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Magnetic Resonance in Medicine 41:1221–1229 (1999) Resistive Homogeneous MRI Magnet Design by Matrix Subset Selection Patrick N. Morgan,1* Steven M. Conolly,2 and Albert Macovski2 A new technique for designing resistive homogeneous multicoil magnets for magnetic resonance imaging (MRI) is presented. A linearly independent subset of coils is chosen from a user-defined feasible set using an efficient numerical algorithm. The coil currents are calculated using a linear least squares algorithm to minimize the deviation of the actual magnetic field from the target field. The solutions are converted to practical coils by rounding the currents to integer ratios, selecting the wire gauge, and optimizing the coil cross-sections. To illustrate the technique, a new design of a short, homogeneous MRI magnet suitable for low-field human torso imaging is presented. Magnets that satisfy other constraints on access and field uniformity can also be designed. Compared with conventional techniques that employ harmonic expansions, this technique is flexible, simple to implement, and numerically efficient. Magn Reson Med 41:1221–1229, 1999. r 1999 Wiley-Liss, Inc. Key words: resistive; homogeneous; magnet design For the decade following the first demonstration of MRI in 1973 (1), resistive magnets were most often used to create the main magnetic field, B0 (2–4). Technical advances over the last 20 years have contributed to the proliferation of superconducting and permanent magnet systems in MRI. Today MRI scanners based on resistive main magnets are relatively scarce in the marketplace. However, cost pressures have encouraged the reconsideration of resistive B0 magnets in MRI scanners. Even with the additional costs of a power supply with precision regulation and a cooling system, resistive magnets are generally much cheaper and simpler to build than superconductors (5). Precision-regulated power supplies with ratings up to 50 kW can be purchased commercially (6) or created by externally stabilizing a standard power supply with feedback circuitry and a current sensor (7,8) or a field probe (4,9–11). If the supply cost is high, efficiency may be improved by incorporating ferrite materials into the electromagnet (12). Some examples of clinical whole-body scanners successfully operated with resistive magnets are the 0.15 T scanner at the University of Cincinnati (13), the 0.02 T scanner made by Instrumentarium (14), and the 0.08 T scanner at the University of Aberdeen (15). Resistive main magnets 1Department of Electrical Engineering, Texas A&M University, College Station, Texas. 2Department of Electrical Engineering, Stanford University, Stanford, California. Grant sponsors: National Science Foundation, Air Defense Systems Department of the Johns Hopkins University Applied Physics Laboratory, Whitaker Foundation, and GE Medical Systems. *Correspondence to: Patrick N. Morgan, 208A Zachry Engineering Center, Department of Electrical Engineering, Texas A&M University, College Station, TX 77843-3128. E-mail: morgan@ee.tamu.edu Received 10 July 1998; revised 3 February 1999; accepted 5 February 1999. r 1999 Wiley-Liss, Inc. have also been employed by researchers in low-field MRI (16), Overhauser imaging (8), and prepolarized MRI (17,18). Various techniques have been used to design a resistive B0 magnet. The most venerable of these relies on a spherical harmonic expansion of the magnetic field (19). For magnets consisting of multiple coils, the coil currents, radii, and locations are determined by setting the harmonic coefficients to zero. These equations have a linear dependence on the coil currents and a much more complicated nonlinear dependence on the coil radii and locations. Historically, the equations have been solved by making assumptions such as equal coil currents or radii (20,21). Hand calculation or nonlinear optimization may also be used (22,23). Although they permit a solution to be obtained, these assumptions limit the ability of this design approach to satisfy geometric constraints such as maximum or minimum coil sizes or gaps. For example, the equal current assumption creates solutions where the radii and locations of the coils approximate