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Journal of the Magnetics Society of Japan

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Vol.48 No.3
JournalElectronic Journal URL: https://www.jstage.jst.go.jp/browse/msjmag

2024

Magnetic Recording
Inﬂuence of Design Parameters on Adjacent Track Interference in Heated-Dot Magnetic Recording
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T. Kobayashi and I. Tagawa…40

M. Washino, K. Nomura, T. Matsuda, S. Seino, T. Nakagawa, T. Kiwa, A. Tanaka, and T. Sakane…47

Biomagnetism / Medical Applications
Viscosity Eﬀects on Relaxation Time Diﬀerences of Magnetic Nanoparticles

Design Method of Surface Receive Coil with High SNR for Various Field Intensities in MRI

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M. Takahashi, M. Fushimi, S. Yabukami, M. Sekino, and A. Kuwahata

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Journal of the Magnetics Society of Japan
Vol. 48, No. 3

Electronic Journal URL: https://www.jstage.jst.go.jp/browse/msjmag

CONTENTS
Magnetic Recording
Influence of Design Parameters on Adjacent Track Interference in Heated-Dot Magnetic Recording…………………………………………………………………………… T. Kobayashi and I. Tagawa

40

Biomagnetism / Medical Applications
Viscosity Effects on Relaxation Time Differences of Magnetic Nanoparticles…………………………M. Washino, K. Nomura, T. Matsuda, S. Seino, T. Nakagawa, T. Kiwa,
		

A. Tanaka, and T. Sakane

47

Design Method of Surface Receive Coil with High SNR for Various Field Intensities in MRI……………………… M. Takahashi, M. Fushimi, S. Yabukami, M. Sekino, and A. Kuwahata

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Copyright ©2024 by the Magnetics Society of Japan.
This article is licensed under the Creative Commons Attribution International License (CC BY 4.0)
http://creativecommons.org/licenses/by/4.0/

J. Magn. Soc. Jpn., 48, 40-46 (2024)<Paper>Influence of Design Parameters on Adjacent Track Interferencein Heated-Dot Magnetic Recording
T. Kobayashi and I. Tagawa*

Graduate School of Engineering, Mie Univ., 1577 Kurimamachiya-cho, Tsu 514-8507, Japan*Electrical and Electronic Engineering, Tohoku Institute of Technology,35-1 Yagiyama-Kasumicho, Sendai 982-8577, Japan

We discuss the influence of design parameters on adjacent track interference (ATI) in 4 Tbpsi heated-dot magnetic
recording where the parameters are the mean Curie temperature, Curie temperature variation, anisotropy constantratio, dot size variation, Gilbert damping constant, writing field magnitude, and writing field angle. We calculate thedot height to achieve a bit error rate of 10-3 after adjacent track writing as a function of the design parameters. The
dot height must be increased when we choose a lower mean Curie temperature, since the thermal gradient decreases
simultaneously. The adjacent track temperature is related to the Curie temperature variation via the writing
temperature. ATI is strongly affected by the anisotropy constant ratio. The dot height must be increased as the dot
size variation increases, since the probability of a small dot appearing increases. The Gilbert damping constant has
an effect on ATI. Since a writing field magnitude of 10 kOe is relatively small against the anisotropy field, the increasein the dot height is relatively small when the writing field magnitude increases from 10 to 15 kOe or the writing field
angle changes from 180 to 135 deg.
Key words: HDMR, ATI, mean Curie temperature, Curie temperature variation, anisotropy constant ratio, dot size
variation, Gilbert damping constant, writing field magnitude, writing field angle
1. Introducti on

Many magnetic recording methods have been
proposed to solve the trilemma problem1) of conventional
magnetic recording (CMR) on granular media. Thesemethods include shingled magnetic recording (SMR),
microwave-assisted magnetic recording (MAMR), heatassisted magnetic recording (HAMR), bit patterned
media (BPM), and three-dimensional magnetic
recording (3D MR).
The challenges facing the design of MR media are(1) information stability during 10 years of archiving,known as the 𝐾𝐾! 𝑉𝑉 ⁄(𝑘𝑘𝑘𝑘) problem1), where 𝐾𝐾! , 𝑉𝑉, 𝑘𝑘, and𝑇𝑇 are respectively the grain or dot anisotropy constant,volume, Boltzmann constant, and temperature,(2) information stability in an adjacent track during
writing, known as the adjacent track interference (ATI)
problem, and(3) the writing field dependence of the bit error rate(bER), namely writability.
Micromagnetic calculation is useful for examining (2)in SMR and (3). However, this is not practical due to thelong calculation time required for subjects (1) and (2) in
CMR because of the 103-104 times rewrite in the adjacent
track. We have proposed a model calculation employing
the Néel-Arrhenius model with a Stoner-Wohlfarth
grain or dot. This model is applicable to all threesubjects2) including SMR and CMR.
The above three subjects, namely (1), (2), and (3), must
be dealt with simultaneously, since they are in a tradeoff relationship. For example, if the design parameter of

the anisotropy constant ratio is larger, the informationin (1) and (2) is more stable, but (3) the writability will
be worse even for HAMR. The anisotropy constant ratio
𝐾𝐾! /𝐾𝐾"!#$ , which we introduced3), is the intrinsic ratio of
the medium anisotropy constant to the bulk FePt
anisotropy constant. The 𝐾𝐾! /𝐾𝐾"!#$ value is independentof the Curie temperature 𝑇𝑇% , and is constant for any
temperature from zero Kelvin to 𝑇𝑇% . The design
parameters are related to each other in a complex
manner. It is necessary to examine the influence of thedesign parameters on the above three subjects when
designing the medium.
Akagi et al. reported (3) the recording performance ofheated-dot magnetic recording (HDMR)4), namely
HAMR on BPM, employing micromagnetic calculation.
We have previously discussed information stability (1)during 10 years of archiving and (2) during adjacent
track (AT) writing for HDMR5) employing our model
calculation, in which we have calculated the dot heightto achieve a bER of 10-3 after AT writing as a function of
the thermal gradient for the cross-track direction.
In this paper, as a first step in examining the trade-off
relationship between (2) ATI and (3) the writability, we
discuss the influence of the design parameters on (2) ATI
in 4 Tbpsi HDMR where the parameters are the mean
Curie temperature, Curie temperature variation,
anisotropy constant ratio, dot size variation, Gilbert
damping constant, writing field magnitude, and writing
field angle. We calculate the dot height to achieve a bit
error rate of 10-3 after adjacent track writing as afunction of the design parameters.

Corresponding author: T. Kobayashi (e-mail: kobayasi@phen.mie-u.ac.jp).

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2 . C a l c u l a t i o n C o n d i t i o n a n d Me t h o d
2 . 1 Do t a r r a n g e me n t a n d me d i u m s t r u c t u r e
Figure 1 shows the dot arrangement and medium
structure in 4 Tbpsi HDMR where 𝐷𝐷& , 𝐷𝐷' , and ℎ are
the dot sizes for the down-track and cross-track
directions, and the dot height, respectively. The bit
length 𝐷𝐷( and track width 𝐷𝐷) were both 12.7 nm. We
assumed that the mean dot size 𝐷𝐷* and mean dot
spacing Δ+ are the same for both the down-track andcross-track directions, namely 𝐷𝐷* = Δ+ = 6.35 nm. Theℎ values were 5.1 and 2.8 nm for the standard values inconventional and shingled HDMR, respectively.
There are two cases for the dot sizes 𝐷𝐷& and 𝐷𝐷'
according to the dot manufacturing method. (1) In one
case, the 𝐷𝐷& and 𝐷𝐷' sizes are the same, and the 𝐷𝐷& =𝐷𝐷' size fluctuates. (2) Another case is that the 𝐷𝐷& and𝐷𝐷' sizes fluctuate independently. We examined (1) the
𝐷𝐷& = 𝐷𝐷' case, since the bER is larger for the same ℎvalue5). We generated a random number 𝐷𝐷& = 𝐷𝐷'
according to a log-normal distribution with a standard
deviation 𝜎𝜎+ . We used a 𝜎𝜎+ /𝐷𝐷* value of 15 % for the
standard value.
2 . 2 Ma g n e t i c p r o p e r t i e s
The temperature dependence of the medium
magnetization 𝑀𝑀, was calculated by employing mean
field analysis6), and that of the 𝐾𝐾! value was assumed
to be proportional to 𝑀𝑀,- 7). 𝑀𝑀, (𝑇𝑇% = 770 K, 𝑇𝑇 = 300 K) =1000 emu /cm3 was assumed for FePt. Based on thisassumption, the 𝑀𝑀, value can be calculated for all
values of 𝑇𝑇% and 𝑇𝑇.
𝐾𝐾! (𝑇𝑇% = 770 K, 𝐾𝐾! /𝐾𝐾"!#$ = 1, 𝑇𝑇 = 300 K) = 70 Merg/cm3
was assumed for bulk FePt. Using this assumption, we
can calculate 𝐾𝐾! for all values of 𝑇𝑇% , 𝐾𝐾! /𝐾𝐾"!#$ , and 𝑇𝑇.
No intrinsic distribution of 𝐾𝐾! was assumed. However,there was a fluctuation in 𝐾𝐾! caused by 𝑇𝑇% variation.
The 𝑇𝑇% value of each dot can be adjusted by changing
the Cu composition 𝑧𝑧 for (Fe..0 Pt ..0 )123 Cu3 .
With a 𝑇𝑇% value of 750 K and a 𝐾𝐾! /𝐾𝐾"!#$ value of 0.8,
in this work we obtain a 𝐾𝐾! value of 51 Merg/cm3 andan anisotropy field 𝐻𝐻$ of 107 kOe at a readout

temperature of 330 K.
2 . 3 T e mp e r a t u r e p r o f i l e
The writing temperature 𝑇𝑇4 for the dot was assumed
to be

as shown in Fig. 1 where 𝑇𝑇%* and 𝜎𝜎)% are the mean
Curie temperature and the standard deviation of 𝑇𝑇%* ,respectively, taking account of the 𝑇𝑇% variation. The 𝑇𝑇%
distribution was assumed to be normal. Based on thisassumption, 99.9 % of dots in the writing track areheated to above their 𝑇𝑇% values during the writing
period. We used 𝑇𝑇%* and 𝜎𝜎)% /𝑇𝑇%* values of 750 K and
2 %, respectively, for the standard values.
For simplicity, the thermal gradient d𝑇𝑇/d𝑦𝑦 in the
cross-track direction was assumed to be constantanywhere. The thermal gradient in the down-track
direction was zero, since the exposure time for writing
has little effect on the results as shown below in 3.1.
Since the d𝑇𝑇/d𝑦𝑦 value can be adjusted by changing the
medium structure, we used a d𝑇𝑇/d𝑦𝑦 value of 14 K/nm
for the standard value.
When the 𝑇𝑇%* value decreases from high Curie
temperature 𝑇𝑇%*5 to low 𝑇𝑇%*6 , the thermal gradients
also decrease from d𝑇𝑇5 (𝑦𝑦)/d𝑦𝑦 to d𝑇𝑇6 (𝑦𝑦)/d𝑦𝑦 as
explained below. If the medium structure is the same,the difference between the medium temperature 𝑇𝑇7 (𝑦𝑦)and ambient temperature 𝑇𝑇8*" is proportional to thelaser power 𝑃𝑃47 for heating regardless of the medium
position 𝑦𝑦 where 𝑖𝑖 = H for media with 𝑇𝑇%*5 and 𝑖𝑖 = L
for 𝑇𝑇%*6 . Therefore, we can obtain the following equation.
!! (#)%!"#$!% (#)%!"#$

=

&&!

