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Electromagnetic design of a 1.5T open MRI superconducting magnet
Physica C: Superconductivity and its applications 570 (2020) 1353602
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Physica C: Superconductivity and its applications journal homepage: www.elsevier.com/locate/physc
Electromagnetic design of a 1.5T open MRI superconducting magnet a
a,b,⁎
Yaohui Wang , Qiuliang Wang a b c
a
a
a
c
, Lei Wang , Hongyi Qu , Yang Liu , Feng Liu
T
Division of Superconducting Magnet Science and Technology, Institute of Electrical Engineering, Chinese Academy of Sciences, Beijing 100190, China School of Electronic, Electrical and Communication Engineering, University of Chinese Academy of Sciences, Beijing 100049, China School of Information Technology and Electrical Engineering, University of Queensland, St Lucia, Brisbane, QLD 4072, Australia
ARTICLE INFO
ABSTRACT
Keywords: MRI Magnet Superconducting wire Stress Open bore
The open magnetic resonance imaging (MRI) system is more patient-friendly and is easier for interventional therapy than the cylindrical MRIs owing to the large patient bore. In this work, a 1.5T open MRI magnet was designed. To evaluate the split-bore magnet design, several system parameters are presented that include magnetic field performance, stress, sensitivity, superconducting wire and operating current margin. The critical design parameters of the open magnet were carefully evaluated which include a maximum hoop stress 85.1 MPa, operating current 481A with margin 29.10%, inductance 103.60H and storage energy 11.99 MJ. The improved magnet optimization method has the capability to tackle complicated magnet design and the proposed 1.5 T open magnet design has a safe hoop stress and current margin. However, some potential risks such as large Lorentz force should be specially evaluated during the magnet support structure design and fabrication.
1. Introduction The main magnet is a key component in a magnetic resonance imaging (MRI) system, which is used to generate a highly-homogeneous static magnetic field in the region of interest (ROI) [1,2]. Generally, the homogeneity of the magnetic field profile and the stray field intensity should be constrained. For example, in the following 1.5 T open magnet design, the magnetic field inhomogeneity is 8 ppm over a 45 cm sphere, and the 5 Gauss line range is 5 m by 4 m. Owing to the strict engineering constraints of the current superconducting technology, it is rather challenging to design a high-performance, low-cost superconducting magnet for MRI applications, particularly for those with split MRI configurations. An open MRI superconducting magnet is a type of magnet that has a large gap between the opposite-placed magnet poles [3–6]. The splitmagnet configuration significantly augments the design difficulties in terms of magnetic field homogeneity, electromagnetic force balance, superconducting wire performance, and so on. Although it is challenging to develop an open MRI superconducting magnet, it does offer a number of benefits compared with conventional cylindrical systems. For example, it can substantially reduce the patients’ feeling of claustrophobia because of the larger patient bore and shorter system length [7]. It is also feasible for image extremity by moving the patient horizontally to expose the corresponding body part in the imaging area. In addition, an open MRI system provides a superb operating platform for
⁎
interventional therapy, which is convenient for the healthcare workers needing access to the patient [8]. Previously, building open superconducting MRI systems with lower field strengths has been attempted. Those commercially-available systems include the 0.7 T OpenSpeed from GE [9], the 1.0 T Philips Panorama [10] and the 1.2 T Oasis from Hitachi [11]. Because of the high fabrication cost and low performance, the 0.7 T and 1.0 T system have been discontinued and 1.2 T Hitachi Oasis is the only open superconducting scanner in commercial production at this time. Besides, several thousand non-superconducting open MRI are still in production or being installed. In recent years, to gain a higher SNR, higher field open MRI systems are increasingly demanded [12]. In this paper, a 1.5 T open MRI superconducting magnet was designed using an improved optimization scheme. The mechanical property, sensitivity characteristics and superconducting wire performances were further evaluated. 2. Method 2.1. Improved linear optimization computation Fig. 1 illustrates a diagram of a meshed magnet space, which is a sketch of the linear optimization principle of the initial estimation of a magnet design. Fig. 1(b) shows the current approximation of the existing methods [13–18], where the thick solenoid of each element is
Corresponding author. E-mail addresses: [email protected] (Y. Wang), [email protected] (Q. Wang).
https://doi.org/10.1016/j.physc.2020.1353602 Received 12 November 2019; Accepted 14 January 2020 Available online 15 January 2020 0921-4534/ © 2020 Elsevier B.V. All rights reserved.
Physica C: Superconductivity and its applications 570 (2020) 1353602
Y. Wang, et al.
inside the enclosure of the gradient assembly and magnet. Considering the coil bobbin, liquid helium vessel, cold shield and warm bore, the gap of the patient bore is estimated at about 0.45 m after the cryostat installation. 2.3. Magnetic field constraint The magnet design generates a 8 ppm (peak-peak value) uniform magnetic field profile in a 0.45 m sphere with the stray fields being actively-shielded to be less than 5 Gauss in a space 5 m × 4 m along horizontal (r) and vertical (z) directions, respectively. The target is to minimize the superconductor volume function within the magnetic field constraints. For simplicity, during the superconducting wire volume calculation, it is assumed that the wires have the identical current density Jmax, and thus, the real wire usage will be a ratio between the calculated and the designated current density. In the linear procedure, the optimization problem is expressed as: Fig. 1. Diagram of the linear optimization in the magnet design: (a) meshed magnet space, (b) filament-loop approximation for magnetic field calculation and, (c) magnetic field calculation using a thick solenoid.
