1

Coil Design for Magnetic Particle Imaging: Application for aPre-clinical Scanner

Gael Bringout, Member, IEEE, and Thorsten M. Buzug, Member, IEEEA method to design electromagnetic coils for magnetic particle imaging (MPI) scanners is presented. Three different coils for a

rabbit sized scanner are designed, namely a cylindrical drive coil, a biplanar drive coil, and a quadrupole. Based on the inverseboundary elements method, the optimization of all coils used in an MPI scanner is possible. The inverse problem is solved using theTikhonov regularization and a quadratic programming with quadratic constraints algorithm. Depending on the chosen objectives,the coils can be designed to precisely generate a given field, having a small inductance, resistance, or having a smooth windingpattern. The properties of a cylindrical drive coil are then validated on a prototype. The use of this technique allows the optimizationof the scanner design, to enhance the imaging capacity, and to easily gain control over the key properties of the coils in order toscale up scanners to human size.

Index Terms—MPI, coil design, inverse boundary elements, drive coil, quadrupole, quadratic programming, Tikhonov regular-ization.

I. INTRODUCTION

Magnetic particle imaging (MPI) is a new imaging tech-nique to visualize a volume containing a distribution of mag-netic nanoparticles [1]. The first in vivo experiments performedwith mice have demonstrated that this technique is able toacquire data with high sensitivity, high resolution, and highspeed [2].

In order to detect the nanoparticles in a volume, a homoge-neous sinusoidal magnetic field, the drive field, is generatedas illustrated in Fig. 1. Due to the non-linear magnetization ofthe nanoparticles, the received signal is altered, i.e. the signalspectrum contains harmonics, with the energy of the harmonicsbeing proportional to the nanoparticles concentration.

To spatially encode the signal, a magnetic gradient field,the selection field, is superimposed to the drive field. Thisfield contains a field free space (FFS). In close vicinity of theFFS, the nanoparticle magnetization is affected by the drivefield and harmonics are created. Particles further away fromthe FFS are saturated and do not create any higher harmonics(see Fig. 2). In order to encode a volume, the FFS is movedin space by varying the amplitude of the drive field.

Two types of FFS are used. It may be either a field freepoint (FFP) or a field free line (FFL). In an FFP scanner, theselection field is constant in time and drive fields are used withdifferent frequencies to move the FFP along trajectories suchas a Lissajous or a Cartesian curve [3], [4]. In an FFL scanner,the selection field is rotated at a much lower frequency thanthe drive fields. This rotation allows the use of an acquisitionscheme similar to the one used in a computed tomographyscanner [5]–[8]. Up to now, different implementation conceptsexist. An overview of these concepts is pesented in [9].

The drive coil generates the drive field, which varies si-nusoidally in time with a single frequency. This frequency isfixed for each scanner, ranging from 1 kHz to 150 kHz and hasan amplitude of a few tenth of millitesla to a few millitesla.The selection coil, which generates the selection field, is

G. Bringout and T. M. Buzug are with the Institute of Medical Engineering,Universitat zu Lubeck, Lubeck, Germany. e-mail: bringout,buzug@imt.uni-luebeck.de.

1 3 5 ...

M M u Signal

HD

t

H t t

f/f0

Signalspectrum

u

Fig. 1. In the bottom left, the drive field HD has a sinusoidal shape with afrequency f0. In the presence of nanoparticles with a magnetization curve, asshown in the top left part, the resulting magnetization M has the profile shownin the middle part of the figure. The voltage u induced by the nanoparticlesin the receive coil is shown in the top right part of the figure. The spectrumof the signal u, shown in the bottom right figure, will thus include the groundfrequency f0, but also harmonics, which are characteristic of the presence ofnanoparticles and proportional in amplitude to the nanoparticle concentration.

1 3 5 ...

M M u Signal

HS +HD

t

H t t

f/f0

Signalspectrum

u

Fig. 2. When a selection field HS is added on top of the drive field, the fieldamplitude may be high enough to saturate the nanoparticles, as shown onthe left part of the figure. Therefore, the signal induced by the nanoparticlesin the receive coil, shown on the right part of the figure, will not containmuch signal. Moreover, even harmonics are now present. Thus, by adding aselection field in form of a gradient, a given volume around the FFS of theselection field can be used (see Fig. 1) for the evaluation of the nanoparticleconcentration, whereas the remainder of the volume will not contribute to thesignal.