&&%

.

Since at the center of the track,!! (#)%!"#$!% (#)%!"#$=𝑇𝑇6 (𝑦𝑦) − 𝑇𝑇8*" =we can obtain/!! (#)=

Fig. 1 Dot arrangement and medium structure.

(1)

𝑇𝑇4 = 𝑇𝑇%* + 3𝜎𝜎)%/#=!&! %!"#$!&% %!"#$=!'#! '()('! %!"#$!'#% '()('% %!"#$,

𝑇𝑇cmL +3𝜎𝜎TcL −𝑇𝑇amb∙ (𝑇𝑇5 (𝑦𝑦) − 𝑇𝑇8*" ),𝑇𝑇cmH +3𝜎𝜎TcH −𝑇𝑇amb!'#! '()('! %!"#$!'#% '()('% %!"#$

𝑇𝑇cmL (1+3×0.02)−330× 14,
750×(1+3×0.02)−330∙/!% (#)/#,(2)

for 𝑇𝑇%*5 = 750 K, where 𝑇𝑇45 and 𝑇𝑇46 are the writing
temperatures for media with 𝑇𝑇%*5 and 𝑇𝑇%*6 ,respectively, and 𝜎𝜎)%5 and 𝜎𝜎)%6 are the standard
deviations for media with 𝑇𝑇%*5 and 𝑇𝑇%*6 , respectively.
We assumed that the 𝑇𝑇8*" value was 330 K.
Although we do not deal the dependence of the 𝑇𝑇0variation on the Cu composition in this paper, we pointout this in the following, since this will be important in
actual HAMR and HDMR. When a third element is

Journal of the Magnetics Society of Japan Vol.48, No.3, 2024

41

INDEX

taking account of the effective anisotropy field 𝐻𝐻$:<< andwriting field angle 𝜙𝜙 as shown in Fig. 3 where 𝛾𝛾, 𝛼𝛼,and|𝐻𝐻4 |
are respectively the
𝑉𝑉 = 𝐷𝐷& 𝐷𝐷' × ℎ,gyromagnetic ratio, Gilbert damping constant, dot
volume, and writing field magnitude. 𝐾𝐾; is the thermal
stability factor given by
𝐾𝐾; =Fig. 2 Curie temperature distribution for various mean
Curie temperatures 𝑇𝑇%* .

added to FePt to reduce its 𝑇𝑇%* , some dots contain moreor less atoms of a third element than a mean number.
Reducing 𝑇𝑇%* by adding a third element intrinsicallyresults in 𝑇𝑇% variation and the 𝑇𝑇% variation may lead toan increase in bER. Figure 2 shows the 𝑇𝑇% distribution
for various 𝑇𝑇%* values, in which the third element
variation was calculated statistically and 𝑇𝑇% was
calculated by employing mean field analysis for an Fe
site number 𝑛𝑛9: of 1000 in a dot. The 𝑇𝑇% distribution ofcourse becomes zero for FePt (𝑇𝑇% ≈ 770 K) with no third
element. The 𝑇𝑇% variation increases as the third
element number increases and the 𝑇𝑇%* value decreases.
The 𝑇𝑇% standard deviation 𝜎𝜎)% is inversely proportional
to L𝑛𝑛9: , namely √𝑉𝑉. We used a 𝜎𝜎)% /𝑇𝑇%* value of 2 % for
the standard value. This problem is a subject for future
study.

2 . 4 AT I e v a l u a t i o n me t h o d
The information stability for 10 years of archiving has
been discussed employing the Néel-Arrhenius modelwith a Stoner-Wohlfarth grain or dot. The attempt
period 1/𝑓𝑓. has a value in picoseconds for FePt in heatassisted magnetic recording. Since the magnetization
direction attempts to reverse with a certain probability
at each attempt period, the information stability for 10
years of archiving is extrapolated as a stack ofphenomena in picoseconds. Therefore, the NéelArrhenius model is valid for any time from the order ofa picosecond to more than 10 years. Therefore, we havealso applied the Néel-Arrhenius model to phenomena
with a short time, and examined information stability
during AT writing.
The magnetization reversal number 𝑁𝑁𝑁𝑁 for the dot
from time 0 to 𝑡𝑡 is expressed as

𝑁𝑁𝑁𝑁 = 𝑓𝑓. 𝑡𝑡 expS−𝐾𝐾; T,(3)

employing the Néel-Arrhenius model where 𝑓𝑓. is theattempt frequency8). We assumed 𝑓𝑓. as
𝑓𝑓. =

42

3

𝛾𝛾𝛾𝛾 U𝑀𝑀s 𝐻𝐻keff 𝑉𝑉|𝐻𝐻w |cos𝜙𝜙|𝐻𝐻w |cos𝜙𝜙 V1
+
W
X1
−
V
W Y,
2
𝐻𝐻keff
𝐻𝐻keff2𝜋𝜋𝜋𝜋𝜋𝜋1+𝛼𝛼(4)𝐸𝐸1 −𝐸𝐸0
,
𝑘𝑘𝑘𝑘

(5)

where 𝐸𝐸1 − 𝐸𝐸. is the energy barrier. The 𝑓𝑓. 𝑡𝑡 value
gives an attempt number for magnetization reversal,
and the Boltzmann factor expS−𝐾𝐾; T is interpreted as
the probability of magnetization reversal.
We have reported an approximate equation9) for 𝐸𝐸1 −𝐸𝐸. in the Stoner-Wohlfarth dot for angles 𝜙𝜙 of 0 to 180
deg, taking account of Pfeiffer’s approximation10) andshape anisotropy energy. When |𝐻𝐻4 | = 0, 𝐸𝐸1 − 𝐸𝐸.
becomes 𝐾𝐾!:<< 𝑉𝑉 where 𝐾𝐾!:<< is the effective anisotropy
constant, taking account of the shape anisotropy. The
approximate equations for
0 ≤ 𝜙𝜙 ≤ 90 deg
aresummarized as follows,
D: %D;

E<=>> F= !1 + 2 !cos𝜙𝜙 −(|𝐻𝐻4 |⁄𝐻𝐻$:<< ≤ 𝐻𝐻,4 /𝐻𝐻$:<< )G |I& |/I?=>>
H"I@& /I?=>>(6)

𝑥𝑥 = 2.0(𝐻𝐻,4 /𝐻𝐻$:<< ),and for 90 ≤ 𝜙𝜙 ≤ 180 deg,
D: %D;

E<=>> F= !1 −

𝑥𝑥",|I& |/I?=>>
I@& /I?=>>(|𝐻𝐻4 |⁄𝐻𝐻$:<< ≤ 𝐻𝐻,4 /𝐻𝐻$:<< )

𝑥𝑥

",
(7)

𝑥𝑥 = 0.86 + 1.14(𝐻𝐻,4 /𝐻𝐻$:<< ),where

𝐾𝐾!:<< = 𝐾𝐾! +2(4𝜋𝜋−3𝑁𝑁𝑧𝑧 )𝑀𝑀s,
4

𝑁𝑁3 = 8 arctan l
2𝐾𝐾

𝐻𝐻$:<< = 𝑀𝑀ueff ,
s
I@&

I?=>>

=

𝐷𝐷𝑥𝑥 𝐷𝐷𝑦𝑦222ℎ>𝐷𝐷𝑥𝑥 +𝐷𝐷𝑦𝑦 +ℎ

(8)
(9)

m,

G

(|PQRS|E/G '|TUPS|E/G )G/E

(10)
.

(11)

𝐻𝐻,4 and 𝑁𝑁3 are respectively the magnetization
switching field and demagnetizing factor.
The dot error probability 𝑃𝑃 from time 0 to 𝑡𝑡 is wellknown as
𝑃𝑃 = 1 − exp n−𝑓𝑓. 𝑡𝑡 expS−𝐾𝐾; To.

(12)

𝑃𝑃 = 𝑁𝑁𝑁𝑁 = 𝑓𝑓. 𝑡𝑡 expS−𝐾𝐾; T.

(13)

If 𝑓𝑓. 𝑡𝑡 expS−𝐾𝐾; T ≪ 1, Eq. (12) becomes

Journal of the Magnetics Society of Japan Vol.48, No.3, 2024

INDEX

Fig. 3 Definition of angles of magnetization 𝑴𝑴 andwriting field 𝑯𝑯𝐰𝐰 vectors.

(a)Table 1 Standard calculation conditions.

(b)Fig. 4 (a) Bit error rate (bER) as a function of mean Curie
temperature 𝑇𝑇%* after adjacent track (AT) writing forvarious exposure times 𝑡𝑡 for writing and (b) dot heightℎ to achieve a bER of 10-3 as a function of mean Curie
temperature 𝑇𝑇%* .