m
min:
vi· i= 1
xi Jmax
(1)
subject to:
modeled as a current loop with no cross-sectional area, and the coordinate of the loop is at the rectangle center, namely ((z1+z2)/2, (r1+r2)/2) [19]. The linear optimization target is to find the unknown currents of the rectangular elements. This method uses a concentrated current to represent the distributed one, neglecting the dimensional content of the current element. Thus, it has to apply a dense mesh on some challenging design cases, such as the ultra-short magnet. In our proposed method, the filament-loop approximation is discarded, but a direct computation of a thick solenoid is adopted. At each element, the rectangular area has a uniform current density [20–22]. Fig. 1(c) depicts such a design process, where the coordinate of the solenoid is (z1, z2, r1, r2). To calculate the magnetic field produced by each thick solenoid, an angular integral was derived with the parameters z1, z2, r1, r2 and a current density J [23]. In a numerical computation, the angular integral was transformed as a Gaussian-quadrature rule with a summation of only some integration points multiplying the corresponding weights. In this work, a 10-order Gaussian integral was used. After completing the above improved linear optimization procedure, a follow-up global nonlinear optimization algorithm was applied to search for the final block dimensions.
AzX
TBz TBz
BX X
SB Jmax
(2)
By introducing a new variable yi with |xi| ≤ yi, the above nonlinear constraints can be transformed into a linear model [25]:
TBz (1 + Az 0 TBz (1 Az 0 S B Bz 0 z S Bz Bz 0 Br 0 [ S X Y ] Br Br 0 S Br E E 0 E E 0 E0 Jmax E0 Jmax
) )
(3)
where xi is the unknown current density at each rectangular element, m is the rectangular element quantity, Jmax is the maximum current density, Az and B are the primary magnetic field and shielding magnetic field matrices, respectively, TBz is the target field intensity, ɛ is the magnetic field deviation, SB is the maximum stray field intensity and E is the diagonal matrix with ones. In the step-2 involved nonlinear optimization problem, the target function is expressed as:
2.2. Arbitrary mesh generation For the 2D rectangular element meshing one can resort to a commercial mesh generator, if the irregular magnet geometry is considered. Using the proposed method, the rectangular elements are not necessarily identical with each other. In those magnet sections containing the sparse coil block distribution, a large-sized mesh can be applied; and in the dense magnet sections, a fine mesh structure can be employed. In this work, a 1.5 T whole-body planar magnet design was exemplified. Fig. 2(a) reveals the magnet-space mesh of the cross-section for a planar magnet, where the structure supports were denoted by slashes. The magnet configuration was referred to in [24]. Owing to the symmetric configuration, only a quarter was used in the design process, as shown in Fig. 2(b). The magnet coil space has a dimension 2.4 m × 2.0 m (diameter × height), which takes an overall consideration over a 2.0 T split magnet design [12] and 1.0 T panorama open MRI system. For the 1.5 T design, the magnet configuration is between the two open systems, and the outer diameter is estimated to be larger than 1.2 T system and smaller than 2.0 T one [5,12]. The open bore has a gap 0.0.72 m and diameter 1.5 m, which is larger than a 0.7 T open MRI magnet design [14]. Here, we considered the groove structure to mount the gradient assembly with the patient space being
M
min:
Vi
(4)
i=1
subject to: t Bz
TBz TBz
s Bz 2 + s B r 2 R1 Z1
r1 < r2 z1 < z2
SB R2 Z2
(5)
Apart from the coil block dimension constraint, the magnetic field constraints in the above nonlinear optimization cannot be linearized, and thus the nonlinear function should be defined. During the implementation of the linear optimization in step-1, SBz and SBr are both set to be 5.0 Gauss [26]. ɛ is 4 ppm for both the linear and nonlinear optimization procedures. The maximum current density is 100 MA/m2. Here, the maximum current density is expected to be 2
Physica C: Superconductivity and its applications 570 (2020) 1353602
Y. Wang, et al.
Fig. 2. The mesh structure of the magnet-space: (a) the rectangular mesh for a full magnet model and, (b) a quarter of the full mesh model. (b) was applied for the magnet design, because of its symmetrical property of the magnet space.
well below the critical current density of the selected superconductive wire. The matlab built-in linear optimization function linprog and the nonlinear optimization function patternsearch were used for the magnet designs presented in this work. In the implementation of the linprog function, a medium-scale algorithm was adopted.
In the nonlinear optimization, we firstly find a solution with uniform current density 100 MA/m2, where the maximum magnetic field strength in the coils is 4.3 T. Considering the current-carrying capability of superconducting wire and proper current margin, we selected a wire gauge 2.94 mm × 1.59 mm. Afterwards, the coil blocks were discretized with real-wire winding. The wire-wire gap was estimated as 10 μm and fiber-glass cloth was laminated between coil layers with a thickness 0.04 mm. A further optimization was implemented with the acquired wire counts by slightly moving the block position around. The optimized magnet coil blocks are illustrated in Fig. 4, where red and blue lines indicate opposite current direction. The coil block dimensions and corresponding current directions are listed in Table 1, where only a quarter of the full model with four coil blocks is displayed. The “+” represents the current flowing in a positive direction and the “-” denotes the opposite direction. The wire counts for each coil block are also listed, where the wire gap and fiber-glass cloth thickness were considered. The corresponding operating current in the superconducting wire is 481A. The space from the coils to the patient bore gap is occupied by coil bobbin (25 mm), liquid helium vessel (5 mm), vacuum (5 mm), heat insulation foil (15 mm), cold shield (5 mm), vacuum (5 mm) and warm bore (3 mm), with an estimated patient gap 45 cm.