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mainly a solenoid or a dipole in the case of FFP scanners andquadrupoles in case of FFL scanners. The gradient amplitudevaries from a few hundreds of millitesla per meter to severaltesla per meter. The selection field is either static or may havea low rotation speed in case of FFL scanners.

Moreover, the field amplitude tends to be increased. Indeed,the higher the field amplitudes, the higher is the maximalresolution and the field of view [10], [11]. But, differentchallenges remain. The upscaling of the coils toward humansized MPI scanners and the use of magnetic fields with highfrequencies and/or high amplitudes encourage the design ofoptimized coils. Such an optimization has multiple objectives.It may either keep the dissipated power at a reasonable levelto simplify the cooling of the system, increase the thermalstability of the scanner, limit the currents amplitude to reducethe constraints on the other elements of the system, lowerthe voltage amplitude, or simplify the scanner construction.Besides, in an FFL scanner the existence and quality of theline is strongly related to the fields properties. The linearityand homogeneity of those fields has thus to be consideredduring any coil optimization for FFL scanners.

To date, different methods exist to model, parametrize,andsolve such optimization problems [12]–[18]. We have decidedto use a technique based on the boundary element method(BEM) [14]–[18], which uses a meshed surface as support,thus allowing any type of form as support. The same designtechnique can be used to optimize any type of coil needed in anMPI scanner and is more versatile as other design techniquescurrently used for MPI scanner [19], [20]. In order to fulfillthe quasi-static hypothesis used by those techniques, the coilswere designed to use litz wire when needed.

In this paper, three coils for a rabbit sized MPI scannerare designed, namely a cylindrical drive coil, a cylindricalquadrupole for an FFL scanner, and a planar drive coil for anFFP scanner. The simulated properties for the cylindrical drivecoil are validated based on measurements with a prototype.

II. METHODS

The aim of the coil design technique presented here, is tofind the most suitable coil, which generates a desired magneticflux density B in a volume Vtarget.

This is done in three steps. First, the optimal current densityJ is obtained through a single optimization problem, whichcan be regularized using a Tikhonov regularization [18], [21]scheme and solved using a linear or quadratic programmingalgorithm [22]. Once an optimal J is found, the wire loops areapproximated and a wire path is created. This step takes otherconstraints into consideration, as the wire width, flexibility, thecooling methods, the specification of other parts as filter or thecoil construction techniques. If some construction constrainscannot be met, the optimization problem have to be adapted. Ifall the construction constrains and minimal coil performancesare met, the most suited coil can be build and validated.

In this section, all steps from the definition of a givengeometry and magnetic field to the coil validation will beexplained in detail. In subsections II-A to II-E the problemis described using continuous functions describing a current

density. In II-F to II-I the problem is discretized and solved.Then from II-J to II-K, the solution is reworked to obtain thecoil centroids and important physical properties are calculated.Finally, the coils can be built and their properties validated.

A. Forward problem

A conducting volume Vcond with a conductivity ρ and athickness d, supporting J, produces in a different volume withan electric permittivity ε and a magnetic permeability µ, amagnetic flux density B and an electrical field E accordingto the Maxwell relation [23]

curl(

B

µ

)= J +

∂(εE)

∂t. (1)

The design technique is done under the quasi static hypotheses.Thus the time dependent part of (1) can be ignored. J is chosenconstant in time. This leads to

curl(

B

µ

)= J, (2)

which can be reformulated as the Biot-Savart law [23]

B(r0) =µ

4π

∫R3

r− r0‖r− r0‖3

× J(r)dV, (3)

with r = [x y z]ᵀ ∈ Vcond and r0 = [x0 y0 z0]ᵀ in thetarget field volume. The magnetic field is given as a functionof the current density. By inversion of (3), J will be given asa function of B.

B. Representation of the current density as series

J has to be reformulated to facilitate its later expression ona mesh. It can be expressed as the series [22]

J(r) =

∞∑n=1

Jn(r), (4)

with (Jn(r))n∈N being a suitable set of independent basisfunctions, which guarantees that

div Jn = 0. (5)

This means that J is source-free, i.e. it represents a physicallyplausible solution, where no charges are created nor destroyed.

