Geometric distortion and shimming considerations in a rotating MR-linacdesign due to the influence of low-level external magnetic fields

K. Wachowicza)

Department of Medical Physics, Cross Cancer Institute, University of Alberta, 11560 University Avenue,Edmonton, Alberta T6G 1Z2, Canada and Department of Oncology, University of Alberta, 11560 UniversityAvenue, Edmonton, Alberta T6G 1Z2, Canada

T. TadicDepartment of Physics, University of Alberta, 11560 University Avenue, Edmonton, Alberta T6G 1Z2, Canada

B. G. FalloneDepartment of Medical Physics, Cross Cancer Institute, University of Alberta, 11560 University Avenue,Edmonton, Alberta T6G 1Z2, Canada and Departments of Oncology and Physics, University of Alberta,11560 University Avenue, Edmonton, Alberta T6G 1Z2, Canada

(Received 12 October 2011; revised 13 March 2012; accepted for publication 14 March 2012;

published 19 April 2012)

Purpose: This work investigates with simulation the effect of external stray magnetic fields on a

recently reported MRI-linac hybrid, which by design will rotate about the patient axis during ther-

apy. During rotation, interactions with magnetic fields from the earth or nearby ferromagnetic struc-

tures may cause unacceptable field distortions in the imaging field of view. Optimal approaches for

passive shimming implementation, the degree and significance of residual distortion, and an analy-

sis of the active shimming requirements for further correction are examined.

Methods: Finite element simulations were implemented on two representative types of biplanar

magnet designs. Each of these magnet designs, consisting of a 0.2 T four-post and a 0.5 T C-type

unit, was simulated with and without an external field on the order of the earth’s field (0.5 G) over a

range of rotated positions. Through subtraction, the field distribution resulting from the external

field alone could be determined. These measured distributions were decomposed into spherical har-

monic components, which were then used to investigate the effect of their selective removal to sim-

ulate the effects of passive and active shimming. Residual fields after different levels of shim

treatment were measured and assessed in terms of their imaging consequence.

Results: For both magnet types, the overall success of a passive shim implementation was highly de-

pendent on the orientation for which it was based. If this orientation was chosen incorrectly, the pas-

sive shim would correct for the induced fields at that location, but the overall maximal distortion at

other locations was exacerbated by up to a factor of two. The choice of passive shim orientation with

the least negative consequence was found to be that where the magnet B0 axis and transaxial compo-

nent of the external field were aligned. Residual fields after passive shimming and frequency offset

were found to be low in the simulated scenarios, contributing to <1 mm of distortion for most stand-

ard imaging sequences (based on a 0.5 G external field). However, extremely rapid single-shot

sequences could be distorted by these residual fields to well over 5 mm. These residuals when ana-

lyzed were found to correspond primarily to second-order spherical harmonic terms. One term in par-

ticular was found to account for the vast majority of these residual fields, defined by the product of

the two axes perpendicular to the axis of rotation. The implementation of this term would allow the

resulting geometric distortion to fall to the order of 1 mm, even for single-shot sequences.

Conclusions: After appropriate passive shimming, the imaging distortion due to an external field of

0.5 G was found to be important only in rapid single-shot sequences, which are especially suscepti-

ble to field inhomogeneity. Should it be desirable to use these sequences for real-time tracking,

made conceivable due to the lower susceptibility concerns at low field, these residual fields should

be addressed. The ability to use only one second-order term for this correction will reduce the cost

impact of this decision. VC 2012 American Association of Physicists in Medicine.

[http://dx.doi.org/10.1118/1.3702591]

Key words: rotating MRI, linear accelerator, distortion, external magnetic field

I. INTRODUCTION

Magnetic resonance (MR) imaging devices in the clinical

environment have reliably been stationary units over their

last half-century of use. That is, their position with respect to

the imaging suite does not change. This stationary position-

ing allows assumptions of a static magnetic environment,

which have implications on magnet design and the

approaches used to maintain a uniform imaging field. Of

course, there are always concerns with transient positioning

2659 Med. Phys. 39 (5), May 2012 0094-2405/2012/39(5)/2659/10/$30.00 VC 2012 Am. Assoc. Phys. Med. 2659

of large metallic equipment surrounding the suite, which can

influence the field inside the bore. However, these transient

changes can be controlled with both building design and the

watchful eye of MR staff to ensure that a large piece of ma-

chinery has not found a convenient parking location immedi-

ately adjacent the MR suite. If, on the other hand, the

assumption of a stationary magnet is no longer valid, the

external magnetic environment (which may include stray

fields such as that produced by the earth) will have a variable

influence on the uniform imaging field. This influence

depends on the position and, more importantly, the orienta-

tion of the MR unit.

