AbstractmdashA novel approach of designing gradient coil for superconducting MRI is presented in this paper This design approach which is based on the well-known target field method is capable of designing shielded gradient coil of finite length by describing the current density distribution on the surface of the coil in the form of Fourier series Asymmetric coil with constraints such as torque balance can also be designed using this approach Design examples of a shielded X gradient coil for whole body imaging and a shielded torque-balanced asymmetric X gradient coil for head imaging were given in this paper The design process and results prove that this design approach is efficient and practical

I INTRODUCTION IGH field and ultra-high field superconducting MRI have higher SNR and image resolution than those of

low field permanent MRI Gradient coils of high field superconducting MRI which are commonly of cylindrical shape other than biplanar shape should have higher strength slew rate and shielding efficiency than those of lower field MRI to realize advanced functions like DWI and BOLD

Many methods have been proposed to design gradient coils for superconducting MRI Some are based on certain optimization technique such as simulated annealing [1 2] genetic optimization [3 4] and conjugate gradient descent [5] Using these methods the positions of the coil wires are adjusted iteratively according to certain optimizing rules until an optimal solution is found in the design variable space Numerical methods as another kind of gradient coil design methods include finite element method [6] and boundary element method [7 8] These methods almost set no limit to the shape of the coils for they deal with the coil as a combination of elements Target field method one of the most classic gradient coil methods was proposed by Turner [9] which is widely used and proved to be very efficient in designing electromagnetic coils Compared with the target field method methods based on optimization or numerical methods need a large amount of computation for mass data which is time-consuming

In this paper a novel target method based on the Turnerrsquos is presented which is proved to be efficient and practical

Manuscript received 30 June 2011 This work was supported by

National Natural Science Foundation of China (Grant No 50707033 and 50807050)

All authors are with Institute of Electrical Engineering Chinese Academy of Sciences Beijing 100190 China (corresponding author Wen Hui Ynag phone +86-10-8254-7038 fax +86-10-8254-7164 e-mail yangwenh mailieeaccn)

Xiao Fei You is also with Graduate University of Chinese Academy of Sciences (e-mail youxiaofei mailieeaccn)

The classical target field method is based on the Fourier-Bessel transformation which overcomes the ill-condition of the design problem but is unable to restrict the length of the coils The method presented in this paper is based on Fourier series expansion which is able to limit the coil length within a reasonable range by taking the advantage of the periodicity of Fourier series expansion The corresponding shield coil can be designed together with the primary gradient coil through this method As for local coils with asymmetric wire pattern such as gradient coil for head imaging whose DSV locates close to one end of the coil this method is also applicable By using this method certain constrains of the coil can also be considered during the design procedure so that gradient coils with required features such as low inductance and torque balance can be obtained The design process will be specified in part II Some design results and further discussion will appear in part III and conclusion comes in part IV

II THEORY As shown in Fig 1 gradient coil is fixed on the surface of

a cylinder The radius of the cylinder is a and the length of the cylinder is 2L

Fig 1 The coordinate system of the gradient coil

According to Fig 1 the surface current density on the

gradient coil can be expressed in a polar coordinate system (ρ θ z) as

1

1

( ) sin[ ( ) (2 )]sin( )

( ) (2 )cos[ ( ) (2 )]cos( )

Q

z qq

Q

qq

J z U k q z L L k

J z U q a L q z L L kθ

θ π θ

θ π π θ

=

=

⎧= +⎪

⎪⎨⎪ = +⎪⎩

sum

sum

(1)

A Novel Approach of Torque-balanced Asymmetric Gradient Coil Design for Head Imaging

Xiao Fei You Wen Hui Yang Tao Song Li Li Hu and Hui Xian Wang

H

115

In (1) Uq are undetermined coefficients and k is a constant integer depending on the gradient direction of the coil For Z gradient coil let k = 0 and for X gradient coil let k = 1 Y gradient coil can be obtained by rotating the X gradient 90 degrees anticlockwise along the Z axis (1) have the form of Fourier series expansion and the number of the expansion terms Q needs to be pre-determined The larger Q the more accurate gradient magnetic field can be achieved but the more computational amount and coil manufacturing difficulty So Q should be assigned an appropriate integer value The form of the current density also satisfies the current continuity equation

