MEDICAL magnetic resonance (MR) imaging during interventional

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1 3382 IEEE TRANSACTIONS ON MAGNETICS, VOL. 40, NO. 5, SEPTEMBER 2004 Permanent Conical Magnet for Interventional Magnetic Resonance Imaging Manlio G. Abele and Henry Rusinek Abstract We describe an open structure of permanently magnetized material for interventional magnetic resonance imaging of the brain. We transformed the ideal magnet, contained between two coaxial cones, into a practical device by a series of geometrical steps that minimize field perturbations. By taking advantage of the quasi-linear demagnetization characteristics of rare-earth materials, we could analyze the structure with an exact mathematical model. The magnet, built of material of remanence 1.38 T, generates a field of 0.45 T within its central gap 30 cm wide. Our numerical computations show a remarkable 0.25% field uniformity in the imaging region. The large opening makes the conical magnet suitable for interventional and surgical imaging. Index Terms Biomedical imaging, boundary-element methods, ferrites, Laplace equations, magnetic fields, magnetic resonance imaging, neodymium alloys, permanent magnets, surgery. I. INTRODUCTION MEDICAL magnetic resonance (MR) imaging during interventional and surgical procedures is an emerging field that has gained importance in recent years [1] [5]. Development in this field is hampered by limitations of existing magnets in combining adequate magnetic performance with sufficient access to a patient. The magnet in a conventional MR unit consists of a superconductive coil that is well suited for the generation of a strong and highly uniform magnetic field. The long cylindrical cavity of such a magnet, however, allows only limited access to the imaging region. The majority of currently used open MRI units employ permanent magnets. The most commonly used magnetic material is the ferrite, with a remanence of the order of 0.4 T and the rare-earth material such as neodymium iron boron with a remanence of the order of 1.3 T [6]. While the use of ferrite limits the field strength to approximately 0.2 T, rare-earth alloys extend the application of permanent-magnet technology to fields that are several times greater. In a traditional permanent-magnet design, the imaging region is contained in the gap between two cylindrical blocks of magnetized material [7], [8]. However, the required field uniformity within the imaging region dictates that the transversal dimensions for the blocks be large compared to the gap, resulting in a large and heavy magnet that provides inconvenient, lateral access to the patient. This paper describes the design principle for a conical magnet, a novel open structure of permanently magnetized Manuscript received February 22, 2004; revised June 28, The authors are with the Department of Radiology, New York University School of Medicine (NYUMC), New York, NY USA ( hr18@nyu.edu). Digital Object Identifier /TMAG Fig. 1. Conical magnet. material. We begin with basic equations that define the design geometry, then present the considerations relevant to the practical implementation of the open configuration. The paper concludes with key numerical computations and parameter optimization of the conical magnet. The new structure is shown to achieve a field level, field uniformity, weight, and open access that is attractive when compared with currently available permanent magnets. II. IDEAL MAGNETIC CONE Fig. 1 shows the basic conical structure of magnetized material contained between two coaxial cones of half angles and. The material is assumed to be magnetized with a uniform polarization density (remanence) parallel to the axis of the cone. The medium surrounding the external cone is assumed to be nonmagnetic, and the medium inside the internal cone is assumed to be a ferromagnetic material with infinite magnetic permeability. Assume a system of spherical coordinates where is the distance of a point from the center of the cones, is the angle between and axis of the cones, and is the angle between the planes formed by a point and the axis and the arbitrary plane that contains the axis. Assume the limit of infinite radial dimension of the cones and consider the special case of a field configuration independent of the angular coordinate. Let (1) /04$ IEEE