a sphere around the sample. If access is required between the inner-most coils in the arrangement, the design must either be scaled in size or redesigned using a different number of coils. Either way, the harmonic design technique does not guarantee an optimal solution that satisfies constraints on both access and field homogeneity. Another technique is to employ a ‘‘target field’’ approach whereby the design is optimized based on a desired magnetic field profile within a user-defined region. This approach has been previously used to design MRI gradient coils (24), NMR field cycling magnets (25), and shim coils (26). In those approaches, no attempt was made to optimize coil radii or locations, so a fixed magnet geometry is required. Pissanetzky (27) presented a target field approach that does optimize these parameters along with the coil currents using quadratic programming. The optimization causes most, if not all, of the currents to attain their upper or lower bounds, resulting in clustering of similar current values. In the case of relatively few magnet coils, the clustering may be desirable, but in other cases it may prove overly restrictive. Simulated annealing is another numerical approach that has recently been applied to open access superconducting magnet design (28,29). Linear programming has also been used to optimize coil locations and radii (30). Here we present an alternative technique to design a resistive B0 magnet. Ferrite materials have not been modeled, so the technique is suitable for designing air core electromagnets only. Our approach involves selecting an optimal subset of coils from a user-defined feasible set prior to determining the coil currents using the target field. All computations require only standard matrix decompositions and linear matrix operations. Once the optimal coil locations and currents have been calculated, the solutions 1221 1222 Morgan et al. are converted to practical coils by rounding the currents to integer ratios, selecting the wire gauge, and optimizing the coil cross sections. Compared with previous techniques, this technique involves relatively few calculations, often less than 10 million floating point operations (flops), which can typically be executed in less than 1 sec on a SUN workstation. Using this technique, the magnet designer is required to specify the bounds of the feasible set as well as the location, shape, and size of the target field. This approach has the advantage of enabling the design to meet constraints on both access and field homogeneity. Furthermore, the designer can trade field homogeneity for power dissipation by changing the number of selected coils. This flexibility is particularly valuable since closely spaced coils in the feasible set often produce largely redundant magnetic fields. By changing the bounds of the feasible set or other design parameters, variation of power dissipation, field homogeneity, access, or other quantities can be explored interactively. Following presentation and discussion of the theory, we illustrate use of the technique with a new design of a short B0 magnet. This magnet may be integrated into a low-cost MRI scanner designed to image the human torso with minimal patient claustrophobia. THEORY Our magnet design technique is based on minimizing the residual ␦ ⫽ 0A I ⫺ b0, [1] where 0 · · · 0 denotes vector length (sometimes called ‘‘2norm’’), A denotes the magnetic field created by the set of feasible coils, I denotes the coil currents, and b denotes the target field points. The residual can be minimized by calculating the coil currents using the ‘‘pseudo-inverse’’ of A, as shown previously (26). If the target field constraints can be satisfied exactly, so ␦ ⫽ 0, then the pseudo-inverse provides the solution that minimizes the sum of the squares of the coil currents (31). In either case, this approach generally requires all coils in the feasible set to carry a nonzero amount of current. Rather than using all coils in the feasible set, we select an optimal subset of coils prior to minimizing the residual using the pseudo-inverse. Our implementation of this technique and some of the relevant tradeoffs are discussed in this section. FIG. 1. A feasible set of N coaxial