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Optimal design of a self shielded magnetic resonance imaging magnet
M. Souza, C. Vidigal, J. Taquin, M. Sauzade
To cite this version:
M. Souza, C. Vidigal, J. Taquin, M. Sauzade. Optimal design of a self shielded magnetic res- onance imaging magnet. Journal de Physique III, EDP Sciences, 1993, 3 (11), pp.2121-2132.
�10.1051/jp3:1993265�. �jpa-00249071�
Classification Physics Abstracts
06.90 07.58
Optimal design of a self shielded magnetic resonance imaging magnet
M. Souza (*. **), C. Vidigal (*> **), J. Taquin and M. Sauzade
Institut d'Electronique Fondamentale, Universitd Paris XI, Bit. 220, 91405 Orsay Cedex, France (Received 26 April J993, accepted J£' September J993)
Rksum4. Cet article prdsente une configuration optimale d'un aimant rdsistif trds homogdne pour
I.R.M. Un matdriau magndtique perrnet h la fois de parfaire l'homogdndit6 au centre et de rdaliser le blindage de l'ensemble. les calculs ont dtd effectuds ~ l'aide d'un code utilisant [es notions de
potentiels scalaire et vecteur. Le rdsultat obtenu concerne un systkme de rdvolution avec des
cylindres de fer entourant les bobines. Con&u pour un imageur corps entier, il prdsente une grande
accessibilitd du fait de son faible encombrement en longueur. Les dimensions des diffdrents dldments et [es cartes de champs sont prdsentdes.
Abstract. This paper describes an optimal design of a highly homogeneous resistive coil system for Magnetic Resonance Imaging (MRI). Magnetic material is used to improve field uniformity at the central region and to shield the magnet. The influence of magnetic material is calculated by using a code based on the solution of scalar and vector potentials equations. The obtained result is
an axisymmetric coil configuration enveloped by iron whose optimization was made by fixing one of the criteria the weak stray field near the magnet. It presents a great accessibility to the homogeneous area and satisfies the bore's required dimensions. Dimensions and field charts are given.
1, Introduction,
In MRI systems the homogeneous field area is usually situated at quite a distance from the magnet borders. In superconducting magnets employed in imaging human subjects, central
region that represents imaging zone is located usually at more than one meter from the borders.
So, it is necessary for patients to traverse a long and narrow tunnel whose diameter must be as small as practically possible for operating cost reasons.
(*) M. Souza and C. Vidigal have scholarships of CAPES-Brazil.
(**) M. Souza and C. Vidigal have as permanent address Faculdade de Engenharia da UFJF 36 loo Juiz de Fora. Brazil.
This geometry has the following disadvantages
some claustrophobic patients (5 §l of the population) become anxious once inside this
long and narrow tunnel and refuse to be examinated
images of human body parts situated in the boundary of uniform field area are often distorted, requiring, for these parts, local antennas difficult to use
patient must be placed entirely inside the magnet although the member to be imaged is, for instance, only his leg or arm. Also, the magnet must have big dimensions although the
volume to be imaged does not require a big uniform field area.
In a lesser extent, a conventional electromagnet presents the same disadvantages. In
addition, its weight is a supplementary constraint.
We propose in this paper a configuration of a MRI magnet having a very uniform field and
reduced dimensions that provides an easy access to the homogeneous field area. Main
magnetic field is obtained by a set of resistive coils and field uniformity is assured by a
convenient arrangement of axisymmetric magnetic material pieces. In addition, it is passively
shielded thus providing a stray field of 5 G at less than 3 meters from the central region.
In the following, we consider only the main field producing set, excluding field gradient systems and radiofrequency coils.
To date, many approaches have been made to design new MRI systems. Some of them propose passively shielded magnets [1, 2]. Others present actively shielded coil configurations [3, 4] as well as hybrid systems, combining active and yoke shields [5].
2, The optimization method.
Generally, the design of a magnet is guided by the specification of a set of characteristics to be met by the configuration. They may be field intensity, field homogeneity, useful volume, bore size, power dissipation and the surrounding stray field at a remote region. The more
stringent is the specification the fewer will be the configurations that will meet it. Dimensions
and currents are the degrees of freedom of the system. Among them, we must choose a set of
parameters that will be allowed to vary about some limits that constitute a set of constraints.
The solution of the problem consists of varying the parameters while evaluating the results.
That means that each time we vary a set of parameters, we must calculate mathematical
expressions to verify if we are closer or farther from an acceptable solution. In a system having
coils and iron parts, this implies that we must have appropriate magnetic field calculation methods. This is often the most time consuming operation whose repetition thus requires fast
methods. Methods and codes like GFUN, BEM and the finite-element method have been
already described [6-8].
3. The configuration choice.
Optimal design methods applied to resistive and superconducting coil systems have been described elsewhere [9, 10]. They show that it is very difficult, if not impossible, to build short magnets using only coils, if power consumption has to remain reasonable. However, the introduction of iron parts to the coil system has many advantages because they provide a spatial magnetic contribution as opposed to that of the coil. Although this is true only for certain
configurations yet it has been a directive of this work since its begining.
