Perturbation Method for Magnetic Field Calculations of Nonconductive Objects Mark Jenkinson,* James L. Wilson, and Peter Jezzard Magnetic Resonance in Medicine 52:471 477 (2004) Inhomogeneous magnetic fields produce artifacts in MR images including signal dropout and spatial distortion. A novel perturbative method for calculating the magnetic field to first order (error is second order) within and around nonconducting objects is presented. The perturbation parameter is the susceptibility difference between the object and its surroundings (for example, 10 ppm in the case of brain tissue and air). This method is advantageous as it is sufficiently accurate for most purposes, can be implemented as a simple convolution with a voxel-based object model, and is linear. Furthermore, the method is simple to use and can quickly calculate the field for any orientation of an object using a set of precalculated basis images. Magn Reson Med 52:471 477, 2004. 2004 Wiley- Liss, Inc. Key words: susceptibility; distortion; simulation; field calculation Theoretical calculation of the magnetic B field, given a distribution of tissue, allows modeling of various phenomena, such as MRI signal dropout, geometric distortion, interaction of B field and motion effects, manipulation of the B field using active and weakly magnetic passive shims, and respiration effects. Existing methods for calculating the B field use full finite element calculations (1,2) or approximate solutions to Maxwell s equations given either surface models of matter interfaces (3 5), voxelbased elements (6), spherical elements (7,8), or Fourier representations (9). By using a perturbation approach to solving Maxwell s equations (10), a linear first-order solution can be found which is fast and appropriate for most MR imaging applications. In addition, the perturbation method allows the magnitude of the errors to be calculated, and hence the accuracy and appropriateness of the method to be estimated for various applications. THEORY Assuming the object is nonconductive (so J 0), the relevant Maxwell s equations (ƒ H 0 and ƒ B 0) can be reduced to a single equation by using the magnetic scalar potential (11): H ƒ. This gives: 0 ƒ 1 ƒ 0, [1] Oxford Centre for Functional Magnetic Resonance Imaging of the Brain (FMRIB), University of Oxford, Department of Clinical Neurology, John Radcliffe Hospital, Headington, Oxford OX3 9DU, UK. rant sponsors: UK MRC, laxosmithkline, UK EPSRC. *Correspondence To: M. Jenkinson, Oxford Centre for Functional Magnetic Resonance Imaging of the Brain (FMRIB), University of Oxford, Department of Clinical Neurology, John Radcliffe Hospital, Headington, Oxford OX3 9DU, UK. E-mail: mark@fmrib.ox.ac.uk Received 3 November 2003; revised 17 March 2004; accepted 3 April 2004 DOI 10.1002/mrm.20194 Published online in Wiley InterScience (www.interscience.wiley.com). 2004 Wiley-Liss, Inc. 471 where is the tissue dependent susceptibility. Let the susceptibility,, be expanded as: 0 1, [2] where 0 is the susceptibility of air (4 10 7 ) and is a constant, equal to the difference in the susceptibility of the tissue under consideration and air (e.g., 9.5 10 6 in the case of brain tissue and air). Note that 1 is a scaled version of the susceptibility difference and can take on continuous values when using a complex tissue model, but will take values of 0 or 1 for a simple two-tissue model. Similarly, expand in a series: 0 1 2 2... [3] This perturbation expansion in can be substituted back into Eq. 1 to give: 0 1 0 ƒ 2 0 0 [4] 1 0 ƒ 2 1 ƒ 1 ƒ 0 0, [5] for the zeroth and first-order terms in. The first-order equation is a 3D Poisson equation: with solution: ƒ 2 1 1 ƒ 1 1 ƒ 0 [6] 0 1 x x x f x dx, [7] where (x) (4 r) 1 is the reen s function, r x x 2 y 2 z 2 and f 1 ƒ 1 1 ƒ 0. 0 This convolution can also be more concisely written as 1 f. The z-component of the B field is given by: B z H z z 0 1 0 0 z 0 0 1 z 1 0 1 O z 2. [8]
