Shimming procedure that includes determination of the target field by optimization in a parameter space of reduced dimensionality

Abstract

A method for homogenizing the static magnetic field with a distribution B0(r) in the active volume of a magnetic resonance apparatus having a number N of shim coils defines a target field distribution B0T(r) using a filter method in which a norm of the shim currents is influenced by means of filter factors. An optimization procedure works in a parameter space having M control parameters, wherein 2≤M<N. One of the control parameters is used as a weighting parameter for modification of a spatial weighting function and another control parameter is used to control the filter factors. Using this method the hardware limitations can be taken into account when determining the target field distribution, without a significant increase in the computational effort to determine the target field distribution during optimization.

Claims

1. A method for homogenizing a static magnetic field having a distribution B0(r) in an active volume of a magnetic resonance apparatus, the magnetic resonance apparatus having a number N of shim coils, the method comprising the steps of: a) mapping the magnetic field distribution B0(r) of the static magnetic field; b) defining a target field distribution B0T(r) using an optimization procedure to optimize a numerical quality criterion for the target field distribution B0T(r), the optimization procedure supplying values for currents through the N shim coils, wherein a spatial weighting function is used in the optimization procedure, the optimization procedure further comprising a filter method in which a norm of the currents in the shim coils is influenced by means of filter factors, wherein the optimization procedure works in a parameter space having M control parameters with 2≤M<N, one of the control parameters being used as a weighting parameter for modification of a spatial weighting function, wherein an additional control parameter controls the filter factors; and c) generating, following step b), the target field distribution B0T(r) in the active volume by setting the currents in the shim coils.

2. The method of claim 1, wherein an NMR spectrometer, an MRI scanner, an EPR instrument or an ion cyclotron resonance apparatus is used as the magnetic resonance apparatus.

3. The method of claim 2, wherein the magnetic resonance apparatus is an NMR spectrometer in which a sample is rotated about one or more axes.

4. The method of claim 3, wherein the axes are inclined with respect to a direction of the static magnetic field.

5. The method of claim 1, wherein in step (a), a gradient-echo method or a spin-echo method is used for mapping the magnetic field distribution B0(r) of the static magnetic field.

6. The method of claim 1, wherein, in defining the target field distribution in step (b), an adjustment range of currents in the shim coils and power consumed by all the shim coils are taken into account.

7. The method of claim 1, wherein the filter method used in step (b) includes one of Tikhonov regularization, Tikhonov-Phillips regularization, truncated singular value decomposition and damped singular value decomposition.

8. The method of claim 7, wherein the optimization procedure optimizes a numerical quality criterion in step (b) and a calculation method is used to calculate the quality criterion, wherein the calculation method uses as input the static magnetic field B0(r), the influence of the currents in the shim coils on the magnetic field, a weighting parameter and a regularization parameter, the calculation method outputting the quality criterion and a list of current settings, wherein the optimization procedure comprises the steps of: i) selecting a list of values for a first control parameter, wherein the first control parameter is the weighting parameter; (ii) selecting a list of values for a second control parameter, wherein the second control parameter is the regularization parameter; (iii)forming pairs of weighting parameters and regularization parameters from the list of values from (i) and the list of values from (ii) and calculating the quality criterion with input of those pairs of weighting parameters and regularization parameters; (iv) evaluating whether a list of current settings comprises feasible current settings; and (v) selecting an optimum pair of weighting parameters and regularization parameters on a basis of the quality criterion, while excluding non-feasible current settings.

9. The method of claim 8, wherein a simulated magnetic resonance spectrum is used in calculation of the quality criterion.

10. An electronically readable non-transitory data medium on which a computer program is stored, that computer program containing instructions for executing the method of claim 1.