C. Objective and constraint functions

In order to optimize the current density, objective andconstraint functions have to be evaluated. In this work, fourquantities are used for this purpose. Two are quadratic func-tions of J, namely the dissipated power Pdis and the storedenergy Estored. The two others are linear functions of J, namelythe Laplacian of the current density, L J , and the magneticfield amplitude, B. The latter is already presented in (3), thethree others are described in detail here.

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1) Stored energy - mutual inductanceAn operator able to evaluate the mutual inductance between

the elements of J is defined based on the stored energy inVcond. The stored energy Estored of a current I through a coilwith an inductance L is given by

Estored =1

2LI2. (6)

The inductance of Vcond is given by the mutual inductancebetween the basis functions Jm and Jn. It can be expressedas [14]

Mmn =µ0

4π

∫Vm

∫V ′n

Jm(r) · Jn(r′)

‖rm − r′n‖dV ′ndVm. (7)

Thus, minimizing the stored energy in the coil will alsominimize the inductance.

2) Dissipated power - mutual resistanceSimilar to the inductance, the resistance is estimated from

the dissipated power Pdis as

Pdis = RI2, (8)

with R being the resistance of the coil. The mutual resistancebetween the basis function Jm and Jn is given by

Rmn =ρ

d(r)

∫V

Jm(r) · Jn(r)dV, (9)

if the two basis functions Jm and Jn share a commonelement [22], [24]. Otherwise, the mutual resistance is zero.Thus, minimizing the dissipated power will also minimize theresistance of the coil.

3) LaplacianThe Laplacian of the current density is a scalar field defined

for every point on a surface, that evaluates the smoothness ofthe current density. Minimizing the Laplacian will reduce thegeometrical variation of the surface current density. This fa-vorises straight wire paths over curved ones, hence facilitatingthe fabrication of the coil. The Laplacian of the current densityis expressed as

L J(r) = ∇2J(r). (10)

D. From volume to surface

To simplify the representation of the conducting volume,the thickness d is assumed to be small compared to theother dimensions. Thus, the conducting volume Vcond can berepresented by a surface S. J is then replaced by the surfacecurrent density j

j(r) = d(r)J(r), with r ∈ S. (11)

E. Stream function

To accurately calculate the conductor path on the surface,j should be expressed in terms of a stream function ψ [14],[15]. It follows that

j(r) = curl(ψ(r)n(r)), (12)

with n(r) being the normal of the surface S at position r.

F. DiscretizationIn order to discretize the problem on a mesh with N nodes,

a finite set of basis stream functions (ψn(r))Nn=1, which arepiecewise continuously differentiable on S [22], is introduced.ψ is represented by a set of basis functions,

ψ(r) =

N∑n=1

snψn(r), (13)

with sn being the stream function amplitude at position r.Considering N nodes with coordinates rn, n = 1, . . . , N onS, the function ψn at node n is defined as

ψn(rj) =

1 if j = n,

0 if j 6= n,j, n = 1, . . . , N. (14)

and ψn decreases linearly on the edges connected to the noden. The vector s = [s1 s2 . . . sN ]ᵀ is called the stream functionvector.

In this work, S will be formed by K triangular elementsand obtained via the modelization software Blender (Blender2.68a, Blender Foundation, Netherlands).

The surface current density on the triangle k connectedto the nodes (r1, r2, r3) with the basis (ψ1, ψ2, ψ3) is givenby [22]

jk =s1ψ1(r3 − r2) + s2ψ2(r1 − r3) + s3ψ3(r2 − r1)

‖(r2 − r1)× (r3 − r1)‖ .

(15)The surface current density can thus be expressed by

j(r) =

K∑k=1

jk, (16)

which permits the reformulation of the objective and constraintfunctions as matrices in the set of basis functions, (ψn(r))Nn=1,on a mesh.

G. Matrix formulationThe objective and constraint functions needed to optimize

the coil have to be expressed on the mesh. Various challengesexist for their calculation. A review and possible solutions isgiven in [24].