0nabla sdot =J (2) in which

z zJ Jθ θ= +J e e (3) Next the target points of the gradient magnetic field need

to be chosen According to the law of the magnetic field distribution only the target points on the surface of the DSV need to be chosen Further considering the symmetry of the gradient magnetic field distribution and the form of the current density equation for X gradient only the target points in the first quadrant are necessary and the target points located in the XZ plane for the Z gradient coil as shown in Fig 1 The number of the target points should not be less than Q

The magnetic vector B at the target point (xt yt zt) in Fig 1 can be calculated through Biot-Savart law

03( )

4t t t Sx y z dμ σ

πtimes prime= int

J RBR

(4)

in which μ0 is the permeability of vacuum R is the vector from the source point to the field point and S denotes the whole cylindrical surface of the coil According to (4) and (1) the Z component of the vector B at the target point can be expressed as

1( )

Q

z t t t q qq

B x y z U D=

=sum (5)

And the expression for Dq is 20

30

( cos sin )4

L q t tq L

S a x yD d dz

Rπ θ θμ θ

π minus

minus minus= int int (6)

in which 2 cos[ ( ) 2 ]cosqS q a L q z L L kπ π θ= + (7)

and 1

2 2 2 2 2[ 2 ( cos sin ) ( ) ]t t t t tR x y a a x y z zθ θ= + + minus + + minus (8) If M target points are selected then an MtimesQ matrix equation can be derived

11 12 1 1 1 1

21 22 2 2 2 2

1 2

1 1

q Q

q Q

m m mq mQ q m

M M Mq MQ Q M

D D D D U BD D D D U B

D D D D U B

D D D D U B

⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥

=⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥

⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎣ ⎦⎣ ⎦ ⎣ ⎦

(9)

in which the only unknown column is composed by the coefficients from (1) and the column on the right side of the equation is the target value of the selected magnetic field

point By solving the matrix equation the unknown coefficients can be derived and so does the current density distribution that can generate the target magnetic field

To generate the winding pattern of the gradient coil the corresponding stream function Ψs of the current density is needed which is related to the current density through the equation

sψ= nabla times ρJ e (10)

The contour map of the stream function distribution on the surface of the coil will be the wire layout of the gradient coil Set the stream function equal to N different equal-interval constant values between its maximum and minimum value and then draw the function curves of each function equation Those function curves will form the whole winding pattern of the gradient coil The number N need to be determined according to the number of the coil turns As for the real coil some connecting wire must be added to ensure the electrical connection of the whole wire path

Considering the gradient coil with shield coil of radius b and length 2Ls located outside the main coil as shown in Fig 2 the current density on the surface of the shield coil can be expressed similarly as

1

1

( ) sin[ ( ) (2 )]sin( )

( ) (2 )cos[ ( ) (2 )]cos( )

Psz p s s

p

Ps

p s s sp

J z V k q z L L k

J z V pb L q z L L kθ

θ π θ

θ π π θ

=

=

⎧ = +⎪⎪⎨⎪ = +⎪⎩

sum

sum

(11)

in which Vp are as Uq in (1)

Fig 2 The primary coil and the shield coil

In such case two kinds of target field points need to be

considered Ones are those on the surface of the DSV and the others which should be set equal to zero are those located close to the outside surface of the shield coil as shown in Fig 2 Similar to the target points inside the coil for X gradient coil the target points outside the shield coil only need to be chosen on one quarter of the surface of a cylinder with a radius slightly larger than the shield coil in the first quadrant and for Z gradient coil they only need to be located along one line parallel to the Z axis in the same position as shown in Fig 2

116

Then (5) will be expanded as

1 1( )

Q P

z q q p pq p

B x y z U D V E= =

= +sum sum (12)

If there are still totally M target points an Mtimes(Q+P) matrix equation which has the same form as (9) can be derived By solving this equation the wire layout of the primary and shield coil can be obtained together

The method described above can also be applied to asymmetric local gradient coil The main design process remains unchanged and the only difference is that the location of the target points move close to one end of the coil along with the DSV as Fig 2 shows As for such asymmetric gradient coil it may rotate while working in a high static magnetic field due to the large unbalanced torque acting on it Therefore torque balance constraint should be considered during the design process The torque acting on the coil can be expressed as

( ( ))l

la dzd

π

πθ

minus minus= times timesint int 0T r J B (13)

in which B0 is the magnetic vector of the static field A Z gradient coil is torque-free because it is azimuthally symmetric But an asymmetric X gradient coil may experience a torque causing it rotate along the Y axis which can be written as

