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Transactions on Biomedical Engineering
Abstract— Objective: With advantages of reduced coupling and
compact structure, Matrix Coils (MCs) design extension to
approximate multiple target inhomogeneities is necessary to
improve its performance in shimming applications. Methods: A
Spherical Harmonic Decomposition Method (SHDM) is proposed
for the multi-target MCs optimization problem. The magnetic
field generated by the MCs is represented in form of SHs’
orthogonal basis, based on which the MCs pattern is opti mized to
adapt to multiple SH targets. Results: With multi-target SHs of the
1st, 3rd, and mixed 1st&2nd degrees in Halbach magnet shimming,
MCs structure optimizations were successfully performed.
Comparisons with regular interleaved MCs show the optimized
coil structure provides better performance, including reduction of
power dissipation, maximum current amplitude, and total current
requirement. Conclusion: This w ork proposed a simple and
intuitive way of irregular MCs optimization, which is of high
benefits in compact MR systems based on permanent magnets.
Significance: This methodology may also be translated into local
gradient & shimming matrix coils designs for conventional
magnetic resonance device.
Index Terms—Matrix coils, shimming, Spherical Harmonics,
Halbach magnet.
I. INTRODUCTION
n traditional MR (magnetic resonance) active shimming, the
main magnetic field is decomposed using a Spherical
Harmonic (SH) orthogonal basis characterized, and
shimming coils are designed for each SH term. However,
impurities exist within these SH shimming coil groups. For
example, the Z4 shimming coil is often contaminated by the Z2
field component [1]. Besides, a displacement between multiple
sets of coils during assembly may also introduce high-degree
magnetic field residuals, which may result in a time-consuming
shimming process [1-2]. Furthermore, tens of layered SH
shimming coils may also occupy too much magnet inner space
[3].
Matrix Coils (MCs, also called ‘multi-coils’) technique
utilizes a set of coils distributed on a cylindrical surface to
directly generate a target magnetic field distribution [4]. To date,
numerous publications on MCs have concerned themselves
with superconducting magnets. MCs structure composed of 24
circular coils has successfully generated X2-Y2, Z2, Z2X
This work was funded by Grants 11805263 from NSFC, 2018K058B from
Jiangsu Post doctor funds, 2018ZY013 from Wenzhou Science and Technology
Plan, and in part by the German Research Foundation (DFG) ZA 422/5-1.
Yajie Xu, Peng Yu, Ya Wang, *Xiaodong Yang (correspondence e-mail:
xiaodong.yang@sibet.ac.cn) are with Imaging department, Suzhou Institute of
magnetic field [5]. An array of 48-circle coils was proposed to
verify the possibility of SH field generation from 1st to 4th
degrees [6], and the experiment was executed in 4 T MRI
scanner. In vivo experiments in human and rat brains were
performed at 7 T [7], 9.4 T [8], which has verified the
superiority of MCs technology. In 2013, Han proposed a
specialized MCs system called iPRES [9] that combines
shimming, RF receiving, and transmitting, together. An 84-
channel plug-in MCs structure for nonlinear gradient field
imaging was designed and implemented on a 3 T scanner [10,
11]. MCs technology was shown to combine shimming,
gradient functions together, and implement high degree
shimming with tight space requirements.
In ordinary applications, number and positions of MCs based
on regular type of coil elements, such as circular or saddle
structures were optimized to improve its performance [4-9].
However, the actual applications may more focus on some
specified components. Thus, MCs geometrical shape
optimizations are necessary for given scenarios. Chen [12],
Zivkovic [13] have proposed polygon coil shape for single MC
element targeting at a specified inhomogeneous field B in the
region of interest, but the coils’ flexibilities for multi-target field
shimming and precision for complicated target field
approximation are limited. While [14] has applied the stream
function method as a basis for each SH to optimize the MCs
shape with the axial Bz field design demonstrated. But the
optimization of stream function was up to the approximation
and error evaluation for a composite field pattern, which is not
straightforward. A more direct MCs design to improve MCs
performance in practical use is required.
Halbach array magnet is of great potential in desktop NMR
applications [15]. Due to the stringent requirements of field
homogeneity in MR application, active shimming is mandatory.
Traditional shimming executions with groups of coils is limited
in the compact magnet, for which MCs technology is especially
appropriate, but till now there are no publications about this. In
this paper, MCs design is demonstrated based on Halbach
magnet structure. Different from the cylindrical
superconducting magnet with the main magnetic field in the
axial direction, field in Halbach has a certain transversal
distribution which we denote as By. Thus, we provide a
transversal magnetic field shimming design method in this
paper. Meanwhile, this approach could also be extended to Bx
or Bz field applications.