Although the bER value is calculated using the 𝑃𝑃
values of the grains in a bit for HAMR, the bER value isequal to the 𝑃𝑃 value for HDMR, since 1bit consists of 1
dot.
The criterion determining whether or not informationis stable was assumed to be a bER of 10-3. The bER in
this paper is useful only for comparisons.
The standard calculation conditions are summarizedin Table 1. We used an exposure time 𝑡𝑡 of 1 𝜇𝜇s forwriting in conventional HDMR, taking account of 103
times rewrite. A 𝑡𝑡 value of 1 ns was used in shingled
HDMR. The |𝐻𝐻4 | and 𝜙𝜙 values were 10 kOe and 180
deg, respectively.
3. Cal cul at i on Resul t s

3 . 1 Me a n C u r i e t e mp e r a t u r e
The 𝑇𝑇%* dependence of the bER after AT writing isshown in Fig. 4 (a). A 𝑡𝑡 value of 1 ns was used inshingled HDMR. However, the results for 1 ns and 0.5

ns are almost the same, since 𝑡𝑡 is not a variable of the
exponential function as shown in Eq. (13).
The AT temperature 𝑇𝑇8@A can be calculated as
d𝑇𝑇

𝑇𝑇8@A = 𝑇𝑇%* + 3𝜎𝜎)% − 𝐷𝐷) d𝑦𝑦,

(14)

where 𝐷𝐷) is the track width. We assumed a d𝑇𝑇/d𝑦𝑦
value of 14 K/nm for 𝑇𝑇%* = 750 K and lowered the
d𝑇𝑇/d𝑦𝑦 value indicated in Fig. 4 according to Eq. (2) as
the 𝑇𝑇%* value decreased. We adjusted the ℎ value to
2.8 nm so that the bER value reached 10-3 for 𝑇𝑇%* =750 K and 𝑡𝑡 = 1 ns as shown in Fig. 4 (a). As a result,the bER value increases when we choose the lower 𝑇𝑇%*value, since the temperature difference 𝑇𝑇%* − 𝑇𝑇8@A
decreases from 133 to 98 and 64 K as the d𝑇𝑇/d𝑦𝑦 value
decreases from 14 to 10.8 and 7.62 K/nm, respectively.
The 𝑇𝑇 value is a parameter with considerable impact,
since 𝑇𝑇 is a variable of the exponential function via 𝐾𝐾; .
Figure 4 (b) shows the ℎ value needed to achieve abER of 10-3 after AT writing as a function of 𝑇𝑇%* . The ℎvalue must be increased strongly as the 𝑇𝑇%* value
decreases, since the 𝑇𝑇%* and d𝑇𝑇/d𝑦𝑦 values are closely
related to each other.

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3.2 𝑻𝑻𝐜𝐜 standard deviation
Figure 5 shows the ℎ value as a function of 𝜎𝜎)% /𝑇𝑇%*
for 𝑇𝑇%* = 750 K. When the 𝜎𝜎)% /𝑇𝑇%* value increases, theprobability of a low 𝑇𝑇% dot appearing increases.
Furthermore, the 𝑇𝑇4 and d𝑇𝑇/d𝑦𝑦 values increase as the𝜎𝜎)% /𝑇𝑇%* value increases according to Eqs. (1) and (2),respectively. The resultant 𝑇𝑇8@A value calculated with
Eq. (14) increases as the 𝜎𝜎)% /𝑇𝑇%* value increases as
indicated in Fig. 5. Therefore, the ℎ value must be
increased as the 𝜎𝜎)% /𝑇𝑇%* value increases.

3 . 4 Do t s i z e v a r i a t i o n
When the 𝜎𝜎+ /𝐷𝐷* value increases, the probability of asmall dot appearing increases. Therefore, the ℎ value
must be increased as the 𝜎𝜎+ /𝐷𝐷* value increases asshown in Fig. 7.
3 . 5 Gi l b e r t d a mp i n g c o n s t a n t
The 𝑃𝑃 value is determined by 𝑓𝑓. and 𝐾𝐾; as shown in
Eq. (13). If the 𝑓𝑓. value becomes 10 times larger, the 𝐾𝐾;

3. 3 Ani sot ropy const ant rat i o
We also examined the 𝐾𝐾! /𝐾𝐾"!#$ dependence of ℎ .
When 𝐾𝐾! /𝐾𝐾"!#$ is halved from 0.8 to 0.4, 𝐾𝐾!:<< is also
almost halved, since the shape anisotropy energy issmall. Furthermore, 𝐻𝐻$:<< is almost halved and the 𝐾𝐾;
value is reduced by less than half as
𝐾𝐾

𝑉𝑉|𝐻𝐻 | -

𝐾𝐾; = ueff
V1 − 𝐻𝐻 w W .
𝑘𝑘𝑘𝑘
keff

(15)

Therefore, the ℎ value for a bER of 10-3 must be morethan doubled for a decrease in the 𝐾𝐾! /𝐾𝐾"!#$ value from
0.8 to 0.4 as shown in Fig. 6.

Fig. 5 Dot height ℎ to achieve a bER of 10-3 as a functionof the standard deviation 𝜎𝜎)% /𝑇𝑇%* of the Curie
temperature after AT writing.

Fig. 6 Dot height ℎ to achieve a bER of 10-3 as a functionof anisotropy constant ratio 𝐾𝐾! /𝐾𝐾"!#$ after AT writing.

44

Fig. 7 Dot height ℎ to achieve a bER of 10-3 as a functionof the standard deviation 𝜎𝜎+ /𝐷𝐷* of the dot size after AT
writing.

(a)(b)
Fig. 8 Dot height ℎ to achieve a bER of 10-3 as a functionof the Gilbert damping constant 𝛼𝛼 after (a) 10 years ofarchiving and (b) AT writing.

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value must increase by 2.3 to obtain the same 𝑁𝑁𝑁𝑁 valueas𝑓𝑓. expS−𝐾𝐾; T = 10 𝑓𝑓. exp (−𝐾𝐾; ′),𝐾𝐾; ′ = 𝐾𝐾; + ln(10) ≈ 𝐾𝐾; + 2.3.

(16)

Furthermore, the 𝛼𝛼 value is considered to be smaller
than 0.1. Therefore, 𝑓𝑓. is almost proportional to 𝛼𝛼,
since
𝑓𝑓. ∝

𝛼𝛼
.
1+𝛼𝛼2

(17)

We assumed the storage temperature 𝑇𝑇,CD to be 350 K
for 10 years of archiving. We took a certain margin into
account. The value of 𝐾𝐾; is around 120 at 𝑇𝑇,CD , and thatis much larger than the value of 2.3 seen in Eq. (16).
Therefore, the 𝛼𝛼 value has little effect on 10 years ofarchiving as shown in Fig. 8 (a). However, since the 𝐾𝐾;
value becomes small due to the temperature increasing
to 617 K during AT writing, the 𝛼𝛼 value has an effect on
ATI as shown in Fig. 8 (b).
3 . 6 Wr i t i n g f i e l d ma g n i t u d eFigure 9 shows the ℎ value as a function of |𝐻𝐻4 |. The
𝐾𝐾; value decreases as the |𝐻𝐻4 | value increases
according to Eq. (15) where the 𝐻𝐻$:<< value is about 67
kOe. Therefore, the ℎ value must be increased as the|𝐻𝐻4 | value increases.

3 . 7 Wr i t i n g f i e l d a n g l e
When 𝜙𝜙 decreases from 180 to 135 deg, the 𝐻𝐻,4value is halved from 1.0 to 0.5 according to Eq. (11). Thenthe 𝐾𝐾; value decreases according to Eqs. (5) and (7), andthe ℎ value must be increased as shown in Fig. 10 (a).
Figure 10 (b) shows the (𝐸𝐸1 − 𝐸𝐸. )/(𝐾𝐾!:<< 𝑉𝑉) value as afunction of |𝐻𝐻4 |/𝐻𝐻$:<< for various 𝜙𝜙 values. Althoughthe 𝐻𝐻,4 value is halved, the decrease of (𝐸𝐸1 − 𝐸𝐸. )/(𝐾𝐾!:<< 𝑉𝑉), namely 𝐾𝐾; , from 𝜙𝜙 = 180 deg to 135 deg isrelatively small, since the |𝐻𝐻4 | value of 10 kOe isrelatively small against the 𝐻𝐻$:<< value of 67 kOe.
Therefore, the increase of ℎ from 𝜙𝜙 = 180 deg to 135
deg is small.

Fig. 9 Dot height ℎ to achieve a bER of 10-3 as a functionof writing field magnitude |𝐻𝐻4 | after AT writing.

(a)

(b)
Fig. 10 (a) Dot height ℎ to achieve a bER of 10-3 as afunction of writing field angle 𝜙𝜙 after AT writing and(b) energy barrier (𝐸𝐸1 − 𝐸𝐸. )/(𝐾𝐾!:<< 𝑉𝑉) as a function of thewriting field magnitude |𝐻𝐻4 |/𝐻𝐻$:<< for various 𝜙𝜙values.
4. Concl usi ons

We discussed the influence of the design parameters
on ATI in 4 Tbpsi HDMR. We calculated the ℎ value toachieve a bER of 10-3 after AT writing as a function of
the design parameters.
(1) Mean Curie temperature 𝑇𝑇%*
The ℎ value must be increased strongly as the 𝑇𝑇%*value decreases, since 𝑇𝑇%* and thermal gradient areclosely related.
(2) Standard deviation 𝜎𝜎)% /𝑇𝑇%*
In addition to the increased probability of a low 𝑇𝑇% dot
appearing, the adjacent track temperature is related tothe 𝜎𝜎)% /𝑇𝑇%* value via the writing temperature.
(3) Anisotropy constant ratio 𝐾𝐾! /𝐾𝐾"!#$The ℎ value must be more than doubled for a decreasein the 𝐾𝐾! /𝐾𝐾"!#$ value from 0.8 to 0.4.
(4) Standard deviation 𝜎𝜎+ /𝐷𝐷*
The probability of a small dot appearing increases as
the 𝜎𝜎+ /𝐷𝐷* value increases.
(5) Gilbert damping constant 𝛼𝛼
The 𝛼𝛼 value has little effect on 10 years of archiving
but has an effect on ATI.
(6) Writing field magnitude |𝐻𝐻4 |Journal of the Magnetics Society of Japan Vol.48, No.3, 2024

45

INDEX

The ℎ value must be increased as the |𝐻𝐻4 | value
increases.
(7) Writing field angle 𝜙𝜙
Since the |𝐻𝐻4 | value of 10 kOe is relatively small, theincrease in the ℎ value is relatively small when the 𝜙𝜙value changes from 180 to 135 deg.
Acknowledgement We acknowledge the support of theAdvanced Storage Research Consortium (ASRC), Japan.
Ref erences
1) S. H. Charap, P. -L. Lu, and Y. He: IEEE Trans. Magn., 33,
978 (1997).
2) T. Kobayashi, Y. Nakatani, and Y. Fujiwara: J. Magn. Soc.
Jpn., 47, 1 (2023).