3. Results and discussions 3.1. Coil pattern The linear optimization provides a sparse solution with the available current density elements concentrated together. The current density distribution of a quarter of the magnet space is shown in Fig. 3(a). The current directions are marked with different colors. The coil configuration depends on the parameter constraint used in the optimization implementation, for example, the diameter of spherical volume (DSV) size, field homogeneity, 5-Gauss line range, etc. The variation of these parameters may result in different magnet coil pattern, superconducting wire volume and magnet dimension. After the elements with a near-zero current density are removed, the coil clusters are intuitively displayed. As shown in Fig. 3(a), some coil clusters located at the left corner are quite small. Too many coil blocks and the alternation of the current densities would make the magnet quite complex; it could also potentially increase the fabrication costs and induce system instability. Therefore, these small coil clusters were merged into block at the leftmost region. Fig. 3(b) displays the initial rectangular space of the coil blocks. There are a total of four coil blocks with two positive and one negative current. The discrete elements with rectangular enclosures are illustrated. From Step 1, the coil block boundaries were determined and transferred to the next nonlinear optimizations (Step 2). This determined the final coil blocks for which the maximum current density was input as the operational current density of the magnets.
3.2. Magnetic field The magnetic field deviations over the ROI are illustrated in Fig. 5(a), where the 4 ppm error contours (absolute deviation) are marked. Inside the ROI, a homogenous magnetic field distribution was achieved, and a circle indicates where the ROI resides. Fig. 5(b) displays the stray field profile, and the 5-Gauss line is well-constrained within the designated regions (the elliptic area). In a magnet design, the superconducting wire performance is strongly affected by the inner magnetic field intensity. Fig. 6 illustrates the magnetic field intensity in and around the magnet's coil blocks,
Fig. 3. Current density distributions and determination of initial coil blocks after the linear-optimization implementation: (a) magnet coil clusters with current density direction and, (b) initial magnet block boundary by enclosing the discrete elements. 3
Physica C: Superconductivity and its applications 570 (2020) 1353602
Y. Wang, et al.
Fig. 4. Magnet coil block distributions after nonlinear optimization.
Fig. 6. Magnetic field intensity distribution around the magnet space.
Table 1 Coil block dimensions and current directions.
Coil Coil Coil Coil
1 2 3 4
Coil block (Unit: m) r1 r2
z1
z2
0.175682 0.400021 0.795948 1.148415
0.474958 0.462076 0.288710 0.447956
0.492648 0.497466 0.359500 0.719346
0.200092 0.466811 1.082788 1.197275
Wire count
Current direction
15 × 6 41 × 12 176 × 24 30 × 92
+ + + –
outmost coil is shown in Fig. 7(c). With a 5 mm steel trip binding, the maximum hoop stress of the outmost coil was reduced to 85.1 MPa as shown in Fig. 7(d), which is 23.9 MPa less than the maximum value and the maximum strain was reduced to 0.078%. Comparatively, the stress in the steel strip amounted to154 MPa, bearing the main stress after binding. Appropriate pre-stress on coil is expected to reduce the electromagnetic stress and strain further. The electromagnetic force applied on the magnet coils is illustrated in Fig. 8 with the marked force directions and values. It can be clearly seen that the corresponding coils at the separate magnet poles have an opposite z-direction force and an identical r-direction force. Coils 1, 2, and 4 have overall expansion radial forces, but coil 3 is applied with an overall compressive radial force. For the longitudinal direction, Coils 1, 2 and 3 are attracted to each other by the counterparts at the other pole. However, there is a repulsive force on coil 4 and its corresponding counterpart.
where the module value of the magnetic field is presented. As shown in Fig. 6, a relatively weak and homogeneous magnetic field is produced inside the magnet coils and the maximum magnetic field intensity is 4.3 T, located in coil 3, and coils 1–3 are exposed in a magnetic field intensity of less than 4 T. 3.3. Mechanical property
3.4. Sensitivity analysis
Stress level is also a very important factor for consideration during the magnet design, especially for the hoop stress levels [27]. Excessive stress on the superconducting wires may induce a magnet quench and it can also, potentially, damage the magnet. The hoop stress distribution on the magnet coil blocks is displayed in Fig. 7. For the mechanical analysis, uniform material properties were used for both the coils and steel strips. The maximum magnetic field intensity, hoop stress and strain over the magnet coils are listed in Table 2 and the material properties are displayed in Table 3. The maximum stress is109 MPa, which occurs at the outermost coil blocks, namely, coil 4 and the maximum strain 0.100% is less than acceptable strain 0.2% [28]. In order to reduce the hoop stress, steel trip was bound on the outer surface of the magnet coils to restrain the coil expansion due to electromagnetic force. The hoop stress distribution after binding on the
A sensitivity analysis has been conducted on the magnet coils. Since the superconducting wire has maximum dimension deviation ± 0.015 mm, this will lead to winding errors during the coil fabrication. Taking coil 4 as an example, it has 30 layers and each layer has 92 turns, so the realistic winding counts were randomly allocated as either 92 + 0.015 × 92/2.94 turns, exactly 92 turns or 92–0.015 × 92/2.94 turns and the radial deviation was either 0.015 mm, 0 mm, or −0.015 mm. Fig. 9(a) is the sensitivity analysis of magnetic field homogeneity over the imaging volume and Fig. 9(b) is about the sensitivity analysis on stray field strength on the theoretical 5-Gauss contour. At each random setting of wire counts, both peak-peak value of the magnetic
Fig. 5. Magnetic field profiles of the designed open-bore magnet: (a) magnetic field deviation over the ROI and, (b) stray field profile around the magnet. 4
Physica C: Superconductivity and its applications 570 (2020) 1353602
Y. Wang, et al.
Fig. 7. Hoop stress distributions over the magnet coil blocks: (a) 2D cross section of the magnet coils and, (b) 3D magnet coils, (c) magnet coils with 5 mm steel trip binding and (d) magnet coils after binding. Table 2 Maximum magnetic field intensity and hoop stress over the magnet coils.