In this paper, a Spherical Harmonic Decomposition Method
(SHDM) is proposed for the multi-target MCs design. In section
Biomedical Engineering and Technology. Yingcong Yu is in Wenzhou people’s
hospital. Feng Jia and Maxim Zaitsev (correspondence e-mail:
maxim.zaitsev@uniklinik-freiburg.de) are with Dept. of Radiology, Medical
Physics, Medical Center University of Freiburg, Faculty of Medicine,
University of Freiburg, Germany.
A Spherical Harmonics Decomposition Method
(SHDM) for irregular Matrix Coils design
Yajie Xu, Peng Yu, Feng Jia, Ya Wang, Yingcong Yu, Xiaodong Yang*, Maxim Zaitsev*
I
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Transactions on Biomedical Engineering
2, the magnetic field of MCs in SH orthogonal basis is derived,
and then the multi-target optimization problem within SHs
space is established to simplify calculations. In section 3
implementation based on the Halbach prototype in our lab is
introduced and in section 4, examples to implement the MCs
optimization technology are demonstrated for 1st, 3rd, and mixed
1st &2nd degree SHs, and comparisons with standard interleaved
MCs are performed. Finally, a MCs prototype of the first degree
was constructed and tested to verify the performance of the
optimized MCs structure.
II. METHOD
A. SHDM Algorithm for MCs construction
In the region of interest (ROI), magnetic field distribution By
produced by the current flowing on a cylinder surface can be
calculated with a differential form of Biot-Savart’s la w.
Generally, with source point (,,) and field
point (,,), magnetic field result from the current density
is calculated by its cross product with Green’s function [16].
=
||×d, (1)
where is the permeability of free space, d is a differential
surface element. For the cylindrical current-carrying surface,
d=d, in which R is the radius of the cylinder carrying
the coils.
Fig. 1. a) 16 blocks multi-layer Halbach magnet, and b) current-carrying
surface. In matrix coils design, the surface is split into Q×K units (here 4×4 is
selected to show the structure), and the blue sphere indicates the ROI.
With condition ||<||, Green’s function can be
expanded in spherical coordinates [16].
||=
()!
()!
(cos )
cos ()
(cos ) (2)
Where n and m represent the degree and order of spherical
harmonic functions, = 1 for m = 0 and 2 for m > 0,
() is
the associated Legendre functions. For a cylindrical current-
carrying space, the current can be separated into azimuthal
component and axial component , such that magnetic field
By can be expressed as:
=
||
||d
=
(cos )[((+ 1)(cos )cos
sin )cos ( )(sin )((+
1)(cos )sin +
cos ) cos cos ()+
(cos )sin ( )sin
=
(cos )[cos ()sin
cos cos ()+si n ( )sin ] (3)
where
=()!
()! ,
=
( +
1),(cos )(+ 1)cos (cos ). And intermediate
variables are written as
=(+ 1)(cos )cos
sin ;
=(+ 1)(cos )sin +
cos ;
=
(cos );
Hence, magnetic field could be further simplified as
equation 4 shows, with the source points and target points
variables separated.
=
(cos )(cos sin )
co s sin cos cos sin sin
sin sin sin cos + cos sin
(4)
Magnetic field in the bore of the MR system satisfies
Laplace’s equation. Therefore, to solve equation (4) in spherical
coordinates, it is helpful to represent the magnetic field with SH
decomposition as shown in equation (5). This equation is valid
under the condition that the radius of the target spherical
volume is far smaller than the radius of the current-carrying
surface [16].
(,,) =
( )(,
,)
(), (5)
where ,, , denotes the basis value for SH (n, m). Hence,
relationship between SHs basis decomposition in ROI and the
current density on the current-carrying surface can be achieved.
For MCs with × units, the size of each coil unit can be
defined as:
= 2;
= 2,
where L is the half-length of the cylinder surface. With the
above definitions, the SH amplitude resulting from MCs can
be derived as:
=
( cos cos
sin cos
sin sin
)d; (6)
=
(
cos sin
sin sin
+ sin cos
)d, (7)
, , are intermediate variables related to the
source point coordinate
and the q, k index matrix coils
elements.
To confine coil winding to each coil element, boundary
conditions
and
are set to zero on the angular and axial
boundary of the (k, q) region, respectively. Because of the
symmetric property of SHs, evenly distributed coils layouts
were selected.
From equation (6-7), when the dimensions of the current-
carrying surface are fixed, the values of and due to
the azimuthal and axial current density distribution can be
calculated. In the following section, a current density model is
established for the matrix coil element. Referring to While [17],
By
ROI
(q, k)
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Transactions on Biomedical Engineering
current density functions are expressed as the Fourier series
decomposition e.g. weighted summation of sinusoidal and
cosinusoidal periodic functions. To account for the SH degrees
in azimuthal direction, we introduce a term of phase shift ()
as
=
+(), 0,
.