There is some precedent for nonstationary MR devices in

the clinical world. For example, the IMRIS systems1 incor-

porate a moveable magnet on rails that can provide imaging

for patients undergoing various interventional or surgical

procedures and then recede from the suite without disturb-

ance to patient position. However, all the imaging in a given

suite is performed at a similar translational position and con-

sistent orientation angle, meaning that the magnet has a

fairly consistent view of the magnetic environment surround-

ing the unit. On the other hand, the concept of an MRI inte-

grated with a linear accelerator (linac) for radiotherapy as

presented by Fallone et al.2 will require the device to provide

imaging at arbitrary orientations over a complete rotation

within a single treatment session (Fig. 1). In this design, a

linac rotates in tandem with a low-field biplanar MR imager

about a stationary patient. This particular design for an MRI-

linac hybrid allows for the delivery of the treatment beam in

a direction either perpendicular or parallel to B0 (i.e., refer-

ring to the orientations in which the linac can be fixed to the

rotating magnet), which has been shown to offer some dosi-

metric advantages.3 Assuming that the hardware can be

made to rotate in a sufficiently rigid manner, there is no rea-

son to expect that the acquisition of images should proceed

with any difficulties beyond those in the standard static re-

gime. However, given that the imaging from this device is to

be used for the guidance of a radiotherapy beam, one has to

ensure that the variable magnetic nonuniformity induced by

interactions with external fields at different rotational posi-

tions is of a negligible order or that there is an effective

means to compensate for them.

Traditionally, the influence of static low-level external

magnetic fields can be effectively negated when an MR unit

is passively shimmed on site. However, in this instance, a

passive shimming approach will be unable to treat field con-

tributions from external sources except at one particular rota-

tional position: despite the fact that the passive shims rotate

with the magnet, the static external sources will induce vary-

ing field distributions in the magnet FOV as the device

rotates, making any one passive shim solution incapable of

correcting for distortion at all orientations. This angular-

dependent distortion of Bz originates not only from the influ-

ence of the external field component that is parallel to the

magnetic axis (B0) at any given orientation, but when a mag-

netic yoke is utilized even components of the external field

perpendicular to this axis can distort Bz. (Bz refers to the por-

tion of the B0 vector field that is parallel to the magnetic

axis. Since the B0 vector field is so dominated by Bz, influen-

ces on the transverse components Bx and By will have a neg-

ligible effect on the resulting Larmor frequency and are not

considered in this work.)

Active shimming techniques, where gradient fields of

various spatial orders are generated by room-temperature

coils to compensate for field inhomogeneities, are means by

which these variable field effects could be compensated. Of

course, the practicality of this approach will be determined

by the orders of distortion seen upon changes in orientation,

as well as the significance of the distortion fields they

would correct. This work explores these issues through ex-

amination of a rotating biplanar unit as modeled through fi-

nite element simulations. Two types of biplanar designs (a

four-post design and one with a C-type support structure)

are simulated as representative of this class of magnet,

which may include yokeless magnets such as the “double-

donut” design.4

FIG. 1. (a) Biplanar 0.2 T magnet model. (b) C-type 0.5 T magnet model. The B0 field is oriented between the magnet poles, as illustrated on the C-type mag-

net. When implemented in a rotating MR-linac design, the patients would be aligned along the axis of rotation as marked on the diagrams, and the entire appa-

ratus including a linear accelerator rigidly fixed to the gantry would rotate about them.

2660 Wachowicz, Tadic, and Fallone: External magnetic field influence on rotating MRI-linac hybrids 2660

Medical Physics, Vol. 39, No. 5, May 2012

II. METHODS

II.A. Magnetic models

A permanent 0.2 T biplanar magnet was modeled in

COMSOL MultiphysicsTM (Version 3.5). The magnet model

was based on a four-post permanent whole-body magnet

design [Fig. 1(a)]. A model for a 0.5 T magnet was also

developed using a C-type design [Fig. 1(b)]. The span

between pole structures was 70 cm for the biplanar magnet

and 75 cm for the C-type. These models were intended to be

representative of common design concepts, with the knowl-

edge that the gap size and the shape in general will vary on

actual systems. Nonferromagnetic hardware such as the gra-

dient coils was not included in the simulations. A virtual

cubic enclosure with an edge dimension of 8 m was used to

surround the device for the definition of boundary condi-

tions. At this distance, the model enforces the assumption

that the magnetic field normal to the boundary will have

fallen to zero. In total, three different materials were incor-

porated into the magnetic model. The majority of the magnet

structure comprised 1020 carbon steel, including the posts

and upper and lower support plates for the four-post design

and the “C” support structure for the C-type magnet. The

pole plates and rose rings nominally comprised Armco ingot

iron. The B-H curves used for these simulations were based

on curves reported in the literature5,6 and are displayed in

Fig. 2. Finally, the magnet core material was simply mod-

eled as a volume with a constant magnetization in the Z axis.

The magnetization was set as 8� 105 A/m for the 0.2 T mag-

net and 1.9� 106 A/m for the 0.5 T magnet.

II.B. Simulations

Since it can be assumed that no free electric currents exist

in these static models, the problem was solved using the

magnetic scalar potential, U, as the solution variable. COM-

SOL uses a finite element method (FEM) to optimize this so-

lution on the basis of applicable Maxwell’s equations,

namely r � �B ¼ 0 and r� �H ¼ 0 (in this current-free case).

