using fmincon function in MATLAB, and we could get a result in 10 minutes.

3. Result

Electromagnet used for human brain imaging is relatively big and expensive to be manufactured, in order to verify this method, we designed and made a scaled down prototype of electromagnet, the length is 350 mm; the inner diameter is 200 mm; and the ROI is a sphere with diameter of 80 mm. The magnetic field homogeneity is under 10 ppm; the maximum current, energizing electromagnet, is 10 A; and the maximum magnetic flux density in ROI is around 30 mT.

With the hybrid method in this paper, we obtained an electromagnet structure (structure parameters were presented in Table 1). We also calculated the magnetic field distribution of this electromagnet with finite elements method (FEM) simulation, which is implemented in Ansys Maxwell (Ansys Inc.); the simulation model and magnetic field distribution are shown as Fig. 7. It could be found that the magnetic field distribution in the center of the electromagnet is very homogeneous. Figure 8 shows the homogeneity distribution in a square with side length of 100 mm, which is in the center of the cross section of the electromagnet.

Table 1
Design result of electromagnet

r _in r _out z _left Z _right

(mm) (mm) (mm) (mm)

Coil pair 1 100 103 2.1/−25.1 25.1/−2.1

Coil pair 2 100 104 31.1/−49.1 49.1/−31.1

Coil pair 3 100 105 63.3/−85.3 85.3/−63.3

Coil pair 4 100 107 119/−165 165/−119

	r _in	r _out	z _left	Z _right
Coil pair 1	100	103	2.1/−25.1	25.1/−2.1
Coil pair 2	100	104	31.1/−49.1	49.1/−31.1
Coil pair 3	100	105	63.3/−85.3	85.3/−63.3
Coil pair 4	100	107	119/−165	165/−119

Fig. 7.

FEM simulation model and result.

Fig. 8.

The homogeneity distribution of electromagnet.

With the result acquired by commercial software (Ansys Maxwell), the correctness and accuracy of the method in this paper is testified, in order to further study the negative influence on electromagnet performance, introduced by manufacturing, we also made an electromagnet prototype. The prototype consists of two parts, enameled copper wire with diameter of 1 mm and metal framework made by aluminium. We machined 4 pairs of grooves on cylindrical framework surface based on the calculation results in Table 1, and winded the copper wire on the cylindrical framework along the grooves, shown as Fig. 9.

Fig. 9.

The homemade prototype of electromagnet.

Fig. 10.

FEM simulation result, which shows magnitude distribution of magnetic field in ROI (Viewed in two maps).

We energized our electromagnet with a DC current source (Agilent, 6654A); with considering the power limitation of the this power source, the current output was set to 2.5 A. we use gaussmeter (F. W. Bell, 8030) to measure the magnetic field distribution in a square with side length of 100 mm, which is in the center of the cross section of electromagnet. Figure 11 shown the magnitude and contour maps. The FEM simulation was also implemented and the result was presented in Fig. 10. We defined the inhomogeneity of the magnetic field in ROI with δ_m which is calculated with Eq. (9). Where, B _max and B _min are the maximum and minimum magnetic flux density in the ROI, respectively. $\begin{eqnarray}\displaystyle \frac{(B_{\max }-B_{\min })}{(B_{\max }+B_{\min })/2}={\delta}_{m}. & & \displaystyle\end{eqnarray}$ (9) The inhomogeneity of simulated magnetic field is <10 ppm, while that of measured magnetic field is around 120 ppm. According to the result shown in Fig. 10 and Fig. 11, the contour map of simulated result and measurement is different, which indicates that negative influence, introduced by electromagnet manufacturing, can decrease the magnetic field homogeneity drastically.

Fig. 11.

Measurement result, which shows magnitude distribution of magnetic field in ROI (Viewed in two maps).

During the manufacturing process, the performance of the electromagnet could be influence by the machining precision of the framework and the wire-winding precision. The framework was processed by numerically-controlled machine tool, whose processing error is around 10 μm; its negative influence on electromagnet performance could be ignored. Whereas, for wire winding, all the wire loops are in series which is different from the concentric loops, simulated in Maxwell; moreover, the inhomogeneous insulation on the surface of enamelled wire makes it hard to estimate the diameter of the wire; thus, thus it is difficult to ensure the shape of each coil bundle is the same as that described in Table 1. We believe this is the main reason of performance decline of the electromagnet prototype.