=
sin ()
cos(
)
(8)
=
cos ()
sin(
)
(9)
[1, ], [1, ] .
where
is the current density coefficient of the coil
element (k, q) with (v, h) mode of Fourier series. Substituting
equation (8-9) into (6-7), we could acquire the sensitivity
matrix =
for each MCs element in which
contains all source related variables other than current
coefficient
. Hence, when mode (v, h) of Fourier series in
each subregion (k, q) and its axial and angular sizes are fixed,
could be obtained directly.
According to the continuity equation J = 0, scalar stream
function of the current density can be introduced J =
× S(r)n(r) [18]. Then stream function could be denote as
=
sin ()
sin(
)
, (10)
Hence
reflects the complexity of stream functions. With
the combined stream function, the coils winding loops can be
approximated as level lines of the stream functions.
B. MCs optimization for multi-target SHs
In general, shimming with MCs consists of searching for the
best driven currents fitting multiple target field patterns. We set
up the multi-target field with proportions of the required SH
basis. They could be the maximum compensation capability
required in actual application, or prior analysis of the field
composition ratio to be compensated. However, this
optimization is not trying to restrict the coils performance
following the exact field composition, but an attempt to
improve the MCs capability for certain SH basis.
Concretely, MCs optimization can be divided into two parts:
I. MCs element structure optimization, which is relevant to the
coefficient
in equation (8-10). II. Current optimization
targeting at multi-target SHs under the established MCs
structure and sensitivity matrix . To build a general coils
unit group for SH (,) , (n + 1)2(m + 1) (rows and
columns) evenly distributed coils are adequate to cover all the
orders of a degree. The actual field compensation may be
complex, and the proper MCs number can be chosen according
to the highest required shimming degree.
To solve the multi-target SHs optimization problem, MCs
design need to be sought to approximate the target fields with
the selected performance factors, such as the current
consumption, shimming accuracy, coil complexity, wire width,
etc. Considering the compact Halbach magnet, the target
function is combined with the current sum, power dissipation,
and shimming accuracy. Here the power dissipation is
calculated as P
=
(
+
)
[19], in which
is the conductor thickness, and is the electrical conductivity.
Shimming accuracy is qualified with the homogeneity after coil
compensation. In multi-target field shimming, for each target
SH it is qualified with =max (B B). This
objective function was selected according to the actual goal of
achieving suitable MCs designs.
=arg min
(), (11)
() = P
++
S. t. >;
where , , are the weighting factors selected manually
referring to the value of the power dissipation, driven currents
and shimming accuracy. denotes the distance between coil
windings loops and
is minimal distance chosen by the
designer. According to the aforementioned analysis, the
problem generally includes a joint optimization of
and
,
with the latter term relying on the value of the former one.
Therefore, when the MCs positions and the current density
coefficients are fixed, the magnetic fields generated by the coil
elements per ampere can also be confirmed. Then the MCs
shimming problem is simplified to a variant of the generalized
assignment problem [21].
In traditional MCs problem, assuming M points are selected
in the ROI, sensitivity matrix is of M×QK size. Normally, the
target point number is chosen to be quite large to promote the
approximation accuracy, which leads to a long calculation time.
Therefore, a quick and accurate current optimization algorithm
is necessary. In the next section, we propose a solution based
on optimization in SHs space to cut down the computation
burden.
C. Current Optimization in SH space
Take Gy gradient field in MR imaging device as an example.
In physical space coordinates, tens of thousands of target points
may be included to describe the field distribution precisely.
Whereas, the field can be projected into SHs space, in which Gy
pertains only to a single SH component (1, -1). Notably, to
avoid influence from other orthogonal basis functions, SH
components with higher degrees should be considered
simultaneously. Take the 4th degree as an example, 24
components are involved. The 0th-degree component can be
omitted here considering the frequency shift caused can be
compensated by the main frequency adjustment. Consequently,
the underdetermined problem of a computational complexity
M×QK can be simplified to P(nm)×QK, in which P(nm)
corresponds to the n-degree, m-order SHs basis involved, such
that computation of equation (11) is largely accelerated.
=
(12)
For the solution of the multivariable problem linear and
nonlinear optimization algorithms, for instance, Levenberg -
Marquardt method [8], Simulated Annealing [20], Genetic
Algorithm [13], can be applied with objective function as a
combination of parameters mentioned above. In this paper, a
pseudo-inverse of the sensitivity matrix is implemented directly
to accelerate the computation speed.
III. IMPLEMENTATION
The structure of MCs (e.g., number and coil shape) is closely
associated with the target field complexity, which correspond to
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Transactions on Biomedical Engineering
high degree in form of SH basis. The problem we are trying to
solve is targeting a set of magnetic SH field compositions to
improve the MCs geometry. Based on the parameters of
Halbach magnet in our lab [21], the MCs optimization is
implemented to demonstrate the design process.
Field patterns of the 1st, 2nd, and 3rd degree SH basis have
been taken as examples. All the results are compared with a
standard coils structure defined with only the first term (
) of
current density coefficients set as 1, others 0, as shown in fig. 2.