The H field can then be determined from the gradient of this

solution, i.e., �H ¼ rU. Given that the B-H curves of these

modeled nonlinear materials are known a priori, the result-

ant B field can be easily inferred.

The effect of a stationary uniform field such as that pro-

duced by the earth (hereafter referred to as the external field)

was reproduced by altering the boundary conditions on two

opposing faces of the outer cubic enclosure. Rather than forc-

ing the normal magnetic field to zero on these two opposing

sides, both sides were assigned a normal field of half a gauss,

pointing out of the enclosure on one side and into the cube on

the other. For each magnet type, this cubic enclosure was then

rotated to a number of different orientations around the mag-

net model, simulating the rotation of the MR device about its

patient axis in the presence of a magnetic field. Orientations

of the cubic boundary relative to the model were varied

between 0� and 90� with 22.5� intervals. At each rotation

angle, two simulations were performed: one with the uniform

external field and one without, but both with the same finite

element mesh. In this way, a subtraction of the two resulting

field maps could give a detailed spatial view of the magnetic

contribution resulting from the applied uniform field. Symme-

try principles were then used to generate field maps resulting

from the remaining angular positions.

The ferromagnetic materials used in magnet construction

clearly have a nonlinear magnetic permeability, as can be

seen in Fig. 2. However, it can be expected that the magnetic

response will be approximately linear over a small range of

applied field, particularly since the magnet materials are al-

ready under a large magnetic influence and, as such, are

operating outside of their most nonlinear regime. A series of

simulations were performed on the C-type magnet with

applied uniform fields of 0, 0.5, 2, 4, 6, and 8 G to test this

expectation and to determine a range of acceptable linear

extrapolation for our results evaluated with an external field

of 0.5 G. The 0 G simulation was subtracted from those at

other fields to obtain distributions originating from the influ-

ence of the external field alone.

II.C. Harmonic decomposition

At each rotational magnet orientation, the Bz field contri-

butions in the imaging ROI that result from interactions

between the magnet structure and the external field were an-

alyzed to determine what strengths and orders of shim cor-

rections would be required to negate them. This is an

important consideration, since unlike traditional MR suites

where the impact of external fields can be dealt with using

passive shim techniques, here, passive shims (anchored to

the rotating device) will only be able to fully address a single

magnet orientation. Field interactions at all other orienta-

tions will have to be addressed by active shim techniques,

and unlike passive shims, each increasing order of active

field correction adds a dramatically increasing level of com-

plexity and expense. This analysis was carried out by decom-

posing the spatial magnetic field perturbations into spherical

harmonic basis functions.

The Bz component of the magnetic field distribution (the

component of interest for MRI) will satisfy the Laplace

equation:FIG. 2. B-H curves for Armco ingot iron and 1020 steel, implemented in

magnetic simulation models.

2661 Wachowicz, Tadic, and Fallone: External magnetic field influence on rotating MRI-linac hybrids 2661

Medical Physics, Vol. 39, No. 5, May 2012

r2BZ ¼ 0: (1)

The solution to this equation in spherical coordinates

involve the spherical harmonic functions, allowing Bz to be

described as7,8

BZ ¼XN

n¼0

Xn

m¼�n

Cn;mrnYmn h;/ð Þ; (2)

where the spherical harmonic functions Ymn h;/ð Þ ¼

Fmn � Pm

n cos hð Þeim/ �1ð Þm. The normalization factor Fmn is

defined asffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2nþ1

4p �n�mð Þ!nþmð Þ!

q, and Pm

n xð Þ is the associated Legendre

polynomial. This particular normalization factor is chosen to

enforce the principleðX

Ymn

�� ��2dX ¼ 1; (3)

which simplifies extraction of the orthogonal basis set compo-

nents from an arbitrary distribution. The spherical harmonic

basis set as defined above consists of complex functions, and

as such, field distributions cannot be physically created to be

representative of this basis set. However, the functions

described by Ymn h;/ð Þ can be manipulated through linear

combination to define a new all real basis set Smn h;/ð Þ.9 Since

it can be shown that �Ymn h;/ð Þ ¼ �1ð Þm�Y�m

n h;/ð Þ, then

Smn h;/ð Þ ¼

1ffiffiffi2p Ym

n h;/ð Þ þ �1ð Þm�Y�mn h;/ð Þ

� �for m > 0

Y0n h;/ð Þ for m ¼ 0

1

iffiffiffi2p Y mj j

n h;/ð Þ � �1ð Þm�Y� mj jn h;/ð Þ

� �for m < 0:

8>>>>><>>>>>:

(4)

The 1� ffiffiffi

2p

factor is derived from applying the same normal-

ization to Smn as was defined in Eq. (3). Therefore, Bz can

now be described as

BZ ¼XN

n¼0

Xn

m¼�n

An;mrnSmn h;/ð Þ: (5)