463) T. Kobayashi, Y. Isowaki, and Y. Fujiwara: J. Magn. Soc. Jpn.,
39, 8 (2015).
4) F. Akagi, M. Mukoh, M. Mochizuki, J. Ushiyama, T.
Matsumoto, and H. Miyamoto: J. Magn. Magn. Mater., 324,
309 (2012).
5) T. Kobayashi and Y. Nakatani: J. Magn. Soc. Jpn., 47, 57(2023).
6) M. Mansuripur and M. F. Ruane: IEEE Trans. Magn., MAG22, 33 (1986).
7) J. -U. Thiele, K. R. Coffey, M. F. Toney, J. A. Hedstrom, andA. J. Kellock: J. Appl. Phys., 91, 6595 (2002).
8) E. D. Boerner and H. N. Bertram: IEEE Trans. Magn., 34,
1678 (1998).
9) T. Kobayashi and I. Tagawa: J. Magn. Soc. Jpn., 47, 128(2023).
10) H. Pfeiffer: Phys. Status Solidi (a), 118, 295 (1990).
Received Oct. 28, 2023; Revised Jan. 24, 2024; Accepted Feb.
26, 2024

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Copyright ©2024 by the Magnetics Society of Japan.
This article is licensed under the Creative Commons Attribution International License (CC BY 4.0)
http://creativecommons.org/licenses/by/4.0/

J. Magn. Soc. Jpn., 48, 47-52 (2024)<Paper>Viscosity Effects on Relaxation Time Differences ofMagnetic Nanoparticles
M. Washino, K. Nomura, T. Matsuda, S. Seino*, T. Nakagawa*, T. Kiwa**, A. Tanaka***, and T. Sakane***Mitsubishi Electric Corp., 8 -1 -1 Tsukaguchi-Honmachi, Amagasaki, Hyogo 661 -8661, Japan*Osaka Univ., 2 -1 Yamadaoka, Suita, Osaka 565 -0871, Japan**Okayama Univ., 1 -1 -1 Tsushimanaka, Okayama Kita-ku, Okayama 700 -0082, Japan***Kobe Pharmaceutical Univ., 4 -19 -1 Motoyamakitamachi, Higashinada-ku, Kobe-shi, Hyogo, 658 -8558, Japan

Increased fluid viscosity in vivo is associated with diseases such as hypertension, atherosclerosis, and cancer.
Fortunately, viscosity distribution enables pathological diagnosis. Magnetic-particle imaging (MPI), which detects thehigh-frequency magnetic field response of magnetic nanoparticles (MNPs) for obtaining highly sensitive images, canbe applied to viscosity mapping to obtain a solvent viscosity distribution based on the relaxation time of MNPs. In thisstudy, we assessed the relaxation time differences of MNPs in media with varying viscosities, detected as phasedifferences in the signal using an MPI system. The differences, which increases with more viscosity, differ dependingon the particle size and the magnetic properties of the magnetic nanoparticles.
Keywords: magnetic particle imaging, magnetic nanoparticles, Brownian relaxation, Neel relaxation, viscosity
mapping
1. Introduction
Magnetic-particle imaging (MPI), which detects andvisualizes the nonlinear response of magnetic
nanoparticles (MNPs) to an external magnetic field, isattracting attention as a new medical imaging
technique1). Recently, the dynamic properties of MNPs,
such as their alternating current (AC) magnetization
characteristics in solution, have been studied intensively.
For example, the magnetic relaxation time generally
changes when MNPs are present in media with different
viscosities3). In vivo fluid viscosity mapping has been
proposed as an MPI application that utilizes the viscosity
effect on the magnetic relaxation behavior of MNPs3-4).
Elevated fluid viscosity levels in vivo are closely
associatedwithseveraldiseases,
including
hypertension5), atherosclerosis6), and cancer7), and thepossibility of viscosity mapping using MPI is highly
anticipated for the diagnosis of such pathologicalconditions.
We developed an MPI system capable of detecting
differences in the magnetic properties and relaxation
times of MNPs in different media as signal-phasedifferences8). Arbitrary MNPs can be distinguished basedon phase differences and only image arbitrary MNPs via
signal processing.
In this study, we evaluated the relaxation timedifferences among three distinct MNPs in media withdifferent viscosities, detecting it as a phase difference inthe signal through an MPI system. We detail the resultsof our investigation on the changes in the magnetic
relaxation times of MNPs owing to viscosity effects andtheir influence on signal intensity.

                     Corresponding author: M. Washino(e-mail: Washino.Masaomi@ap.MitsubishiElectric.co.jp).

2. Theory
When an alternating magnetic field is applied tomagnetic nanoparticles, two relaxation mechanisms
exist for the magnetic moment: Brownian and Néel.
In the Brown relaxation mechanism, the magnetic
moment relaxes owing to the particle’s rotation.
Therefore, in a highly viscous medium, the rotationof MNPs is restricted, lengthening the relaxation
time. By contrast, in the Néel relaxation mechanism,
the relaxation time is independent of the medium’s
viscosity because of the magnetic moment’s rotationin the particle. Magnetic relaxation time 𝜏𝜏E of the
MNP can be expressed as Eq. (3) using Brown
relaxation time 𝜏𝜏B and Néel relaxation time 𝜏𝜏N 9):

Fig. 1 Relaxation times numerically calculated asfunction of viscosity

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INDEX𝜏𝜏B =3𝜋𝜋𝜂𝜂𝑑𝑑H
2𝑘𝑘B 𝑇𝑇𝜋𝜋𝜋𝜋𝑑𝑑c3𝜏𝜏N = 𝜏𝜏0 𝑒𝑒𝑒𝑒𝑒𝑒 ()
6𝑘𝑘B 𝑇𝑇𝜏𝜏E =𝜏𝜏N 𝜏𝜏B𝜏𝜏N + 𝜏𝜏B

(1)

(a)

(2)
(3)

where 𝑑𝑑c denotes the core particle size of the MNPs, 𝑑𝑑H
denotes the hydrodynamic particle size of MNPs, 𝜂𝜂denotes the viscosity of the medium, 𝐾𝐾 denotes themagnetic anisotropy energy constant, 𝑘𝑘B denotes
Boltzmann's constant, and 𝑇𝑇 denotes absolutetemperature. As observed from the three equations above,the magnetic relaxation time varies for different MNP
particle sizes, magnetic properties, and medium
viscosities. It increases with more viscosity based on the
Brown relaxation mechanism and converges to the Néel
relaxation time as the viscosity increases (Fig. 1).

(b)3. Material and Methods
3.1 Field-free line MPI System
Fig. 2 present an overview and a diagram of ourdeveloped MPI system8). It employs a field-free line (FFL)
method, which forms a zero-field region on a line byarranging opposing magnets that combine a U-shaped
iron core with permanent magnets. The FFL method has
a wider signal acquisition area and a higher detectedsignal intensity than the field-free point (FFP) method,which forms a point-shaped zero-field region. Therefore,
the FFL method is suitable for acquiring signals fromobjects with a small number of magnetic nanoparticles10).
In this system, the sample is simultaneously translatedusing a coil system that integrates the excitation andreceiver coils, and the sample is rotated and scanned toacquire projection data and reconstruct a cross-sectional
image. The gradient field strength can be continuously
varied from 1 to 4 T/m by changing the distance between
the opposing magnets. In this study, the value was set to
1.5 T/m. The AC magnetic field generated by the
excitation coil had an amplitude of approximately
30 mTp-p (30 Ap-p) at 500 Hz. A main feature of this systemis that the phase difference of the measured MPI signal
can be calculated by referring to the energizing currentof the excitation coil using a lock-in amplifier. Since thephase difference information is determined by the time
constant of the measurement device and the relaxation
time of the magnetic nanoparticles, phase differences canbe used to discriminate from noise and evaluate therelaxation time change of the MNPs. However, phasemeasurement with a lock-in amplifier is affected by thesignal propagation characteristics of the measurement
system in addition to the signal response from the MNPs.
Therefore, it is difficult to only evaluate the phase change
caused by the relaxation time of the MNP as an absolute48

Fig. 2 MPI system using FFL generated bypermanent magnet: (a) its appearance and (b) its
diagramvalue11). However, if the signals of different MNPs arecompared without changing the configuration of themeasurement system, the phase difference caused byonly the relaxation time of the MNP can be extracted
from the phase change, enabling a relative evaluation
based on the phase difference and the detection
frequency. For example, relaxation time difference ∆𝜏𝜏E
between samples 1 and 2 can be calculated using Eq. (4)8):
𝜃𝜃1 − 𝜃𝜃2∆𝜃𝜃∆𝜏𝜏E = −=−(4)(360𝜔𝜔r ⁄2𝜋𝜋)
360𝑓𝑓rwhere 𝜃𝜃1 and 𝜃𝜃2 denote the signal phase of each sample
and 𝑓𝑓r denotes the frequency of the harmonic. Since thephase difference measured here is expressed in terms ofa 360° period, a relaxation time difference exceeding one
period cannot be properly evaluated. The system uses a
low AC field frequency of approximately 1 kHz. For
example, when an AC field frequency of 500 Hz is used,
one period of the third harmonic is more than 600 µsec,
which is sufficiently longer than the relaxation time of
the magnetic particles, enabling the evaluation ofrelaxation time differences due to phase differences.

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Fig. 3 Enlarged view of numerically assumed MH curve for MPI measurement range.
3.2 Measurement samples
We prepared three types of magnetic nanoparticles toinvestigate the effects of particle size and the magnetic
properties on the magnetic relaxation time. One was
Resovist (commercially distributed by FUJIFILM RI
Pharma, Japan) and the other two were MNPs with core
particle sizes of 25 nm (SA25) and 15 nm (SA15) (SigmaAldrich). The hydrodynamic particle sizes of SA25 andSA15 were 35.9 and 62.5 nm by dynamic light scattering(DLS)8). The core particle size and the hydrodynamic
particle size of Resovist were obtained from T. Yoshida etal.9) Fig. 3 shows the DC susceptibility characteristics ofeach MNP8). In the range of AC field strengths used inthis study, the DC magnetization of Resovist is thelargest; that of SA25 is the smallest.
Subsequently, MNP samples with different viscosities
were prepared. Viscosity was adjusted by adding athickening agent to three different MNP solutions (900µL each). Hydroxypropyl cellulose (HPC) was used as athickener in amounts (sealed in a 460 µL vial) that variedin four ways: 0 (no addition), 1, 2, and 3% of the solution
volume. Fig. 4 shows the 12 samples. Fig. 5 shows theresults of the viscosity measurements of the 12 samples.
The viscosity increased with the amount of additional
HPC, and samples with different viscosities wereobtained in the range of 1~500 mPa-s.