Coil Coil Coil Coil
1 2 3 4
Maximum magnetic field intensity (T)
Maximum hoop stress (MPa)
Maximum strain (%)
2.0 2.6 4.3 3.9
29.6 59.6 −8.1 109
0.027 0.054 0.005 0.100
Table 3 Material properties for the mechanical analysis.
Magnet coil Steel strip
Young's modulus (GPa)
Poisson’ ratio
110 210
0.30 0.30
field homogeneity and maximum stray field strength were calculated. There are total 1000 repetitions on the sensitivity evaluation and the minimum, maximum and average magnetic field homogeneities are 16.2 ppm, 784.1 ppm and 197.2 ppm, respectively. However, some other factors may deteriorate the magnetic field homogeneity further, such as winding gap, low-temperature contraction, assembling deviation, etc. As for the stray field, the maximum magnetic field strength exceeds 5 Gauss on the 5 m × 4 m (r × z) region on the 1000 random cases, ranging from 4.99 Gauss to 5.75 Gauss, with an average value of 5.14 Gauss. When extending the shielding region to 5.0 m × 4.1 m (r × z), the maximum, minimum and average stray field strength are 4.72 Gauss, 5.06 Gauss and 4.83 Gauss, respectively, as shown in Fig. 9(c). For the magnetic field correction, both active superconducting shim coils inside the magnet and passive iron pieces in the gradient assembly are planned to be applied to compensate the fabrication deviations [29,30].
Fig. 8. Electromagnetic force applied on the magnet coils.
ability and higher stability. The coil dimension is 2.94 mm × 1.59 mm with insulation. There are 31 NbTi filaments in the wire. The copper-tosuperconductor (Cu:Sc) ratio is 12 and the residual resistance ratio (295 K/10 K, RRR) is larger than 80. The superconducting wire specification is listed in Table 4. Fig. 10 shows the loading current performance of the superconducting wire. The cross-point of the load line and the critical line is 678.4A at 6.1T and the operating point is 4.3 T and 481A, where 481A
3.5. Superconducting wire selection Rectangular superconducting wire was selected to wind the magnet coils because rectangular conductor generally has larger copper-to-superconductor ratio than round wire, which has stronger current-sharing 5
Physica C: Superconductivity and its applications 570 (2020) 1353602
Y. Wang, et al.
Fig. 9. Sensitivity analysis: (a) is the sensitivity analysis in terms of the magnetic field homogeneity over the imaging volume, (b) is the sensitivity analysis in terms of the stray field strength on the shielding region 5 m × 4 m (r × z) and (c) is the sensitivity analysis in terms of the stray field strength on the shielding region 5 m × 4.1 m (r × z).
6
Physica C: Superconductivity and its applications 570 (2020) 1353602
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Table 4 Superconducting wire specification. Number of filaments
Nominal filament size (μm)
Nominal wire size (mm) Nominal bare Nominal bare size corner radius
Insulated size
Nominal Cu:Sc
Typical reported Ic
RRR (295 K/ 10 K)
Yield strength
31
103
2.71 × 1.38
2.94 × 1.59 ± 0.015
12
≥825A@5T
≥80
≥[email protected]% strain
0.35
Fig. 10. Loading performance of the superconducting wire.
unconventional magnet design. However, the high Lorentz forces on the magnet coils make a potential threat on the support structure, which should be considered carefully during the design and fabrication stages.
Table 5 Design parameters of a 1.5 T open bore superconducting magnet. Magnetic field strength
1.5 T
DSV Magnetic field homogeneity Stray field range (5 Gauss line) Operating current Peak field Maximum hoop stress with binding Inductance Storage energy Conductor length Conductor weight
450 mm 8 ppm 5 m × 4 m (r × z) 481 A 4.30 T 85.1 MPa 103.60 H 11.99 MJ 93.43 km 3913.35 kg
Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. The authors declare the following financial interests/personal relationships which may be considered as potential competing interests Acknowledgement This work is funded by CAS Pioneer Hundred Talents Program(grant no. Y8402A1C31), Beijing Science and Technology Plan (grant no. Z181100003818020), International Partnership Program of Chinese Academy of Sciences(grant no. 182111KYB20170067, no. 182111KYSB20170039).