In this work, the magnetic field distributions as determined

by simulation were decomposed into the function set

rnSmn h;/ð Þ. Coefficients for this set of basis functions were

evaluated up to sixth-order for completeness (n � 6),

although it was expected that the higher-order functions

would comprise an increasingly small portion of the overall

B0 distortion resulting from interactions between the magnet

structure and an external uniform field. The weighting con-

stants An;m were determined by taking advantage of the or-

thogonal nature of the rnSmn h;/ð Þ function set. Thus,10

An;m ¼1

R2n

ðX

BFESðh;/;RÞ � Rn �Smn h;/ð Þ dX; (6)

where BFESðh;/;RÞ is the magnetic field data derived from

finite element simulation, evaluated at radius R. In practice,

the field distributions were evaluated at the mesh points

from the finite analysis procedure and spline-interpolated to

points on a 30 cm radius sphere centered at the magnet iso-

center. Field values were sampled from rings on the sphere

at a constant polar angle, h. In total, 179 rings were sampled

at polar angle increments of 1�. A single point was also

sampled at both h¼ 0� and h¼ 180�. The sampling density

on each ring was based on the nearest integer value of

180 sin hð Þ, giving an arbitrary D/ interval of 2� at the equa-

tor and larger as one approached the poles, maintaining a

roughly uniform sampling density over the sphere surface.

The smallest number of samples permitted on any ring was

four (near the poles) to avoid any regional bias in the sam-

pling pattern. A numerical implementation of Eq. (6) was

used to evaluate the weightings of the harmonic components

present in our results.

The An;m coefficients as described in Eqs. (5) and (6) have

dimensions of Flux DensityDistancen . Although these have the same

dimensionality as the field patterns generated by gradients

and shim coils, they will have to be scaled appropriately to

have physical meaning in and of themselves. (This is due to

the fact that the largest value in the each Smn h;/ð Þ function,

to which the coefficients are applied, is not in general unity.)

To this end, the values Gn;m defined below refer to the rate

change of Bz along the radial direction for which that rate is

a maximum:

Gn;m ¼ An;m �max Smn h;/ð Þ

�� ��� �: (7)

II.D. Analysis

Each magnetic field simulation was sampled with the

spherical distribution as described above, in addition to a

three-dimensional Cartesian grid with 1 cm resolution. For

each simulation scenario, a subtraction of both these numeri-

cal samplings was implemented between cases with and

without the presence of the external field. This subtracted

Cartesian array was taken as a representation of the induced

field distribution. The corresponding subtracted spherical

samplings were then used to decompose the distribution in

terms of spherical harmonic functions.

Frequently in this work, certain harmonic terms were iden-

tified as targets for shimming, either passively or actively. In

this case, those harmonic terms were reconstructed onto a

Cartesian grid and then subtracted from the appropriate

2662 Wachowicz, Tadic, and Fallone: External magnetic field influence on rotating MRI-linac hybrids 2662

Medical Physics, Vol. 39, No. 5, May 2012

Cartesian distributions—from all rotation angles in the pas-

sive case and one particular angle for active shimming.

III. RESULTS AND DISCUSSION

III.A. Linearity tests

The results from the linearity tests are displayed in Fig. 3.

Plot 3(a) displays spherical harmonic components derived

from simulated C-type magnet field distributions, which

through subtraction represent the isolated influence of uni-

form external fields ranging between 0.5 and 8 G. Rather

than expressing them in absolute terms, they are normalized

to the results in the 0.5 G case. In this way, all components

should vary along the same linear path, and deviations from

this path are easy to visualize. Plot 3(b) further illustrates

these deviations and quantifies them in terms of a percent

difference from the linear path. As is seen here, the harmonic

components exhibit less than 1% error when the 0.5 G simu-

lation is linearly extrapolated to a higher field up to 4 G,

with the majority of components deviating less than 0.5%.

The extrapolation error increases further with higher external

fields—in the case of 8 G, the deviation for some compo-

nents is nearing 2%. While most concerns of environmental

magnetic fields interacting with the rotating MR device are

likely to come from the earth’s magnetic field, other sources

can exist, including static fields generated by nearby equip-

ment or even asymmetrical distributions of steel within and

immediately surrounding the magnet vault. Should examina-

tion of these generated fields reveal a largely dc component

at the site of the device, the results and trends gleaned from

this work will translate and scale with the appropriate field

level. Of course, in a typical scenario, the effect of these

fields could be effectively nulled by passive shimming tech-

niques, but as is demonstrated in later discussion, this

becomes a complicated issue when the angular position of

the MR imager is not fixed.

III.B. Harmonic breakdown of induced fields

As most of our conclusions from this work are drawn

from the spherical harmonic decomposition of the induced

magnetic field within the imaging ROI of the magnet, it was

FIG. 3. The response of the magnetic field distribution resulting from the

inclusion of external fields in the simulation of the C-type magnet. The

external fields are varied up to 8 G. (a) The spherical harmonic components

(up to fourth order) of the induced distributions are displayed here,

expressed as multiples of the solution for the case of 0.5 G. (b) This graph

more clearly shows the deviation from a linear response as the external field

strength increases. At an external field strength of 4 G, the most deviating

components show less than 1% drift from a linear extrapolation of the 0.5 G

simulation.