The so-called standard coils are of the interleaved shapes
similar to a circular shape in ordinary MCs setting yet with
more turns to take up the effective current-carrying space. All
simulations and calculations were performed on a 64-bit
Windows system, with Intel Core i7-2600 CPU @3.4G Hz. All
program of numerical computation is implemented in Matlab
software (MathWorks, Natick, Massachusetts, USA).
Fig.2. Standard coils structure
TABLE 1
PARAMETER OF HALBACH MAGNET
Inner diameter
78 mm
Outer diameter
118 mm
Height
(3 layers)
(70,25.179,70) mm
Strength
0.5 T
ROI
R 2.5 mm
IV. RESULTS
A. MCs targeting SH (1, -1/0/1)
In this section MCs design for the 1st degree SHs including
three orders SH (1, -1/0/1) [22] known as the y, z, x gradient
field in the MR Imaging system is executed. The current density
mode parameter (V, H) in equation (7-9) is chosen as 6
(similarly hereinafter), and (Q, K) is set as (2, 4) according to
the zonal and tesseral freedom of target SHs. Equivalent
weights for the three targets SHs are applied. Current-carrying
cylinder height and radius are set as 0.06 m and 0.03 m to fit
the parameters of the Halbach magnet listed in Table 1.
Due to the symmetric property of the target SH field, the coil
elements may also tend to structures with similarity. Thus,
design of the coils could be simplified to calculate only the
shape parameters, specifically current density coefficients
for one quadrant elements. With this setting, optimized MCs
structure for the 1st degree SHs is shown in Fig. 3, with the
standard MCs’ first element also plotted to exhibit the
differences. From the optimized MCs structure, aggregations of
coils windings to /2 , 3/2 imply the high-efficiency current
density positions in the specified target SHs shimming.
Driven currents are computed for SH (1, -1/0/1) with field
strength set as 1 ppm for each SH function to verify
performance of the improved MCs, compared to the standard
MCs. As shown in fig.4, the dot solid line are results for
optimized MCs, and square dash line for standard MCs
(similarly hereinafter).
Generally, the resulting currents are of the same direction for
optimized and standard MCs, which makes sense because all
coils winding directions are fixed clockwise. Take SH (1, 1)
term as an example, driven currents’ directions of coil1~coil8
are [--++--++] (‘-’ negative current, ‘ + ’ positive current),
consistent with regular shimming coils structure results listed in
[23]. Comparing with the standard MCs, currents of SH (1, -1)
term decrease prominently in optimized MCs, while for terms
SH (1, 0/1) currents’ amplitudes are also slightly reduced. The
maximal and total currents in optimized MCs are accordingly
reduced to 15.8% and 29.9%, which makes it possible to cut
down the coil wire width from 1.4 mm (standard MCs with coil
winding loops Nc = 6) to 0.61 mm (optimized MCs, same turns)
for the target field. Meanwhile, the requirements on the current
amplifiers can also be relaxed, cast off the hardware burden
which stays to be a big problem considering the tens of channels
in MCs operation.
Fig.3. The optimized MCs with 6 turns for the target SH (1, -1/0/1) field. In
the plotting, all matrix coils element windings are set clockwise. Element 1 of
standard MCs (dashed line) is also plotted while the other elements of the
standard MCs are of the same pattern.
Fig.4. Driven currents for SH (1,-1/0/1), in which the dashed line are results
of standard MCs, and solid lines imply results of optimized MCs. Notably, the
driven currents of SH (1, -1) have been improved prominently in the optimized
MCs.
Further comparisons of the power dissipation and shimming
accuracy between the optimized and standard MCs are shown
in fig. 5. Similarly, the power consumption for SH (1, -1) is also
-0.2
0
coil1 coil2 coil3 coil4 coil5 coil6 coil7 coil8
Current (A)
for SH(1, -1)
Optimized Standard
-0.03
0.03
coil1 coil2 coil3 coil4 coil5 coil6 coil7 coil8
Current (A)
for SH(1, 0)
-0.03
0.03
coil1 coil2 coil3 coil4 coil5 coil6 coil7 coil8
Current (A)
for SH(1, 1)
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Transactions on Biomedical Engineering
reduced while the other terms maintains almost the same
performance. Total power dissipation and current also show an
improvement by an order of magnitude. As in the compact
magnet, the temperature rise caused by coil power consumption
is likely to introduce serious consequences, e.g., heating
samples, magnetic field shift, such that the proposed optimized
MCs can be beneficial at this point in the operation. In SH (1,
0) terms compensation, the initial 1 ppm inhomogeneity is
decreased to comparable 0.0408 ppm and 0.0349 ppm for the
optimized and standard coils, respectively. As for the SH (1, 1)
term, the optimized MCs reduce the target 1 ppm to 0.0313 ppm
comparing with the standard MCs achieving 0.009 ppm, that
the latter sustain a better performance. Whereas for SH (1, -1),
0.0214 ppm is achieved by the optimized coils in contrast to
0.202 ppm reached by the standard MCs.