coils. Multiple coils may exist at the same location or with the same radius. When connected in series, any subset of the coils forms a multiple coil magnet. matrix, A, written as A ⫽ (A1A2A3 . . . AN). If the cross-sectional dimensions of the coils are small compared with the coil radii, then the magnetic field can be accurately calculated by modeling the coils as infinitesimal current loops. For the coaxial coils shown in Fig. 1, the magnetic field at the location (r, z) is given by (32, p. 237) Br(an, zn; r, z) ⫽ 冑 (z ⫺ zn) 2r (an ⫹ r)2 ⫹ (z ⫺ zn)2 3 ⭈ ⫺K() ⫹ Bz(an, zn; r, z) ⫽ an2 ⫹ r 2 ⫹ (z ⫺ zn)2 (an ⫺ r)2 ⫹ (z ⫺ zn)2 4 E() 冑 2 (an ⫹ r)2 ⫹ (z ⫺ zn)2 3 ⭈ K() ⫹ an2 ⫺ r 2 ⫺ (z ⫺ zn)2 (an ⫺ r)2 ⫹ (z ⫺ zn)2 E() 4 [3] where Br and Bz are the radial and axial components of the field and r ⫽ 冑x2⫹ y 2. The functions E( ) and K( ) are elliptic integrals where is defined as Feasible Set The feasible set consists of coils that are sampled over a range of allowable coil radii and locations. In this paper, we consider feasible sets consisting of N coaxial coils, as shown in Fig. 1. Multiple coils may exist at the same location or with the same radius. When connected in series, any subset of the coils forms a multiple-coil magnet. The total current carried by the nth coil, in amp-turns, is denoted by In, the average radius of the coil is denoted by an, and the average location is denoted by zn. The magnetic fields produced by the feasible coils carrying unit current are collected as columns of the [2] ⫽ 1 4anr 2 (an ⫹ r)2 ⫹ (z ⫺ zn)2 1/2 . [4] We model the elliptic integrals using polynomial approximations that are rapid to calculate and accurate to 2 ⫻ 10⫺8 (33, pp. 591–592). In MRI, the magnetic moments respond to the magnitude of the field, 0 B 0 ⫽ 冑Br2 ⫹ B2z . In the vicinity of the uniform magnetic field created by coaxial current loops, the radial component is much smaller than the axial component. Hence 0 B 0 ⬇ Bz and the columns of the feasible matrix can MRI Magnet Design by Subset Selection 1223 be calculated by An ⫽ Bz(an, zn; r, z). [5] For magnets consisting of loops with even symmetry in zn, the columns may be calculated by An ⫽ 5 Bz(an, zn; r, z) ⫹ Bz(an, ⫺zn; r, z) Bz(an, 0; r, z) zn ⫽ 0 zn ⫽ 0. [6] Designs producing fields that have odd symmetry in zn would have the fields subtracted since the currents are equal and opposite on either side of zn ⫽ 0. Since the feasible set of coils is restricted to zn ⱖ 0, this approach is desirable to maximize computational efficiency, but it is not required to create symmetric magnet designs. Our experience indicates that nearly symmetric coils are always selected when symmetric target points are specified. Designing magnets that produce other field profiles, such as a shim coil or gradient coil, may require consideration of field components other than Bz. ment since the series is convergent for R ⬍ Rn for all n. Note that as the order increases the magnetic field becomes more uniform around the origin. Constant target field values can be specified by locating the points at the zeros of the Legendre function, PL(cos ). A good approximation to the zeros is given by (33, p. 787) l ⫽ (4l ⫺ 1) 4L ⫹ 2 ⫹ 1 8L 2 cot 1 2 12 (4l ⫺ 1) 4L ⫹ 2 ⫹O 1 L3 , [9] for 1 ⱕ l ⱕ L. The error in the approximation is on the order of 1/L3. For symmetric arrangements that calculate the field using Eq. [6], the number of target points is M ⫽ L/2 arranged over a quarter circle for 0 ⬍ ⬍ /2. Subset Selection We apply an algorithm suggested by Golub that selects a subset of columns of A based on linear independence according to the following steps (35, pp. 590–595). 