1

Q

y q qq

T U H=

=sum (14)

in which

[ ]2 20 cos cos ( )

l

q lH qca B z qc z l dzd

π

πθ θ

minus minus= +int int (16)

If (9) is written as DU = B (17)

and (14) is written as yTTH U = (18)

in which D U B and H are matrix or vectors composed of Dmq Uq Bm and Hq as defined in the previous part of the paper then (17) and (18) will form an (M+1)timesQ matrix equation as

yTλλ⎡ ⎤⎡ ⎤⎢ ⎥⎢ ⎥ timestimes⎣ ⎦ ⎣ ⎦

T

BDU =

H (19)

In (19) λ is the weighting coefficient and Ty should be set to zero By solving (19) the design of a torque-balanced coil can be obtained

III RESULTS A shielded X gradient coil for whole body imaging was

designed using this method To allow the coil to accommodate the whole body and leave enough space for RF coil the diameter of the coil should be not less than 60cm Table I lists the design parameters of the coil

Fig 3 is the winding patterns of the X gradient coil The current flow of the dashed line part in Fig 3 is reversed and so does the other coil maps in this paper Fig 4 shows the contour map of the magnetic field in XZ plane in which the circle of dashed line shows the region of DSV and the two irregular dashed lines show the region of gradient field with error less than 3 and 5 respectively

TABLE I PARAMETERS OF WHOLE BODY X GRADIENT COIL

Parameter X gradient coil

Diameter of main coil[mm] 620

Length of main coil[mm] 1600

Diameter of shield coil[mm] 750

Length of shield coil[mm] 1870

Sensitivity[μTmA] 525

Inductance[μH] 315

DSV[mm] 360

(a)

(b)

Fig 3 The winding pattern of the whole body X gradient coil (a) the primary coil (b) the shield coil

Fig 4 The contour map of the magnetic field generated by the whole body X gradient coil in XZ plane

117

A shielded asymmetric head coil was also designed using this method Head coil with small diameter has small induction and thus high slew rate To ensure the size of DSV the coil can not be too short However the shoulder hinders the head from reaching the isocenter Therefore asymmetric design has to be adopted The design parameters are listed in Table II The weighting coefficient λ was set equal to 1 The distance between isocenter and one end of the coil is about 18cm which is enough for general head imaging Fig 7 shows the winding pattern of the coil and Fig 8 is the contour map of the gradient magnetic field The dashed lines in Fig 8 play the same role as those in Fig 4

Though Ty in (19) was set to zero in the design process there was still a net torque of 20 Nm acting on the coil in a 3 Tesla magnetic field due to the discretization of the current density which was acceptable

TABLE II PARAMETERS OF ASYMMETRIC X GRADIENT COIL

Parameter X gradient coil

Diameter of main coil[mm] 405

Length of main coil[mm] 885

Diameter of shield coil[mm] 650

Length of shield coil[mm] 1055

Sensitivity[μTmA] 68

Inductance[μH] 124

DSV[mm] 220

(a)

(b)

Fig 7 The winding pattern of the asymmetric X gradient coil (a) the primary coil (b) the shield coil

Fig 8 The contour map of the magnetic field generated by the asymmetric X gradient coil in XZ plane

The sensitivity of the asymmetric X gradient coil is 68 μTmA and the inductance is 124 μH which means that the coil can realize advanced function such as BOLD and DWI for head imaging with available power supply The inductance was not considered in the design process because it was small enough But as a matter of fact the inductance of the coil can also be calculated according to (1) just like the torque Using certain optimization technique in the design process the inductance of the coil can be minimized Thus the gradient coil with small inductance and high slew rate can be obtained This indicates that this design method is very flexible and can be applied according to different design requirements

IV CONCLUSION This paper presents a novel approach of designing

torque-balanced asymmetric gradient coil Based on the classic target field method this approach is able to design coil of a finite length Special gradient coil and coil with specific design requirements can also be designed using this approach Design results show that this method is practical and efficient

REFERENCES [1] S Crozier and D M Doddrell Gradient-coil design by simulated

annealing Journal of Magnetic Resonance Series A vol 103 pp 354-354 1993

[2] S Crozier et al The design of transverse gradient coils of restricted length by simulated annealing Journal of Magnetic Resonance Series A vol 107 pp 126-128 1994

