J Supercond Nov Magn (2011) 24: 1037–1043DOI 10.1007/s10948-010-0877-7

O R I G I NA L PA P E R

A Ferromagnetic Shimming Method for NMR/MRI MagnetsAdopting Two Consecutive Optimization Techniques:Linear Programming and Evolution Strategy

Sangjin Lee · Seungyong Hahn · Kideok Sim

Received: 6 September 2010 / Accepted: 7 September 2010 / Published online: 30 September 2010© Springer Science+Business Media, LLC 2010

Abstract Shimming is very important for nuclear mag-netic resonance (NMR) magnets because image resolution ishighly dependent on the homogeneity of the magnetic field.There are two types of shimming: active and passive. Activeshimming is done using coils with adjustable current. Pas-sive shimming involves pieces of steel with good magneticqualities. The steel pieces are placed near a superconductingmagnet. They are magnetized and produce their own mag-netic field. Additional magnetic fields (produced by coils orsteel) add to the original magnetic field of the superconduct-ing magnet in such a way that the total field becomes morehomogeneous.

In this paper, we developed a passive shimming methodadopting consecutive optimization techniques, i.e., linearprogramming (LP) and evolution strategy (ES). The LP isrelatively fast and mostly guarantees a global minimum for alinear problem, whereas the ES is easy to formulate and candigitize design variables. So we suggested an optimizationmethod combining both the LP and the ES consecutively forpassive shimming of NMR magnets.

Keywords NMR/MRI magnet · Ferromagnetic shimming ·Passive shimming · Linear programming · Evolutionstrategy

S. Lee (�)Department of Electrical Eng., Uiduk University, Gyeongju780-713, Koreae-mail: sjlee@uu.ac.kr

S. HahnFrancis Bitter Magnet Laboratory, Massachusetts Institute ofTechnology, Cambridge, MA, 02139, USA

K. SimSuperconductivity Research Center, KERI, Changwon 641-120,Korea

1 Introduction

A shimming technique that corrects inhomogeneous mag-netic fields due to the surrounding environment or manu-facturing error is essential to some devices requiring highlyhomogeneous magnetic fields such as NMR/MRI supercon-ducting magnets. Typical shimming methods include activeshimming with electromagnetic coils and passive shimmingor ferromagnetic shimming using metal sheets. The activeshimming method employs a set of coils that can adjusteach harmonic field gradient. While the active shimmingcan function independently on each inhomogeneous mag-netic field gradient, it requires large volume and relativelyhigh manufacturing costs for shimming of high-order mag-netic field gradients. In contrast, a passive shimming methodusing thin metal sheets removes nonhomogeneities by over-lapping second magnetic fields created by metal sheets mag-netized by a superconducting magnet. This method is widelyused because it has no additional power requirement and lowcost.

To implement this passive shimming method, severalprocesses should be performed that include: 1) a process tocalculate the magnetic field density generated by the satu-rated metal sheet, 2) a process to estimate the number andlocation of metal sheets, and 3) a process to fix the metalsheet at the specific target position near the magnet. Mag-netic field homogeneity is the most important element thatdetermines the performance of NMR/MRI magnets. For thepassive shimming method to improve magnetic field ho-mogeneity, this paper will introduce the process to calcu-late the magnetic field density generated by the metal sheet,and the process to estimate the location and thickness of themetal sheet using linear programming and an evolution strat-egy.

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Fig. 1 Principle of ferromagnetic shimming. The ferromagneticpieces are magnetized by the field from the original winding and pro-duce their own magnetic field in such a way that the total field becomesmore homogeneous

2 Optimization

2.1 Principle and Structure of Passive Shimming

If a metal sheet is placed on a room temperature bore (orwarm bore) of an NMR/MRI magnet as shown in Fig. 1, themetal sheet is magnetized by the magnetic fields from themagnet. A magnetically saturated metal sheet creates Bf inproportion to its thickness. The magnetic field Bf from eachmetal sheet is added to the original inhomogeneous mag-netic fields Bs to form a new magnetic field. If the locationand number (thickness) of the metal sheet are adjusted, it ispossible to correct the inhomogeneous magnetic field withina target space of the magnet; this entire process is called pas-sive shimming. In addition, the magnetic field homogeneitycan be more precisely adjusted depending on the thicknessof the metal sheet, and shimming for each field gradient isalso possible if the magnetic field density can be expressedin separate gradients for space. This function is very usefulif a shimming coil for a specific gradient is attached to themagnet.