1. Calculate the singular value decomposition (SVD) of A, defined as (31, p. 442) Target Field We denote the target field by a vector, b, that consists of M discrete points. All the magnetic fields in the matrix, A, are sampled at the target locations, so the dimension of A is therefore M ⫻ N. The matrix element occupying the mth row and the nth column can be written as A mn ⫽ Bz(an, zn; rm, zm) 1 ⱕ m ⱕ M, 1 ⱕ n ⱕ N. [7] Assuming the target volume contains no current sources, both the divergence and the curl of the field are zero, i.e., ⵜ · B ⫽ 0 and ⵜ ⫻ B ⫽ 0. Hence specification of a uniform target field along a boundary is sufficient to guarantee a uniform field inside. For the coaxial set of loops shown in Fig. 1, the target field can be specified along a semicircular boundary in the r ⫺ z plane. For symmetric arrangements that calculate the field using Eq. [6], the points at z ⬍ 0 are redundant, reducing the boundary to a quarter of a circle. Other target field shapes, such as an ellipsoid, can be specified using an appropriate boundary. For a spherical boundary around a uniform field, the exact location of the target field points can be obtained by expanding the axial component using spherical harmonics, Bz(R, ) ⫽ A ⫽ U S V T, [10] where U is M ⫻ M and orthogonal, S is M ⫻ N and diagonal, and V is N ⫻ N and orthogonal. The diagonal elements of S are the singular values of A. The SVD is calculated so the singular values are ordered in decreasing magnitude. 2. Calculate the QR decomposition with column pivoting of the first N̂ columns of V, defined as (35, p. 271), V (:, 1 : N̂)T ⫽ Q R P T, [11] where N̂ denotes the actual number of coils ultimately used to construct the magnet. The matrix, Q, is N̂ ⫻ N̂ and orthogonal, R is N̂ ⫻ N and upper triangular, and P is the N ⫻ N square permutation matrix that orders the diagonal elements of R in decreasing magnitude. 3. Calculate the columns of A to be selected by collectˆ, ing them into a matrix, A ˆ ⫽ A P (:, 1 : N̂), A [12] ⬁ C R P (cos ) 2兺 l l l l⫽0 ⬇ B(0) ⫹ 2 CLRLPL(cos ), [8] where Cl are the spherical harmonic coefficients, B(0) is the field strength at the origin, and (R, ) are spherical coordinates. The functions Pl( ) are Legendre functions of degree l and Pl1( ) are associated Legendre functions of the first kind of degree l and order 1 (34, p. 1014). The approximation is valid for magnets that produce a field of order L over regions extending to the innermost coils in the arrange- ˆ is M ⫻ N̂. The columns of A are sorted by where A multiplying it with the permutation matrix, P, obˆ, tained in step 2. The matrix of selected columns, A contains the first N̂ sorted columns of A, corresponding to the subset of coils to be retained. The remaining coils are discarded. To obtain useful solutions with the algorithm, some relations among the matrix dimensions are required. To approximate closely the target field typically requires defining more feasible coils than target points, so M ⱕ N. [13] 1224 Morgan et al. For all matrices, the number of linearly independent columns of A, known as the rank, must be less or equal to the smallest dimension, R ⫽ rank( A ) ⱕ M. [14] If R ⫽ M, the matrix is termed ‘‘full rank’’; otherwise it is termed ‘‘rank deficient.’’ Since there are a total of R independent columns, the minimum residual approximation of the target field requires no more than R coils selected from the feasible set. Hence the rank provides an upper bound to the number of selected coils, N̂ ⱕ R. [15] Provided Eq. [15] is satisfied, Golub’s algorithm selects a subset of columns that is guaranteed to be linearly independent since the QR decomposition performs Gram-Schmidt orthogonalization. Heuristically, the subset tends to be well conditioned. However, the conditioning comes at the expense of discarding some nearly dependent columns that may be required to minimize the residual. Alternative algorithms, such as performing a QR decomposition directly on A, may lead to poorly conditioned subsets containing nearly dependent columns that create homogeneous, high-power magnet designs. Later in this paper, we explore the tradeoff between power and homogeneity by varying the number of selected coils, N̂. constraints. These tradeoffs will be illustrated with some design examples later in this paper. Power Dissipation and Coil Mass For resistive magnets, the total power dissipation can be written as the sum of the power dissipated by the individual coils, P⫽ 2 Since the magnetic field depends linearly on the coil currents, the optimum currents can be calculated as the solution to a linear least squares (LLS) problem (31, p. 156), ˆ )⫺1A ˆ T b, Î ⫽ (Â TA [16] which minimizes the residual between the magnetic field produced by the subset of coils and the target field values, as given by ˆ Î ⫺ b 0 . ␦ˆ ⫽ 0 A [17] ˆ are linearly