4. Discussion

In this paper, we proposed a hybrid method for uMRI electromagnet design. The method consists of integer linear programming and nonlinear optimization, which were bridged by two empirical principles.

Liner programming is a classic method for MRI electromagnet design, which was used to get current density distribution, and the final structure of electromagnet was acquired by repeated iterations. In this paper, we acquired the energized coil distribution of electromagnet with using integer linear programming once. The integer linear programming could be completed in 30 minutes, while it would take 20 minutes for the calculation of sensitive matrix A.

The energized coil distribution, acquired in integer linear programming is complicate; that make it hard to manufacture the electromagnet with high precision. So, we adjust the coil distribution according to two experimental principles. The adjustment makes it easy to be manufactured, and provides initial value for following nonlinear optimization.

Nonlinear optimization was solved by nonlinear programming, with initial value, the nonlinear programming could be completed in 10 minutes. Considering the objective function in nonlinear optimization is total volume of coil bundles, which is determined by the radii of each coil bundle. In Table 1, we could find all the radii of bundles to be the smallest, and that make the electromagnet more compact.

The hybrid method, proposed in this paper, has been testified by FEM simulation. However, the performance of electromagnet depends upon both design and electromagnet processing technology. Usually, the inhomogeneity of electromagnet increases by 1 ∼ 2 order of magnitude because of processing error. The electromagnet, made in this work, is a scaled down prototype, which is more affected by processing error, because the relative error of a small electromagnet is larger than that of a bigger one. The magnetic field inhomogeneity of the scaled down electromagnet is 120 ppm, and we believe the whole-size electromagnet, which is around 3 times as big as the scaled down one, can has an inhomogeneity lower than 120 ppm with same processing technology, applied on the scaled down electromagnet.

Compare to high field MRI, uMRI has a lower requirement for magnetic field homogeneity. As is reported in [6], the brain images, obtained with magnetic-field inhomogeneity of 160 ppm and magnetic field of 6.5 mT, could show the brain structure. The work reported in [16] also present the phantom images obtained with magnetic-field inhomogeneity of 160 ppm and magnetic field of 23 mT. Thus, an electromagnet with inhomogeneity lower than 120 ppm and magnetic field of 10–20 mT can be acceptable for low field imaging. Moreover, in the future, we would study electromagnet shimming, which is supposed to further improve the magnetic field homogeneity of the electromagnet.

The whole-size electromagnet, we are going to build, is around 3 time as big as the scaled down prototype, which would be applied for human brain imaging. The length of whole-size electromagnet would less than 1 m, the diameter would be less than 0.35 m, the ROI would be a spherical volume with diameter larger than 0.2 m. The total weight would be controlled under 150 kg.

5. Conclusion

In this paper, we proposed a hybrid method for homogeneous magnet design, it is a combination of integer linear programming and nonlinear optimization which were bridged by two empirical principles. This method provides a simple way for designing electromagnet for MRI. The design and practice in this paper also provide an instance for uMRI magnet.

Footnotes

Acknowledgements

This work was supported by the National Natural Science Foundation of China (No. 51677008 and 51707028), and Fundamental Research Funds for the Central Universities (No. 106112017CDJQJ158834 and 2018CDJDDQ0017).

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Electromagnet design for ultra-low-field MRI

Abstract

Keywords

1. Introduction

2. Method

2.1. Integer linear programming (ILP)

Table 1 Design result of electromagnet r in r out z left Z right (mm) (mm) (mm) (mm) Coil pair 1 100 103 2.1/−25.1 25.1/−2.1 Coil pair 2 100 104 31.1/−49.1 49.1/−31.1 Coil pair 3 100 105 63.3/−85.3 85.3/−63.3 Coil pair 4 100 107 119/−165 165/−119

5. Conclusion

Footnotes

Acknowledgements

References

Table 1
Design result of electromagnet

r _in r _out z _left Z _right

(mm) (mm) (mm) (mm)

Coil pair 1 100 103 2.1/−25.1 25.1/−2.1

Coil pair 2 100 104 31.1/−49.1 49.1/−31.1

Coil pair 3 100 105 63.3/−85.3 85.3/−63.3

Coil pair 4 100 107 119/−165 165/−119