2 ABELE AND RUSINEK: PERMANENT CONICAL MAGNET FOR INTERVENTIONAL MR IMAGING 3383 Fig. 2. Equipotential surfaces in the ideal conical structure (left). The thick arrows indicate the orientation of the permanent polarizarion (remanence). The opening of the central section of the cone of dimension 2z is needed to create the imaging area (right). The design must also accommodate the human torso of dimension 2z. The magnetostatic potential generated by remanence within the magnetized material of the cone is given by the particular solution of Laplace s equation that satisfies the boundary condition Within the magnet, i.e., for, the solution is given by In the above equation a vacuum, and and (2) (3) denotes the magnetic permeability of The equipotential surfaces within the magnetized material are given by the equations where In the external region potential is (4) (5) (6), the magnetostatic Equation (7) is the potential of a uniform magnetic induction oriented parallel to the axis and of magnitude given by (7) (8) where is the intensity of the magnetic field. and are measured in tesla. We have Consequently, the equipotential surfaces in the external region are planes perpendicular to the axis. An example of the equipotential surfaces generated by the ideal cone is shown in Fig. 2 for the angles. In this example, the uniform field corresponds to a value. Hence, if one assumes rare-earth magnetic material of permanent polarization T, the infinite cone generates in the region a uniform field T. III. PRACTICAL MAGNET FOR INTERVENTIONAL IMAGING To transform the ideal magnetic cone into a device suitable for medical imaging, one needs to eliminate the central section of the cone of dimension to accommodate the human head (see Fig. 2) within the imaging region. Another design requirement is to accommodate the human torso of dimension. Assume now that the surface of potential (Fig. 3) passing through point becomes the new interface that separates the magnetic material and an inner ferromagnetic core of infinite magnetic permeability. Assume also that a block ferromagnetic material of infinite magnetic permeability has been inserted in the external region of the cone as shown in Fig. 3. The new block is adjacent to the magnetic material along the equipotential surface (Fig. 3) passing through point. The new ferromagnetic block is separated from the imaging region by the planar surface. Because of the uniqueness of the solution of Laplace s equation given the boundary conditions, the transformation does not affect the uniform field generated by the original ideal conical structure. A. Conical Magnet of Finite Extent A practical implementation of the proposed design approach must address the termination of the magnetic components. In this section we analyze the field perturbation due to the finite (9)

3 3384 IEEE TRANSACTIONS ON MAGNETICS, VOL. 40, NO. 5, SEPTEMBER 2004 Fig. 3. Equipotential surfaces of potential 8 and 8 become the boundaries between the magnetized material and a medium of infinite permeability. This transformation preserves the uniformity of the magnetic field in the cavity. Fig. 4. Transformation of infinite conical magnet into a finite structure. external dimension along the axis of the conical structure. We impose two requirements on the truncated cone: 1) the magnetized material should maintain the same potential difference between the central and the lateral ferromagnetic blocks as the infinite structure; 2) the potential of the central block should have the value. One way to implement these requirements is shown in Fig. 4. The structure is limited in the direction by the ferromagnetic yoke (shown as a thick line). Thus, at the magnetostatic potential is zero. Adjacent to the yoke is a cylindrical block of thickness. The cylinder is magnetized with a uniform remanence, oriented parallel to the axis. In order to generate the required potential within the central ferromagnetic core, the thickness of the magnetic material must be related to the coordinate of the central gap by the equation (10) The central core of the structure is extended laterally by a ring of thickness. The extended core separates the structure into two independent components, one that generates the potential, and another that generates the potential difference. The annular block of magnetized material of uniform remanence, oriented