FIG. 4. Comparison of external-field induced Bz distortions directly from finite element simulation (a and d) to reconstructions based on spherical harmonic

decomposition to sixth order (b and e). The last column (c and f) are the residual maps resulting from subtraction of columns one and two. The first row (a–c)

corresponds to the C-type magnet, and the second row (d–f) corresponds to the four-post design. The applied uniform field had a strength of 0.5 G in these

examples and was oriented parallel with B0.

2663 Wachowicz, Tadic, and Fallone: External magnetic field influence on rotating MRI-linac hybrids 2663

Medical Physics, Vol. 39, No. 5, May 2012

important to validate the in-house code written to perform

the decomposition. Figure 4 displays representative field

cross-sections from both the four-post and C-type magnets.

In each case, a field distribution taken directly from the FEM

simulations is displayed next to an analytic reconstruction of

the field from harmonic components calculated as described

in Sec. II. Maps of the residuals after subtraction are also

shown. Qualitatively, the slowly varying field distribution

shows good agreement between the FEM simulation and the

harmonic reconstruction. There are discrepancies seen in the

residual maps of up to 8% of the signal span. However, these

differences seem to originate primarily from the coarse mesh

grid used in the finite element modeling (seen as the triangu-

lar shapes in the residual map). Given the general lack of

structured signal in the residuals beyond these local mesh-

related fluctuations, the decomposition process as imple-

mented appears to generate the appropriate harmonic

coefficients.

Tables I and II identify the amplitudes of calculated har-

monic components derived from the four-post and C-type

magnet structures, respectively, while under the influence of

an external field. Results are included over a complete range

of magnet orientations with respect to the external field. The

components that exert a nontrivial influence on the field dis-

tributions are italicized. It is interesting to note how the four-

post model gives rise to no sizable odd-order components

when subjected to an external field, while there are signifi-

cant first- and third-order components arising from the

C-type model. This is likely a result of the nonsymmetric

design of the C-type magnet, in which field distributions

corresponding to flux induced in the single structural column

are not balanced by a similar column on the opposing side of

isocenter.

The exact numbers as displayed in these tables are not as

important as the trends and orders of magnitude that they

show. Each magnet design will produce a different field dis-

tribution when exposed to an external field due to the various

dimensions and magnetic properties of the support structures

can take. However, due to the consistent biplanar nature of

these magnets, certain useful patterns can be extracted that

one may expect to see in the generic case. This will be dis-

cussed in more detail later in the manuscript.

III.C. Passive shimming considerations

First, if there is any flexibility to do so, the rotational axis

of the magnet (Fig. 1) should be oriented as parallel as possi-

ble to the direction of the external field. This would maxi-

mize the component of the external field that falls along the

patient axis, the effects of which will not vary with magnet

angle and, thus, are well suited to compensation with passive

shimming methods. However, even in the case where the

TABLE I. Strengths of fitted spherical harmonic field patterns matching the distribution generated by imposing a magnetic field over the four-post biplanar mag-

net structure. Results are expressed per unit gauss of external field.