Fig.5. Comparison of power dissipation and the after-shimming homogeneity
on the three target SHs of the optimized and standard MCs.
A figure of merit (FoM) is defined as
similar to [14]
including the performance of current efficiency ,
resistance , and shimming accuracy to quantitatively
compare the two MCs structures. Here is calculated based
on the maximum amplitude of SHs per unit current, resistance
is expressed as =
(
+
)
=
.
Shimming accuracy is defined aforementioned. With weight
, , , F as 5×10-9, 100, 0, and 9.2501, FoM of the
optimized and standard MCs are obtained as 3.6×10-4 and 8.9
×10-5, respectively. The improvement of FoM is due to
amelioration of all three parameters especially shimming
accuracy. In brief, the optimized MCs could balance the overall
performance for multi-target SHs shimming with the
outstanding FoM.
TABLE 2
COMPARISON BETWEEN OPTIMIZED MCS AND STANDARD MCS TARGETING
SH (1, -1/0/1) WITH NC=6
Optimized
Standard
Total Power
dissipation(w)
4.16e-03 7.11e-02
Shimming
Accuracy ()
4.08E-02 2.02E-01
Current Amplitude
(A)
2.34e-02/-2.34e-02 0.148/-0.148
Current efficiency
1.05e-05
7.11e-06
FoM
3.6e-04
8.86e-05
To further demonstrate influence of the coil geometry shape,
field on the R2.5 mm sphere is calculated with driven current 1
Ampere for both standard and optimized MCs element coil1 in
fig. 3. Field are fitting with SHs basis up to the 4th degree as
displayed in Fig. 6. We can find more SH (1, -1) term in the
optimized MCs, and a slight increase of SH (1, 0/1) component.
From analysis on the specified SHs we can roughly qualify the
coil capability on field compensation, but the shimming
operations depend also on element number, relative position,
the adjustment of currents’ amplitude and directions.
In multi-target MCs design, term in equation (7-9) is
important for tesseral terms. Take the 1st SH terms as an
example, zero phase shift is believed to be more efficient for the
SH (1, 0/1) terms [23]. Hence, to find whether other phase shifts
can lead to even better performance, we implement phase shift
()=/4 . According to the definition of the objective
function (10), power dissipation, shimming accuracy, and
maximal current amplitude compose the final MCs
performance, which relies strongly on the value of weighting
factors. Hence, a series of weighting values w0, w1, w2 are tested
for phase shift /4 as listed in Table 3. MCs performance for
phase shift of zero and standard coils are also depicted in fig.7,
labeled as solid lines 8, 9, respectively. To exhibit the
differences more clearly, each parameter has been preprocessed
with (Data Data)/Data , and the logarithmic axis of the
radar chart is set in the figure. As seen, the phase shift /4 can
achieve better coils performance comparing with standard MCs
with the carefully adjusted weighting values.
Additionally, the MCs structures resulting from index 4 in
table 3 are depicted in fig.8. Comparing with the results in fig.
3, low current density generally appears in azimuthal
positions and 2 , indicating relatively low shimming
efficiency for the MCs element position arrangement. Though
the total power dissipation of MCs is reduced according to fig.7,
the maximal and total currents increase at the same time, and
shimming accuracy is also reduced. To sum up, the phase shift
is inappropriate for the 1st degree SHs target field comparing
with zero phase shift MCs. As shown by this example, the phase
shift for the MCs design needs to be selected carefully in the
design to a balance among the multi-SHs target fields.
Fig.6. SH functions fitting of R2.5 mm sphere resulting from standard and
optimized MCs element coil1 in fig. 3 up to the 4th degree. More SH (1, -1/0/1)
terms can be found for the optimized coils geometry shape.
TABLE 3
WEIGHTING VALUE OF OPTIMIZED MC WITH PHASE SHIFT 0, /4 AND
STANDARD MC TARGETING AT SH (1, -1/0/1)
index
phase shift
w0
w1
w2
1
=/4
5×10-15
1
5×107
2
5×10-9
1
5×107
3
5×10-8
1
5×107
4
5×10-8
1
5×106
5
5×10-8
1
5×105
6
5×10-8
10
5×105
7
5×10-8
20
5×105
0
0.02
0.04
0.06
0.08
0
0.05
0.1
0.15
0.2
0.25
-1 1 3
Power dissipation (W)
residual homo (ppm)
target SHs
PowerOptimized
PowerStandard
HomoOptimized
HomoStandard
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Transactions on Biomedical Engineering
8
= 0
5×10-9
100
0
9
Standard MCs
-
-
-
Fig.7. Performance comparison of phase shift 0, /4 and standard MCs
including power dissipation, shimming accuracy, and maximal current Imax. The
index of each curve can be checked in column 1, Table 3.