independent, the Since the columns of A inversion can always be calculated. If the number of selected coils equals the rank of A, N̂ ⫽ ˆ spans the range R, the residual is minimized since A of A. All other subsets can produce no smaller residual. Hence, ␦ˆ ⱖ ␦, [18] where ␦ is obtained with no subset selection as given by Eq. [1]. Although reducing the number of selected coils, N̂, may produce a larger residual, the magnet design is simplified since fewer coils are used. In some cases, the magnet power dissipation decreases. Hence this technique enables the magnet designer to trade field homogeneity for lower power by changing N̂. This enables the designer to obtain a low-power design that just satisfies the target field 兺 a I /A , 2 n n n⫽1 [19] n where is the electrical conductivity of the wire material, an is the average radius of the nth coil, In is the nth element of the vector of currents, I, in amp-turns, and An is the coil cross-sectional area. The total mass of the coils is given by ˆ N 兺aA, m ⫽ 2 n [20] n n⫽1 where is the density of the wire material. Given the coil currents, equations [19] and [20] indicate that the power and the mass can be traded off for each other by changing the wire gauge. Our approach to this tradeoff is to specify the wire current, Iw, based on the maximum rating of the power supply. The wire area, Aw, can be determined by Aw ⬇ Iw Linear Least Squares ˆ N 冑 1 1 cp dT/dt . [21] where cp is the specific heat, and dT/dt is an approximation for the uncooled heating rate of the coils (18). Typically we specify a slow heating rate of dT/dt ⫽ 67C/min to permit minimal air cooling. Higher values could be specified if a more sophisticated cooling system is to be designed to manage the heating. The appropriate wire gauge is selected by comparing Aw with a specification list available from the wire manufacturer. MATERIALS AND METHODS We implemented this technique in the mathematical analysis software package, MATLAB. Interactivity is achieved in the MATLAB environment through a text-based user interface and multiple plotting windows. Computationally intensive functions, such as those for field calculation, were developed in C and linked into MATLAB as object files. Magnets were defined as arrays of coil structure elements each containing the wire currents, locations, radii, number of wires, and wire sizes. The magnet structure was passed among custom functions developed for subset selection, coil optimization, field calculation, and display. After subset selection, the solutions were converted to practical coils by quantizing the loop currents to integer ratios based on the desired wire current, Iw. We implemented a constrained nonlinear optimization procedure to minimize the RMS inhomogeneity by varying the coil locations and radii over a small range. The optimization was implemented using a sequential quadratic program- MRI Magnet Design by Subset Selection 1225 ming method with the MATLAB function ‘‘constr( ).’’ A similar method was implemented to optimize the coil aspect ratios, defined as the ratio of the build of the coil in r to the width of the coil in z. In our experience, these optimization steps converge quickly since the starting values determined by subset selection are close to the optimum values. To minimize calculation time, we perform ‘‘economy’’ decompositions that return the first R columns of V and return the permutation matrix as a vector. For numerical stability, we calculate the pseudo-inverse based on the ˆ . For M ⫽ 20 target field points arranged on a SVD of A semicircle of radius 10 cm and N ⫽ 1000 feasible coils of radii 25 cm uniformly sampled along a 2 m long solenoid, the total calculation time is about 350 msec, corresponding to about 5.5 million floating point operations (Mflops). Although not fundamentally important to the magnet design problem, rapid computation enables the designer to explore variation of power dissipation, field homogeneity, access, or other quantities by changing the bounds of the feasible set or other design parameters interactively. RESULTS We are most interested in low-field MRI of the human torso with 10–20 ppm field inhomogeneity over a 20 cm DSV and a field strength of 0.07 T, corresponding