472 Jenkinson et al. The zeroth-order term is B z (0) 0 (1 0 ) 0 / z, and the first-order term is: Using the fact that: B z 1 1 1 0 B z 0 1 0 1 z. [9] f f x x x f holds for any and f, together with Eqs. 4, 5, 7 and 9, gives: 2 B 1 z 1 B 1 z 1 0 1 0 2 y z 1 B y 2 Lorentz Correction x z 1 B x 2 z 1 B z. [10] The solution derived above is valid for a continuous medium but not for a medium composed of discrete particles. For MR calculations, however, it is the field that is applied externally to the discrete nuclei that is of interest. This field can be calculated from the continuous media solution using the Lorentz Correction (11,12) (or local field representation). The corrected field is given by B LC B 2 3 0M, where M H is the magnetization of the material. Therefore, the Lorentz Corrected field can be expressed in terms of the scalar potential, (the continuous media solution) as B LC 0 (1 /3)ƒ. Comparing this with the uncorrected (continuous media) solution, B 0 (1 )ƒ, shows that the Lorentz Corrected solution can be found simply by replacing all instances of with /3 in equations involving only and terms. Consequently, the zeroth and first-order corrected fields can be written as: B LC 0 1 1 3 0 ƒ 0 [11] B 1 LC 1 3 0 1 ƒ 0 0 1 1 3 0 ƒ 1 [12] which, together with Eq. 10 gives: 1 B LC,z 1 3 0 B LC,z 2 1 1 0 y z 1 B LC,y 2 x z 1 B LC,x 2 z 2 1 B LC,z. [13] For the remainder of this article only the Lorentz Corrected fields will be used and the LC subscript will be dropped. Single Voxel Solution Equation 13 allows the first-order B z (1) field to be calculated if the zeroth-order field B (0) and the susceptibility distribution 1 are known. The zeroth-order field represents the field which would be present if there were no object in the scanner. For example, with a constant field in the z direction, B x (0) B y (0) 0 and B z (0) B 0, so that Eq. 13 simplifies considerably. The susceptibility distribution represents the object in the scanner and needs to be specified at each point in space. For complicated functions, though, the required convolutions are difficult, if not impossible, to do analytically. For most purposes, however, it is sufficient to approximate the object using small rectangular volume elements (voxels). The advantage of this is that the convolution can be done analytically for a single voxel. Consider a single voxel of dimensions (a, b, c) and, without loss of generality, let it be centered at the origin (0, 0, 0) with a susceptibility of 1 1 within the voxel and 1 0 outside the voxel. iven this, the required convolutions in Eq. 13 can be written as: 2 v z 1 B v 1 x x B v x x 2 v z x dx x a/2 x a/2 y b/2 dx y b/2 dy z c/2 z c/2dz 2 v z x B v, [14] where v stands for either x, y, or z and it is assumed that B v (0) is not spatially varying. The last integral can be easily calculated from the indefinite integral, which we will denote here as F (x). That is: F x 2 v z x B v dx [15] giving the single voxel solution as: H v,z x 2 v z 1 B v F x a 2, y b 2 2, z c F x a 2, y b 2, z c 2 F x a 2, y b 2, z c 2 F x a 2, y b 2, z c 2 F x a 2, y b 2, z c 2 F x a 2, y b 2, z c 2 F x a 2, y b 2, z c 2 F x a 2, y b 2, z c 2, where the following specific solutions can be substituted: For a constant (normalized) field along the z-axis: B (0) (x) (0, 0, 1):
Perturbative B 0 Calculation 473 F x 2 x z dxdydz 1 4 sinh 1 y x 2 z 2. [17] For a constant (normalized) field along the y-axis: B (0) (x) (0, 1, 0): F(x) is the same as for the x-axis case except all x and y terms are swapped. Combining Voxels Due to the linearity of Eq. 13 the single voxel solutions can be added together to give the total field: B 1 z x 1 x H x x, [18] x where x are the source points (locations of the voxel centers) and x is the field point where the field is evaluated. This equation takes the form of a discrete convolution of the voxel-based susceptibility map, 1, with the single voxel solution, H, if both field points and source points lie on the same rectangular grid. Therefore, the calculation can be efficiently implemented by using the 3D Fast Fourier Transform (FFT) as: B z 1 1 Z 1 H, [19] where (... ) is the FFT and Z(... ) is a zero-padding function, used to ensure that there is no periodic wrap-around in the convolution. Note that in previous sections the equations have been general enough so that any field point could be chosen. However, in