Fig. 4 MNP encapsulated cylinder samples(volume:460 µL)

Fig. 5 Sample viscosity versus thickener
addition rate

4. Results and Discussion
4.1 Viscosity dependence of relaxation time differences
The magnetic signals were acquired by scanning the
prepared magnetic nanoparticle samples along the x-axisof the MPI system. A single X-axis scan takes only 5 min.
Fig. 6 shows the Lissajous curve of the third-harmonic
signal in the Resovist sample series at an AC magnetic
field frequency of 500 Hz. The phase angle was adjusted
such that the vertical axis (imaginary part) coincidedwith the Lissajous curve for the case of 0%, which

Fig. 6 Lissajous curve of 3rd harmonic signal of
Resovist sample series at 500 Hz

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coincided with the Lissajous curve for the case of 0%
thickener addition to facilitate comparison of thedifferences in the phase angle. We confirmed that itchanged based on the amount of added thickener(difference in viscosity). Third-harmonic signals werealso obtained for the SA25 and SA15 sample series at an
AC field frequency of 500 Hz. The phase angle changed
based on the amount of added thickener (difference inviscosity).
Fig. 7 shows the calculated phase differences for eachamount of thickener added to each sample using thephase of the Lissajous curve when the amount of
thickener added as a reference is 0%. Fig. 8 shows thecalculated relaxation time difference from the phasedifference shown in Fig. 7 using Eq. (4). The relaxation
time differences increased with viscosity, regardless ofthe type of magnetic nanoparticles. In the low viscosity
range (tens of mPa-s), all the MNP samples show a large
increase in the relaxation time difference as the viscosity
increases. As the viscosity increased further, thedifference in the relaxation time saturated to a constantvalue. This result indicates that in a low viscosity rangeof several tens of mPa-s, the viscosity-dependent Brown
relaxation time is dominant for all MNPs, but in the highviscosity range, the viscosity-independent Néel
relaxation time tends to converge. The SA15 sample
series had the smallest relaxation time difference among
the three MNP sample series; SA15 had a shorter
Brownian relaxation time owing to its smallerhydrodynamic particle size than the other two MNPs.
SA15 also had a shorter Néel relaxation time owing to its
smaller core particle size than the other two MNPs.
Therefore, because the Néel relaxation time wasdominant for SA15 in the viscosity range measured inthis study, the viscosity dependence of the relaxation
time difference is small, and the relaxation timedifference is the smallest among the three MNPs. In
contrast, the Resovist sample series exhibits the largest
relaxation time difference among the three MNP samples.
Since Resovist and SA25 have comparable core andhydrodynamic particle sizes, we infer that the contrast
was caused by differences in their saturation
magnetization. Resovist exhibits a higher saturation
magnetization than SA25 (Fig. 3). Consequently, therelaxation time is prolonged, resulting in acorrespondingly larger relaxation time difference. These
results suggest that among the three MNPs, Resovist,
which has the largest relaxation time difference fordifferent viscosities, is the most suitable for viscosity
mapping in a viscosity range of 1~500 mPa-s measured
in this study.
4.2 Frequency of AC magnetic field dependence ofrelaxation time difference and signal intensity
To investigate the effect of AC field frequency on therelaxation time difference, third-harmonic signals were50

Fig. 7 Phase differences calculated for various
thickener addition levels, using 0% thickener as
reference

Fig. 8 Relaxation time differences calculated fromphase differences at various levels of thickener
addition, with 0% thickener addition as reference
obtained at AC field frequencies of 500, 800, and 1200 Hz.
Fig. 9 shows the viscosity dependence of the relaxation
time differences for each magnetic nanoparticle as theAC field frequency is varied. Figs. 9(a), (b), and (c), whichshow the measurement results of the Resovist, SA25, andSA15 sample series, respectively, indicate that therelaxation time difference tends to decrease as the AC
field frequency increases, regardless of the MNP type.
This result can be attributed to the fact that the magnetic
relaxation behavior of the MNPs cannot follow anincrease in the AC field frequency. Therefore, the lower
the AC magnetic field frequency, the larger therelaxation time difference, which is suitable for viscosity
mapping.
The effects of AC field frequency and viscosity on thesignal intensity were also investigated. Figs. 10(a), (b),and (c) show the results for the Resovist, SA25, and SA15

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(a)

(b)

(c)

Fig. 9 AC field frequency dependence of relaxation time differences as function of viscosity: (a) Resovist sample
series, (b) SA25 sample series, and (c) SA15 sample series.
(b)

(a)

(c)

Fig. 10 Viscosity dependence of signal intensity as function of AC field frequency, using signal strength at 500
Hz as reference: (a) Resovist sample series, (b) SA25 sample series, (c) and SA15 sample series.
sample series. For all the MNP samples, the signal
intensity increases approximately linearly with the AC
field frequency. The ratio of the increase in the signal
intensity to the AC field frequency is constant, regardlessof the amount of thickener added (viscosity). Since thesignal intensity increases linearly with the AC field
frequency, regardless of the viscosity, using a high AC
field frequency is preferable for the viscosity mapping ofobjects with small magnetic signals.
5. Conclusion
In this study, we prepared three types of MNPs withdifferent particle sizes and magnetic properties andinvestigated the effect of viscosity on the magnetic
relaxation times of the MNPs and the signal intensities.
Regardless of the type of MNPs, we confirmed that therelaxation time difference increased with the viscosity inthe viscosity range of 1~500 mPa-s. Among the three
MNPs, Resovist exhibited the largest change in therelaxation time difference as the viscosity increased.
Moreover, we confirmed that the relaxation timedifference tends to decrease as the AC field frequency
increases, regardless of the MNP type; the signal

intensity increases. These results indicate that our MPI
system can measure changes in the magnetic relaxation
time owing to viscosity effects and suggests its potential
application to viscosity mapping. For viscosity mapping,
MNPs must be selected with a dominant Brown
relaxation time in the viscosity range of interest, and alarge relaxation time difference is desirable for
accurately discriminating the viscosity. Although therelaxation time difference increases when a low AC field
frequency is used, it causes a decrease in the signal
strength. To achieve a significant difference in relaxation
time and an enhanced signal strength (yielding a highsignal-to-noise ratio), the choice of AC magnetic field
frequency should be carefully considered based on thedimensions and viscosity of the magnetization under
examination. In the future, we intend to conduct specificviscosity mapping to evaluate the spatial resolution ofthis mapping technique.
Acknowledgments This research was supported by the
Advanced Measurement and Analysis Technology andInstrument Development Program of the Japan Agency
for Medical Research and Development (project number
JP22 hm0102073).

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Phys., 41, 105009 (2008).
11) K. Umeo: Rev. High Press. Sci. Technol., 25, 283 (2015).
Received Oct. 13, 2023; Revised Jan. 16, 2024; Accepted Feb. 26,
2024

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Copyright ©2024 by the Magnetics Society of Japan.
This article is licensed under the Creative Commons Attribution International License (CC BY 4.0)
http://creativecommons.org/licenses/by/4.0/

J. Magn. Soc. Jpn., 48, 53-59 (2024)<Paper>Design method of surface receive coil with high SNR
for various field intensities in MRI
Masaharu Takahashi*, Motofumi Fushimi**, Shin Yabukami*, ***,
Masaki Sekino**, and Akihiro Kuwahata*,***

6-6-05 Aramaki-aza-Aoba, Aoba-ku, Sendai 980-8579, Japan
7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan*** Graduate School of Biomedical Engineering, Tohoku Univ., 6-6-05 Aramaki-aza-Aoba, Aoba-ku, Sendai 980-8579, Japan* Graduate School of Engineering, Tohoku Univ.,** Graduate School of Engineering, The University of Tokyo,

Magnetic resonance imaging (MRI) is a noninvasive medical imaging technology with a high spatial resolution. We
aim to design a single-channel radiofrequency receive coil with an optimized signal-to-noise ratio (SNR) for various field
intensities in MRI. We propose a method that maximizes the SNR through surface current optimization using the target
field method. The proposed method can design a dedicated coil shape by solving an objective function with terms related
to the signal, coil resistance noise and sample resistance noise. We found coil shapes for a rat kidney model in the case ofstatic magnetic field intensity |𝑩𝑩0 | = 7 T, 0.3 T and 6.5 mT as an example. Our method will enable the design of single
channel receive coil for preclinical applications of small animals.
Keywords: magnetic resonance imaging, signal-to-noise ratio, radiofrequency coil, receive coil, target field method
1. Introduction
The fundamental issue in magnetic resonance imaging(MRI) is improving the signal-to-noise ratio (SNR) of thereceived signal to get images with high spatial resolution1). For example, the measurement of glomeruli (sphericalstructures with diameters ranging from 10 – 100 µm) todiagnose kidney disease. In imaging with voxel sizes
around 10 µm, Clinical MRI (scan time ≤ 60 min.) has
insufficient image SNR, making it impossible to measure
the number of glomeruli 2).
The main magnet generates a static magnetic field (𝑩𝑩0 ),inducing the precession of spins at a frequency of 𝑓𝑓0 =𝛾𝛾|𝑩𝑩0 | (𝛾𝛾 is the gyromagnetic ratio of proton) along its
direction. The transmit coil produces a radiofrequency(RF) magnetic field (𝐵𝐵1+ ) rotating at a frequency of 𝑓𝑓0 ,
causing the spin vector to deviate from the direction of
𝑩𝑩0 . Once 𝐵𝐵1+ is deactivated, the magnetization returns
to its initial state with precession, inducing a voltage in
the RF receive coil.
Spatial resolution is the minimum voxel size capable ofdistinguishing adjacent pixels 3). Decreasing the voxel
size results in a lower SNR for the image. This is because
the received signal from a target point 𝑟𝑟0 decrease withthe voxel size V according to the following equation 4):

𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆(𝑟𝑟0 ) = 𝜔𝜔0 𝑉𝑉𝑀𝑀0 |𝐵𝐵1− (𝑟𝑟0 )|,(1)

where 𝜔𝜔0 = 2𝜋𝜋𝑓𝑓0 , 𝑀𝑀0 (∝ |𝑩𝑩0 |) is the magnetization, and|𝐵𝐵1−| is the sensitivity distribution of the receive coil.
Decreasing the voxel size is achieved by reducing the
bandwidth of transmit and receive coils and increasing
the gradients of 𝑩𝑩0 . To obtain sufficient SNR in images
for distinguishing adjacent voxels (enhancing spatial                     Corresponding

author:A.

akihiro.kuwahata.b1@tohoku.ac.jp).