Fig.8. MCs result of SH (1, -1/0/1) corresponding to index 4 in table 3 for
phase shift =/4.
B. MCs target SH(3,-3/-2/-1/0/1/2/3)
Seven orders in the 3rd degree SHs are included to form the
high degree field compensation. The same proportions for all
SH basis except for SH (3, 0) term twice the amount are set,
simply denoted as (1:1:1:2:1:1:1). The cylinder height and
radius are kept the same as section A. Considering the
complexity of the target field order and degree, (Q, K) is chosen
as (4, 8). With weight , , , F as 2.29×10-7, 5.00×10-8, 5
×10-5, and 10.1415, the optimized MCs windings are achieved
as shown in fig. 9 with Nc equates 3. In the optimization, the
symmetric property is also introduced among four quadrants.
Hence, coil patterns are shown only in one quadrant, while
structures in the other three quadrants are of similar patterns.
We could see the optimized MCs are virtually square shapes
with marginal fluctuation on the axial boundaries. Typically, the
regular coil elements may attribute to the symmetric target field
distribution and strength settings of each term.
Fig.9. MCs optimized result with target field SH (3, -3/-2/-1/0/1/2/3). This
figure shows only one quadrant of the coil structure due to its symmetry.
For the optimized coil shapes in fig.9, an incresed density of
windings at the boundaries can be observed. For further
comparison current densities of the first element extracted from
optimized and standard MCs are also depicted in fig.10, where
it becomes obvious that the optimized MCs result in higher
current density distributions at the boundaries for both current
directions. Refering to the coil loops approximation and
currents calculation approach as contour line division of stream
function value differences, the high current density may require
a reduced wire width with the same current amplitude. And this
MCs wire width could be qualified considering the current
carrying capacity of conductor, e.g. copper. For cases of high
inhomogeneity, the coils implementation may become a
problem. When this happens, an increased weighting value of
the current density constraint is necessary in the coil
optimization to avoid surpassing the physical limitations of the
conductor.
Fig.10. Current density in the azimuthal and axial direction for optimized
and standard coils. The shift to boundaries in the optimized coils corresponds
to a more efficient current density distribution in the target field approximation.
Furthermore, driven currents and power dissipation are
calculated and compared with the standard MCs with field
power
dissipation(w)
Imax(A)
shimmed
accuracy(ppm)
1 2 3
4 5 6
7 8 9
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inhomogeneity 0.2 ppm for each proportion. Similar to results
of the 1st degree field shimming, the optimized MCs can reduce
the total power dissipation and current prominently, while both
achieve high shimming accuracy. To make it clear, total current
consumption is defined as a sum of driven currents level under
all target SHs, for which the optimized MCs cut the total current
to 35.8%, and the maximum current is also reduced to 26.2%.
With further analysis of the driven current for each target field,
the maximal current is observed for the SH (3,-3) term, coherent
with power consumption performance in fig.11. Yet, the
optimized coils again fit the worst performance term of standard
MCs, providing a balanced structure for all target field patterns.
The FoM is listed in Table 4, from which we could see that the
optimized MCs improved the overall performance prominently.
Along with power and current efficiency ameliorated,
shimming accuracy introduce several orders of magnitude
differences, which lead to the big improvement of the FoM for
optimized MCs. The above results of current, power dissipation
and FoM indicate the optimized MCs have adapt to the target
SH basis effectively.
Fig.11. Power dissipation comparison of the 3rd SHs between the optimized and
standard MCs. Optimized MCs achieves good performance on SH (3, -3) term
comparing with the standard Mcs.
TABLE 4
COMPARISON BETWEEN OPTIMIZED MCS AND STANDARD MCS TARGETING
SH (3, -3/-2/-1/0/1/2/3) WITH NC=3
Optimized
Standard
Total Power dissipation(W) 5.17 27.4
Shimming Accuracy () 3.26e-15 1.29e-09
Current Amplitude (A) 6.59e-01/ -6.59e-01 2.519/ -2.519
Total Current (A)
30.31
84.69
Current efficiency
2.29e-07
1.5e-07
Minimum Wire Width (mm)
0.5
1.3
FoM
53.5
0.0676
C. MCs target Mixed SH basis
In this section, a mixed target field is assembled, including
all the orders in the 1st and 2nd degree. And the field proportions
for the mixed 7 order SHs are set as (2.5,2.5,2.5,1,1,1,1,1). To
adapt to the high order 2nd degree, the value of (Q, K) is chosen
as (4, 6) and zero phase shift is applied. One quadrant of the
optimized MCs is shown in Fig.12 with weight , , , F
as 8×10-7, 3×10-7, 2×109, and 15.6618, in which the coil shape
shows symmetry in both azimuthal and axial directions.
Fig.12 MCs optimized result with mixed target field of SH (1, -1/0/1) & SH
(2, -2/-1/0/1/2).