[3] B Fisher et al Design by genetic algorithm of az gradient set for magnetic resonance imaging of the human brain Measurement Science and Technology vol 6 pp 904-909 1995

[4] B Fisher et al Design and evaluation of a transverse gradient set for magnetic resonance imaging of the human brain Measurement Science and Technology vol 7 pp 838-843 1996

[5] E C Wong et al Coil optimization for MRI by conjugate gradient descent Magnetic resonance in medicine vol 21 pp 39-48 1991

[6] F Shi and R Ludwig Magnetic resonance imaging gradient coil design by combining optimization techniques with the finite element method IEEE Trans Magnetics vol 34 pp 671-683 1998

[7] R A Lemdiasov et al A stream function method for gradient coil design Concepts in Magnetic Resonance Part B Magnetic Resonance Engineering vol 26 pp 67-80 2005

[8] M Poole and R Bowtell Novel gradient coils designed using a boundary element method Concepts in Magnetic Resonance Part B Magnetic Resonance Engineering vol 31 p 162-175 2007

[9] R Turner A target field approach to optimal coil design Journal of physics D Applied physics vol 19 pp 147-151 1986

118

In (1) Uq are undetermined coefficients and k is a constant integer depending on the gradient direction of the coil For Z gradient coil let k = 0 and for X gradient coil let k = 1 Y gradient coil can be obtained by rotating the X gradient 90 degrees anticlockwise along the Z axis (1) have the form of Fourier series expansion and the number of the expansion terms Q needs to be pre-determined The larger Q the more accurate gradient magnetic field can be achieved but the more computational amount and coil manufacturing difficulty So Q should be assigned an appropriate integer value The form of the current density also satisfies the current continuity equation

0nabla sdot =J (2) in which

z zJ Jθ θ= +J e e (3) Next the target points of the gradient magnetic field need

to be chosen According to the law of the magnetic field distribution only the target points on the surface of the DSV need to be chosen Further considering the symmetry of the gradient magnetic field distribution and the form of the current density equation for X gradient only the target points in the first quadrant are necessary and the target points located in the XZ plane for the Z gradient coil as shown in Fig 1 The number of the target points should not be less than Q

The magnetic vector B at the target point (xt yt zt) in Fig 1 can be calculated through Biot-Savart law

03( )

4t t t Sx y z dμ σ

πtimes prime= int

J RBR

(4)

in which μ0 is the permeability of vacuum R is the vector from the source point to the field point and S denotes the whole cylindrical surface of the coil According to (4) and (1) the Z component of the vector B at the target point can be expressed as

1( )

Q

z t t t q qq

B x y z U D=

=sum (5)

And the expression for Dq is 20

30

( cos sin )4

L q t tq L

S a x yD d dz

Rπ θ θμ θ

π minus

minus minus= int int (6)

in which 2 cos[ ( ) 2 ]cosqS q a L q z L L kπ π θ= + (7)

and 1

2 2 2 2 2[ 2 ( cos sin ) ( ) ]t t t t tR x y a a x y z zθ θ= + + minus + + minus (8) If M target points are selected then an MtimesQ matrix equation can be derived

11 12 1 1 1 1

21 22 2 2 2 2

1 2

1 1

q Q

q Q

m m mq mQ q m

M M Mq MQ Q M

D D D D U BD D D D U B

D D D D U B

D D D D U B

⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥

=⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥

⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎣ ⎦⎣ ⎦ ⎣ ⎦

(9)

in which the only unknown column is composed by the coefficients from (1) and the column on the right side of the equation is the target value of the selected magnetic field

point By solving the matrix equation the unknown coefficients can be derived and so does the current density distribution that can generate the target magnetic field

To generate the winding pattern of the gradient coil the corresponding stream function Ψs of the current density is needed which is related to the current density through the equation

sψ= nabla times ρJ e (10)

The contour map of the stream function distribution on the surface of the coil will be the wire layout of the gradient coil Set the stream function equal to N different equal-interval constant values between its maximum and minimum value and then draw the function curves of each function equation Those function curves will form the whole winding pattern of the gradient coil The number N need to be determined according to the number of the coil turns As for the real coil some connecting wire must be added to ensure the electrical connection of the whole wire path