Description

BRIEF DESCRIPTION OF THE DRAWING

(1) FIG. 1 a flow chart with optimization in a parameter space with M=2 control parameters;

(2) FIG. 2 a flow chart with a shimming method according to the prior art;

(3) FIG. 3 a flow chart with a possible sequence of one variant of the method according to the invention;

(4) FIGS. 4a, b the construction of a set spatial weighting functions for the fit with a varying data width, with (a) functions for varying the data width, so that weighting functions in FIG. 4b are obtained by scaling the measured excitation profile with these functions; (b) in each row the excitation profile scaled with the corresponding functions from FIG. 4a, which is used as a weighting function in the fit;

(5) FIGS. 5a, b the illustration of the optimization of the quality criterion in a two-dimensional parameter space, with (a) 2D plot of the quality criterion for the shimming state versus log regularization parameter (horizontally) and data width (vertically); (b) 2D plot of the quality criterion for the shimming state versus log of the 2-norm of the shim currents (horizontally) and data width (vertically);

(6) FIG. 6 the quality criterion (Q) for the shimming state as a function of the data width for solutions obtained by optimum regularization (“discrepancy principle” or L-plot);

(7) FIGS. 7a-d the creation of spatial weighting functions for the fit, derived from an MRI image, with (a) MRI image recorded in preparation for the shimming procedure, wherein a region (71) selected by the user in which the highest homogeneity requirements are set, additional regions (72) having different signal strengths, represented by different hatching, and regions (73), wherein no signal was received; (b) a spatial weighting function having values between 0 and 1 along the W axis for the fit in the shimming operation, wherein there is a uniform transition from full weight 1 in region 71 to weight 0 in region 73; (c) a spatial weighting function with values between 0 and 1 along the W axis for the fit in the shimming operation, wherein the weighting function from FIG. 7b was raised to the power of k, and wherein 0<k<1, so that regions close to 71 have a greater weight; (d) a spatial weighting function with values between 0 and 1 along the W axis for the fit in the shimming operation, wherein the weighting function from FIG. 7b has an exponent of k and wherein k>1, so that the weight drops rapidly for regions outside of 71;

(8) FIGS. 8a, b additional possibilities of varying the spatial weighting functions with one parameter, namely (a) varying the spatial weighting function for the fit in the form of (B1(r)).sup.k, where k assumes the following values from bottom to top: 0.01, 0.2, 0.5, 1, 2, 5, 10, 20; and

(9) FIG. 8b variation of the spatial weighting function for the fit where the z position of a place of reduced weight is varied.

DESCRIPTION OF THE PREFERRED EMBODIMENT

(10) In performing the method according to the invention, the following steps are incorporated in a manner suitable for the task in an overall sequence which may include repetition of the steps and decisions about achieving a goal: (a) Mapping the magnetic field distribution B0(r) of the static magnetic field (b) Defining a target field B0T(r) (c) Generating the target field distribution B0T(r) in the active volume by setting currents through the shim coils.

(11) One possible sequence is shown in FIG. 3 of the reference [5] (reproduced here in FIG. 2). This sequence includes an internal “fit loop” which has not been discussed previously here and in which only the difference between the prevailing magnetic field distribution B0(r) and the target field distribution B0T(r) is fitted, without defining any new target field distribution. Such a loop makes the method more robust for the case when the calculated change in current does not lead exactly to the precalculated change in the magnetic field distribution. Iterative readjustment without doing the work associated with defining a new target field distribution and the optimization required for it can greatly accelerate the entire process. Ultimately this loop is no longer necessary, if an accurate knowledge of the reaction of the entire measurement setup to changes in current in the shimming systems is achieved, and therefore it is not an obligatory component of the method according to the invention.

(12) Various methods may be considered for mapping the magnetic field distribution in the active volume. Phase-sensitive magnetic resonance imaging methods which use switchable gradient coils, such as the gradient-echo method or the spin-echo method, are especially suitable for performing the mapping of the magnetic field distribution in the active volume. Information about the local magnetic field is obtained from the phase difference of the signals in two images recorded with a different evolution time for the spins.

(13) The advantage of these methods is that mapping of the magnetic field can be performed using the same apparatus (i.e., transmitting/receiving coils, gradient coils) with which the experiment is performed for which homogenization of the magnetic field is a prerequisite. The sample that is ultimately used or the object to be imaged or the patient to be examined may be in the measurement position already, so that their influence on the homogeneity of the magnetic field can already be taken into account and corrected for.