to a Larmor frequency of 3 MHz for protons in water. The minimum bore size is 45 cm. Solenoid Magnet Figure 2 shows the feasible set and target field points for a solenoidal magnet. The feasible set consists of 40 coils uniformly sampled down the length of the magnet. To ensure a sufficiently large bore, all feasible coils have equal radii set to 25 cm. The magnet length is judiciously set to 3 m to minimize the resulting field inhomogeneity. The target field points, denoted by ‘‘o,’’ are set at the zeros of the Legendre function corresponding to a 12th order field and mapped to a semicircle of diameter 20 cm in the r ⫺ z plane. The currents and magnetic field profile obtained after subset selection are shown in Fig. 3. The currents obtained after subset selection are denoted by ‘‘x.’’ For the sake of comparison, the currents and fields for a magnet designed without subset selection, using the pseudo-inverse of A ˆ , are shown by the dashed lines in the figure. instead of A The negative currents correspond to counter-wound coils leading to an ‘‘overshoot’’ of the magnetic field. When subset selection is used, none of the coils are counter- FIG. 3. Coil currents (a) and on-axis magnetic fields (b) for 3 MHz solenoidal magnets designed by subset selection (x and solid line) and designed by pseudo-inverse without subset selection (dashed lines). wound, and the entire magnet is much shorter. In this case, the magnetic field falls off gradually at the edges. Both magnetic fields are comparable in field strength and uniformity over the target DSV. Table 1 shows a more detailed comparison of this magnet design with similar designs obtained using other techniques. For equivalent access, the inner radius was set to 25 cm for all designs. The RMS inhomogeneity was calculated over a 20 cm DSV. The power dissipation and coil mass were calculated assuming an uncooled temperature rise of 67C/min. The pseudo-inverse technique involves no subset selection, so the magnet consists of 40 coils. This approach achieves minimum inhomogeneity but very high power and mass because of the counter-wound coils. The peak current technique, which involves calculating the pseudo-inverse only at the six locations of peak positive current, has lower power dissipation and mass but proTable 1 Comparison of Magnet Design Techniques Method FIG. 2. Feasible set and target field points for a solenoidal magnet. Pseudo-inverse Peak currents Lee-Whiting solenoid Garrett solenoid Golub subset selection RMS Power Coil No. of inhomogeneity dissipation mass coils (ppm) (kW) (kg) 40 6 4 6 6 0.3 770.9 38.3 0.3 8.3 56.554 25.644 6.165 7.023 6.603 1471 667 160 183 172 1226 Morgan et al. FIG. 4. Coil cross sections and simulated magnetic field for a six-coil, 3 MHz solenoidal magnet. Contours are drawn for 1, 10, and 100 ppm field inhomogeneity. duces an excessively inhomogeneous field. The four-coil Lee-Whiting solenoid produces an 8th order field that is also excessively inhomogeneous (20). The six-coil Garrett solenoid produces a 12th order field with moderate power (21). However, some improvement can be gained using the subset selection technique that converges approximately to the Garrett solenoid but achieves somewhat lower power by relaxing the field inhomogeneity closer to our 10 ppm limit. Figure 4 shows the results of this design when converted to practical coils. Contours are drawn around the coil cross sections and for field inhomogeneity values of 1, 10, and 100 ppm. The magnet and coil design parameters are shown in Table 2. Minimal air cooling is required since the coils heat slowly. Designing the magnet to use a more sophisticated cooling system could allow less massive coils at the expense of higher power dissipation. The mass of the support material is neglected. The coil axial locations and radii refer to the centers-of-mass of the turns. Note that some extra turns are required to be placed on top of the others. Figure 5 shows a plot of the power-mass product versus RMS field inhomogeneity for subsets consisting of 1–12 coils selected using the feasible set and target points shown in Fig. 2. The number above the ‘‘x’’ denotes the number of selected coils in the subset. To improve the field homogeneity, the