The hardware for passive shimming consists of a metalsheet (or shim) and a shim pocket that can fasten the sheet[1]. The pocket is a device that can hold metal sheets in theaxial and circumferential directions at the outermost cell ofthe room temperature bore. Several pieces of metal sheetsare inserted to it and fixed.

2.2 Magnetic Field Density of Metal Sheet

The first step of passive shimming is to form a magneticfield map, the collection of values representing the effects ofeach metal sheet located at the outermost cell of the roomtemperature bore on the area of interest at the center. If itis assumed that the metal sheet used for shimming is ex-posed to strong external magnetic fields and completely sat-urated, the N pole and S pole are formed on both ends ofthe metal sheet and act as a magnetic dipole. In all NMRmagnets used, the outermost area of the room temperature

bore with the metal sheet shows a sufficient magnetic fielddensity to saturate the metal sheet [2].

For shimming of the magnetic fields using the thicknessof the metal sheet, we need to calculate the magnetic fielddensity generated by the metal sheet saturated at a certainposition. The room temperature bore where the metal sheetis located can be conveniently indicated using the cylindri-cal coordinate system because its shape is a cylinder. Equa-tion (1) shows the magnetic scalar potential Vm at an arbi-trary point P(ρ,ϕ, z) when the magnetic charge qm is lo-cated at P ′(ρ′, ϕ′, z′).

Vm = qm

4πR= MA

4πR(1)

where

R is distance from P ′ to P

M is magnetization of metal sheetA is cross-sectional area of metal sheet.

Distance R can be calculated as shown in (2), and thus themagnetic field density generated from the metal sheet is asshown in (3).

R2 = (ρ cosϕ − ρ′ cosϕ′)2

+ (ρ sinϕ − ρ′ sinϕ′)2 + (z − z′)2

= ρ2 + ρ′2 − 2ρρ′ cos(ϕ − ϕ′) + (z − z′)2 (2)

B = −μ0(

Δ

Vm) (3)

In addition, if the homogeneity for the axial componentof magnetic field density Bz is improved in the calculationof magnetic field homogeneity, the homogeneity of the ra-dial component Bρ is automatically improved, thus increas-ing the overall magnetic uniformity. For this reason, onlythe axial component of the magnetic field density is typi-cally considered in the design of magnets for generation ofhomogeneous magnetic fields such as in NMR/MRI [3]. Ac-cordingly, the axial component of the magnetic field gener-ated in the metal sheet can be expressed as follows. One canassume that a magnetic field density with the same size andopposite direction appears since the metal sheet acts as onemagnetic dipole in actual cases as explained above.

Bz = −μ0(∇Vm)z

= π0MA(z − z′)2

4π(ρ2 + ρ′2 − 2ρρ′ cos(ϕ − ϕ′) + (z − z′)2)32

(4)

2.3 Optimization Using Linear Programming

Linear programming is a type of optimization method andshould be expressed as the equality constraints, the inequal-

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ity constraints, and the scope of values that the design vari-able can take [4], as shown in (5).

Aeqx = Beq

Aineqx ≤ B ineq (5)

lb ≤ x ≤ ub

where

x is design variable vectorAeq,Aineq are coefficient matrices related to design vari-ablesBeq,B ineq are constant matrices unrelated to design vari-ableslb,ub are lower and upper limits of design variables.

For the optimization discussed here, the difference be-tween the average magnetic field density and the magneticfield density at m points located on the surface of the di-ametric spherical volume (DSV), an area of interest in theNMR magnet, should be within a given error range, whichcan be expressed in the numeric equation as shown in (6).