parallel to the axis, and of thickness is located in the second component (Fig. 4). The thickness is related to the parameter by the equation (11) To complete the transformation, a transition region ABC (Fig. 4) has to be inserted between the annular block and the conical region to maintain continuity of the flux of magnetic induction. The methodology previously developed for the design of two-dimensional transition wedges [9] can be applied to cylindrical geometry to prove that the region ABC requires a uniform remanence oriented along the coordinate and of magnitude (12) By virtue of (3) (6), the equation of the equipotential lines inside the cone can be rewritten in the form (13) where is the distance from of the point of the cone of angular width of potential. In the coordinates, the equations of the equipotential lines in the region (ABC) are Thus (13) (14) provide the equation of the curve AB (14) (15) B. Simplification of Magnet Geometry To simplify the design and lower the cost, an approximate magnet configuration is often considered. In the example of the conical magnet in Fig. 4 one can take advantage of the fact that the transition region ABC is relatively small and is located away from the imaging region. If region (ABC) is assumed to be magnetized with the same remanence as the surrounding magnets, the structure can be implemented using rare-earth material of high remanence. Moreover, practical considerations dictate that the thin ferromagnetic layer between the planes and can be eliminated. The result of these transformations is shown in Fig. 5. Since the departure from the ideal configuration generates field distortion within the imaging region, the practical design incorporates additional geometric parameters that will be used to compensate for these distortions. Fig. 5 indicates two such parameters: the radius of the cylindrical

4 ABELE AND RUSINEK: PERMANENT CONICAL MAGNET FOR INTERVENTIONAL MR IMAGING 3385 TABLE II DISTRIBUTION OF THE MAGNETIC FIELD GENERATED BY THE CONICAL MAGNET FROM FIG. 5 Fig. 5. Practical implementation of the conical structure. TABLE I DISTRIBUTION OF THE MAGNETIC FIELD GENERATED BY THE CONICAL MAGNET FROM FIG. 5 The field is given along the r and z axes of the imaging region, for a range 0 50 mm of thickness D = z 0 z of the ferromagnetic plate. The column D! 0 corresponds to a plate with D approaching zero while maintaining infinite permeability. Also simulated is the magnet with no ferromagnetic plate. It is assumed that r =65mm. The field is in tesla, all spatial dimensions are in millimeters. The field is given along the r and z axes of the imaging region, for a range mm of parameter r. The field is in tesla, all spatial dimensions are in millimeters. hole in the magnetized material, and the external radius the cylindrical structure. C. Results of Numerical Simulations The numerical computations shown in this section are based on the boundary elements method, implemented under the assumption of linear demagnetization characteristics, as is the case for modern magnetic materials [10]. With this assumption, an exact formulation of Laplace s equation is developed by computing the charges induced by magnetic material at all surfaces of (boundaries) that separate magnetic, ferromagnetic, and nonmagnetic media. After computing the surface charges by solving a system of linear equations, the 3-D distribution of the magnetic field can be computed in an efficient manner. Because of its efficiency, the boundary elements method is often used to simulate and optimize the magnet design. The field distribution has been computed for the simplified magnet of Fig. 5 based on the cone angles and. The magnetic material is assumed to have a remanence T. The cavity dimensions were mm, mm. The extent of the structure was assumed to be mm along the dimension and mm radially. The central ferromagnetic piece was terminated at mm, and the thickness of the external ferromagnetic annulus was chosen as mm. Table I shows the values of the magnetic field (in tesla) along the axes and for values of in the range mm. It is seen that the field is more uniform along the axis than along the axis. The value mm results in a field level of 0.42 T at the center and a uniformity of approximately 0.24% over the range of 100 mm of the dimension. There is a negligible (0.05%) field distortion along the axis. The effect of the thickness of the ferromagnetic annulus is illustrated in Table II. In this simulation the radius