number of feasible coils was increased by 1 for odd numbers of selected coils. The general trend of the plot indicates that the field becomes more homogeneous at the expense of higher power-mass as the number of coils increases. In this case, there are two exceptions to this rule. First, the power-mass rapidly increases for N̂ ⫽ 12 because this design requires counter-wound coils that substantially cancel the field. The field inhomogeneity is slightly worse (less than 0.1 ppm) than for N̂ ⫽ 11, perhaps because of numerical roundoff error. Second, two sets of designs (N̂ ⫽ 7, 9) and (N̂ ⫽ 8, 10) appear nearly identical on the plot. In this case, the design with the fewer number of coils is preferable to minimize manufacturing cost. Other criteria such as sensitivity to coil positions could also be considered. Short Magnet Figure 6 shows the feasible set and target field points for a solenoidal magnet. Each intersection point defines the Table 2 Design Parameters for the Six-Coil, 3 MHz Solenoidal Magnet Shown in Fig. 4 Magnet parameters Value No. of coils Wire current Field strength Power dissipation Uncooled heat rate Coil mass Bore Length Wire gauge RMS inhomogeneity 6 100 A 0.07 T 6.665 kW 6°C/min 171 kg 0.441 m 0.608 m AWG #5 11 ppm Coil parameters Outer Middle Inner Axial locations Radii No. of turns ⫾0.273215 m 0.251312 m 13 ⫻ 13 ⫹ 2 ⫾0.116104 m 0.249898 m 8⫻7⫹1 ⫾0.036940 m 0.252810 m 6⫻7⫹3 FIG. 5. Plot of power-mass product versus RMS field inhomogeneity for subsets consisting of 1–12 coils. The number above the x denotes the number of selected coils in the subset. MRI Magnet Design by Subset Selection FIG. 6. Feasible set and target field points for a short solenoidal magnet. Each intersection point defines the radius and location of each coil in the feasible set. radius and location of each coil in the feasible set. In this case the maximum length is restricted to 30 cm. To include the possibility of larger coils at the same locations as smaller coils, both dimensions are included in the feasible set. To ensure a sufficiently large bore, the minimum coil radius is set to 25 cm. The number of feasible coil locations was set to 40, and the number of feasible coil radii was set to 3. In this case, the best results were obtained when the number of target field points was increased to 20, as denoted by ‘‘o.’’ As before, the target DSV is set to 20 cm. Increasing the target points beyond the desired field order may be required for magnets designed with restrictive length constraints. Figure 7 shows the results of this design when converted to practical coils. Contours are drawn around the coil cross sections and for field inhomogeneity values of 1, 10, and 100 ppm. The magnet and coil design parameters are shown in Table 3. For comparable heating and field homogeneity, this design requires 3.7 times more power than the previous design. The top coils carry most of the current and are counter-wound relative to all the other coils. 1227 points or less are sufficient to obtain useful magnet designs. To strictly enforce constraints such as minimum power, mass, or inductance, other subset selection or optimization algorithms may be developed. Although not analyzed here, successful magnet design often requires consideration of other factors such as the extent of fringe fields, Lorentz forces, insulation, mechanical stresses, or use of water cooling. These factors become most significant at a high field strength or with numerous counter-wound turns. The sensitivity of field homogeneity to inevitable coil winding or positioning errors is also a consideration. However, for a realistic position tolerance of 100 m, the field degradation for the magnets considered here was simulated to be less than 100 ppm, so homogeneity could be restored by low-power shim coils. Ferrite materials have not been modeled, so the technique is suitable for designing air core electromagnets only. Shielded magnets may be designed with our technique by incorporating remote target field points. It may also be desirable to design the coils with equal inner radii to reduce manufacturing costs or to improve mechanical stability. To avoid the nonlinear optimization steps when converting the designs to practical