Orientation of external field with respect to primary field direction

Order Field pattern 0� 22.5� 45� 67.5� 90� 180� 270� Units

Zero S00 : dc 102.27 94.50 72.31 39.13 0.00 �102.27 0.00 lT

rS10 : Z 0.00 �0.02 �0.01 �0.01 0.02 0.00 0.02

First rS11 : X 0.00 0.00 0.01 0.00 �0.01 0.00 �0.01 lT/m

rS1�1 : Y 0.00 0.00 0.01 0.03 �0.01 0.00 0.01

r2S20 : Z2 � ðX2 þ Y2Þ=2 �4.91 �4.56 �3.37 �1.83 0.00 4.91 0.00

r2S21 : ZX �0.02 �0.02 �0.10 0.04 0.04 0.02 0.04

Second r2S2�1 : ZY 0.00 �24.01 �44.34 �57.95 �62.73 0.00 62.73 lT/m2

r2S22 : X2 � Y2 �0.01 0.00 0.02 0.06 �0.04 0.01 �0.04

r2S2�2 : 2XY 0.00 0.01 �0.04 0.06 0.01 0.00 �0.01

r3S30 0.00 �0.13 �0.16 0.55 0.13 0.00 0.13

r3S31 �0.02 �0.08 0.46 0.08 �0.05 0.02 �0.05

r3S3�1 0.04 0.17 �0.32 0.14 �0.34 �0.04 0.34

Third r3S32 �0.04 �0.05 �0.07 0.45 �0.22 0.04 �0.22 lT/m3

r3S3�2 �0.01 �0.10 0.09 0.09 0.12 0.01 �0.12

r3S33 �0.02 �0.03 �0.27 0.03 0.05 0.02 0.05

r3S3�3 �0.02 �0.01 0.00 0.37 0.12 0.02 �0.12

r4S40 22.77 20.21 16.47 9.29 �1.69 �22.77 �1.69

r4S41 0.08 �0.64 0.53 �0.12 �0.02 �0.08 �0.02

r4S4�1 �0.05 33.28 59.19 79.63 85.42 0.05 �85.42

r4S42 0.10 0.62 �1.44 0.01 �1.16 �0.10 �1.16

Fourth r4S4�2 0.07 0.17 0.68 1.76 0.99 �0.07 �0.99 lT/m4

r4S43 �0.06 �0.41 �0.21 �1.08 0.13 0.06 0.13

r4S4�3 �0.13 �0.54 �0.61 �0.15 �0.93 0.13 0.93

r4S44 2.29 1.85 2.61 0.95 0.02 �2.29 0.02

r4S4�4 0.06 �0.15 0.45 �0.99 0.54 �0.06 �0.54

2664 Wachowicz, Tadic, and Fallone: External magnetic field influence on rotating MRI-linac hybrids 2664

Medical Physics, Vol. 39, No. 5, May 2012

room layout is forgiving enough to allow the patient axis to

be sited at will, any remaining external field component per-

pendicular to the patient axis will remain to affect homoge-

neity during rotation. Further, the orientation of the

MR-linac treatment device is likely to be constrained pre-

dominantly by the room dimensions and access points, mak-

ing this approach of limited use.

There are two passive shimming approaches that may be

applied in this rotating magnet scenario, both of which have

merits. The first simply involves the implementation of pas-

sive shims based on a single magnet orientation. This will

clearly create the optimal field conditions for that particular

magnet position. However, as the magnet rotates away from

this shimmed orientation (together with the implemented pas-

sive shims) to different angles, the interaction with the exter-

nal field will evolve so that it is no longer compensated by

this passive approach. In fact, the position directly opposite to

the shimmed orientation (after a 180� rotation) will actually

exhibit twice as much distortion due to the presence of the

passive shims. However, a judicious selection of the orienta-

tion angle for passive shimming will minimize the impact of

this distortion-enhancing drawback while still providing the

optimal homogeneity at the selected angle. Figure 5 illustrates

the impact of an improperly selected orientation for shim-

ming. The two solid-line plots follow the maximum field dis-

tortion within a 40 cm FOV after being treated with passive

shims optimized at two different angles, 0� and 90�. These

two angles were chosen because they resulted in the most

extreme difference in residual field distortion through a mag-

net rotation. (These angles refer to the separation between the

external field direction and the B0 axis and are not in reference

to vertical or horizontal. Further, they relate only to the posi-

tion at which the shim was optimized and pose no restriction

on angular positions possible for imaging.) The two different

passive shims result in distinctly different distortion levels

during magnet rotation. The results in the first column include

the effects of zero-order (dc) field shifts and are therefore of

least significance since the compensation for dc field offsets

are trivial as the magnet rotates from one angle to another.

The second and third columns, which assume corrections for

zero-, and zero- and first-order distortions, respectively, both

show that minimal distortion through rotation occurs when

the passive shims are calculated based on the 0� orientation

(at alignment of B0 and the external field). Apart from the

zero-order term, the interaction with the external field seems

to have the smallest consequence at this orientation, and there-

fore, when it is passively shimmed, there is little distortion to

exaggerate when the magnet rotates to the opposite position.

An alternate approach would be to calculate a passive

shim arrangement based on an average field measurement

throughout a complete rotation. This procedure would result

in a solution similar to that obtained in the absence of any

TABLE II. Strengths of fitted spherical harmonic field patterns matching the distribution generated by imposing a magnetic field over the C-type biplanar mag-

net structure. Results are expressed per unit gauss of external field.