is the running current and 4.3 T is the maximum magnetic field intensity in the magnet coils. The current margin is 29.10% and the estimated temperature margin is 1.98 K. The magnet coils have a total volume of 0.4368 m3, with a superconducting wire length of 93.4 km. The magnet design parameters were summarized in Table 5. The inductance of the magnet coils is 103.60 H and the storage energy is 11.99 MJ at the operating current 481A. The conductor weight is estimated around 4 t
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4. Conclusion This work has proposed a design scheme for a 1.5 T open MRI magnet. The magnetic field performance, mechanical property, coil dimension sensitivity and superconducting wire characteristics were analyzed. The magnetic field homogeneity in the imaging volume was optimized to 8 ppm, the maximum hoop stress was controlled to 85.1 MPa, and the superconducting wire was operated with a current margin of 29.10%. An improved optimization algorithm was implemented in the magnet design that had no requirement for uniformly meshing the magnet space, which is particularly flexible on 7
Physica C: Superconductivity and its applications 570 (2020) 1353602
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8
Contents lists available at ScienceDirect
Physica C: Superconductivity and its applications journal homepage: www.elsevier.com/locate/physc
Electromagnetic design of a 1.5T open MRI superconducting magnet a
a,b,⁎
Yaohui Wang , Qiuliang Wang a b c
a
a
a
c
, Lei Wang , Hongyi Qu , Yang Liu , Feng Liu
T
Division of Superconducting Magnet Science and Technology, Institute of Electrical Engineering, Chinese Academy of Sciences, Beijing 100190, China School of Electronic, Electrical and Communication Engineering, University of Chinese Academy of Sciences, Beijing 100049, China School of Information Technology and Electrical Engineering, University of Queensland, St Lucia, Brisbane, QLD 4072, Australia
ARTICLE INFO
ABSTRACT
Keywords: MRI Magnet Superconducting wire Stress Open bore
The open magnetic resonance imaging (MRI) system is more patient-friendly and is easier for interventional therapy than the cylindrical MRIs owing to the large patient bore. In this work, a 1.5T open MRI magnet was designed. To evaluate the split-bore magnet design, several system parameters are presented that include magnetic field performance, stress, sensitivity, superconducting wire and operating current margin. The critical design parameters of the open magnet were carefully evaluated which include a maximum hoop stress 85.1 MPa, operating current 481A with margin 29.10%, inductance 103.60H and storage energy 11.99 MJ. The improved magnet optimization method has the capability to tackle complicated magnet design and the proposed 1.5 T open magnet design has a safe hoop stress and current margin. However, some potential risks such as large Lorentz force should be specially evaluated during the magnet support structure design and fabrication.
1. Introduction The main magnet is a key component in a magnetic resonance imaging (MRI) system, which is used to generate a highly-homogeneous static magnetic field in the region of interest (ROI) [1,2]. Generally, the homogeneity of the magnetic field profile and the stray field intensity should be constrained. For example, in the following 1.5 T open magnet design, the magnetic field inhomogeneity is 8 ppm over a 45 cm sphere, and the 5 Gauss line range is 5 m by 4 m. Owing to the strict engineering constraints of the current superconducting technology, it is rather challenging to design a high-performance, low-cost superconducting magnet for MRI applications, particularly for those with split MRI configurations. An open MRI superconducting magnet is a type of magnet that has a large gap between the opposite-placed magnet poles [3–6]. The splitmagnet configuration significantly augments the design difficulties in terms of magnetic field homogeneity, electromagnetic force balance, superconducting wire performance, and so on. Although it is challenging to develop an open MRI superconducting magnet, it does offer a number of benefits compared with conventional cylindrical systems. For example, it can substantially reduce the patients’ feeling of claustrophobia because of the larger patient bore and shorter system length [7]. It is also feasible for image extremity by moving the patient horizontally to expose the corresponding body part in the imaging area. In addition, an open MRI system provides a superb operating platform for
⁎
interventional therapy, which is convenient for the healthcare workers needing access to the patient [8]. Previously, building open superconducting MRI systems with lower field strengths has been attempted. Those commercially-available systems include the 0.7 T OpenSpeed from GE [9], the 1.0 T Philips Panorama [10] and the 1.2 T Oasis from Hitachi [11]. Because of the high fabrication cost and low performance, the 0.7 T and 1.0 T system have been discontinued and 1.2 T Hitachi Oasis is the only open superconducting scanner in commercial production at this time. Besides, several thousand non-superconducting open MRI are still in production or being installed. In recent years, to gain a higher SNR, higher field open MRI systems are increasingly demanded [12]. In this paper, a 1.5 T open MRI superconducting magnet was designed using an improved optimization scheme. The mechanical property, sensitivity characteristics and superconducting wire performances were further evaluated. 2. Method 2.1. Improved linear optimization computation Fig. 1 illustrates a diagram of a meshed magnet space, which is a sketch of the linear optimization principle of the initial estimation of a magnet design. Fig. 1(b) shows the current approximation of the existing methods [13–18], where the thick solenoid of each element is
Corresponding author. E-mail addresses: [email protected] (Y. Wang), [email protected] (Q. Wang).
https://doi.org/10.1016/j.physc.2020.1353602 Received 12 November 2019; Accepted 14 January 2020 Available online 15 January 2020 0921-4534/ © 2020 Elsevier B.V. All rights reserved.