Considering the gradient coil with shield coil of radius b and length 2Ls located outside the main coil as shown in Fig 2 the current density on the surface of the shield coil can be expressed similarly as

1

1

( ) sin[ ( ) (2 )]sin( )

( ) (2 )cos[ ( ) (2 )]cos( )

Psz p s s

p

Ps

p s s sp

J z V k q z L L k

J z V pb L q z L L kθ

θ π θ

θ π π θ

=

=

⎧ = +⎪⎪⎨⎪ = +⎪⎩

sum

sum

(11)

in which Vp are as Uq in (1)

Fig 2 The primary coil and the shield coil

In such case two kinds of target field points need to be

considered Ones are those on the surface of the DSV and the others which should be set equal to zero are those located close to the outside surface of the shield coil as shown in Fig 2 Similar to the target points inside the coil for X gradient coil the target points outside the shield coil only need to be chosen on one quarter of the surface of a cylinder with a radius slightly larger than the shield coil in the first quadrant and for Z gradient coil they only need to be located along one line parallel to the Z axis in the same position as shown in Fig 2

116

Then (5) will be expanded as

1 1( )

Q P

z q q p pq p

B x y z U D V E= =

= +sum sum (12)

If there are still totally M target points an Mtimes(Q+P) matrix equation which has the same form as (9) can be derived By solving this equation the wire layout of the primary and shield coil can be obtained together

The method described above can also be applied to asymmetric local gradient coil The main design process remains unchanged and the only difference is that the location of the target points move close to one end of the coil along with the DSV as Fig 2 shows As for such asymmetric gradient coil it may rotate while working in a high static magnetic field due to the large unbalanced torque acting on it Therefore torque balance constraint should be considered during the design process The torque acting on the coil can be expressed as

( ( ))l

la dzd

π

πθ

minus minus= times timesint int 0T r J B (13)

in which B0 is the magnetic vector of the static field A Z gradient coil is torque-free because it is azimuthally symmetric But an asymmetric X gradient coil may experience a torque causing it rotate along the Y axis which can be written as

1

Q

y q qq

T U H=

=sum (14)

in which

[ ]2 20 cos cos ( )

l

q lH qca B z qc z l dzd

π

πθ θ

minus minus= +int int (16)

If (9) is written as DU = B (17)

and (14) is written as yTTH U = (18)

in which D U B and H are matrix or vectors composed of Dmq Uq Bm and Hq as defined in the previous part of the paper then (17) and (18) will form an (M+1)timesQ matrix equation as

yTλλ⎡ ⎤⎡ ⎤⎢ ⎥⎢ ⎥ timestimes⎣ ⎦ ⎣ ⎦

T

BDU =

H (19)

In (19) λ is the weighting coefficient and Ty should be set to zero By solving (19) the design of a torque-balanced coil can be obtained

III RESULTS A shielded X gradient coil for whole body imaging was

designed using this method To allow the coil to accommodate the whole body and leave enough space for RF coil the diameter of the coil should be not less than 60cm Table I lists the design parameters of the coil

Fig 3 is the winding patterns of the X gradient coil The current flow of the dashed line part in Fig 3 is reversed and so does the other coil maps in this paper Fig 4 shows the contour map of the magnetic field in XZ plane in which the circle of dashed line shows the region of DSV and the two irregular dashed lines show the region of gradient field with error less than 3 and 5 respectively

TABLE I PARAMETERS OF WHOLE BODY X GRADIENT COIL

Parameter X gradient coil

Diameter of main coil[mm] 620

Length of main coil[mm] 1600

Diameter of shield coil[mm] 750

Length of shield coil[mm] 1870

Sensitivity[μTmA] 525

Inductance[μH] 315

DSV[mm] 360

(a)

(b)

Fig 3 The winding pattern of the whole body X gradient coil (a) the primary coil (b) the shield coil

Fig 4 The contour map of the magnetic field generated by the whole body X gradient coil in XZ plane

117

A shielded asymmetric head coil was also designed using this method Head coil with small diameter has small induction and thus high slew rate To ensure the size of DSV the coil can not be too short However the shoulder hinders the head from reaching the isocenter Therefore asymmetric design has to be adopted The design parameters are listed in Table II The weighting coefficient λ was set equal to 1 The distance between isocenter and one end of the coil is about 18cm which is enough for general head imaging Fig 7 shows the winding pattern of the coil and Fig 8 is the contour map of the gradient magnetic field The dashed lines in Fig 8 play the same role as those in Fig 4