1) Matrix MThe mutual inductance can be expressed by

Estored =1

2

N∑m=1

N∑n=1

Mmnsnsm. (17)

The elements Mmn are assembled into a positive definitematrix M ∈ RN×N , which is used to rewrite (17) as

Estored =1

2sᵀMs. (18)

2) Matrix RThe mutual resistance can be expresses as

Pdis =

N∑m=1

N∑n=1

Rmnsnsm. (19)

The elements Rmn are assembled into the positive definitematrix R ∈ RN×N , which is used to rewrite (19) as

Pdis = sᵀRs. (20)

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3) Matrix CThe relation (3) can be split into its three components,

giving one matrix per component.

Bx(r0) =µ

4π

∫R2

(z − z0)jy(r)− (y − y0)jz(r)

‖r− r0‖3dS

=

N∑n=1

Cxt(r0)sn,

(21)

By(r0) =µ

4π

∫R2

(x− x0)jz(r)− (z − z0)jx(r)

‖r− r0‖3dS

=

N∑n=1

Cyt(r0)sn,

(22)

Bz(r0) =µ

4π

∫R2

(y − y0)jx(r)− (x− x0)jy(r)

‖r− r0‖3dS

=

N∑n=1

Czt(r0)sn.

(23)

The matrices Cx, Cy , and Cz are then assembled from the val-ues Cxt, Cyt, and Czt, respectively. Cx, Cy , and Cz ∈ RT×N

with T being the number of target points on which themagnetic field amplitude is calculated. Relations (21)-(23) canbe expressed in matrix form as

B =

Bx

By

Bz

=

Cx

Cy

Cz

s. (24)

4) Matrix LThe Laplacian of the surface current density at the point m

can be expressed by

L j(rm) =1

4πh2

K∑k=1

Area(k)

3

N∑n=1

e−‖rn−rm‖

4h (j(rn)−j(rm)),

(25)with h being a positive quantity defining the size of theneighborhood [25].

The elements L j can be assembled into the matrixLS ∈ RN×N to form the system of linear equations

L = LSs. (26)

5) Target fieldThe field, which has to be produced by the surface current

density, is called the target field Btarget and is given by therelation

Btarget =

Bxtarget

Bytarget

Bztarget

, (27)

with Bxtarget ,Bytarget ,Bztarget ∈ RT being the x, y, and zcomponents of the target field.

All functions of the forward problem are now expressedas matrices. But, the boundary conditions have still to beintegrated, in order to accurately represent the functions.

H. Boundary condition

The boundary condition applies on all the boundaries ofthe surface. In order to guarantee that j is divergence free, thestream function ψ has to be constant and equal to zero on eachclosed sub-boundaries of S [22].

The forward problem is now completely expressed on themesh, can be inversed, and solved.

I. Solving the inverse problem

In order to solve the inverse problem, two different methodsare used. For the sole minimization of the field error, theill-posed problem is solved using a Tikhonov regularizationscheme. For minimization problems involving quadratic re-lation of s, as the stored energy or the dissipated power, aquadratic programming scheme is used.

1) Tikhonov regularizationConsidering relations (24) and (27), the Tikhonov regular-

ization leads to the minimization problem [21]

mins‖B−Btarget‖2 + λ2‖Γs‖2, (28)

with the matrix Γ ∈ RN×N being the Tikhonov matrix. TheTikhonov matrix gives a preference to solutions based on thenorm of the linear relation Γs. When using the matrix LS

as the Tikhonov matrix, the solution will be smoother. Whenusing the identity matrix as the Tikhonov matrix, the solutionwith a small norm will be preferred. The factor λ is calledthe regularization parameter and is chosen experimentally.This means that the surface current density to be build isnot necessarily the true solution of the minimization problem.A specialized Matlab (Matlab R2010b, MathWorks, USA)toolbox [26], [27] has been used to solve the minimizationproblem (28).

2) Quadratic programmingThe optimization problem can also be formulated as a

quadratic programming problem with quadratic constrains(QPQC). This will be, for example, the minimization of thestored energy, while keeping B into a 10 % error margincompared to Btarget and having dissipated power below 1000W. This can be written as

mins1

2sᵀMs

subject to: BTinf ≤

Cx

Cy

Cz

s ≤ BTsup

sᵀRs ≤ 1000

(29)

and solved using the Matlab toolbox OPTI TOOLBOX [28]and the solver IPOPT [29].