Orientation of external field with respect to primary field direction

Order Field pattern 0� 22.5� 45� 67.5� 90� 180� 270� Units

Zero S00 : dc 17.36 16.02 12.28 6.62 �0.01 �17.36 �0.01 lT

rS10 : Z �0.04 0.04 0.07 0.08 �0.04 0.04 �0.04

First rS11 : X 8.31 7.66 5.84 3.13 �0.07 �8.31 �0.07 lT/m

rS1�1 : Y 0.02 �0.02 0.00 0.01 0.04 �0.02 �0.04

r2S20 : Z2 � ðX2 þ Y2Þ=2 �26.91 �25.00 �19.08 �10.50 0.23 26.91 0.23

r2S21 : ZX 0.02 0.08 �0.13 0.10 0.06 �0.02 0.06

Second r2S2�1 : ZY 0.01 60.11 111.09 145.29 156.81 �0.01 �156.81 lT/m2

r2S22 : X2 � Y2 �2.89 �2.70 �2.09 �1.14 0.07 2.89 0.07

r2S2�2 : 2XY 0.04 �0.03 �0.06 0.05 �0.07 �0.04 0.07

r3S30 0.18 0.02 �0.49 �0.15 �0.19 �0.18 �0.19

r3S31 �31.95 �29.60 �23.27 �12.77 0.67 31.95 0.67

r3S3�1 �0.11 �0.43 0.44 0.22 �0.17 0.11 0.17

Third r3S32 �0.04 �0.17 �0.47 �0.07 0.03 0.04 0.03 lT/m3

r3S3�2 �0.04 10.11 18.08 24.52 26.24 0.04 �26.24

r3S33 �0.21 �0.35 0.00 0.15 0.62 0.21 0.62

r3S3�3 0.01 �0.08 �0.08 �0.04 �0.23 �0.01 0.23

r4S40 81.78 75.79 60.68 32.79 0.93 �81.78 0.93

r4S41 �0.15 1.53 �0.71 0.76 1.65 0.15 1.65

r4S4�1 0.53 0.82 1.07 3.08 �0.37 �0.53 0.37

r4S42 3.65 3.45 3.96 4.00 1.19 �3.65 1.19

Fourth r4S4�2 0.12 0.07 1.63 2.09 0.23 �0.12 �0.23 lT/m4

r4S43 �0.45 �0.28 �1.11 �0.24 0.04 0.45 0.04

r4S4�3 �0.05 �3.61 �7.37 �8.77 �8.46 0.05 8.46

r4S44 2.28 2.28 1.23 0.88 0.72 �2.28 0.72

r4S4�4 0.17 �0.39 �0.03 0.30 �0.70 �0.17 0.70

2665 Wachowicz, Tadic, and Fallone: External magnetic field influence on rotating MRI-linac hybrids 2665

Medical Physics, Vol. 39, No. 5, May 2012

external field. The results of this approach are represented in

Fig. 5 by the dotted-line plot. As one can see from this fig-

ure, this passive shimming tactic creates a marginally

improved homogeneity over the implementation of passive

shims based on a single orientation, at least for the majority

of magnet angles. However, the level of improvement is

quite small, and this method does not provide the benefit of

an optimal shim for one of the orientations.

III.D. Active shimming considerations

Clearly, since rotation of the MR device will create vari-

able field distributions when interacting with an external

field, active shimming techniques are required for compensa-

tion. The zero-order field changes can simply be addressed

by a shift in resonance frequency from one orientation to

another, particularly if a calibration is performed in advance.

Similarly, the mandatory presence of linear imaging gra-

dients will render all MR devices capable of correcting

induced first-order distortions. Corrections for higher-order

terms would require extra room-temperature shim coils, with

each coil generating a field distribution that corresponds to

one term of the spherical harmonic basis set. For example,

five extra coils would be required to completely define (and

correct for) second-order variations. It is not uncommon to

find second-order shim sets in clinical imaging systems, par-

ticularly in units 3 T or higher. Third-order compensating

fields and higher are uncommon in clinical systems but are

found in spectroscopy systems and in some animal imagers.

Unfortunately for this scenario, harmonic analysis of the

field distributions incurred upon rotation of the MR device

(Tables I and II) revealed little contribution from first-order

terms. As such, it would seem that beyond the trivial correc-

tion of induced dc offsets, the second-order terms are the

most important consideration. Figure 6 illustrates the relative

consequence of active correction to first and then second

order in columns 2 and 3. As suspected above, the inclusion

of first-order corrections alone have very little impact on

reducing field distortion levels. A much greater impact came

from the inclusion of second-order correction.

Of course, the addition of a biplanar second-order shim

set11 would add a sizable incremental cost to the system.

This is a cost that under most imaging situations would be

difficult to justify given the relatively low magnitude of the

induced magnetic distortions per gauss of external field—

roughly 8 and 3 lT for the C-type and four-post magnet,

FIG. 5. Comparison of passive shim methodologies and their performance, with and without minimal (first order) active correction. Plots show the maximum

field (Bz) distortion in lT within a 40 cm FOV centered at the isocenter of the four-post magnet (a–c), and the C-type magnet (d–f). The angular component of

the graph represents the relative orientation of the B0 field and the external field. In the first column (a and d), there is no active correction, in the second (b

and e), there is compensation for a dc field offset, and in the third (c and f), there is a first order correction optimally configured for each angular position. The

three different plot-types represent results for different approaches to the passive shimming of the rotating device. The dashed plot corresponds to a passive

shim calculation based on mean field measurements taken over a 360� range of orientation. The other two plot-types correspond to the situation where the pas-

sive shim calculation is based on measurements at a single orientation (0� and 90� in these examples).