order to take advantage of the efficiency of the discrete convolution the generality of the field point locations must be restricted to a grid (with spacing equal to the source voxel size and with the points located at the source voxel centers). radient of the Perturbed Field FI. 1. Spherical object model (a), analytical field (b) and perturbation field calculation (c) for the case of a sphere of radius R 0 8 voxels. The displayed scale of the field maps is 0.4 ppm. F x 2 z 2 dxdydz 1 4 atan xy. [16] zr For a constant (normalized) field along the x-axis: B (0) (x) (1, 0, 0): The exact analytical gradient of the perturbed field B z (1) can be calculated by simply replacing the convolution kernel, H, with its gradient: where q stands for x, y or z. 1 B z q 1 x H q x x x [20] Changing Orientation The field produced by the object in another orientation can be calculated by rotating the applied B (0) field. Note that this calculation is in the reference frame of the object, not the scanner, and so calculating the scanner defined z component of the perturbed field requires projection of the full B (1) field vector onto the scanner frame z-axis unit vector. Hence the desire to express Eqs. 10 and 13 in a general
474 Jenkinson et al. form containing B x (0) and B y (0) as well as B z (0). Furthermore, since B (1) is a linear function of B (0), a rotation can be reconstructed using a basis set of B (1) images resulting from B (0) fields in the x, y and z directions: 1 B x 1 1,0,0 B 1 x 0,1,0 B 1 x B 0 01 R B 1 z 1 y 1,0,0 B 1 y 0,1,0 B 1 y B 1 z 1,0,0 B 1 z 0,1,0 B 1 z 0,0,1 0 1 R 0, [21] where R isa3 3 matrix (13) that represents the rotation from the scanner coordinates, x sc, to the object coordinates, x ob, so that x ob Rx sc, and B p (1) (qˆ) is the field calculated in the object coordinate system with the p direction (x, y, or z) specifying the component of the field resulting from an applied field B (0) qˆ, where qˆ is a unit axis vector either xˆ (1, 0, 0), ŷ (0, 1, 0) or ẑ (0, 0, 1). For example, p y and q x gives B y (1) (xˆ), which represents the y component of the field due to a unit applied field along the x direction (all in the object coordinate system). Note that R. [001] T represents the scanner unit z-axis vector (the direction of the applied field) as represented in the object coordinate system. This vector specifies the direction of the applied B 0 field in the object coordinate system. The matrix of perturb fields, [B p (1) (qˆ)], represents a set of nine basis images, which can be precalculated and then combined as specified above to give the desired field at any orientation. This does not involve further approximation; it is precisely the same perturbed field that would be calculated for the object in the new orientation. In addition, although the perturbed field is linear in the basis images, it is not linear in the rotation angles, since the elements of R are nonlinear functions of these angles. A similar calculation can be done for the gradients of the field, ƒb (1), although this requires 27 basis images. VALIDATION AND RESULTS Analytical Sphere The analytical magnetic field (including Lorentz Correction) produced by a spherical object of radius R 0 and susceptibility i inside a medium of susceptibility e is given analytically by (12) as: B B 0ẑ i R e 0 3 3 e r B 0 3 cos rˆ ẑ r R 0 B 0 ẑ r R 0, [22] where the angles and are defined by rˆ (cos sin, sin sin, cos ) and ẑ (0, 0, 1). This expression gives all three components of the field: B (B x,b y,b z ). Figure 1 shows, qualitatively, the z component of the field distribution (less B 0 ) for a sphere, comparing the analytical solution given in Eq. 22 with the solution calculated with the perturbation method for the case of 9.5 10 6. Figure 2 shows plots of this field taken along the z-axis through the centers of these spheres for a range of different radii, R 0 8, 16, 32 voxels, where in each case FI. 2. Plots of the B z distribution along the z-axis for spherical objects of radius R 0 8, 16, 32 voxels. Results are shown for the analytical calculation (theory), the perturbation calculation, and the difference between the two.