(14) Other known methods for measuring magnetic fields with spatial resolution use a displaceable magnetic field sensor, with which the magnetic field can be measured at different positions, or they use an array of magnetic field sensors. Hall sensors or small NMR probes are suitable here. Such a method may be used during the installation of a magnet for generating a basic homogeneity of the magnetic field, for example. After homogenization of the magnetic field, however, the displaceable magnetic field sensor must be replaced by a different measurement apparatus and, further, the sample to be examined must not be in the active volume during the mapping of the magnetic field.

(15) In defining the target field distribution B0T(r), one or more quality criteria are used, from which a target variable whose optimum value (minimum or maximum) defines the target field distribution B0T(r) may be derived. One target variable, which may be used to define the target field distribution, is the root of the mean of the squared deviations from the constant ideal field (root mean squared deviation, RMSD).

(16) This variable is not always suitable for ensuring the quality of the target field. This is the case, for example, with spectra where the highest possible resolution of the spectral lines must be achieved, and at the same time a split into closely adjacent lines that might be confused with multiplets must be avoided. In such a case, a method according to reference [4] is suitable, where the target function is calculated by analyzing a prediction of the resulting spectrum. Properties such as the half-width of spectral lines and of envelope curves around spectral lines are determined on this predicted spectrum and are combined to yield a variable, which serves as a target variable for an optimization algorithm.

(17) This procedure of using a simulated spectrum makes it possible to formulate the quality criterion in the language of the spectroscopist as well as to explicitly take into account the properties of the sample that are relevant for the planned experiment, the measurement method used and its specific dependence on the homogeneity of the magnetic field. Possible quality criteria may include, for example, the full width at half height (FWHH), the width at another height or the half-value width of an envelope (as described in reference [4]) of a spectral line. The natural line width of the specimen to be examined is an important experimental parameter which, when set correctly, avoids inefficient homogenization on a scale that is too fine. Different pulse sequences, having different sensitivities to inhomogeneities in the magnetic field, may be used on the same apparatus. This effect can also be taken into account during simulation of the spectrum.

(18) Another quality criterion that is particularly useful in evaluating homogeneity in MRI images is obtained by squaring the local gradient of the magnetic field (gradient of all three directions in space) and adding them up over all voxels. By minimizing the resulting value, the entire signal is maximized inasmuch as it depends on the homogeneity of the magnetic field in that the intravoxel dephasing caused by local gradient is minimized.

(19) The decisive feature of the present invention is the optimization in a parameter space using M control parameters, where 2≤M<N, for the determination of the target field distribution B0T(r). At least one of the control parameters, the weighting parameter, is used for modification of a spatial weighting function. This parameter serves to find possible solutions in the parameter space of the shim currents with a high dimensionality by varying the relative weighting of subareas in the measurement data.

(20) One possible implementation of such a weighting parameter is given in reference [5] using the parameter k:

(21) A k-dependent weighting function W(r, k)=(B1(r)).sup.k is constructed, based on the normalized RF excitation profile B1(r). The effect of the choice of k is that with k=1, the weighting of the excitation profile is accepted unchanged, with 0<k<1, the edge of the excitation profile is weighted more strongly, and with k>1, the edge of the excitation profile is attenuated in the fit.

(22) The local effect of k depends on the value of the excitation profile at the respective position. The choice of a spatial weighting function that strongly resembles on the form of the excitation profile of the RF coil is particularly suited in the case of a spectroscopic method, in which a homogeneous specimen is analyzed. In this case, the differential treatment of the regions in the sample, that contribute greatly toward the received signal, and regions in the sample, that are at the edge of the excitation signal and make only weak contributions and that potentially suffer from measurement artifacts, is important. FIG. 8a shows a range of such weighting functions generated with variation of the exponent k from the measurement excitation profile B1(z).

(23) Another possibility for obtaining a variation in the solutions by means of the weighting function in the fitting procedure is achieved when a measured excitation profile is scaled with a function that modulates the effective width of the signal. This function is also taken into account in analysis of the scaled data.

(24) FIG. 4b shows how a set of weighting functions can be generated from a measured excitation profile, by means of the 14 different functions, which serve to vary the data width as shown in FIG. 4a.