Physica C: Superconductivity and its applications 570 (2020) 1353602
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inside the enclosure of the gradient assembly and magnet. Considering the coil bobbin, liquid helium vessel, cold shield and warm bore, the gap of the patient bore is estimated at about 0.45 m after the cryostat installation. 2.3. Magnetic field constraint The magnet design generates a 8 ppm (peak-peak value) uniform magnetic field profile in a 0.45 m sphere with the stray fields being actively-shielded to be less than 5 Gauss in a space 5 m × 4 m along horizontal (r) and vertical (z) directions, respectively. The target is to minimize the superconductor volume function within the magnetic field constraints. For simplicity, during the superconducting wire volume calculation, it is assumed that the wires have the identical current density Jmax, and thus, the real wire usage will be a ratio between the calculated and the designated current density. In the linear procedure, the optimization problem is expressed as: Fig. 1. Diagram of the linear optimization in the magnet design: (a) meshed magnet space, (b) filament-loop approximation for magnetic field calculation and, (c) magnetic field calculation using a thick solenoid.
m
min:
vi· i= 1
xi Jmax
(1)
subject to:
modeled as a current loop with no cross-sectional area, and the coordinate of the loop is at the rectangle center, namely ((z1+z2)/2, (r1+r2)/2) [19]. The linear optimization target is to find the unknown currents of the rectangular elements. This method uses a concentrated current to represent the distributed one, neglecting the dimensional content of the current element. Thus, it has to apply a dense mesh on some challenging design cases, such as the ultra-short magnet. In our proposed method, the filament-loop approximation is discarded, but a direct computation of a thick solenoid is adopted. At each element, the rectangular area has a uniform current density [20–22]. Fig. 1(c) depicts such a design process, where the coordinate of the solenoid is (z1, z2, r1, r2). To calculate the magnetic field produced by each thick solenoid, an angular integral was derived with the parameters z1, z2, r1, r2 and a current density J [23]. In a numerical computation, the angular integral was transformed as a Gaussian-quadrature rule with a summation of only some integration points multiplying the corresponding weights. In this work, a 10-order Gaussian integral was used. After completing the above improved linear optimization procedure, a follow-up global nonlinear optimization algorithm was applied to search for the final block dimensions.
AzX
TBz TBz
BX X
SB Jmax
(2)
By introducing a new variable yi with |xi| ≤ yi, the above nonlinear constraints can be transformed into a linear model [25]:
TBz (1 + Az 0 TBz (1 Az 0 S B Bz 0 z S Bz Bz 0 Br 0 [ S X Y ] Br Br 0 S Br E E 0 E E 0 E0 Jmax E0 Jmax
) )
(3)
where xi is the unknown current density at each rectangular element, m is the rectangular element quantity, Jmax is the maximum current density, Az and B are the primary magnetic field and shielding magnetic field matrices, respectively, TBz is the target field intensity, ɛ is the magnetic field deviation, SB is the maximum stray field intensity and E is the diagonal matrix with ones. In the step-2 involved nonlinear optimization problem, the target function is expressed as:
2.2. Arbitrary mesh generation For the 2D rectangular element meshing one can resort to a commercial mesh generator, if the irregular magnet geometry is considered. Using the proposed method, the rectangular elements are not necessarily identical with each other. In those magnet sections containing the sparse coil block distribution, a large-sized mesh can be applied; and in the dense magnet sections, a fine mesh structure can be employed. In this work, a 1.5 T whole-body planar magnet design was exemplified. Fig. 2(a) reveals the magnet-space mesh of the cross-section for a planar magnet, where the structure supports were denoted by slashes. The magnet configuration was referred to in [24]. Owing to the symmetric configuration, only a quarter was used in the design process, as shown in Fig. 2(b). The magnet coil space has a dimension 2.4 m × 2.0 m (diameter × height), which takes an overall consideration over a 2.0 T split magnet design [12] and 1.0 T panorama open MRI system. For the 1.5 T design, the magnet configuration is between the two open systems, and the outer diameter is estimated to be larger than 1.2 T system and smaller than 2.0 T one [5,12]. The open bore has a gap 0.0.72 m and diameter 1.5 m, which is larger than a 0.7 T open MRI magnet design [14]. Here, we considered the groove structure to mount the gradient assembly with the patient space being
M
min:
Vi
(4)
i=1
subject to: t Bz
TBz TBz
s Bz 2 + s B r 2 R1 Z1
r1 < r2 z1 < z2
SB R2 Z2
(5)
Apart from the coil block dimension constraint, the magnetic field constraints in the above nonlinear optimization cannot be linearized, and thus the nonlinear function should be defined. During the implementation of the linear optimization in step-1, SBz and SBr are both set to be 5.0 Gauss [26]. ɛ is 4 ppm for both the linear and nonlinear optimization procedures. The maximum current density is 100 MA/m2. Here, the maximum current density is expected to be 2
Physica C: Superconductivity and its applications 570 (2020) 1353602
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Fig. 2. The mesh structure of the magnet-space: (a) the rectangular mesh for a full magnet model and, (b) a quarter of the full mesh model. (b) was applied for the magnet design, because of its symmetrical property of the magnet space.
well below the critical current density of the selected superconductive wire. The matlab built-in linear optimization function linprog and the nonlinear optimization function patternsearch were used for the magnet designs presented in this work. In the implementation of the linprog function, a medium-scale algorithm was adopted.
In the nonlinear optimization, we firstly find a solution with uniform current density 100 MA/m2, where the maximum magnetic field strength in the coils is 4.3 T. Considering the current-carrying capability of superconducting wire and proper current margin, we selected a wire gauge 2.94 mm × 1.59 mm. Afterwards, the coil blocks were discretized with real-wire winding. The wire-wire gap was estimated as 10 μm and fiber-glass cloth was laminated between coil layers with a thickness 0.04 mm. A further optimization was implemented with the acquired wire counts by slightly moving the block position around. The optimized magnet coil blocks are illustrated in Fig. 4, where red and blue lines indicate opposite current direction. The coil block dimensions and corresponding current directions are listed in Table 1, where only a quarter of the full model with four coil blocks is displayed. The “+” represents the current flowing in a positive direction and the “-” denotes the opposite direction. The wire counts for each coil block are also listed, where the wire gap and fiber-glass cloth thickness were considered. The corresponding operating current in the superconducting wire is 481A. The space from the coils to the patient bore gap is occupied by coil bobbin (25 mm), liquid helium vessel (5 mm), vacuum (5 mm), heat insulation foil (15 mm), cold shield (5 mm), vacuum (5 mm) and warm bore (3 mm), with an estimated patient gap 45 cm.