Figures I and 2 show respectively the field lines for a coil and when the later is placed near a
ferromagnetic cylinder. The representation illustrates only the first quadrant of the field
configuration. The presence of the cylinder improved field uniformity on the central regions,
but unfortunately this improvement is to be in detriment of the field strength.
P
a
z
Fig. I. First quadrant view of the magnetic field lines of a pair of coils having z axis as common axis.
The coils are symmetrically placed in the R axis.
z
Fig. 2. A cylinder of ferromagnetic material is placed near to the coils of figure I to produce a
relatively uniform magnetic field in the central region. Division of the tore into elements is made in order to calculate its magnetization by the method explained below.
Another geometry, proposed by Romeo and Hoult Ill, consists of an iron ring of cross- sectional area A, magnetized in the direction of its axis of symmetry (Fig. 3).
p
P J
z
Fig. 3. The variables involved in the expression of the magnetic field due to a ring of ferromagnetic
material.
The z component H~ of the magnetic field due to this element in the central region is :
~~
~
~ )) ~ ~~ " L ~~ ~ j)( I ~ ~ ~
~
(~" ~ ~
~ + 2 (~" " j
(
n 0
~
where the P~~s are Legendre polynomials and M~ is the element magnetization. All terms in this equation have a factor that is independent of the point coordinates named here coefficient of spherical harmonics. Figure 4 shows the variation of the first 4 coefficients of equation (I)
as a function of the position of the elementary ring shown in figure 3.
The equivalent representation for a coil is shown in figure 5, where the phase difference between corresponding terms previously pointed out is obvious.
It must be noted that the ring placed in an azimuthal symmetric magnetic field has a radial component of magnetization.
The field H~ due to this component M~ is given by
H~
=
~ ~~~ " M~~f)
sin a f ~~ ~ ~~
P~(cos P
~
~(cos a (2)
2 f
n =o
f~ +
Figure 6 shows the variation of the first 4 coefficients of spherical harmonics of equation (2).
4. CALMAG3D method [12].
We give below the principles of the Code we developed for the magnetic field computation.
Let us suppose that a homogeneous ferromagnetic body of volume V~ be placed in a region
subject to a magnetic field H~ produced by conventional currents. The volume V~ is divided
into N elements of volume V, over which the components of the magnetization density are
assumed constant. Using scalar and vector potentials definitions, the magnetic field H at the I-th element is :
H~ = H~~~
,
V M~ V'( dv' (3)
4 "
v, jr r'(
Bn
20.
lo.
-20.
0.0 0.20 0.40 0.60 0.80 1.00
cosa
Fig. 4. Representation of the first 4 coefficients due to M~ of the ring of figure 3,
B~ = ~ ~~ ~ ~~~ ~ "~ "
P
~ ~
~(cos « ).
2 f
H,
= H~~~, + V x lM~ XV'( dv' (4)
4 "
i~,
jr r'(
where r is a vector from the origin to a point in the element in which M, is calculated (sampling point), r' is the integration variable taken all over volume V,, and the primed differential
operators acting on the primed coordinates. H~~~,, represents the actual field at the cavity that would exist if the volume V, was removed (Fig. 7) and has as sources :
conventional or true currents giving rise to H~ and ;
the N I elementary volumes that form, with V~, the magnetic body.
Then we can write
N
~ext, " ~0.
,
~ i ~j,
,
(5)
0 j #
where H~,, represents the magnetic field produced on the I-th element by the j-th element.
By combining equations (5) with (3) or (4) the magnetic field may be written as
iH,i
t iHo,~i + £ ( ifji imji £ iD~i iM,1 (6)
j #,
Bn
Bo
B~
86
0.0 0.20 0.40 0.60 0.80 1.00
cosa
Fig. 5. Representation of the equivalent coefficients of the pair of coils in figure I. For a given cos «, the coils and the ring coefficients have opposite signs.
where the elements of [F~ and [D, depend only on spatial variables. We can use (3) or (4) to calculate these elements and this enables us to get simpler expressions. Furthermore, with a proper selection of the sampling point [D~] becomes a diagonal matrix.
From the magnetization curve of the ferromagnetic material which shows the M -H
relationship
M,
= Y (H,) (7)
we can determine M, (or H,) of each element by iteration and finally, the magnetic field in all space.
5. Optimal design.
Systems normally employed in MRI have axially symmetric coils. If we add magnetic material parts to improve homogeneity and shield the magnet, these parts will have the same cylindrical
symmetry. That means that, if we decompose these pieces in elementary volumes as required by the method presented above, the resulting forms are elementary rings and then, the system
can be treated as a distribution of circular coils and ferromagnetic rings of known
magnetization. To each of them we can write the coefficients of the spherical harmonics of B- as [I Ii
~~
Bn
20.
~~ 84
Bo
82
-10.