(25) A result is achieved by means of an efficient linear analysis, as shown in FIGS. 5a and 5b, from which the best solution is then selected. The axial direction l1 in these figures denotes the direction in which the data width is varied.

(26) A third possibility for varying the spatial weighting function is beneficial in cases where the fit result is greatly influenced by a localized distortion of the magnetic field distribution. Instead of varying the width of the distribution as above, the spatial position of a zone in which the weighting function is to be reduced locally is to be varied. FIG. 8b shows a set of such functions.

(27) In the case of imaging applications, it is necessary to choose a spatial weighting function, which takes into account the spatial arrangement of the objected to be imaged. A suitable weighting function can be derived, for example, based on an image recorded in preparation and a region therein marked by the user in which the highest homogeneity requirements are set. Such a region may be an organ of a patient to be examined, for example.

(28) A compromise between global homogeneity in the entire image and local homogeneity in the selected region can be found through a parameter, which has a continuous influence on the weighting function. One such parameter can be used as the first dimension in the optimization in step (b) of the shimming method. FIGS. 7b, 7c and 7d show a set of weighting functions constructed in this way, which fit with the MRI image represented schematically in FIG. 7a.

(29) Another control parameter controls the filter factors of the filter method in which a norm of the shim currents is influenced by means of these filter factors. The norm of the shim currents thereby influenced may be, for example, the total power of all shim coils.

(30) The norm of the shim currents is thereby influenced indirectly. The term “indirect influence” here means that the precise value of the norm of the shim currents may not be specified directly but, starting with a solution to a parameter value, a solution whose norm is less than or greater than the starting value can be found in a targeted manner by choosing a greater or smaller value for the control parameter that controls the filter factors.

(31) A suitable parameter is the regularization parameter of a regularization method in which the following methods may be used as regularization methods: for example, discretization, truncated singular value decomposition (TSVD), damped singular value decomposition (DSVD), Tikhonov regularization, Tikhonov-Phillips regularization (see reference [6]). With a norm of the shim currents, the entire list of currents in the N shim coils is combined into a single characteristic value.

(32) If the power is of primary interest, then the norm is formed as the root of the total of squares of the shimming currents (the “2-norm”), if need be with a weighting that corresponds to the different resistances of the shim coils.

(33) If the sum of the currents is important as a limiting variable, then as an alternative the sum of the amounts of the currents (the “1-norm”) may also be taken into account.

(34) The maximum norm is the suitable variable if one wants to avoid extreme loads on individual shim coils.

(35) A very efficient implementation of the method is achieved by utilizing the regularization parameter λ from the Tikhonov method to achieve solutions, which have a steadily declining shimming power with an increase in the value of λ.

(36) Let
K=UWV.sup.T
be the singular value decomposition of the set of shim functions K, then the solution g of the linear analysis of the mapped data s can be written according to the least squares method as

(37) $g = \underset{l_{2}}{.Math.} \frac{1}{w_{l_{2}}} v_{l_{2}} u_{l_{2}}^{T} s$

(38) where the singular values w.sub.l.sub.z are sorted according to decreasing size so that w.sub.1 is the largest singular value and w.sub.N is the smallest singular value, where N is the number of available shim functions. The singular values w.sub.l.sub.z typically decline very rapidly with an increase in the index l2 and are responsible for the numerical problems which must be regularized.

(39) In the Tikhonov regularization, the parameter λ has a filter effect, which is described by the equation

(40) $g = \underset{l_{2}}{.Math.} \frac{w_{l}}{(w_{l_{2}}^{2} + λ)} v_{l_{2}} u_{l_{2}}^{T} s$

(41) However, this equation can also be used to efficiently find solutions in the highly complex parameter space by performing analyses using different values for λ. It is also a major advantage that λ can be varied with any desired precision, which is not possible for all regularization methods.

(42) A two-dimensional representation of the shimming quality as a function of the data width and the parameter λ, achieved from a mapped magnetic field distribution B0(r) in the active volume of the static magnetic field, is shown in FIG. 5a on the basis of one practical example of an NMR probe.