Fig.13 Power dissipation comparison in the mixed 1st and 2nd degree between
the optimized and standard MCs, in which the optimized MCs reduce the power
dissipation a lot on the SH (2, -2) term. Shimming accuracy of the target SHs,
where the SH (1, 1) homogeneity is improved obviously in optimized MCs.
Fig.14 Current of optimized and standard MCs for SH (2, -2), in solid and
dashed line, respectively. A notable reduction of maximum current amplitude
could be found in optimized MCs results.
With field inhomogneity 0.4 ppm for each proportion, power
dissipation for standard and optimized MCs are compared
quantitatively, and a prominent reduction for SH(2, -2) term
appears in fig. 13, which is further analyzed in fig. 14. Generally,
comparison with standard coils results in a 38.5% decrease of
power dissipation, 41.7% reduction of maximum current, and
8.5% cut of the total current. Optimized MCs also achieve
better shimming accuracy, especially for the SH(1, 1) term, with
the FoM also improved from 1.75×10-4 to 5.13×10-4 for standard
and optimized MCs, respectively.
Hence, in the mixed target field shimming performance the
MCs results in SHDM also yield a significant improvement for
the given SH basis. We can arrive to the conclusion that the
proposed approach can provide an efficient optimization
0
10
20
30
Power Dissipation(W)
Target SHs
Optimized Standard
0
0.2
0.4
0.6
0.8
1
1.2
0
0.01
0.02
0.03
0.04
Power Dissipation (W)
residual homo (ppm)
Target SHs
PowerOptimized
PowerStandard
HomoOptimized
HomoStandard
-1
-0.5
0
0.5
1
coil1
coil2
coil3
coil4
coil5
coil6
coil7
coil8
coil9
coil10
coil11
coil12
coil13
coil14
coil15
coil16
coil17
coil18
coil19
coil20
coil21
coil22
coil23
coil24
Current for SH (2, -2)
Optimized Standard
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Transactions on Biomedical Engineering
method for multi-SHs MCs design.
V. EXPERIMENTS
To certify the MCs' performance, a prototype for the first
degree is fabricated with a 3D printed base board shown in fig.
15a. The R0.27 mm enameled copper wire was manually
inserted into the slot and sealed up with epoxy resin. Powered
with a switched power supply (GPD-2303s, Gwinsteck,
Taiwan), field distribution on radius 2.5 mm sphere and three
orthogonal planes were measured with a Tesla meter (CH-3600,
Ch hall inc., Beijing).
For the 1st SH field, driven currents for each order are
already achieved as shown in fig.4. Magnetic field distributions
on R2.5 mm sphere (b.1, c.1, d.1) and planes (b.2-b.3, c.2-c.3,
d.2-d.3) are measured and depicted for SH (1, -1/0/1). Generally,
the field distribution can fit the target SH shape well, on both
sphere and the corresponding planes. However, marginal
nonlinearity could also be found in the planar field maps
especially the SH (1, -1) term.
To further verify the theoretical results, the intensities of the
target SHs are analyzed and transformed to match the driven
currents’ strength in section IV.A, in which field amplitudes of
all targets SHs are set to 1 Gs/m, corresponding to 1ppm in R2.5
mm ROI. The measured intensities of SH (1, -1/0/1) field are
listed in Table 5, and the differences with theoretical value are
computed. The results show a good agreement with theoretical
values, which play an important role in the actual shimming
process.
TABLE 5
COIL EFFICIENCY COMPARISON AND ERROR
Theoretical
(Gs/m)
Measured
Plane, Gs/m
Sphere, Gs/m
SH(1,-1)
1
1.077(+7.7%) 0.936(-6.3%)
SH(1,0) 0.925(-7.1%) 1.025(+2.5%)
SH(1,1)
1.025(+2.5%)
1.040(+4.24%)
However, minor deviations also could be found on both
planar and spherical field distributions. Sources for the field
strength and linearity discrepancy may involve the following
factors: slot deviation, winding loops deflection, coil
interconnection, and precision of the measuring device. And
this could be further improved with better fabrication
technology and measurement conditions.
VI. DISCUSSION
A SHDM Method for the MCs design was proposed in this
paper. In traditional MCs, regular coil shape [4-9] was carried
out lack of pertinence in actual applications. In [14], coils shape
optimization was implemented based on approximation error
reduction for a composite field pattern of 1st and 2nd degree SH,
but the optimizations process and results were not
straightforward. In this paper, the SHDM provide MCs design
to adapt to specified field distribution, and according to the
comparisons with critical standard MCs for the 1st, 3rd, and
mixed 1st &2nd type field, the optimized MCs show prominent
improvement. Further calculations demonstrated that the coil
geometry shape was amended to adapt to the worst performance
SH term of the standard MCs, leading to an overall better
shimming capability. Notably, this improved MCs do not mean
to design for a specified target field shape but try to obtain a
relatively more peculiar construction to get with certain SH
field comparing with ordinary MCs.