3. Results and discussions 3.1. Coil pattern The linear optimization provides a sparse solution with the available current density elements concentrated together. The current density distribution of a quarter of the magnet space is shown in Fig. 3(a). The current directions are marked with different colors. The coil configuration depends on the parameter constraint used in the optimization implementation, for example, the diameter of spherical volume (DSV) size, field homogeneity, 5-Gauss line range, etc. The variation of these parameters may result in different magnet coil pattern, superconducting wire volume and magnet dimension. After the elements with a near-zero current density are removed, the coil clusters are intuitively displayed. As shown in Fig. 3(a), some coil clusters located at the left corner are quite small. Too many coil blocks and the alternation of the current densities would make the magnet quite complex; it could also potentially increase the fabrication costs and induce system instability. Therefore, these small coil clusters were merged into block at the leftmost region. Fig. 3(b) displays the initial rectangular space of the coil blocks. There are a total of four coil blocks with two positive and one negative current. The discrete elements with rectangular enclosures are illustrated. From Step 1, the coil block boundaries were determined and transferred to the next nonlinear optimizations (Step 2). This determined the final coil blocks for which the maximum current density was input as the operational current density of the magnets.
3.2. Magnetic field The magnetic field deviations over the ROI are illustrated in Fig. 5(a), where the 4 ppm error contours (absolute deviation) are marked. Inside the ROI, a homogenous magnetic field distribution was achieved, and a circle indicates where the ROI resides. Fig. 5(b) displays the stray field profile, and the 5-Gauss line is well-constrained within the designated regions (the elliptic area). In a magnet design, the superconducting wire performance is strongly affected by the inner magnetic field intensity. Fig. 6 illustrates the magnetic field intensity in and around the magnet's coil blocks,
Fig. 3. Current density distributions and determination of initial coil blocks after the linear-optimization implementation: (a) magnet coil clusters with current density direction and, (b) initial magnet block boundary by enclosing the discrete elements. 3
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Fig. 4. Magnet coil block distributions after nonlinear optimization.
Fig. 6. Magnetic field intensity distribution around the magnet space.
Table 1 Coil block dimensions and current directions.
Coil Coil Coil Coil
1 2 3 4
Coil block (Unit: m) r1 r2
z1
z2
0.175682 0.400021 0.795948 1.148415
0.474958 0.462076 0.288710 0.447956
0.492648 0.497466 0.359500 0.719346
0.200092 0.466811 1.082788 1.197275
Wire count
Current direction
15 × 6 41 × 12 176 × 24 30 × 92
+ + + –
outmost coil is shown in Fig. 7(c). With a 5 mm steel trip binding, the maximum hoop stress of the outmost coil was reduced to 85.1 MPa as shown in Fig. 7(d), which is 23.9 MPa less than the maximum value and the maximum strain was reduced to 0.078%. Comparatively, the stress in the steel strip amounted to154 MPa, bearing the main stress after binding. Appropriate pre-stress on coil is expected to reduce the electromagnetic stress and strain further. The electromagnetic force applied on the magnet coils is illustrated in Fig. 8 with the marked force directions and values. It can be clearly seen that the corresponding coils at the separate magnet poles have an opposite z-direction force and an identical r-direction force. Coils 1, 2, and 4 have overall expansion radial forces, but coil 3 is applied with an overall compressive radial force. For the longitudinal direction, Coils 1, 2 and 3 are attracted to each other by the counterparts at the other pole. However, there is a repulsive force on coil 4 and its corresponding counterpart.
where the module value of the magnetic field is presented. As shown in Fig. 6, a relatively weak and homogeneous magnetic field is produced inside the magnet coils and the maximum magnetic field intensity is 4.3 T, located in coil 3, and coils 1–3 are exposed in a magnetic field intensity of less than 4 T. 3.3. Mechanical property
3.4. Sensitivity analysis
Stress level is also a very important factor for consideration during the magnet design, especially for the hoop stress levels [27]. Excessive stress on the superconducting wires may induce a magnet quench and it can also, potentially, damage the magnet. The hoop stress distribution on the magnet coil blocks is displayed in Fig. 7. For the mechanical analysis, uniform material properties were used for both the coils and steel strips. The maximum magnetic field intensity, hoop stress and strain over the magnet coils are listed in Table 2 and the material properties are displayed in Table 3. The maximum stress is109 MPa, which occurs at the outermost coil blocks, namely, coil 4 and the maximum strain 0.100% is less than acceptable strain 0.2% [28]. In order to reduce the hoop stress, steel trip was bound on the outer surface of the magnet coils to restrain the coil expansion due to electromagnetic force. The hoop stress distribution after binding on the
A sensitivity analysis has been conducted on the magnet coils. Since the superconducting wire has maximum dimension deviation ± 0.015 mm, this will lead to winding errors during the coil fabrication. Taking coil 4 as an example, it has 30 layers and each layer has 92 turns, so the realistic winding counts were randomly allocated as either 92 + 0.015 × 92/2.94 turns, exactly 92 turns or 92–0.015 × 92/2.94 turns and the radial deviation was either 0.015 mm, 0 mm, or −0.015 mm. Fig. 9(a) is the sensitivity analysis of magnetic field homogeneity over the imaging volume and Fig. 9(b) is about the sensitivity analysis on stray field strength on the theoretical 5-Gauss contour. At each random setting of wire counts, both peak-peak value of the magnetic
Fig. 5. Magnetic field profiles of the designed open-bore magnet: (a) magnetic field deviation over the ROI and, (b) stray field profile around the magnet. 4
Physica C: Superconductivity and its applications 570 (2020) 1353602
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Fig. 7. Hoop stress distributions over the magnet coil blocks: (a) 2D cross section of the magnet coils and, (b) 3D magnet coils, (c) magnet coils with 5 mm steel trip binding and (d) magnet coils after binding. Table 2 Maximum magnetic field intensity and hoop stress over the magnet coils.