J. Wire approximation

To approximate j as one single wire, a set of wire loopssn has to be extracted first. According to the number of setsof loops Nloops ∈ N+, the current flowing in a loop can bedefined with the relation

Ic =max(s)−min(s)

Nloops. (30)

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The centerline of each set of loops sn is then given by

sn = min(s) + (n− 1

2)Ic, n = 1, . . . , Nloops. (31)

Once the centerlines are obtained, Solidworks (SW 2010,Dassault systemes, France) is used to manually assemble themin a single wire path. The results of this process can be seenin Fig. 3.

−0.1 0 0.1−0.2

−0.1

0

0.1

0.2

x (m)

z(m

)

(a)

-0.3 0

−0.1

0

0.1

θ (m)

z(m

)

(b)

-0.3 0

−0.1

0

0.1

θ (m)

z(m

)

(c)

Fig. 3. Wire discretization process from the solution on a cylinder to the wirepath. (a) Top view of a cylinder with a surface current density optimized witha Tikhonov regularization using the identity matrix to produce a drive coil (i.e.a homogeneous By fields). The white color corresponds to a surface currentdensity of zero and the black color of the maximal surface current density.(b) Planar projection of the wire loops centerlines. (c) Planar projection ofthe wire path approximation.

K. Coil properties evaluation

Calculations have been done in Matlab (Matlab 7.11.064bit, Mathworks, Natick, USA). The implementationis available on http://www.imt.uni-luebeck.de andhttp://github.com/gBringout/CoilDesign. Once the wireloops or the wire paths are obtained, the inductance isevaluated using FastHenry (FastHenry 3.32, FastFieldSolversS.R.L, Italy) [30]. The length of the path is then calculatedas the sum of each coil segments length. The resistanceis obtained by multiplying the length by the resistance permeter of the used conductor. The field is calculated from thecenterlines using a Biot-Savart implementation in Matlab.

L. Quasi-static work-around / Litz wire

Due to the fact that a signal in the kilohertz range is usedfor the drive coil, the quasi static hypotheses does not hold forsuch coils. The main effect is the increased impact of the skineffect on the coil resistance [23]. In order to limit this increase,the coil is wound with a litz wire specially designed for thisapplication. A typical litz wire used in MPI is made of manythousands of single isolated wires with a copper diameter of afew tenth of micrometer. For a given wire cross section, the useof litz wire limits the resistance increase as a function of the

frequency. This minimize the power dissipated at the sendingfrequency and receiving frequencies. So, the litz wire designis a trade-off between the single wire diameter, productioncost and high-frequencies (>1 kHz) resistance. Due to theinsulation material of each single wire, the DC resistance ofa litz wire is increased by a factor of approximatively twowhen compared to the one of a copper wire of equivalentcross section.

M. Coil properties validation

In order to validate the numerical simulation, a prototypeof the cylindrical drive coil is build. The wire path ob-tained in 3c are scaled to the inner diameter of the coil,printed and manually engraved on a PVC cylinder. Then,the litz wire is wound and glued on the engraved path. Theinductance and resistance are measured with an high preci-sion LCR meter (E4980A, Agilent Technologies, USA). Theresonance frequency is measured with an impedance analyser(4194A, Hewlett-Packard Company, USA). A DC Currentsource (SM800 7.5-80, Delta Elektronika, Netherlands) is usedto supply an appropriate current for the measurements. Thecurrent amplitude is continuously measured with a currentclamp (A622, Tektronix, USA) connected to an oscilloscope(DPO 3034, Tektronix, USA). A current of 10.0 A was usedto produce a magnetic field amplitude high enough to enableproper measurements and to keep the dissipated power low.The magnetic field was measured with a Hall probe anda Gaussmeter (MMZ-2508-UH and Model 460, Lake shore,USA) using a robot (3D set of linear axis, isel Germany AG,Germany).

III. RESULTS

Three coil sets have been designed using the coil designtechniques proposed in this contribution, one of which havebeen build. Each of the coil sets is an example for a given MPIscanner geometry, with different requirements. The cylindricaldrive coil should have a length smaller than 0.4 m and a fieldhomogeneity in a sphere of 0.08 m of about 10 %. The biplanardrive coil wire, which are located in the middle of the surface,should be as straight as possible, and the field homogeneityin a sphere of 0.05 m has to be about 10 %. The quadrupoleshould have an efficiency as high as possible, for a length of0.25 m. The requirements are summarized in Table I.