5 3386 IEEE TRANSACTIONS ON MAGNETICS, VOL. 40, NO. 5, SEPTEMBER 2004 Fig. 6. Sectional view (left) and an overall view (right) of the conical magnet. of the cylindrical hole is assumed to be 65.0 mm. Decreasing D results in an increased field level at the cost of a marginal decrease in field uniformity. Even with no ferromagnetic annulus, the field uniformity is 0.29% over the 100 mm range of the dimension, with a value of magnetic field equal to 0.45 T at the center. Fig. 6 presents an overview of the simplified conical magnet. When built with a gap cm, the magnet will require 2.85 tons of magnetic material. IV. CONCLUSION The design approach follows a methodology for designing magnetic structures that takes advantage of the quasi-linear demagnetization characteristics of rare-earth material and makes it possible to analyze the structure with an exact mathematical model. The methodology has been applied previously to the design of a small MRI unit Artoscan (Esaote Biomedica, Genova, Italy) dedicated to the imaging of the joints [11], [12]. In a traditional permanent magnet the imaging region is contained within a relatively narrow gap between two cylindrical magnetized blocks, with thick ferromagnetic pole pieces separating the main cavity from the magnetized blocks. The pole pieces serve to improve the field uniformity by shaping the distribution of magnetostatic potential within the cavity. The numerical computations show that a powerful permanent magnet can be developed following the design approach presented in this paper. A remarkably high field uniformity is achieved in the imaging region. The field distortion is of the order of 0.25% and it can be readily compensated using standard shimming techniques [8]. When built with rare-earth material of magnetic remanence T, the magnet will generate a field of approximately 0.45 T. The conical magnet configuration can accommodate the brain and torso in the central gap, while providing a remarkably open access to the imaging region. The large opening makes it possible to integrate the magnet into a surgical suite with minimal interference with the surgical procedures and instrumentation. REFERENCES [1] R. B. Lufkin, Interventional MRI. St. Louis, MO: Mosby, [2] M. Hadani, R. Spiegelman, Z. Feldman, H. Berkenstadt, and Z. Ram, Compact, intraoperative magnetic resonance imaging-guided system for conventional neurosurgical operating rooms, Neurosurgery, vol. 48, no. 4, pp , [3] R. Steinmeier, R. Fahlbusch, O. Ganslandt, C. Nimsky, M. Buchfelder, M. Kaus, T. Heigl, G. Lenz, R. Kuth, and W. Huk, Intraoperative magnetic resonance imaging with the Magnetom open scanner: Concepts, neurosurgical indications, and procedures A preliminary report, Neurosurgery, vol. 43, pp , [4] R. Wutke, F. A. Fellner, C. Fellner, R. Stangl, H. Richter, W. Franck, A. Cavallaro, and W. A. Bautz, Alow-field MR system in acute traumatological imaging in radiology, Rontgenpraxis, vol. 54, no. 2, pp , [5] M. Schulder, D. Liang, and P. W. Carmel, Cranial surgery navigation aided by a compact intraoperative magnetic resonance imager, J. Neurosurgery, vol. 94, no. 6, pp , [6] K. J. Strnat, Modern permanent magnets for application in electrotechnology, Proc. IEEE, vol. 78, pp , June [7] T. Miyamoto, H. Sakurai, and M. Aoki, A permanent magnet assembly for MRI devices using Ne.Fe.B magnet, in Proc. 10th Int. Workshop on Rare Earth Metals and Their Applications, Kyoto, Japan, 1989, pp [8] T. Miyamoto, H. Sakurai, H. Takabayashi, and M. Aoki, Permanent Ne.Fe.B magnet for MRI, J. Mag. Soc. Jpn., vol. 13, no. 2, p. 465, [9] M. G. Abele, Structures of Permanent Magnets. New York: Wiley, 1993, pp [10] M. G. Abele and H. Rusinek, Field computation in permanent magnets, IEEE Trans. Magn., vol. 28, pp , Jan [11] D. Messineo, A. Cremona, M. Trinci, A. Francia, and A. Marini, MRI in the study of distal primary myopathopies and of muscular alterations due to peripheral neuropathies: Possible diagnostic capacities of MR equipment with low intensity field (0.2 T) dedicated to peripheral limbs, Magn. Res. Imag., vol. 16, no. 7, pp , [12] C. Masciocchi, Dedicated MR system and acute trauma of the musculoskeletal system, Eur. J. Radiology, vol. 22, no. 1, pp. 7 10, 1996.