Perturbative B 0 Calculation 475 Table 1 Quantitative Error Measurements of B z for Spherical Objects R 0 Mean Median P90 P95 P99 P99.9 Max 4 0.158 0.0248 0.498 0.733 1.96 3.93 3.93 8 0.085 0.0094 0.221 0.485 1.25 2.03 4.63 16 0.0388 0.0027 0.0505 0.181 0.865 1.77 5.11 32 0.0199 0.0007 0.0111 0.0504 0.549 1.60 5.48 64 0.0096 0.0001 0.0031 0.0108 0.246 1.29 5.72 Quantitative error measurements of the absolute difference in B z (in units of ppm) between the analytical result and the perturbation calculations for a spherical object of radius R 0 voxels, with 9.5 10 6. The maximum field value (less B 0 ) for the analytical solution is 6.33 ppm. P90 represents the 90th percentile, etc. the voxel size is 1 1 1 mm. This illustrates the relationship between spatial extent and size of the error. To investigate the dependence of the error on voxel size more quantitatively, the absolute difference between the analytical and perturbation-based calculated fields were generated for a larger range of radii: R 0 4, 8, 16, 32, 64 voxels. In each case, several measurements of the distribution of this absolute difference (error) were made mean, median, maximum and 90th, 95th, 99th and 99.9th percentiles which are shown in Table 1. Note that the measurements include all field points in the 3D volume, not just along an axis, as in Fig. 2. In addition, to test the ability of the method to calculate other components of the field the same absolute difference calculations for the x component of the field, B x, were generated. The results from this test are shown in Table 2. Note that the x-component of the field was calculated using a form of Eq. 13 derived for B x rather than B z (where the first term becomes zero and the last term interchanges x and z). The tables indicate that the errors are very small overall and that as the resolution of the object improves (i.e., more voxels or, equivalently, increased radius) the averaged errors (mean, median, percentiles) all decrease, such that for R 0 32 voxels, 95% of the voxels, or field points, have an error less than 1% of the maximum field, and 99% of the voxels have an error less than 10% of the maximum field. However, some larger errors still persist at any resolution, indicated by the maximum error values. These errors occur in a very small number of voxels which are typically located near the surface of the object, where the susceptibility changes sharply. These results are very similar to those shown in Refs. 1, 2, 5, 6 despite the range of different object models (e.g., boundary element methods vs. voxels) and approximations used. However, superior performance is shown by Ref. 3, where the object is modeled with piecewise planar patches and a boundary-element style calculation is used. In addition, the method presented in Refs. 7, 8, which modeled the object as a collection of spheres, shows very accurate performance, although a direct comparison is difficult from the available results. However, these methods are more numerically intensive and require significantly greater amounts of computational time. In Vivo Human Head An experimentally acquired field map of an in vivo human head was used to validate the method in practice. The MR field map sequence used a symmetric-asymmetric spinecho pair (14,15) (2.5 ms asymmetry time; 128 256 20 voxels of size 1.5 1.0 6.0 mm). The 3D theoretical field map was calculated using a 3D object susceptibility map 112 164 156 (1.0 1.0 1.0 mm), that was created by segmenting a whole-head CT image into bone, tissue, and air and then assigning a value of 1.0 (for 1 ) to the voxels classified as bone or tissue and a value of 0.0 to those classified as air. This susceptibility map was then registered to the magnitude component of the MR image corresponding to the acquired field map in order for the theoretically calculated field map to be registered to the experimentally acquired map. A full 3D theoretical field map was calculated where the source and field points were taken as the voxel centers from the input susceptibility map (1 mm spacing). Figure 3 shows 2D slices from the 3D CT image used to define the object susceptibility map, plus slices from both the 3D experimentally acquired field map and the 3D field map calculated using the voxel-based perturbation method described above (execution time was 9 min on a 1.8 Hz Athlon, 2 B memory running Linux). Note that both field maps have been masked so that only brain tissue is in- Table 2 Quantitative error measurements of B x for spherical objects R 0 Mean Median P90 P95 P99 P99.9 Max 4 0.141 0.0274 0.428 1.02 1.39 1.60 1.60 8 0.0763 0.0107 0.167 0.413 1.19 1.92 2.20 16 0.0364 0.0031 0.0432 0.146 0.856 1.79 2.04 32 0.0180 0.0009 0.0098 0.0405 0.475 1.62 2.53 64 0.0087 0.0001 0.0025 0.0091 0.200 1.30 2.41 Quantitative error measurements of the absolute difference in B x (in units of ppm) between the analytical result and the perturbation calculations for a spherical object of radius R 0 voxels, with 9.5 10 6. The maximum field value for the analytical solution is 4.75 ppm. P90 represents the 90th percentile, etc.