TABLE IMODEL USED FOR THE DESIGN.

Cylindricaldrive coil

Biplanardrive coil

Quadrupole

Optimization goal Length Buildability EfficiencyDiameter (m) 0.2 0.49 0.33Disk distance (m) 0.317Target field amplitude 15 mT 3 mT 0.5 T/mTarget field direction y x line in zTarget diameter (m) 0.08 0.05 0.04# node 2501 782 2470# triangle 4880 1500 4750# target point 515 515 88

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A. Cylindrical drive coil

The Tikhonov regularization (28) using Γ = I with λ =4.0e−6 has been chosen to solve this problem as it deliverscoils with a small lengths even with an unconstrained mesh.The choice of the regularization parameter has to be experi-mentally determined, as the L-curve is ill-shaped. In Table II,the different properties of s and of the discretized coil shownin Fig. 4 are given. As this is a drive coil used with a signalfrequency of 25 kHz, it is made of litz wire, which explainsthe high DC resistance as shown in Table II. The litz wire hasa width of 7.8 mm, the coil is thus made of 18 loops, whichare spaced by a minimum of 8.5 mm. The obtained magneticfield By in Fig. 4b shows a 10 % homogeneity in a sphere ofroughly 8 cm in diameter. Moreover, this coil generates a drivefield amplitude of 15 mT peak, with a peak current of 295 A,a peak voltage of 1070 V, and an RMS dissipated power of542 W.

To validate those properties, a prototype, shown in Fig. 4cwas build. A 7.8x7.8 mm2 square litz wire made of 10000single filaments of 63 µm in diameter was produced for thispurpose (custom product, Sofilec, France). The coil has anefficiency of 53.0 µT/A, a resistance of 10.1 mΩ at DC andof 13.3 mΩ at 25 kHz and 22C, an inductance of 30.7 µHand a self-resonance frequency of 11 MHz. The magneticfield By was measured on a grid of 41x41 elements coveringa square of 60x60 mm2 with a resolution of 3 mm. Dueto the difficulties to align the robot and the coil coordinatesystem, the measured area is not perfectly centered. The extentof the measured surface was limited by the robot arm sizeand the thickness of the PVC former. To ensure a goodstability, the robot movement was stopped for 5 seconds ateach measurement. The whole measurements took around 200minutes. The results are shown in Fig. 4d.

B. Biplanar drive coil

The Tikhonov regularization (28) using Γ = L withλ = 4.0e−8 has been chosen to solve this problem as theresulting coils had a simpler wire pattern. The choice of theregularization parameter has to be experimentally determined,as the L-curve is ill-shaped. In Table II, the different propertiesof s and of the discretized coil shown in Fig. 5 are given.As this is also a drive coil used with a signal frequency of25 kHz, it is made of litz wire. The foreseen litz wire has awidth of 3.9 mm, the coil should be made of 16 loops, spacedby a minimum of 4.9 mm. The simulated magnetic field By

in Fig. 5b shows a 10 % homogeneity in a sphere of roughly10 cm in diameter. In order to generate a field amplitude of3 mT, a peak current amplitude of 570 A is required.

C. Cylindrical quadrupole

In order to obtain a coil with an high efficiency, theminimization of the stored energy according to (29) has beenchosen as minimization goal. To obtain a quadrupole topology,the matrices Cx and Cy model a gradient field with anamplitude of 0.5 T/m in the x and y direction, respectively. Thematrix Cz models a field with no amplitude. This describes a

−0.1 0 0.1

−0.1

0

0.1

x (m)

z(m

)

(a)

−0.1 0 0.1−0.1

0

0.1

x (m)

y(m

)

(b)

(c)

−0.1 0 0.1−0.1

0

0.1

x (m)

y(m

)

(d)

Fig. 4. Cylindrical drive coil generating an homogeneous magnetic field inthe y direction. The arrows show the current flow direction. (a) Top viewof the wire loop centerlines. (b) Side view of the wire loop centerlines andthe By field in the xy plane at z=0. The dashed lines define the volume inwhich the targeted points were defined. The plain lines show, starting fromthe middle of the figure, the 5, 10, and 20 % homogeneity lines. The currentdirection is inverted in the top part compared with the one in the bottompart. (c) Top view of the coil prototype. (d) Measured By field with 5, 10and 20 % homogeneity lines.