86
0.0 0.20 0.40 0.60 0.80 1.0
cos a
Fig. 6. Representation of the first 4 coefficients due to M~ of the ring of figure 3, AIn + I sin~
« p , (cos «
Bn =
~~n " +
fit;
V0
~'-
T-
o
Fig. 7. Magnetized material representation.
for a coil with L turns ; and
N A~ sin a~ M~~(n + I )(n + 2)
~ nF " I
2 f ~n +
~
n +2~~°~ "J
j J j
MPJ (~ ~ ' ~~~ ~~
p '
)l~~~
~ ~n + ~ ~~ ~~~ ~~
for a ferromagnetic body decomposed in N elementary rings having M~ and M~ as
magnetization components.
The field uniformity will be searched by the minimization of the distribution spherical
harmonics.
The optimal design of a system formed by a current distribution, represented by coils, and
ferromagnetic pieces may be achieved by the variation of properly selected parameters. The system is characterized by currents and dimensions of the coils and by dimensions and
magnetization of the elements of the iron parts. Among these, we have chosen the first 3 as parameters that may vary simultaneously in each step of the process of optimization, As our
goal is to search for a great homogeneity of the magnetic field at the central region, we choose to minimize the coefficients of spherical harmonics of the system [13] up to 10th order. The
coefficients of the spherical harmonics are minimized by using subroutines of numerical solution of nonlinear equations [14]. A flow diagram of the optimization program is shown in
figure 8 which emphasizes the procedure that, for one run, the magnetization of the elements
are fixed being updated out of the loop of evaluation of the variation of the parameters. This procedure has been chosen for computing time safety,
Figure 9 shows the starting configuration where the shapes and localization of the iron parts have been chosen by the qualitative considerations made before. The upper iron part is used for
shielding purposes and the ring improves further the field's homogeneity,
6. Results.
Figure lo shows the final form of the magnet [15]. Four coils are enclosed by the ferromagnetic
material structure. Its radius is greater than 0.70m and it is 0.90 m long. Coefficients of spherical harmonics have been cancelled up to 10th order and power consumption is 14 kW for
a modered field strength of 0,099612 T. The optimization procedure has been accomplished by using the magnetization curve of ARMCO iron. Figures I I and 12 show the characteristics of the system.
7. Conclusion.
The proposed method is applicable to the design of systems having rings of iron and coils designed to produce main field in MRI magnets. The configuration shown in figure lo is given
as an example. The evaluation of the performance made by the coefficient annulment of the
spherical harmonics is effective because we need not calculate the field in many points.
The CALMAG3D, a method employed to determine the magnetization of magnetic material, has demonstrated to be fast enough to allow repetitive calculation inherent to the
proposed process of optimization, Another characteristic of this method, the discretization of iron parts, is important in the determination of the configuration of the magnet because the dimensions of these parts may be chosen as parameters, In addition, at early stages of magnetic design, a low level of discretization is enough to show if a configuration is acceptable or not, We can decompose iron parts in fewer number of elements until a detailed geometry is
established. This obviously ensures computing time safety and makes search very effective.
STARTING GEOMETRY
DETERMINATION oF Jf,j AND Jf~j
VARIATION oF PARAMETERS
DETERAiINATION
OF COEF, oF SPHER. HARM.
cm 1 0?
No
~'E5
ACCEPTABLE SOLUTION? NEW PARAMETERS
No yEs
UPDATE J(j AND Jf~~
HOAIOGENEITY 15
NO ACCEPTABLE?
YES I
~LND
Fig. 8. Flow diagram of the optimizing program. There are two criteria Legendre coefficients MO and geometrically acceptable solution. The magnetization components and field in all space are
calculated using the CALMAG3D method.
We present here an example of MRI magnet having satisfactory characteristics and easy
manufacturing iron parts. The low cost of the magnet makes viable its construction as a reduced scale prototype, It could be used to image human body parts as for example legs and
anus in laboratory researches.
P
SHIELD
H
CYLINDER fi~~#
z
Fig. 9, Initial geometry to be optimized.
0-745
~~~
~om3 cows
3=ISAE+oS A/Mz
P=14KW
~ ~
O.4OO
fir nWC'fi3-o
3~g OA13
0,K48
°m Coef. of spher. harmonics
; :
Bo = 0-996 x 10~~
B~ = ~.629 x
84 = ~-140 x
86 = 0-515 x
Bg = 0.180 x
Bio = ~-303 x 10~~
Bi Bi
Fig. lo. Representation of the obtained final form of the magnet.
20.0
p(cm)
1,ox10~~
20.0
Fig. I I. The homogeneity has been analysed in a sphere with radius 0.20 m. The relative variation in
comparison with the central field is shown with lo-level curves spread by 2.90 x 10~~
5 m
P
37~
843
Z 5m
Fig, 12. The shielding effect produced by the upper parts of the ferromagnetics material. Field values
are given in 10~~ T, The 5 G
curve is about 3 m away from the magnet axis.
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