(43) Conventional procedures in regularization methods to determine the optimum value of λ such as the “discrepancy principle” or L-plot yield a result as shown in FIG. 6 and/or also in FIGS. 5a and 5b by means of hexagonal symbols. These places are optimized in order to eliminate the numerical problems with the smallest possible increase in the deviation, i.e., with a minimal filter effect.

(44) However, at the optimum value of parameter λ, the solution that achieves an almost optimal result with the least possible shimming power is not found.

(45) FIG. 5b shows what is found with the same data if the required shimming power is evaluated for each parameter λ and the corresponding data point is plotted in a diagram with shimming power as the horizontal axis. The two-dimensional space often makes it possible to find solutions of a similar quality at significantly lower shimming powers, such as that found with the discrepancy principle, for example. In the example shown here, this yields solutions which save on shimming power in comparison with the regularized solution by up to two orders of magnitude.

(46) A relatively small selection of points in the two-dimensional space is often sufficient to determine a local optimum and to achieve the requirements. One important point in the selection is to take into account known hardware limits before setting the shims and testing experimentally whether the expected shim state is achieved.

(47) This can save a great deal of time. This mechanism is a great advantage in complex problem situations in particular where the algorithm yields solutions that exceed the hardware limits. In particular the second parameter, the regularization parameter, indicates the direction in which one must seek a solution that does not exceed the hardware limits.

(48) Problems in which there is an increased risk of exceeding the hardware limits include, for example, highly inhomogeneous samples or samples in short containers (for example, MAS rotors).

(49) One concrete implementation of the method according to the invention is illustrated in FIG. 3 as a flowchart. In this case, the data width was selected as a weighting parameter. A second control parameter for the optimization here is the regularization parameter λ of the Tikhonov regularization. The sequence includes the following steps: (a) Mapping the magnetic field distribution B0(r) in the active volume of the static magnetic field, (b) Making a choice of data width variations which are taken into account in scaling the theoretical shimming functions, (c) Fitting the magnetic field distribution B0(r) with the scale shimming functions for all data width variations and for multiple values of λ, (d) Defining a target field distribution B0T(r) in the active volume which corresponds to the requirement for a good shim status with a low shimming power, (e) Generating the target field distribution B0T(r) in the working volume by setting currents through the shim coils, (f) Mapping the magnetic field distribution B0(r) in the active volume of the static magnetic field, (g) Repeating steps (b)-(f) until the target field distribution B0T(r) in the active volume is approximately reached.
Main Fields of Application

(50) The method according to the invention may be used with all known magnetic resonance apparatuses, such MRI scanners, NMR spectrometer or EPR devices, for example, but it is particularly advantageous with high resolution spectroscopy, where the goal, namely to achieve the narrowest possible line form, free of spikes, is often not achieved by minimizing the norm of the target field distribution B0T(r).

(51) The method according to the invention may be used to particular advantage for homogenization of the static magnetic field in the working volume of a magnetic resonance apparatus, where the magnetic resonance apparatus is an NMR spectrometer, an MRI scanner, an EPR instrument or an ion cyclotron resonance apparatus.

(52) Inclined Axis of Rotation

(53) Methods of NMR spectroscopy in which a sample rotates about an axis, which is inclined with respect to the direction of the static magnetic field, require a high homogeneity of the magnetic field along this inclined axis or axes of rotation. Magic angle spinning (MAS), variable angle spinning (VAS) and double rotation (DOR) are examples of such methods. These methods may suffer from the fact that the shim coils are designed for having an efficient influence on the field distribution on an axis parallel to the static magnetic field. Such methods benefit in particular from a shimming procedure according to the invention that takes into account the shimming quality and the shimming power required to achieve it.

(54) The method according to the invention is also preferred for use to homogenize the static magnetic field in the active volume of a magnetic resonance apparatus, where the magnetic resonance apparatus is an NMR spectrometer, in which a sample is rotated about one or more axes which may also be inclined with respect to the direction of the static magnetic field.