Though Ty in (19) was set to zero in the design process there was still a net torque of 20 Nm acting on the coil in a 3 Tesla magnetic field due to the discretization of the current density which was acceptable

TABLE II PARAMETERS OF ASYMMETRIC X GRADIENT COIL

Parameter X gradient coil

Diameter of main coil[mm] 405

Length of main coil[mm] 885

Diameter of shield coil[mm] 650

Length of shield coil[mm] 1055

Sensitivity[μTmA] 68

Inductance[μH] 124

DSV[mm] 220

(a)

(b)

Fig 7 The winding pattern of the asymmetric X gradient coil (a) the primary coil (b) the shield coil

Fig 8 The contour map of the magnetic field generated by the asymmetric X gradient coil in XZ plane

The sensitivity of the asymmetric X gradient coil is 68 μTmA and the inductance is 124 μH which means that the coil can realize advanced function such as BOLD and DWI for head imaging with available power supply The inductance was not considered in the design process because it was small enough But as a matter of fact the inductance of the coil can also be calculated according to (1) just like the torque Using certain optimization technique in the design process the inductance of the coil can be minimized Thus the gradient coil with small inductance and high slew rate can be obtained This indicates that this design method is very flexible and can be applied according to different design requirements

IV CONCLUSION This paper presents a novel approach of designing

torque-balanced asymmetric gradient coil Based on the classic target field method this approach is able to design coil of a finite length Special gradient coil and coil with specific design requirements can also be designed using this approach Design results show that this method is practical and efficient

REFERENCES [1] S Crozier and D M Doddrell Gradient-coil design by simulated

annealing Journal of Magnetic Resonance Series A vol 103 pp 354-354 1993

[2] S Crozier et al The design of transverse gradient coils of restricted length by simulated annealing Journal of Magnetic Resonance Series A vol 107 pp 126-128 1994

[3] B Fisher et al Design by genetic algorithm of az gradient set for magnetic resonance imaging of the human brain Measurement Science and Technology vol 6 pp 904-909 1995

[4] B Fisher et al Design and evaluation of a transverse gradient set for magnetic resonance imaging of the human brain Measurement Science and Technology vol 7 pp 838-843 1996

[5] E C Wong et al Coil optimization for MRI by conjugate gradient descent Magnetic resonance in medicine vol 21 pp 39-48 1991

[6] F Shi and R Ludwig Magnetic resonance imaging gradient coil design by combining optimization techniques with the finite element method IEEE Trans Magnetics vol 34 pp 671-683 1998

[7] R A Lemdiasov et al A stream function method for gradient coil design Concepts in Magnetic Resonance Part B Magnetic Resonance Engineering vol 26 pp 67-80 2005

[8] M Poole and R Bowtell Novel gradient coils designed using a boundary element method Concepts in Magnetic Resonance Part B Magnetic Resonance Engineering vol 31 p 162-175 2007

[9] R Turner A target field approach to optimal coil design Journal of physics D Applied physics vol 19 pp 147-151 1986

118

Then (5) will be expanded as

1 1( )

Q P

z q q p pq p

B x y z U D V E= =

= +sum sum (12)

If there are still totally M target points an Mtimes(Q+P) matrix equation which has the same form as (9) can be derived By solving this equation the wire layout of the primary and shield coil can be obtained together

The method described above can also be applied to asymmetric local gradient coil The main design process remains unchanged and the only difference is that the location of the target points move close to one end of the coil along with the DSV as Fig 2 shows As for such asymmetric gradient coil it may rotate while working in a high static magnetic field due to the large unbalanced torque acting on it Therefore torque balance constraint should be considered during the design process The torque acting on the coil can be expressed as

( ( ))l

la dzd

π

πθ

minus minus= times timesint int 0T r J B (13)

in which B0 is the magnetic vector of the static field A Z gradient coil is torque-free because it is azimuthally symmetric But an asymmetric X gradient coil may experience a torque causing it rotate along the Y axis which can be written as

1

Q

y q qq

T U H=

=sum (14)

in which

[ ]2 20 cos cos ( )

l

q lH qca B z qc z l dzd

π

πθ θ

minus minus= +int int (16)