coils, the columns of A could be calculated using expressions derived for finite coils, such as those provided by Urankar (36). All our calculations have assumed resistive copper wire. Alternatively, the coils could be wound with aluminum wire, but because of differences in the specific heat and electric conductivity, approximately 1.5 times more coil area would be required for the same heating rate. Even with the additional volume the aluminum coils may be cheaper and would weigh less than the copper coils. Even with the slightly greater expansion with temperature, aluminum coils could be more cost effective than copper coils. Other wire materials could also be considered. DISCUSSION Design Technique Selecting the optimal subset of coils based on linear independence has the advantage of generating sparse solutions. The disadvantage is that the coils are selected independently of the target field values, so for some combinations of feasible sets and target fields, or if the feasible set is undersampled, the technique may yield designs that are excessively inhomogeneous. In this case, the feasible set or the target points must be redefined to obtain a solution. In our experience, feasible sets consisting of 1000 coils or less, and target fields consisting of 20 FIG. 7. Coil cross sections and simulated magnetic field for an eight-coil, 3 MHz short solenoidal magnet. Contours are drawn for 1, 10, and 100 ppm field inhomogeneity. The two large coils are counter-wound relative to the others. 1228 Morgan et al. Table 3 Design Parameters for the Eight-Coil, 3 MHz Short Solenoidal Magnet Shown in Fig. 7 Magnet parameters Value No. of coils Wire current Field strength Power dissipation Uncooled heat rate Coil mass Bore Length Wire gauge RMS inhomogeneity 8 100 A 0.07 T 24.911 kW 6°C/min 640 kg 0.460 m 0.416 m AWG #5 16 ppm Coil parameters Top Outer Middle Inner Axial locations Radii No. of turns ⫾0.153777 m 0.426109 m 22 ⫻ 23 ⫹ 9 ⫾0.154710 m 0.248902 m 8⫻9⫹5 ⫾0.086846 m 0.250157 m 5⫻6⫹2 ⫾0.030587 m 0.249711 m 6⫻6⫹3 Short MRI Magnets In our experience, counter-wound coils appear to be required for a homogeneous magnetic field when setting aggressive physical constraints such as a short magnet length or small maximum coil radius. Similar counterwound results have recently been shown by other researchers (12,28). In this regime, the magnet efficiency is low because the opposing currents cancel out the magnetic field. Because of the low efficiency, designs requiring counter-wound coils are likely to be more expensive to manufacture and operate than unconstrained designs producing a comparable magnetic field. For very inefficient designs, permanent material or superconducting wire may be required. Magnets of various other geometries could be considered. For example, the magnet could be designed for the patient to lie within the gap between the innermost coils in the arrangement. In this case the feasible set would consist of a constrained set of coils outside the region of the gap. Asymmetric magnets with offset target fields or projected fields may also be designed. CONCLUSIONS Subset selection is a new design technique based on using matrix manipulations to select an optimal subset of coils from a user-defined feasible set. These manipulations efficiently optimize coil radii and locations over a wide range of geometries based on desired access to the sample, maximum or minimum magnet length or radius, or other factors. The residual obtained after subset selection is always greater than or equal to that obtained without subset selection, but requires fewer coils than if all coils were selected. Using fewer coils can reduce magnet power dissipation, mass, or manufacturing cost. For short magnets, large counter-wound coils may be required. Our results indicate that a 33% reduction in magnet length requires 3.7 times more power for comparable heating and field homogeneity with sufficient access for a human torso. ACKNOWLEDGMENTS Patrick Morgan gratefully acknowledges the support of a National Science Foundation Graduate Research Fellow- ship and the Air Defense Systems Department of the Johns Hopkins University Applied Physics Laboratory. 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