−T

2≤ (Bsk + Bf k) − Bmean ≤ +T

2(6)

where

k = 1,2, . . . ,m

Bsk is magnetic field density generated by the source coilat the kth positionBf k is magnetic field density generated by the metal sheetat the kth positionBmean is average magnetic field density on the given DSVT is allowable error range of magnetic field homogeneity.

The average magnetic field density Bmean in the DSVarea can be expressed as the sum of the magnetic field den-sity generated from the metal sheet and the magnetic fielddensity generated from the magnet itself. Using this, (6) canbe expressed again as follows:

−T

2≤ Bsk + Bf k − 1

m(Bs1 + · · ·

+ Bsm + Bf 1 + · · · + Bf m) ≤ +T

2(7)

where k = 1,2, . . . ,m.

Equation (7) can be separated into the terms related todesign variables and those unrelated, and then summarizedas shown in (8) and (9).

ckBf ≤ −ckBs + T

2(8)

−ckBf ≤ ckBs + T

2(9)

where

k = 1,2, . . . ,m

Bf = [Bf 1 · · ·Bf m]′Bs = [Bs1 · · ·Bsm]′ck = [− 1

m, . . . ,1 − 1

m, . . . ,− 1

m]; row vector as only kth

element is 1 − 1m

.

In (8) and (9), magnetic field density Bf generated bythe metal sheet depends on the thickness of the metal sheetlocated at the n position, and this can be expressed as shownin (10) and (11) using the design variable x.

ckaklxl ≤ −ckBs + T

2(10)

−ckaklxl ≤ ckBs + T

2(11)

where

k = 1,2, . . . ,m

l = 1,2, . . . , n

akl is magnetic field density generated at the kth evalua-tion position when the thickness of the metal sheet at thelth position is 1xl is thickness of metal sheet at lth position.

By evolving ck and akl and expressing it as C and A,respectively, as in (12) and (13), the optimization using themetal sheet can be expressed in the form of inequality con-straints as shown in (5).

CAx ≤ −CB + T

2(12)

−CAx ≤ CB + T

2(13)

2.4 Discretizing of Design Variables Using EvolutionStrategy

Linear programming is a deterministic optimization methodand finds the optimal design variables relatively fast. How-ever, since the thicknesses of metal sheets that can be actu-ally obtained are not continuous values, and also since themetal bits should be manufactured from commercial prod-ucts, one requires the process of discretizing the design vari-ables to the multiples of practical minimum thickness. Inthis process, the homogeneity of the magnet must be re-duced, and another optimization process is needed. In thispaper, we discretized the thickness of metal sheets by ac-cepting the results from linear programming and used thesein the initial values of the evolution strategy, as shown inFig. 2. The minimum unit of metal sheet was determined as0.25 mm, which is commercially available.

The evolution strategy is a nondeterministic optimizationmethod in which a predetermined variable group becomesthe first parent generation. From these, the next generation iscreated by random selection in a certain scope. The createdgeneration is called the child generation, separately from the

1040 J Supercond Nov Magn (2011) 24: 1037–1043

parent generation. The next parent generation is formed byselecting the b variable groups most suitable for the desiredvalue of objective functions from the entire parent genera-tion and the child generation. This process is called transi-tion and reproduction. This process is followed by the nextprocess of adjusting the variable width of transition depend-ing on the results of reproduction. It is the adaptation processof adjusting the evolution to find the absolute minimum. Asthe above process repeats, the transition variable width grad-ually decreases, resulting in a design variable most suitablefor the design goal. This adaptation method is called the(a+b) evolution strategy. In particular, the (1+1) evolutionstrategy is mathematically investigated for several variables,and has a simple algorithm and fast convergence speed, andthus is most widely used [5].