If (9) is written as DU = B (17)

and (14) is written as yTTH U = (18)

in which D U B and H are matrix or vectors composed of Dmq Uq Bm and Hq as defined in the previous part of the paper then (17) and (18) will form an (M+1)timesQ matrix equation as

yTλλ⎡ ⎤⎡ ⎤⎢ ⎥⎢ ⎥ timestimes⎣ ⎦ ⎣ ⎦

T

BDU =

H (19)

In (19) λ is the weighting coefficient and Ty should be set to zero By solving (19) the design of a torque-balanced coil can be obtained

III RESULTS A shielded X gradient coil for whole body imaging was

designed using this method To allow the coil to accommodate the whole body and leave enough space for RF coil the diameter of the coil should be not less than 60cm Table I lists the design parameters of the coil

Fig 3 is the winding patterns of the X gradient coil The current flow of the dashed line part in Fig 3 is reversed and so does the other coil maps in this paper Fig 4 shows the contour map of the magnetic field in XZ plane in which the circle of dashed line shows the region of DSV and the two irregular dashed lines show the region of gradient field with error less than 3 and 5 respectively

TABLE I PARAMETERS OF WHOLE BODY X GRADIENT COIL

Parameter X gradient coil

Diameter of main coil[mm] 620

Length of main coil[mm] 1600

Diameter of shield coil[mm] 750

Length of shield coil[mm] 1870

Sensitivity[μTmA] 525

Inductance[μH] 315

DSV[mm] 360

(a)

(b)

Fig 3 The winding pattern of the whole body X gradient coil (a) the primary coil (b) the shield coil

Fig 4 The contour map of the magnetic field generated by the whole body X gradient coil in XZ plane

117

A shielded asymmetric head coil was also designed using this method Head coil with small diameter has small induction and thus high slew rate To ensure the size of DSV the coil can not be too short However the shoulder hinders the head from reaching the isocenter Therefore asymmetric design has to be adopted The design parameters are listed in Table II The weighting coefficient λ was set equal to 1 The distance between isocenter and one end of the coil is about 18cm which is enough for general head imaging Fig 7 shows the winding pattern of the coil and Fig 8 is the contour map of the gradient magnetic field The dashed lines in Fig 8 play the same role as those in Fig 4

Though Ty in (19) was set to zero in the design process there was still a net torque of 20 Nm acting on the coil in a 3 Tesla magnetic field due to the discretization of the current density which was acceptable

TABLE II PARAMETERS OF ASYMMETRIC X GRADIENT COIL

Parameter X gradient coil

Diameter of main coil[mm] 405

Length of main coil[mm] 885

Diameter of shield coil[mm] 650

Length of shield coil[mm] 1055

Sensitivity[μTmA] 68

Inductance[μH] 124

DSV[mm] 220

(a)

(b)

Fig 7 The winding pattern of the asymmetric X gradient coil (a) the primary coil (b) the shield coil

Fig 8 The contour map of the magnetic field generated by the asymmetric X gradient coil in XZ plane

The sensitivity of the asymmetric X gradient coil is 68 μTmA and the inductance is 124 μH which means that the coil can realize advanced function such as BOLD and DWI for head imaging with available power supply The inductance was not considered in the design process because it was small enough But as a matter of fact the inductance of the coil can also be calculated according to (1) just like the torque Using certain optimization technique in the design process the inductance of the coil can be minimized Thus the gradient coil with small inductance and high slew rate can be obtained This indicates that this design method is very flexible and can be applied according to different design requirements

IV CONCLUSION This paper presents a novel approach of designing

torque-balanced asymmetric gradient coil Based on the classic target field method this approach is able to design coil of a finite length Special gradient coil and coil with specific design requirements can also be designed using this approach Design results show that this method is practical and efficient

REFERENCES [1] S Crozier and D M Doddrell Gradient-coil design by simulated

annealing Journal of Magnetic Resonance Series A vol 103 pp 354-354 1993

[2] S Crozier et al The design of transverse gradient coils of restricted length by simulated annealing Journal of Magnetic Resonance Series A vol 107 pp 126-128 1994

[3] B Fisher et al Design by genetic algorithm of az gradient set for magnetic resonance imaging of the human brain Measurement Science and Technology vol 6 pp 904-909 1995

[4] B Fisher et al Design and evaluation of a transverse gradient set for magnetic resonance imaging of the human brain Measurement Science and Technology vol 7 pp 838-843 1996