Coil Coil Coil Coil
1 2 3 4
Maximum magnetic field intensity (T)
Maximum hoop stress (MPa)
Maximum strain (%)
2.0 2.6 4.3 3.9
29.6 59.6 −8.1 109
0.027 0.054 0.005 0.100
Table 3 Material properties for the mechanical analysis.
Magnet coil Steel strip
Young's modulus (GPa)
Poisson’ ratio
110 210
0.30 0.30
field homogeneity and maximum stray field strength were calculated. There are total 1000 repetitions on the sensitivity evaluation and the minimum, maximum and average magnetic field homogeneities are 16.2 ppm, 784.1 ppm and 197.2 ppm, respectively. However, some other factors may deteriorate the magnetic field homogeneity further, such as winding gap, low-temperature contraction, assembling deviation, etc. As for the stray field, the maximum magnetic field strength exceeds 5 Gauss on the 5 m × 4 m (r × z) region on the 1000 random cases, ranging from 4.99 Gauss to 5.75 Gauss, with an average value of 5.14 Gauss. When extending the shielding region to 5.0 m × 4.1 m (r × z), the maximum, minimum and average stray field strength are 4.72 Gauss, 5.06 Gauss and 4.83 Gauss, respectively, as shown in Fig. 9(c). For the magnetic field correction, both active superconducting shim coils inside the magnet and passive iron pieces in the gradient assembly are planned to be applied to compensate the fabrication deviations [29,30].
Fig. 8. Electromagnetic force applied on the magnet coils.
ability and higher stability. The coil dimension is 2.94 mm × 1.59 mm with insulation. There are 31 NbTi filaments in the wire. The copper-tosuperconductor (Cu:Sc) ratio is 12 and the residual resistance ratio (295 K/10 K, RRR) is larger than 80. The superconducting wire specification is listed in Table 4. Fig. 10 shows the loading current performance of the superconducting wire. The cross-point of the load line and the critical line is 678.4A at 6.1T and the operating point is 4.3 T and 481A, where 481A
3.5. Superconducting wire selection Rectangular superconducting wire was selected to wind the magnet coils because rectangular conductor generally has larger copper-to-superconductor ratio than round wire, which has stronger current-sharing 5
Physica C: Superconductivity and its applications 570 (2020) 1353602
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Fig. 9. Sensitivity analysis: (a) is the sensitivity analysis in terms of the magnetic field homogeneity over the imaging volume, (b) is the sensitivity analysis in terms of the stray field strength on the shielding region 5 m × 4 m (r × z) and (c) is the sensitivity analysis in terms of the stray field strength on the shielding region 5 m × 4.1 m (r × z).
6
Physica C: Superconductivity and its applications 570 (2020) 1353602
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Table 4 Superconducting wire specification. Number of filaments
Nominal filament size (μm)
Nominal wire size (mm) Nominal bare Nominal bare size corner radius
Insulated size
Nominal Cu:Sc
Typical reported Ic
RRR (295 K/ 10 K)
Yield strength
31
103
2.71 × 1.38
2.94 × 1.59 ± 0.015
12
≥825A@5T
≥80
≥[email protected]% strain
0.35
Fig. 10. Loading performance of the superconducting wire.
unconventional magnet design. However, the high Lorentz forces on the magnet coils make a potential threat on the support structure, which should be considered carefully during the design and fabrication stages.
Table 5 Design parameters of a 1.5 T open bore superconducting magnet. Magnetic field strength
1.5 T
DSV Magnetic field homogeneity Stray field range (5 Gauss line) Operating current Peak field Maximum hoop stress with binding Inductance Storage energy Conductor length Conductor weight
450 mm 8 ppm 5 m × 4 m (r × z) 481 A 4.30 T 85.1 MPa 103.60 H 11.99 MJ 93.43 km 3913.35 kg
Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. The authors declare the following financial interests/personal relationships which may be considered as potential competing interests Acknowledgement This work is funded by CAS Pioneer Hundred Talents Program(grant no. Y8402A1C31), Beijing Science and Technology Plan (grant no. Z181100003818020), International Partnership Program of Chinese Academy of Sciences(grant no. 182111KYB20170067, no. 182111KYSB20170039).
is the running current and 4.3 T is the maximum magnetic field intensity in the magnet coils. The current margin is 29.10% and the estimated temperature margin is 1.98 K. The magnet coils have a total volume of 0.4368 m3, with a superconducting wire length of 93.4 km. The magnet design parameters were summarized in Table 5. The inductance of the magnet coils is 103.60 H and the storage energy is 11.99 MJ at the operating current 481A. The conductor weight is estimated around 4 t
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