476 Jenkinson et al. FI. 3. Corresponding axial 2D slices from 3D images of an in vivo human head, showing the CT image used to derive the susceptibility map (a), the experimentally acquired field map (b), and the field map from the voxel-based perturbation calculation (c). The displayed scale of the field maps is 0.4 ppm. cluded (although the simulation included all tissues present, with brain bone 1.0) and have had the first and second-order spherical harmonics removed in order to factor out the effect of the shims on the field maps. Qualitatively it can be seen that the match is good. Quantitatively (see Table 3) the mean absolute difference between the field maps is 0.0465 ppm, while the typical range of the field values (used for the display range in Fig. 3) is 0.4 ppm. Furthermore, 90% of the voxels were in error by less than 0.1 ppm. The calculated error can be compared with the neglected second-order terms in the perturbation expansion. These second-order terms have an approximate magnitude of 2 10 10 0.0001 ppm, which is two orders of magnitude less than the observed errors. Therefore, it is likely that the observed errors are due to inaccuracies in the voxel-based modeling, as investigated in the previous section. DISCUSSION In this article we present a perturbation method for calculating the magnetic B field for an object with varying spatial susceptibility. A fast, first-order calculation is presented for voxel-based objects, using the analytical voxel solution. The accuracy of this method was tested using the analytical solution for a sphere as well as by empirical comparison with a human head dataset. These results indicate that highly localized errors of less than 1 ppm are achieved generally, which is very similar to other calculation methods, and sufficient for most MR imaging purposes. This implementation is available as a free download Table 3 Quantitative error measurements on in vivo human brain Mean Median P90 P95 P99 P99.9 Max 0.0465 0.0270 0.105 0.156 0.316 0.589 3.27 Quantitative error measurements of the absolute difference in B z (in units of ppm) between the experimentally measured field map and the perturbation calculations for the in vivo human brain after removal of first- and second-order spherical harmonics. P90 represents the 90th percentile, etc. from www.fmrib.ox.ac.uk/ mark/b0calc and will be distributed with future versions of FSL (www.fmrib.ox.ac.uk/ fsl). There are several main contributions of this work. The first is the use of a principled, perturbation method for arriving at the field approximation. This is useful in that it allows the magnitude of the error terms (second-order and higher) to be estimated, which then permits the relevant applicability of the method to be assessed. For instance, the method cannot be used for metallic objects where J 0 or objects where the susceptibility difference,, is large, but it can be used for some substances with slightly higher susceptibility than biological tissues, such as graphite (15). It is also possible, although potentially analytically intractable, to extend the approximation to higher orders to increase the accuracy. In addition, the formulation of the perturbation equations is separate from the object model specification and could be used with other object models, such as boundary element methods. Another significant contribution of this work is the ability to calculate more than just the z component of the field. In particular, the x and y components can be calculated just as easily (although separately) as well as the gradients of the fields (evaluated at the voxel centers), and formulations are provided for all these cases. More interestingly, it is possible to calculate the field for different object orientations by linearly combining precalculated basis images. This allows the field to be determined, without further approximation, at any orientation and in a very efficient manner. Such calculations will allow the interaction between susceptibility fields and motion artifacts to be explored more easily, a current research interest of the authors. In the field calculations used here there are two main sources of approximation beyond the requirement for zero conductivity: 1) neglecting all perturbation terms beyond first-order, and 2) representing the object by a voxel-based model. The first approximation limits the range of objects for which this method could be applied, as discussed above. The second approximation is potentially more limiting, as the use of a voxel-based model for the object will cause errors that are not as easily estimated as the pertur-