[5] E C Wong et al Coil optimization for MRI by conjugate gradient descent Magnetic resonance in medicine vol 21 pp 39-48 1991

[6] F Shi and R Ludwig Magnetic resonance imaging gradient coil design by combining optimization techniques with the finite element method IEEE Trans Magnetics vol 34 pp 671-683 1998

[7] R A Lemdiasov et al A stream function method for gradient coil design Concepts in Magnetic Resonance Part B Magnetic Resonance Engineering vol 26 pp 67-80 2005

[8] M Poole and R Bowtell Novel gradient coils designed using a boundary element method Concepts in Magnetic Resonance Part B Magnetic Resonance Engineering vol 31 p 162-175 2007

[9] R Turner A target field approach to optimal coil design Journal of physics D Applied physics vol 19 pp 147-151 1986

118

A shielded asymmetric head coil was also designed using this method Head coil with small diameter has small induction and thus high slew rate To ensure the size of DSV the coil can not be too short However the shoulder hinders the head from reaching the isocenter Therefore asymmetric design has to be adopted The design parameters are listed in Table II The weighting coefficient λ was set equal to 1 The distance between isocenter and one end of the coil is about 18cm which is enough for general head imaging Fig 7 shows the winding pattern of the coil and Fig 8 is the contour map of the gradient magnetic field The dashed lines in Fig 8 play the same role as those in Fig 4

Though Ty in (19) was set to zero in the design process there was still a net torque of 20 Nm acting on the coil in a 3 Tesla magnetic field due to the discretization of the current density which was acceptable

TABLE II PARAMETERS OF ASYMMETRIC X GRADIENT COIL

Parameter X gradient coil

Diameter of main coil[mm] 405

Length of main coil[mm] 885

Diameter of shield coil[mm] 650

Length of shield coil[mm] 1055

Sensitivity[μTmA] 68

Inductance[μH] 124

DSV[mm] 220

(a)

(b)

Fig 7 The winding pattern of the asymmetric X gradient coil (a) the primary coil (b) the shield coil

Fig 8 The contour map of the magnetic field generated by the asymmetric X gradient coil in XZ plane

The sensitivity of the asymmetric X gradient coil is 68 μTmA and the inductance is 124 μH which means that the coil can realize advanced function such as BOLD and DWI for head imaging with available power supply The inductance was not considered in the design process because it was small enough But as a matter of fact the inductance of the coil can also be calculated according to (1) just like the torque Using certain optimization technique in the design process the inductance of the coil can be minimized Thus the gradient coil with small inductance and high slew rate can be obtained This indicates that this design method is very flexible and can be applied according to different design requirements

IV CONCLUSION This paper presents a novel approach of designing

torque-balanced asymmetric gradient coil Based on the classic target field method this approach is able to design coil of a finite length Special gradient coil and coil with specific design requirements can also be designed using this approach Design results show that this method is practical and efficient

REFERENCES [1] S Crozier and D M Doddrell Gradient-coil design by simulated

annealing Journal of Magnetic Resonance Series A vol 103 pp 354-354 1993

[2] S Crozier et al The design of transverse gradient coils of restricted length by simulated annealing Journal of Magnetic Resonance Series A vol 107 pp 126-128 1994

[3] B Fisher et al Design by genetic algorithm of az gradient set for magnetic resonance imaging of the human brain Measurement Science and Technology vol 6 pp 904-909 1995

[4] B Fisher et al Design and evaluation of a transverse gradient set for magnetic resonance imaging of the human brain Measurement Science and Technology vol 7 pp 838-843 1996

[5] E C Wong et al Coil optimization for MRI by conjugate gradient descent Magnetic resonance in medicine vol 21 pp 39-48 1991

[6] F Shi and R Ludwig Magnetic resonance imaging gradient coil design by combining optimization techniques with the finite element method IEEE Trans Magnetics vol 34 pp 671-683 1998

[7] R A Lemdiasov et al A stream function method for gradient coil design Concepts in Magnetic Resonance Part B Magnetic Resonance Engineering vol 26 pp 67-80 2005

[8] M Poole and R Bowtell Novel gradient coils designed using a boundary element method Concepts in Magnetic Resonance Part B Magnetic Resonance Engineering vol 31 p 162-175 2007

[9] R Turner A target field approach to optimal coil design Journal of physics D Applied physics vol 19 pp 147-151 1986

118