2666 Wachowicz, Tadic, and Fallone: External magnetic field influence on rotating MRI-linac hybrids 2666

Medical Physics, Vol. 39, No. 5, May 2012

respectively. The corresponding geometric impact on imag-

ing would depend on the strength of both the external field

and the frequency-encoding gradient. As an example, assum-

ing an external field of 0.5 G (roughly the strength of the

earth’s magnetic field) and a common clinical frequency-

encoding gradient of 5 mT/m, the maximum geometric dis-

tortion would be less than 1 mm. Moreover, for tracking pur-

poses, rapid sequences would likely be implemented with

gradient settings larger than 5 mT/m, minimizing the impact

further. However, since one of the primary advantages of an

MR-linac system would be the real-time tracking of mobile

tumors, it is possible that extremely rapid single-shot scans

would be considered. These sequences, such as single-shot

EPI or spiral readouts, are conceivable for this purpose at

low field due to the reduced impact of susceptibility distor-

tions. Unfortunately, they have extremely small effective

encoding gradients in the phase encode and radial directions

for EPI and spiral readouts, respectively. The time between

successive readout “sweeps” in these sequences is typically

on the order of 1 ms,12 resulting in an effective phase gradi-

ent well below 1 mT/m for most fields of view. This, in the

example above, would lead to spatial misregistration or ra-

dial blur on the order of 5 mm and above.

Should one wish to retain the flexibility of using this type

of sequence for tracking, it may be advisable to consider

second-order corrections. Fortunately, there is a straight-

forward way to simplify the associated hardware require-

ments. In the spherical harmonic analysis corresponding to

the two biplanar magnets simulated in this work seen in

Tables I and II, one can see that the ZY term is dominant for

the second-order breakdown. Thus, it may be possible to pro-

duce the vast majority of required second-order corrections

with only one shim coil. The last column of Fig. 6 displays

the residual distortion if only the ZY term is included in the

second-order corrections. This single-term correction appears

to perform nearly as well as the one with all five terms

included, as seen in the column to the left. When imple-

mented, this approach would require one fifth of the coils and

power supplies that would be normally associated with a

second-order shim set, significantly reducing the potential

cost of including this capability in a rotating system.

IV. CONCLUSIONS

The MR-linac design based on a biplanar magnet rotating

in tandem with a linear accelerator has some specialized

design considerations regarding interactions with external

fields. For most MR devices, the field distributions that result

from such interactions can be simply negated by means of

passive shimming techniques. However, in the case of a

rotating MRI, the interactions result in variable magnetic

distributions that cannot be addressed by a single passive

shimming implementation. Although no one passive shim-

ming design can accommodate the distorted field at all rota-

tion angles, this work revealed the importance of basing the

passive shim on the magnet orientation where the B0 axis

and the exterior field most closely align. This choice of ori-

entation for passive shim optimization minimized the field

distortion seen as the magnet rotates. Other choices were

FIG. 6. Effect of higher-order active correction on rotational distortions. Plots show the maximum field (Bz) distortion in lT within a 40 cm FOV centered at

the isocenter of the four-post magnet (a–d) and the C-type magnet (e–h). The angular component of the graph represents the relative orientation of the B0 field

and the external field. The first three columns show residual distortions after active correction to zero, first, and second order, respectively. The last column (d

and h) similarly shows corrections to second order, but with only one of the 5 s order terms in use (the ZY term). The two different plot-types represent results

for two different acceptable approaches to the passive shimming of the rotating device.

2667 Wachowicz, Tadic, and Fallone: External magnetic field influence on rotating MRI-linac hybrids 2667

Medical Physics, Vol. 39, No. 5, May 2012

found to enhance the maximum magnetic distortion due to

the external field by up to a factor of two.

If the field distortion that remains after the implementa-

tion of a passive shim is to be addressed, active techniques

must clearly be used. The simulations in this work indicate

that these field distortions are dominated by zero- and

second-order spherical harmonic terms. (First-order terms

were found to play a minor to nonexistent role, seemingly

depending on the symmetry of the MR device.) The zero-

order terms are trivial to compensate by adjusting the operat-

ing frequency of the magnet, f0. On the other hand, active

compensation of the second-order terms would in general

require a set of five room-temperature shim coils to be

included in the system. Second-order distortions induced by

an external field such as that originating from the earth

(�0.5 G) were found to be fairly small for most sequences

(<1 mm over a 40 cm FOV). However, single-shot sequen-

ces, which may be considered for tracking purposes at low

field, are much more sensitive to field variation and could

result in distortions of over 5 mm in the earth-field example

above. Should this be a concern, the simulations from this

work indicate that the vast majority of the second-order con-

tributions may be corrected by one term alone rather than

the complete set of five, greatly simplifying the hardware

requirements of this option. This term in question will be

defined by the product of the two axes perpendicular to the

axis of rotation, which was ZY in the geometry used herein.

Though the most-likely source of external field will origi-

nate from the earth and low-level contributions from sur-

rounding equipment, field contributions from the anisotropic

distribution of magnetic structures in the environment may

well be dominated by steady-state terms at the magnet site.

Under this condition, much of theory and results from this

work may be applicable, although this would have to be

investigated in future work.

ACKNOWLEDGMENT

The authors gratefully acknowledge the Alberta Cancer

Foundation for financial support.

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