Perturbative B 0 Calculation 477 bation approximation errors and appear to dominate, as demonstrated in the results on the human head. Quantitative investigation of the object modeling errors was conducted on a range of spherical objects and the results are shown in Fig. 2 and Tables 1 and 2. These indicate that the spatial extent of larger errors is limited to a few voxels, which lie near the surface of the sphere (Fig. 1). This is to be expected, as this is closest to the area where the voxel-based object model deviates from the real, continuous object. It also suggests that using higher-resolution images leads to both smaller and more spatially localized errors overall, although at the cost of extra computational effort (the number of computations for N voxels is proportional to N log N). For models of the human head with air-filled cavities, the significant errors are therefore only likely to occur within a few voxels of the air tissue boundaries. Alternative models, such as boundary element methods (1 5,7,8), are likely to be physically accurate in capturing the object shape, but have two main disadvantages. One is that boundary meshes are more difficult to instantiate from typical voxel-based images (although one possible solution to this problem is given in Ref. 3) and the second is that they require more computation for the field calculation as each element (triangle of the mesh) is normally unique and requires separate calculations. In contrast, voxel-based models (6,10) are easy to instantiate and very efficient to calculate (using Fast Fourier Transforms). Furthermore, the numerical results on the spherical object indicate that similar errors are obtained by many of the proposed methods. Finally, all of these object models have an advantage over finite Fourier representations (9) since they can ensure that the object has finite spatial extent, which is not possible with the Fourier method. ACKNOWLEDMENT We thank Dr. Bob Cox for suggesting that a perturbation calculation could be useful for magnetic field calculations in notes from a workshop presentation on motion in FMRI. REFERENCES 1. Li S, Dardzinski BJ, Collins CM, Yang QX, Smith MB. Three-dimensional mapping of the static magnetic field inside the human head. Magn Reson Med 1996;36:705 714. 2. Collins CM, Yang B, Yang QX, Smith MB. Numerical calculations of the static magnetic field in three-dimensional multi-tissue models of the human head. Magn Reson Imag 2002;20:413 424. 3. Hwang SN, Wehrli FW. The calculation of the susceptibility-induced magnetic field from 3D NMR images with applications to trabecular bone. J Magn Reson B 1995;109:126 145. 4. Balac A, Caloz. Magnetic susceptibility artifacts in magnetic resonance imaging: calculation of the magnetic field disturbances. IEEE Trans Magn 1996;32:1645 1648. 5. Munck JD, Bhagwandien R, Muller SH, Verster FC, Herk MB. The computation of MR image distortions caused by tissue susceptibility using the boundary element method. IEEE Trans Med Imag 1996;15: 620 627. 6. Yoder D, Changchien E, Paschal CB, Fitzpatrick JM. MRI simulator with static field inhomogeneity. In: SPIE Proc Med Imag. Image Proc San Diego, February 2002; vol. 4684. 7. Case TA, Durney CH, Ailion DC, Cutillo A, Morris AH. A mathematical model of diamagnetic line broadening in lung tissue and similar heterogeneous systems: calculations and measurements. J Magn Reson 1987;73:304 314. 8. Christman RA, Ailion DC, Case TA, Durney CH, Cutillo A, Shioya S, oodrich KC, Morris AH. Comparison of calculated and experimental nmr spectral broadening for lung tissue. Magn Reson Med 1996;35:6 13. 9. Marques JP, Bowtell R. Evaluation of a Fourier based method for calculating susceptibility induced magnetic field perturbations. In: Proc 11th Annual Meeting ISMRM. Toronto, 2003. p 216. 10. Jenkinson M, Wilson J. Jezzard P. Perturbation calculation of B0 field for non-conducting materials. In: Proc 10th Annual Meeting ISMRM, Honolulu, 2002. p 2325. 11. Schwinger J, DeRaad L Jr, Milton KA, Tsai WY. Classical electrodynamics. New York: Perseus Books; 1998. 12. Haacke EM, Brown RW, Thompson MR, Venkatesan R. Magnetic resonance imaging: physical principles and sequence design. New York: Wiley-Liss; 1999. 13. Foley J, Dam A, Feiner S, Hughes J. Computer graphics: principles and practice in C. Boston: Addison Wesley; 2003. 14. Jezzard P, Balaban R. Correction for geometric distortion in echo planar images from B0 field variations. Magn Reson Med 1995;34: 65 73. 15. Wilson JL, Jenkinson M, Jezzard P. Optimization of static field homogeneity in human brain using diamagnetic passive shims. Magn Reson Med 2002;48:906 914.