−0.2 0 0.2

−0.2

0

0.2

x (m)

z(m

)

(a)

−0.2

0

0.2

−0.200.2

y(m

)

x (m)

(b)

Fig. 5. Biplanar drive coil generating an homogeneous magnetic field in thex direction. The arrows show the current flow direction. (a) Top view of thewire loop centerlines. (b) Side view of the wire loop centerlines and the Bx

field in the xy plane at z=0. The dashed lines define the volume in which thetargeted points were defined. The plain lines show, starting from the middleof the figure, the 5, 10 , and 20 % homogeneity lines. The current directionis inverted in the top part compared with the one in the bottom part.

quadrupole field, with a field free line in z direction. The fieldtopology is thus different from the one of a Maxwell-gradientcoil set. Moreover, a field error of 10 % and a dissipated powersmaller than 10 kW are required. This results in a smoothers, on which more loops were placed. Thus the efficiency wasincreased in comparison to other minimization goals. However,it is expected that the actual dissipated power of the coil willbe higher, as a discrete wire will be used to wound the coil.In Table II, the different properties of s and of the discretizedcoil shown in Fig. 6 are given. As this is an FFL selection

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−0.1 0 0.1

−0.1

0

0.1

x (m)

z(m

)

(a)

−0.1 0 0.1

−0.1

0

0.1

x (m)

y(m

)(b)

Fig. 6. Cylindrical quadrupole generating a field free line in the z directionmade of 4 segments. The arrows show the current flow direction. (a) Top viewof the wire loop centerlines. (b) Side view of the wire loop centerlines andthe gradient of the absolute value of the magnetic flux density in the xy planeat z=0. The dashed lines define the volume in which the targeted points weredefined. The plain lines show, starting from the middle of the figure, the 5,10 , and 20 % linearity lines. The current direction is inverted in the top andbottom part compared with the one in the left and right part.

field coil used with a signal frequency of about 100 Hz, itcan be made with an hollow copper conductor. The increasedresistance due to the skin effect is negligible and the heat willbe more easily extracted. The chosen conductor is square witha width of 6 mm, the coil is made out of 44 loops, each spacedby a minimum of 6.3 mm. The sum of the gradients of themagnetic field absolute values illustrated in Fig. 6b shows a5 % linearity in a sphere of roughly 5 cm in diameter. Togenerate a gradient amplitude of 0.5 T/m, a current amplitudeof 1302 A and a voltage of 70 V are needed, with the RMSdissipated power reaching 16 kW.

TABLE IIPROPERTIES OF THE SURFACE CURRENT DENSITY (ON THE TOP) AND OF

THE WIRE LOOPS (ON THE BOTTOM). VALUE IN PARENTHESES AREMEASURED ONE.

Drive coil cylin-drical

Drive coilbiplanar

Quadrupole

Optimization Tikhonov Tikhonov QPQCPdis (W) 284 47 10000Estored (J) 1.5 0.05 60.5Ic (A) 291.9 87.5 998.7Nominal field 15 mT 3 mT 0.5 T/mEfficiency 51.2 (53.0) µT/A 5.2 µT/A 0.4 mT/(m·A)Current (A) 295 570 1302Wire length (m) 10.5 13.4 23.3RDC (mΩ) 8.8 (10.1) 22.4 13.3L (µH) 23.1 (30.1) 18.1 89.3PRMS (W) 542 (753) 5146 15942