In the MCs structure, coils’ number, and size of the current-
carrying surface play an important role in shimming
applications. Generally, number of coil elements should cover
the maximal target SHs. If the degrees of freedom provided by
the coils are lower than the target SH complexity, shimming
accuracy will be impeded substantially. On the contrary, if too
many coil elements are selected, field shape degrees of freedom
will not be utilized efficiently, which is likely to lead to a
decreased current efficiency. In practice, the field components
can be even more complicated, such that inappropriate coils
Fig.15 Implementation for SH (1, -
1/0/1) target field coils. a) the 3D
printed MCs prototype, b) field
distribution of SH (1, -1) degree on
R2.5 mm sphere, and the field on yoz
and xoy plane are illustrated in b.2-
3; c)-d were results of SH (1,0) and
SH (1,1) target field, and their
corresponding orthogonal slice field
distributions.
a
b.1
c.1
d.1
b.2
c.2
d.2
b.3
c.3
d.3
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Transactions on Biomedical Engineering
number selection may result in a bad MCs construction. In
general, n+1 zonal (m=0) and 2(m+1) tesseral (m≠0)
harmonics according to the actual target SHs is a relatively good
choice [16]. Current-carrying surface are pertinent to the
magnet size, sample position, and the target field to be shimmed.
Inadequate boundary conditions of MCs may lead to a coil-
splitting resulting from the algorithm’s attempts, as fig.8 shows.
Hence, it is a comprehensive problem to select proper current-
carrying surface and MCs’ element boundary. Take the cylinder
current-carrying surface in this paper as an example, FoM
relying on the diameter and length of the cylinder can be
evaluated to search for proper dimensions. As for the azimuthal
boundary division, it is relevant to the shape and strength of the
tesseral harmonics. Some interleaved MCs have been proposed
to adapt to multiple tesseral harmonics [5,24], but these
attempts have no detailed analysis for the coil layout. In the
SHDM algorithm, the phase shift of MCs elements can be
iteratively searched in the MCs optimization to achieve the best
azimuthal phase value.
Additionally, design with overlapping neighboring coils can
be applied with the benefits of decoupling [25], and this could
be classified as a multi-layer design problem. Appropriate
element size definition can also be sought with the SHDM
algorithm for this kind of problem. For the compact cylinder
Halbach magnet, a crowded coil structure may be generated due
to the limited space. This problem may potentially be addressed
by categorizing SH terms with different phase merits to several
groups, to obtain a multi-layer MCs structure. It is a
compromised way similar to a traditional SHs shimming and
further discussion about this is out of the scope in this paper.
When the MCs structure is fixed, the contribution for each
SH term of coil element is also determined. Hence, in the actual
MCs operation, driven currents optimization could also be
accelerated due to the reduced number of variables in SHs space,
and this may be beneficial in dynamic shimming applications
[7].
Many works on MCs have applied circular [4-9] or saddle
[24] coil elements, mostly for field shimming. In non-linear
imaging application [10,11], MCs targeting at strong local
gradients and low eddy current were of more complicated
double-spiral coil structure. For shimming purposes in this
paper, no available MCs could be found for Halbach shimming.
So quasi-circular shape standard coil (current density
coefficients
= 1) has been chosen as a contrast structure.
Exact comparison with the circular, saddle or any other existing
MCs is not considered in this paper.
The idea of analysis and optimization in SH space brings in
the transformation problem between the magnetic field and
spherical harmonic polynomials. The forward process has a
strict requirement of the number of sampling points because of
the periodic nature of SH polynomials [26]. Therefore, in the
original inhomogeneity measurement, the sampling rate and
transform accuracy should be high enough to guarantee the
target field approximation precision.
Generally, the SHDM method is based on cylindrical current
carrying surface and the analysis between current density and
SHs are pertinent to the geometry shape. Hence for other type
of magnet, planar for example, current density model and
relationship between current and target field type should be
reanalyzed [27]. For irregular shape of current-carrying surface,
the current density model could not simply express with Fourier
series expansions, that other more flexible source point model
could be considered to adapt to the design, such as boundary
element method [28], magnetic dipole method [18] etc. And this
kind of design will be considered in our future work.
Despite the advantages of MCs including flexible field
shaping ability, low inductance due to the smaller loops, the
local coils construction sacrifices the current efficiency
comparing with traditional SH shimming coils, to a certain
content [6]. Hence, the application for MCs could focus on local
shimming, compact magnet or dynamic shimming problem. For
whole-body, static shimming circumstance, specially designed
SH shimming coils will be a better choice.
VII. CONCLUSION
Our work proposes a high efficiency MCs optimization
technology and a simplified current optimization algorithm to
implement the MCs shimming, which is applicable for limited
space applications. It may be also meaningful for other
applications of MCs to simplify the shimming process.
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Transactions on Biomedical Engineering
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