(55) Computer Program

(56) All the procedures involved in operating modern magnetic resonance spectrometers and MRI scanners, such as sending RF pulses, switching currents in gradients, adjusting shim currents and receiving and digitizing signals today are deployed under computerized control. The shimming procedure according to the invention is therefore ideally implemented as a computer program, which directly deploys the required actions on the hardware when implemented on the control computer of the magnetic resonance spectrometer or the MRI scanner. To users of the spectrometer, such a program is sent simply on an electronically readable data medium containing a computer program, which, when executed, executes the method according to the invention.

LIST OF REFERENCE NUMERALS

(57) 10 Two entangled loops with counters l1 and l2 11 Step (i): selection of a list of values for a first control parameter, the weighting parameter; select a list {k(1), . . . , k(l1.sub.max)} of weighting parameters. 12 Step (ii): select a list of values for a second control parameter, the regularization parameter; select a list {λ(1), . . . , λ(l2.sub.max)} of regularization parameters. 13 Step (iii): forming pairs of weighting parameters and regularization parameters from the list of values from (i) and the list of values from (ii) and calculating the quality criterion with input of this pair of weighting parameters and regularization parameters; the current parameter pair is {k(l1), λ(l2)}; calculate the quality criterion Q(l1, l2) and the associated current setting J(l1, l2). 14 Step (iv): evaluating whether the list of current settings is a feasible current setting; decision as to whether J is adjustable. If so, set F(l1, l2)=1, otherwise F(l1, l2)=0. 15 Step (v): selecting the optimum pair of weighting parameters and regularization parameters on the basis of the quality criterion and excluding current settings that are not feasible; l1.sub.opt, l2.sub.opt, so that Q(l1.sub.opt, l2.sub.opt) is minimal under the secondary condition that F(l1.sub.opt, l2.sub.opt)=1. The magnetic field calculated from B0(r) and J(l1.sub.opt, l2.sub.opt) is the target field distribution B0T(r). 16 Input of B0(r). 17 Output of B0T(r). 21 Step in the shimming procedure: mapping the magnetic field distribution B0(r) of the static magnetic field. 22 Step in the shimming procedure: definition of a target field distribution B0T(r). 23 Step in the shimming procedure: creating the target field distribution B0T(r) in the working volume by setting currents through the shim coils. 51 Curve in the 2D plot on which solutions obtained by optimum regularization (“discrepancy principle” or L plot) are located. 52 Region of optimum values for the quality criterion found in the search along 51. 53 Additional solutions with similar values for the quality criterion and lower shimming power than in 52. 71 Region in the MRI image that should achieve the greatest possible homogeneity after shimming. 72 Remaining region in the MRI image, having a signal strength greater than zero. 73 Region in the MRI image that does not contain any signal and need not be taken into account in shimming. l1 Axial direction of the data width in the 2D plot Q Axial direction of the quality criterion W Axial direction of the function value for weighting function w1 Data width of the first weighting function w14 Data width of the fourteenth weighting function x Spatial x direction (in the MRI image) y Spatial y direction (in the MRI image) z Spatial z direction (at weighting functions) λ Regularization parameter (lambda)

LIST OF REFERENCES

(58) [1] Richard R. Ernst: “Measurement and Control of Magnetic Field Homogeneity”; The Review of Scientific Instruments 39 (1968) 998-1012 [2] R. E. Gang: U.S. Pat. No. 3,287,630 [3] Dong-Hyun Kim, Daniel M. Spielman, Gary H. Glover, Elfar Adalsteinsson: U.S. Pat. No. 6,529,002 [4] Markus Weiger, Michael Fey, Thomas Speck: EP 1 662 270 B1 [5] Markus Weiger, Thomas Speck, Michael Fey: “Gradient shimming with spectrum optimization”; Journal of Magnetic Resonance 182 (2006) 38-48 [6] P. C. Hansen: “Rank-Deficient and Discrete III-Posed Problems”; SIAM, Philadelphia, 1998 [7] Han Wen, Farouc A. Jaffer: “An in-vivo Automated Shimming Method Taking into Account Shim Current Constraints”; Magnetic Resonance in Medicine 34 (1995) 898-904

Shimming procedure that includes determination of the target field by optimization in a parameter space of reduced dimensionality

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