IV. DISCUSION

The numerical optimization of three different coils for MPIscanners has been carried out. The validation of the numericalcalculation has been done on a cylindrical drive coil. TheAC, DC resistance and inductance differences between thewire loops obtained and the actual coil are of 51 %, 15 %,and 30 %, respectively. The higher resistances and inductancecomes from the imperfection of the litz wire and the extralength of wire used to wound the coil. Those have been keptduring the construction in order to integrate the coil in ascanner. The 3 % efficiency increase of the build coil may

be associated to the actual thickness of the coil. Also, the750 W RMS of dissipated power and 1 kV peak voltage ofthis coil present no technical difficulties and can be easilyintegrated into a scanner. The 11 MHz self-resonance is abovethe expected tracer signal bandwidth, which is expected to gofrom 0 to 2 MHz. The homogeneity profile shows satisfactoryagreement with the simulated one, considering the way theconnection between the loops have been done. However, as theprecise requirement in terms of homogeneity and linearity forthe FFL scanner are yet undetermined, it is hard to say moreabout the homogeneity profile. But, as the presented techniquepermits the design of coils with controlled field properties, animpact study of the field topology on the image quality willbe easier to realize.

In the case of an upscaled system, the integration of biplanardrive coil seems to be too challenging, as the efficiency ismuch smaller as the one of cylindrical coils and the obtainedwire path still presents sharp variation which are unlikely tobe build.

Regarding the quadrupole, the use of an hollow conductorallows the dissipation of the 16 kW. But in the case of anMPI scanner, the other components have also to be takeninto account. Particularly, the filters design is affected by thecurrent amplitude. The required 1302 A being too demanding,the coil will have to be split into two or three layers in order toreduce the current amplitude, which will reduce the dissipatedpower. Taking into account the connection of each segment,a 2 layers design is more suitable for the coil fabrication.Indeed, the connection of each dual-layer segment are thenplaced on one side, enabling the connection of each double-layer segment in order to reflect the current flow directionshown in Figure 6a

Additionally, it has to be noted that the obtained cylindricaldrive coil can easily be used to produce a high sensitivityreceive coil by simply increasing Nloops and using a smallerwire during the fabrication. In a similar way, a given surfacecurrent density can be discretized into Nloops to achieve agiven inductance, resistance or wire length. This is of mainimportance, especially for the cooling of the coils. Indeed,the thermal stability of the coils has to be guaranteed duringthe measurement process, to ensure repeatable results. Thisthermal stability can be influenced by varying the amplitudeof the current and the length of the cooling path.

Also, we are able to produce coils with small resistanceand/or impedance or even with a given quality factors. This isparticularly important for the upscaling of the scanner, as thevoltage and current amplitude will sharply increase comparedto the rabbit sized coil presented in this contribution.

More generally, two main approaches have been used tosolve the inverse problem of coil design based on a targetfield. The Tikhonov approach seems to be less versatile asthe QPQC approach. Indeed, as the minimization is done onthe field error, this method will tend to deliver coils, whichproduce fields similar to the ideal one, but often infeasible toconstruct. The QPQC put the incentive of the minimization onproperties, which are more important for MPI scanners thanthe field quality.

8

V. CONCLUSION

In this paper, the inverse BEM method has been used todesign different coils needed in MPI scanners and results werevalidated through the construction and characterization of oneof the designed coils. The use of this well known techniquein the MRI community allows us to further optimize MPIcoils for upscaled MPI scanner, taking various constrain intoaccount. Nevertheless, the properties associated to the streamfunction used to constrains the numerical optimization are notnecessarily representing the final coil properties. Indeed, thediscretization of the stream function into the wire centroidsand the underlying selection of the fabrication method andmaterial may significantly modify the optimized properties.The whole design optimization of a coil is in fact an iterativeprocess.

From the three designed coils only the cylindrical drivecoil fulfill all the required properties of a rabbit MPI scanner.Moreover, the optimized quadrupole will help to alleviate themain challenge to be faced in the upscaling process of FFLscanner: the dissipated power [31], [32].

This method is also a suitable tool to study the fieldquality needed in order to produce a high quality line forFFL scanners. It will also help to develop shimming coils tofinely tune the fields. This can be used to create or destroymultiple FFPs in a scanner, increase the line quality or evenreduce the induced voltage in the patient, which help to solvethe challenges linked to the upscaling of magnetic systems tohuman size.

ACKNOWLEDGMENT

The authors would like to thank A. Mang, J. Overweg,and M. Kleine for their support and understanding during theelaboration of this work. The authors gratefully acknowledgethe support of the German Federal Ministry of Education andResearch (BMBF) under grant number 13N11090 and of theEuropean Union and the State Schleswig-Holstein with theProgram for the Future-Economy under grant number 122-10-004.

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