Kuwahata(e-mail:

resolution) despite reducing the pixel size, an
improvement in the SNR of received signals is necessary.
Enhancing the SNR of received signals can be achievedby increasing 𝑩𝑩0 (𝜔𝜔0 ∝ |𝑩𝑩0|, 𝑀𝑀0 ∝ |𝑩𝑩0|，𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆 ∝ |𝑩𝑩0 |2 ),increasing the number of imaging sequences, and using
receive coils with high SNR 5),6). Improving the SNR of
receive coils is effective to decrease installation effort ofhigher 𝑩𝑩0 scanner and to make scan time shorter.
For 𝑩𝑩0 along the z-axis, 𝐵𝐵1− is expressed as thefollowing equation7):∗
(𝐵𝐵1,x − 𝑖𝑖𝐵𝐵1,y )(2)𝐵𝐵1− ＝,2
where 𝐵𝐵1,x and 𝐵𝐵1,y are the respective x- and ycomponents of the RF magnetic field (𝑩𝑩1 ) produced by thecoil with unit current, 𝑖𝑖 is the imaginary unit, and theasterisk denotes a complex conjugate. The noise isevaluated using the following equation 4):(3)
𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁 = √4𝑘𝑘𝑘𝑘𝑘𝑘𝑘𝑘𝑘𝑘,
where 𝑘𝑘 is the Boltzmann constant, 𝑇𝑇 denotes theeffective temperature of a system, 𝑅𝑅 is the equivalent
resistance, and 𝛥𝛥𝛥𝛥 is the frequency bandwidth of the RF
coil. As 𝑘𝑘, 𝑇𝑇 and 𝛥𝛥𝛥𝛥 are not determined only by the coilshape, we simplify the noise 𝑎𝑎𝑎𝑎 √𝑅𝑅. In MRI system, 𝑅𝑅
originates from the coil resistance 𝑅𝑅coil and sample
resistance 𝑅𝑅sample as the following: 𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁 ∝ √𝑅𝑅 =√𝑅𝑅coil + 𝑅𝑅sample . 𝑅𝑅coil and 𝑅𝑅sample are expressed as thefollowing equations 8)–11):
𝑅𝑅coil =𝑙𝑙 𝜌𝜌𝜇𝜇0𝑓𝑓0
√
,
𝜋𝜋
𝜑𝜑

𝑅𝑅sample = 𝜎𝜎 ∫ 𝑬𝑬 ∙ 𝑬𝑬 𝑑𝑑𝑑𝑑 = 𝜎𝜎𝜔𝜔0 2 ∫ 𝑨𝑨 ∙ 𝑨𝑨 𝑑𝑑𝑑𝑑,

(4)

(5)

where 𝜌𝜌 is the resistivity of the coil’s windings, 𝑙𝑙 is thetotal length of the windings, 𝜑𝜑 is the diameter of thewinding, 𝜇𝜇0 is the magnetic permeability of the freespace, 𝑬𝑬 is the electric field, 𝑨𝑨 is the magnetic vector

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potential, 𝑣𝑣 is the volume of the sample, and 𝜎𝜎 is the
conductivity of the sample. For eq. (5), the units of 𝑬𝑬, 𝑨𝑨
and 𝑅𝑅sample are (V/m)/A, (Wb/m)/A and W/A2 = Ω,respectively, because the values of 𝑬𝑬 and 𝑨𝑨 are
calculated based on the unit current flowing the coil.
Please refer to details in the references 4), 9). Thus, wedefine the simpliﬁed SNR by the following equation:|𝑩𝑩0|2|𝐵𝐵1−(𝒓𝒓𝟎𝟎 )|(6).
𝑆𝑆𝑆𝑆𝑆𝑆(𝒓𝒓𝟎𝟎 ) =√𝑅𝑅coil + 𝑅𝑅sample
An ultra-high field scanner (for |𝑩𝑩0 | ≥ 7 T, 𝑓𝑓0 ≥ 298
MHz) is often used to enhance the SNR in preclinicalresearch on organs such as kidney 12). Previously, we had
focused on the numerical design of a single channel
receive coil for rat kidney imaging at 7 T 10). Most
conventional coils dedicated to small animal organ were
designed using the existing knowledge of the traditional
forward problem, which does not always enable an
optimized SNR 2),13)–15). The target ﬁeld method is a wellknown numerical method in the optimization ofelectromagnetic coils based on the inverse problem
approach, presented by Turner 16) for a cylindrical MRI
gradient coil. This method calculates the magnetic fieldusing direct current (DC) analysis for the defined basis
surface currents and find surface currents to achieve theminimum error between the target field and produced
field. Several research applied the method to the designof RF coils 17)–21). However, some of them aimed to
improve the homogeneity of the magnetic field
distribution, not SNR 17)–19). Others aimed to maximize
SNR but the definition of the noise is not clear 20),21).
Therefore, we previously proposed a method to maximize
the SNR through the surface current optimization using
target field method based on the inverse problem
approach 10).
For MRI applications, DC inverse problem approach lacks thecalculation of the phase difference of current, eddy current, anddisplacement current, in contrast to forward problem approachbased on alternative current (AC) using electromagnetic
simulation such as the finite-difference time-domain (FDTD)
method 13)–15), the finite-element method 22), and the method of
moments 23). However, to formulate the relation between RF
current and RF field at target region for inverse problem analysis
is computationally too complex, suggesting the uncertainty of thesolution. We had confirmed that our previous proposed coilachieved an SNR of 1.05-fold compared with the conventionalcoil. Although the relative SNR of proposed coil to theconventional coil calculated by the DC analysis was 1.13 and theerror of relative SNR between DC and AC analysis of 8 % exists,
we verified a coil that exhibits higher SNR in DC analysis canyield higher SNR in AC analysis in the range of the target object
size (Depth from coil surface: 5 – 15 mm), frequency (298 MHz)and electrical parameters (Electrical conductivity 𝜎𝜎 = 0.55 (S/m),
Relative permittivity: εr = 51.95), suggesting that the method iseffective for design of RF receive coil for 7 T MRI.
However, in the case of standard clinical MRI devices (|𝐵𝐵0| =1.5 T and 3 T in the superconducting magnets and |𝐵𝐵0| =0.2 –0.5 T in the permanent magnets 24),25)), 𝑅𝑅coil is dominantcompared to 𝑅𝑅sample due to the lower frequency of 𝑓𝑓0 .

54

Additionally, there is significant ongoing exploration into ultralow magnetic fields (|𝐵𝐵0 | < 0.2 T) 26). Shen et al. 27) had proposeda method to maximize the SNR through the coil winding
combination optimization using target field method for humanbrain at 6.5 mT. This method calculates 𝐵𝐵1− for the defined
windings and find a winding combination to achieve the balanceof the minimum error between the target field and produced field(i.e. higher 𝐴𝐴𝐴𝐴𝐴𝐴. (|𝐵𝐵1−(𝑅𝑅𝑅𝑅𝑅𝑅)|)) and lower length of windings (i.e.
lower 𝑅𝑅coil ). The drawback of this method is limited degree ofobtained coil shape’s freedom.
The purpose of this study is to design a single-channel RFreceive coil with an optimized SNR for various field intensities in
MRI. We propose an expanded method of our previous
study 10) to maximize the SNR through the surface
current optimization using target field method. In theobjective function, we clarify terms related to
𝐴𝐴𝐴𝐴𝐴𝐴. (|𝐵𝐵1−(𝑅𝑅𝑅𝑅𝑅𝑅)|) , 𝑅𝑅sample and 𝑅𝑅coil , and explore thebalance of terms related to 𝑅𝑅sample and 𝑅𝑅coil . We designcoil shapes for a rat kidney model in the case of |𝑩𝑩0 | = 7

T, 0.3 T and 6.5 mT as example.

2. Methods

We design the coil shape by defining the magnetic
field distribution for the target region. The magnetic field
produced by a current in the free space is described by
the following equation:
𝜇𝜇0𝑗𝑗
𝒓𝒓−𝒓𝒓𝟎𝟎(7)
𝑩𝑩 =× 𝑑𝑑𝒍𝒍 = 𝑪𝑪′ 𝑱𝑱,∮
4𝜋𝜋 𝑃𝑃 |𝒓𝒓−𝒓𝒓0 |3

where 𝑗𝑗 is a current flowing on a closed path (𝑃𝑃), 𝑑𝑑𝒍𝒍 isthe unit vector along 𝑃𝑃, 𝒓𝒓 is the position vector on 𝑃𝑃, 𝒓𝒓0
is the position vector on a target point, 𝑪𝑪′ is the field
coupling matrix of the coil surface to the target points,
and 𝑱𝑱 is the surface current vector. Using the stream
̂ , which is defined as the following
function 𝜓𝜓 = 𝒔𝒔𝑇𝑇 𝒉𝒉
̂, where 𝒏𝒏
̂ is the normal vector toequation: 𝑱𝑱 = 𝜵𝜵𝜓𝜓 × 𝒏𝒏
the surface, and 𝒔𝒔 is the vector of the weight of thê ), we can obtain 𝑩𝑩 = 𝑪𝑪 𝒔𝒔 ,surface harmonics vector (𝒉𝒉where 𝑪𝑪 is the field coupling matrix of the surface
harmonics to the target points.
We used a software package: bfieldtools 28),29) in Python
3.7 for the inverse problem analysis. In this tool, 𝜓𝜓 isrepresented on a mesh-by-mesh basis of the specified
surface, allowing for inverse problem analysis on any
arbitrary coil surface. The analysis was divided to three
processes. First, we define coil surface mesh, target̂
points, and field distribution at target points ( 𝑩𝑩t ). 𝒉𝒉(length: 𝑁𝑁h ) is defined on the coil surface. The target
points are set in two regions, Region Signal (number ofpoints: 𝑀𝑀signal ) and Region Noise (number of points:
𝑀𝑀noise ). For the field distribution in Region Signal( 𝑩𝑩t,signal (shape: 𝑀𝑀signal × 3)), one of the components
orthogonal to 𝑩𝑩0 is 1 and others are 0. For the field
distribution in Region Noise (𝑩𝑩t,noise (shape: 𝑀𝑀noise ×3)),
all components are 0. We aim to maximize
𝐴𝐴𝐴𝐴𝐴𝐴(|𝐵𝐵1−(𝑅𝑅𝑅𝑅𝑅𝑅)|), which is the average of the signal in theregion of interest (ROI), and to minimize 𝑅𝑅sample ( ∝∫ 𝑩𝑩1 ∙ 𝑩𝑩1 𝑑𝑑𝑑𝑑). We calculate 𝑪𝑪signal (shape: 𝑀𝑀signal ×3× 𝑁𝑁h )̂ and target
and 𝑪𝑪noise (shape: 𝑀𝑀noise ×3× 𝑁𝑁h ) using 𝒉𝒉