3 Verification and Some Examples

We applied the above shimming method to a virtual solenoidmagnet that has a central magnetic field of 5 T with a z2

second-order axial gradient inhomogeneous magnetic fieldand an x first-order radial gradient inhomogeneous mag-netic field, as shown in Fig. 3. The winding size of this vir-tual magnet is 800 mm for the inner radius, and the initialmagnetic field homogeneity at 30 mm DSV was calculated

Fig. 2 Two consecutive optimizations. As the optimized design vari-able X0 changes to its discretized variables Xs , the homogeneity of themagnet is reduced. So another optimizing method is performed

to be 628 ppm. The width and height of one metal sheetto be located at the room temperature bore of the magnetwere determined as 20 mm and 5 mm, respectively, andit was set to arrange 15 and 8 metal sheets at the axialand azimuthal direction of the magnet, respectively. In ad-dition, the thickness of the metal sheet was up to 20 mm,and such values can be adjusted during the optimizationprocess.

Figure 4 shows the results of passive shimming. Thegraph of Fig. 4(b) shows a magnetic field homogeneity of7 ppm before the evolution strategy is applied, and a mag-netic field homogeneity of 8.75 ppm if it is discretized tothe thickness of the metal bit that can actually be man-ufactured using the evolution strategy. Figure 4(a) showsthe location and thickness of the metal sheet that offsetsthe inhomogeneous magnetic field gradient. Figure 5 showsthe comparison with magnetic field homogeneity by lin-ear programming in order to evaluate the discretizing re-sults using the evolution strategy. While the optimum val-ues obtained from linear programming are continuous val-ues, the magnetic field homogeneity slightly falls in thecourse of discretizing. Figure 5 shows the field map at acylinder with a radius of 15 mm, it is slightly differentfrom the magnetic field homogeneity evaluated at 30 mmDSV.

Commercially available NMR/MRI magnets often con-tain a shimming coil for the specific magnetic field gra-dient. In this case, it should have a function to performshimming on gradients other than the specific field gradi-ents. Figure 6 shows the results of such passive shimming.Only the x inhomogeneous field gradient was removed inFig. 3, so that the amount of metal bits can be greatly re-duced.

Fig. 3 5 T virtual magnet forverification. This magnet has z2

second-order axial gradient, xfirst-order radial gradient, and628 ppm of initial magneticfield homogeneity

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Fig. 4 Results from theferromagnetic shimming.(a) The location and thicknessof metal bits after optimization;(b) Field mapping aftershimming with 8.75 ppm

(a)

(b)

Fig. 5 Comparison of resultsfor linear programming andevolution strategy

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Fig. 6 Ferromagneticshimming for x gradient only.(a) The location and thicknessof metal bits to eliminate xgradient; (b) Field mappingwith z2 gradient only afterferromagnetic shimming

(a)

(b)

We also applied the algorithm mentioned above to a realmagnet. The magnet was manufactured at Francis BitterMagnet Laboratory (FBML) as the insert of an NMR mag-net. The homogeneity can be improved dramatically by us-ing the ferromagnetic shimming method presented in thispaper, as shown in Fig. 7.

4 Conclusion

Magnetic field homogeneity is the most critical factor indetermining the performance of NMR/MRI magnets, andthis paper discussed a passive shimming method for im-proving this homogeneity. We induced a formula for mag-netic field density generated by magnetized metal sheets,

optimized the location and thickness of metal sheets us-ing linear programming, and discretized the practical thick-ness value of metal sheets by consecutively applying theevolution strategy. In particular, we could convert the de-sign variable to a realizable value without a big loss ofmagnetic field homogeneity in the course of discretizingit. In addition, a function to remove or do shimming ononly specific inhomogeneous magnetic field gradients wasadded.

Further implementation of this passive shimming methodrequires the process of actually measuring the magnetizationvalue of metal sheets and forming the field map. In addition,follow-up studies are required on the method of fasteningthe metal sheets on the magnet with a strong magnetic field

J Supercond Nov Magn (2011) 24: 1037–1043 1043

Fig. 7 Ferromagneticshimming for real magnet. Themagnet was manufactured inMIT-FBML, and thehomogeneity can be improveddramatically using theferromagnetic shimmingmethod presented in this paper

at the room temperature bore area of the NMR/MRI mag-nets.

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