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Fig. 1 Initial setup for inverse problem analysis, calculated stream functions, and coil shapes with
𝜆𝜆sample , 𝜆𝜆coil = 1e-5, 1e0 and 1e5. Coil surface, Region Signal and Region Noise in xy-plane (a) and in xyzspace (b). Orange area, pink cuboid and blue cylinder corresponds to coil surface, Region Signal and Noise,respectively. (c) Calculated stream function. Color map range of stream function 𝜓𝜓 is [𝑀𝑀𝑀𝑀𝑀𝑀(𝜓𝜓), 𝑀𝑀𝑀𝑀𝑀𝑀(𝜓𝜓)]and corresponds to [blue, red]. (d) Coil shapes with 𝑁𝑁c = 4. Red and blue winding are right- and lefthanded loop, respectively. Size of coil surface in (c) corresponds to that in (a) and (b). The scale in (d) issame as (c).
points in Region Signal and Noise, respectively.
Second, we find 𝒔𝒔 (length: 𝑁𝑁h ) that minimizes theobjective function. In our previous method, we defined
the objective function to increase 𝐴𝐴𝐴𝐴𝐴𝐴. (|𝐵𝐵1−(𝑅𝑅𝑅𝑅𝑅𝑅)|)and to
decrease 𝑅𝑅sample :(8)
𝑓𝑓 ′ (𝒔𝒔) = |𝑩𝑩t − 𝑪𝑪𝑪𝑪|2 ,where the shape of 𝑩𝑩t is (𝑀𝑀signal + 𝑀𝑀noise ) ×3 and theshape of 𝑪𝑪 is (𝑀𝑀signal + 𝑀𝑀noise ) × 3 × 𝑁𝑁h . The function
reaches its minimum when 𝒔𝒔 = 𝑪𝑪−1𝑩𝑩t . In the proposed
method, we define the objective function:(9)
𝑓𝑓(𝒔𝒔) = 𝐿𝐿signal + 𝜆𝜆sample 𝐿𝐿sample + 𝜆𝜆coil𝐿𝐿coil ,2(10)

𝐿𝐿signal = |𝑩𝑩t,signal − 𝑪𝑪signal 𝒔𝒔| ,2(11)
𝐿𝐿sample = |𝑩𝑩t,noise − 𝑪𝑪noise 𝒔𝒔| ,(12)
𝐿𝐿coil = 𝒔𝒔𝑻𝑻 𝑷𝑷𝑷𝑷,
where 𝜆𝜆sample and 𝜆𝜆coil is a scalar parameter to define
weight of the 𝐿𝐿sample and 𝐿𝐿coil , respectively, and 𝑷𝑷(shape: 𝑁𝑁h × 𝑁𝑁h ) is the ohmic energy matrix. We aim to
maximize 𝐴𝐴𝐴𝐴𝐴𝐴(|𝐵𝐵1−(𝑅𝑅𝑅𝑅𝑅𝑅)|) , to minimize 𝑅𝑅sample withproper weight, and to minimize 𝑅𝑅coil with proper weight.
The function reaches its minimum when−1

𝒔𝒔 = (𝑪𝑪signal𝑻𝑻 𝑪𝑪signal + 𝜆𝜆sample 𝑪𝑪noise𝑻𝑻 𝑪𝑪noise + 𝜆𝜆coil 𝑷𝑷)(13)(𝑪𝑪signal 𝑻𝑻 𝑩𝑩t,𝐬𝐬𝐬𝐬𝐬𝐬𝐬𝐬𝐬𝐬𝐬𝐬 + 𝜆𝜆sample 𝑪𝑪noise 𝑻𝑻𝑩𝑩t,noise ).
Third, we determine the coil path design based on thecalculated surface current as follows: 𝑃𝑃i = 𝑀𝑀𝑀𝑀𝑀𝑀(𝜓𝜓) +𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 ∙ {𝑖𝑖 − (1/2)} (𝑖𝑖 = 1,2, … , 𝑁𝑁c ),𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 = {𝑀𝑀𝑀𝑀𝑀𝑀(𝜓𝜓) −𝑀𝑀𝑀𝑀𝑀𝑀(𝜓𝜓)}/𝑁𝑁c , where 𝑁𝑁c defines how finely the surface
current is represented. Note that contour lines of thesurface current correspond to the coil path and thepolarity of 𝜵𝜵𝜓𝜓 represents the direction of the current
flowing on the coil path.
Optimal values of 𝜆𝜆sample and 𝜆𝜆coil depend on |𝑩𝑩0 | ofinterest because of the characteristics of 𝑅𝑅sample and𝑅𝑅coil . At this perspective, we obtain the coil shape for eachcalculated stream function with different 𝜆𝜆sample and𝜆𝜆coil and evaluate the SNR of each shape to find the
optimal values.
In this study, we employed homogeneous sample. Some
research used anatomically accurate body model to assume an
actual MR experiment 14),15),30). For example, Kim et al. 14)
conducted the simulation with FDTD using rat model to evaluate
the coil for deep region imaging such as liver (Depth from coilsurface: ~ 10 – 40 mm) and they include the regions such as
backbone and muscle in the main part of the field of view (FOV)
region. However, our target is a kidney and our main part of FOV
is homogeneous and small compared to those of Kim et al.
because the kidney is located the side of the backbone and nearsurface of the body (Depth from coil surface: ~ 5 – 15 mm) 2),
suggesting the acceptability to use the homogeneous model.
3. Results and Discussion
Figure 1 shows the configuration for the inverse
problem analysis, the calculated stream functions andcoil shapes. In Figs. 1(a) and (b), the cylinder with adiameter of 48 mm was the imaging object (as rat body)and a cuboid of 10 × 10 × 20 mm3 located at a depth in
the range of 4 – 14 mm from the surface of the body was
the ROI (as rat kidney). The coil surface was half of thecylindrical side (diameter 50 mm, length 40 mm) and thelocation of the coil was 1 mm above the surface of the
body. Region Signal of the target points was the ROI, a
cuboid with dimensions of 10 × 10 × 20 mm 3 located at a
depth in the range of 5 – 15 mm from the coil surface (2mm interval, 𝑀𝑀signal = 396). Region Noise of the target
points was a cylinder (diameter 48 mm, length 50 mm)
except for Region Signal (2 mm interval, 𝑀𝑀noise =11,348). The field distribution in Region Signal and Noisê + 0 ∙ 𝒛𝒛̂ (a.u.) and 𝑩𝑩t,noise = 𝟎𝟎
was 𝑩𝑩t,signal = 1 ∙ ̂
𝒙𝒙 + 0 ∙ 𝒚𝒚(a.u.), respectively. Note that the direction of 𝑩𝑩0 was zaxis.

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Fig. 2 |𝐵𝐵1−| maps of coil designed with 𝜆𝜆sample , 𝜆𝜆coil = 1e-5, 1e0 and 1e5. (a) Region where x = [0, 24], y= [-24, 24], z = 0. (b) Region where x = [0, 24], y = 0, z = [-25, 25]. Unit of color map is µT. Plotted rectangularregions correspond to the region of interest (Region Signal in Fig. 1).
We calculated the stream functions with different

𝜆𝜆sample and 𝜆𝜆coil (= 1e-5, 1e-4, …, 1e5) in the case of

𝑁𝑁h = 20 (Fig. 1(c)). We obtained the coil shapes with𝑁𝑁c = 4 (Fig. 1(d)). As #1 for an example, the higher 𝜆𝜆coil
yielded the shorter winding length and larger open angle(the angle formed by OQmax and OQmin, where Qmax
and Qmin are points on the coil with maximum andminimum y-coordinates respectively, and O represents
the origin). As #9 for an example, the higher 𝜆𝜆sample
promoted the shorter winding length and smaller open
angle. For #7, the shape was unique compared to others
and was excluded from further discussion due to its
extremely low SNR (Fig.3(d), Fig.4(d), and Fig.5(d) as
mentioned later).
Figure 2 shows |𝐵𝐵1− | ( = (𝐵𝐵1,x 2 + 𝐵𝐵1,y2 )1/2, where 𝐵𝐵1,x
and 𝐵𝐵1,y are calculated using Eq. (5)) maps of coils
designed with 𝜆𝜆sample , 𝜆𝜆coil = 1e-5, 1e0 and 1e5. The
lower 𝜆𝜆sample and 𝜆𝜆coil enhanced |𝐵𝐵1−| homogeneity,
resulting in the lower 𝐴𝐴𝐴𝐴𝐴𝐴. (|𝐵𝐵1−(𝑅𝑅𝑅𝑅𝑅𝑅)|) because of the2
objective function 𝑓𝑓(𝒔𝒔) ~ 𝐿𝐿signal = |𝑩𝑩t,signal − 𝑪𝑪signal𝒔𝒔| .
For example, the homogeneity of # 4 and # 3 was 51.7 %and 33.0 %, respectively.
Figure 3 shows the dependencies of evaluation factors,
such as the signal, noise, and SNR, on 𝜆𝜆sample and 𝜆𝜆coil ,in the case of |𝑩𝑩0 | = 7 T (𝑓𝑓0 = 298 MHz). The signal was
evaluated as |𝑩𝑩0 |2 𝐴𝐴𝐴𝐴𝐴𝐴. (|𝐵𝐵1−(𝑅𝑅𝑅𝑅𝑅𝑅)|). 𝑅𝑅coil was calculatedusing Eq. (2) with the following parameters: 𝜑𝜑 = 0.5 mm,
𝜌𝜌 = 1.68 × 10-8 Ω∙m, and 𝜎𝜎 = 0.55 S/m 14). 𝑅𝑅sample wascalculated using Eq. (5) and the magnetic vector potential
as the following:
𝑑𝑑𝒍𝒍
𝜇𝜇0 𝑗𝑗(14)
𝑨𝑨 = { } ∮ ().
4𝜋𝜋 𝑃𝑃 |𝒓𝒓−𝒓𝒓𝟎𝟎 |The signal intensity was higher at the left-center regions( 𝜆𝜆coil = 1 and 𝜆𝜆sample < 1) compared to other regions(Fig.3(a)). 𝑅𝑅sample (≤ 28.4 Ω) was dominant compared to
𝑅𝑅coil (≤ 1.68 Ω) (Figs. 3(b) and (c)) and the maximum
SNR was 1305 and at the right-bottom corner region56(𝜆𝜆coil = 1e-5 and 𝜆𝜆sample = 1e+5) in Fig. 3(d). The
corresponding coil shape was # 9 in Fig. 1(d) (winding
length: 285 mm, open angle: 74.8 degrees).
Figure 4 shows the evaluation factors for |𝑩𝑩0 | = 0.3 T.
The dependence of the signal on 𝜆𝜆sample and 𝜆𝜆coil was
similar to that in |𝑩𝑩0 | = 7 T. However, the intensity was
lower because it is proportional to |𝑩𝑩0|2 (Fig. 4(a)). 𝑅𝑅coil(≤ 0.347 Ω) was dominant compared to 𝑅𝑅sample (≤ 52.3mΩ) (Figs. 4(b) and (c)). The maximum SNR was 17.9 andat the center-bottom region (𝜆𝜆coil = 1e-5 and 𝜆𝜆sample =1) in Fig. 4(d). The corresponding coil shape was # 8 inFig. 1(d) (winding length: 358 mm, open angle: 91.3
degrees).
Figure 5 shows the evaluation factors for |𝑩𝑩0 | = 6.5 mT.
The dependence of the signal on 𝜆𝜆sample and 𝜆𝜆coil was
similar to that in other |𝑩𝑩0 | but the intensity was lower(Fig. 5(a)). 𝑅𝑅coil (≤ 51.0 mΩ) was dominant compared to
𝑅𝑅𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 (≤ 24.3 µΩ) (Figs. 5(b) and (c)). The maximum
SNR was 0.0232 and near the center region (𝜆𝜆coil = 1 and𝜆𝜆sample =1e-1) in Fig. 5(d). The corresponding coil shape(winding length: 407 mm, open angle: 108 degrees) was
similar to # 4 (rather than # 5) in Fig. 1(d). Although 𝑅𝑅coil
was dominant compared to 𝑅𝑅sample , the maximum SNR
was not at the upper left region ( 𝜆𝜆coil = 1e5 and𝜆𝜆sample = 1e-5), suggesting that higher 𝜆𝜆coil ( > 1)
decreases |𝐵𝐵1−| and lower 𝜆𝜆sample increases 𝑅𝑅coil .
Inductance and quality factor are also important
indicators for coil. Figure 6 shows the inductance,
𝑄𝑄unloaded (quality factor with unloaded condition), and𝑄𝑄loaded (quality factor with loaded condition) dependenceon 𝜆𝜆sample and 𝜆𝜆coil in the case of |𝑩𝑩0| = 7 T, 0.3 T, and
6.5 mT. We calculated the inductance by the following
equation 31):
𝐿𝐿11 ⋯ 𝐿𝐿1𝑛𝑛⋱⋮ ]) ,(15)
𝐿𝐿 = 𝑠𝑠𝑠𝑠𝑠𝑠 ([ ⋮𝐿𝐿𝑛𝑛1 ⋯ 𝐿𝐿𝑛𝑛𝑛𝑛

Journal of the Magnetics Society of Japan Vol.48, No.3, 2024

INDEX

Fig. 3 Signal, noise and SNR dependence on 𝜆𝜆sample and 𝜆𝜆coil in the case of |𝑩𝑩0 | = 7 T. (a) Signal (a.u.).
(b) Coil resistance (Ω). (c) Sample resistance (Ω). (d) SNR (a.u.). Plotted rectangular region indicated by
the arrow corresponds to the maximum value.

Fig. 4 Signal, noise and SNR dependence on 𝜆𝜆sample and 𝜆𝜆coil in the case of |𝑩𝑩0 | = 0.3 T. (a) Signal(a.u.). (b) Coil resistance (Ω). (c) Sample resistance (Ω). (d) SNR (a.u.). Plotted rectangular region indicatedby the arrow corresponds to the maximum value.

Fig. 5 Signal, noise and SNR dependence on 𝜆𝜆sample and 𝜆𝜆coil in the case of |𝑩𝑩0 | = 6.5 mT. (a) Signal(a.u.). (b) Coil resistance (Ω). (c) Sample resistance (Ω). (d) SNR (a.u.). Plotted rectangular region indicatedby the arrow corresponds to the maximum value.
𝜇𝜇0𝐴𝐴
𝜇𝜇0𝑝𝑝
(𝑙𝑙𝑙𝑙 (
) + 0.25) ,2𝜋𝜋
𝑝𝑝𝑝𝑝
𝑑𝑑𝒓𝒓i ∙ 𝑑𝑑𝒓𝒓j
𝜇𝜇0
𝐿𝐿ij (𝑖𝑖 ≠ 𝑗𝑗) =,∮ ∮4π 𝐶𝐶j 𝐶𝐶i |𝒓𝒓j − 𝒓𝒓i |𝐿𝐿𝑖𝑖𝑖𝑖 (𝑖𝑖 = 𝑗𝑗) =

(16)

(17)

where 𝑝𝑝 is the perimeter of the coil, 𝐴𝐴 is the area enclosedby the coil, 𝑅𝑅 is the wire radius, 𝐶𝐶i is the i-th coil path, 𝒓𝒓i
is the position vector on 𝐶𝐶i, and 𝒓𝒓i is the unit vector along
𝐶𝐶i. Note that in this calculation, the inductance does notdepend on the frequency. We calculated the quality
factors by the following equation 1):2𝜋𝜋𝑓𝑓0 𝐿𝐿(18)
𝑄𝑄unloaded =,𝑅𝑅coil
𝑄𝑄loaded =2𝜋𝜋𝑓𝑓0 𝐿𝐿.
𝑅𝑅coil + 𝑅𝑅sample(19)

The optimal coil shape of 7 T, 0.3 T, and 6.5 mT (# 9, #8, and the shape similar to # 4 in Fig. 1 (d)) had
inductance of 0.30, 0.42, and 0.54 µH, 𝑄𝑄unloaded of 59.08.
24.11, and 154.25 (a.u.), and 𝑄𝑄loaded of 723.46, 24.11,
and 180.62 (a.u.), respectively. Segmented capacitors areused for tuning and matching of a coil in the step offabrication. For the making low-frequency coils, toincrease the inductance considering the coil tuning, we
can increase the number of turns by overwrapping theseveral windings at the location of current pathdetermined by the proposed method.
Specific absorption rate (SAR) is also importantparameter for safety regulation of power transmission
especially for RF transmit coils 1). However, our paper
focuses on the receive coil and most of the papers
regarding the receive only coil do not consider SAR 32).
We confirmed that the optimal
𝜆𝜆coil and𝜆𝜆sample depend on the static magnetic field intensity of|𝑩𝑩0 |. The optimal coil shape for 7 T, 0.3 T, and 6.5 mT (#9, # 8, and the shape similar to # 4 in Fig. 1 (d)) had thewinding length of 285, 358, and 407 mm, and the open
angle of 74.8, 91.3, and 108 degrees, respectively. Thissuggests that the coil shape achieving the optimal SNR
in lower |𝑩𝑩0 | become larger compared to that in higher|𝑩𝑩0 | to enhance 𝐴𝐴𝐴𝐴𝐴𝐴. (|𝐵𝐵1−(𝑅𝑅𝑅𝑅𝑅𝑅)|) by allowing an
increase of 𝑅𝑅coil and 𝑅𝑅sample .
4. Conclus ion

We proposed a novel method based on inverse problem

Journal of the Magnetics Society of Japan Vol.48, No.3, 2024

57

INDEX

Fig. 6 Inductance, Qunload, and Qload dependence on 𝜆𝜆sample and 𝜆𝜆coil in the case of |𝑩𝑩0| = 7 T, 0.3 T, and
6.5 mT. (a) Inductance (µH). (b) Qunload (a.u.) and (c) Qload (a.u.) in the case of |𝑩𝑩0| = 7 T. (d) Qunload (a.u.)and (e) Qload (a.u.) in the case of |𝑩𝑩0 | = 0.3 T. (f) Qunload (a.u.) and (g) Qload (a.u.) in the case of |𝑩𝑩0| = 6.5
mT. Plotted rectangular region indicated by the arrow corresponds to the maximum value.

analysis to design a single-channel RF receive coil with
an optimized SNR in MRI scanner of a variety of static
magnetic field intensity |𝑩𝑩0 |. We defined the objectivefunction with terms related to the signal, coil resistancenoise and sample resistance noise. By minimizing thefunction and evaluating the balance of two terms related
to the noise, we obtained the optimal coil shape for |𝑩𝑩0 |of interest in case that the ROI is rat kidney. The coilshape achieving the optimal SNR in lower |𝑩𝑩0 | became
larger compared to that in higher |𝑩𝑩0 | to enhance
𝐴𝐴𝐴𝐴𝐴𝐴. (|𝐵𝐵1− (𝑅𝑅𝑅𝑅𝑅𝑅)|) by allowing an increase of 𝑅𝑅coil and
𝑅𝑅sample .
In the future, we explore the configuration of our
method to obtain the coil shape more suitable to eachscanner configuration such as the direction of |𝑩𝑩0 |. For
the target region, we will verify the potential expansion
to other regions/organs rather than rat kidney for
preclinical applications of small animals. In addition, we
will conduct the comparison of the coil designed using
proposed method with coils designed using otherconventional methods
by AC analysis
usingelectromagnetic simulation and experimental evaluation.
Acknowledgements
This work was supported by
Suzuken Memorial Foundation and Next Generation
Biomedical Measurement Research Network Program,
The University of Tokyo supported by Nakatani
Foundation, AMED (Japan Agency for Medical ResearchandDevelopment)
under
Grant
Number
23he0422030j0001, and JSPS KAKENHI Grant Number
JP23H03721.

58

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Received Dec. 04, 2023; Revised Jan. 04, 2024; Accepte d Mar. 18,
2024

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INDEX

Editorial Committee Members・Paper Committee Members
S. Yabukami and T. Taniyama (Chairperson), N. H. Pham, D. Oyama and M. Ohtake (Secretary)
H. Aoki
M. GotoT. GotoK. Hioki
S. Inui
M. Iwai
Y. Kamihara
H. Kikuchi
T. Kojima
H. KuraK. Masuda
Y. Nakamura
K. Nishijima
T. Nozaki
T. SatoT. Suetsuna
K. Suzuki
Y. Takamura
K. ThamT. Tanaka
N. Wakiya
S. Yakata
A. YaoS. Yamada
M. Yoshida
N. Adachi
S. Haku
K. Kobayashi
Y. SatoI. Tagawa
S. Yoshimura

K. Bessho
S. Honda
T. MakiS. Seino
K. Tajima

M. DoiS. Isogami
S. Muroga
M. Sekino
M. Takezawa

T. DoiT. Kawaguchi
M. NaoeY. Shiota
T. Takura

T. Hasegawa
T. KawaiT. Narita
T. Shirokura
S. Tamaru

K. ItoA. Kuwahata
E. Shikoh
M. Toko
R. Hashimoto
N. Kikuchi
M. Sakakibara
S. Sugahara
T. Yamazaki

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編 集 委 員・論 文 委 員
薮上 信（理事） 谷山智康（理事） Pham NamHai（幹事）
青木英恵
伊藤啓太
乾
成里
岩井守生
後藤太一
後藤
穣
佐藤
拓
仕幸英治
Kim Kong Tham
仲村泰明
西島健一
山田晋也
吉田征弘
脇谷尚樹

大竹 充（幹事） 小山大介（幹事）
神原陽一
菊池弘昭
藏
裕彰
末綱倫浩
鈴木和也
高村陽太
野崎友大
日置敬子
増田啓介

桑波田晃弘
田中哲郎
八尾
惇

小嶋隆幸
都甲
大
家形
諭

安達信泰
白倉孝典
土井達也
槙
智仁

菊池伸明
田倉哲也
白
怜士

佐藤佑樹
田島克文
別所和宏

塩田陽一
田丸慎吾
本多周太

磯上慎二
菅原
聡
土井正晶
室賀
翔

川井哲郎
清野智史
直江正幸
山崎貴大

川口昂彦
関野正樹
成田正敬
吉村
哲

小林宏一郎
田河育也
橋本良介

榊原
満
竹澤昌晃
長谷川 崇

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