Patent Publication Number: US-9903929-B2

Title: Method and apparatus for acquiring magnetic resonance data and generating images therefrom using a two-point Dixon technique

Description:
BACKGROUND OF THE INVENTION 
     Field of the Invention 
     The invention concerns a method for magnetic resonance (MR) measurement and a magnetic resonance system. In particular, the invention concerns techniques by means of which a determination of first and second spectral components from MR data is possible. 
     Description of the Prior Art 
     In the acquisition of magnetic resonance data, it is possible to separate spectral components that are included in the MR data. The spectral components can designate different spin species, for example nuclear spins in an adipose environment and in an aqueous environment. For this purpose, what are known as chemical shift imaging multi-echo magnetic resonance measurement sequences are often used within the scope of Dixon techniques. Such techniques typically utilize the effect that the resonance frequency of nuclear spins depends on the molecular or chemical environment. This effect is designated as a chemical shift. Different spin species therefore have different resonance frequencies from which the measured spectrum of the MR data is composed. For example, the difference between two resonance frequencies of different spectral components can be expressed in ppm (“parts per million”, i.e. 106 
     The chemical shift between hydrogen nuclear spins in water as a first spectral component and hydrogen nuclear spins in fatty acid chains as a second spectral component is often utilized. In such a case, a water MR image and/or a fat MR image—i.e. individual MR images of the two spectral components—can be determined using MR data. This is of interest in a variety of applications, for example clinical and/or medical applications. 
     In order to be able to separate the spectral components from one another, MR signals are acquired at multiple echo times within the scope of the Dixon technique. The MR signals together form the MR data. The different spectral components have different phase positions at the different echo times. Using this effect, it is be possible to determine the different spectral components separately. 
     For this purpose, a spectral model is generally used that links the measured or acquired MR data with different physically relevant variables. The different variables include the different spectral components to be determined, as well as additional unknowns of the measurement system (depending on precision, scope and complexity of the spectral model). It can then be possible to determine the spectral components considered in the spectral model for each image point of the MR data. 
     In principle, it can be worthwhile to use a relatively complex spectral model, for example such a spectral model which considers a large number of further unknowns in addition to the spectral components to be determined. It can then be possible to determine the spectral components particularly precisely. In this case, however, it can be necessary to acquire particularly many MR signals at different echo times, which can in turn extend a measurement duration and therefore can be disadvantageous. A trade-off situation thus often results between the measurement duration and the precision in the determination of the spectral components. 
     A need therefore exists for techniques that enable a relatively precise determination of spectral components, while simultaneously requiring only a small number of MR signals at different echo times, thus ensure a relatively short measurement duration. 
     In order to satisfy this requirement, techniques are known in which a numerical optimization enables the discovery of solutions to an equation (which forms the basis of the spectral model that is used) to determine the spectral components. However, since the spectral models of the two spectral components can be very similar, the underlying equations can be nearly symmetrical in the spectral components. Therefore, in such scenarios a situation can also occur in which multiple solutions are discovered and it is not clear (or it is clear only to a limited extent) which of the multiple solutions is the physically relevant solution. In other words: the equation underlying the spectral model cannot be uniquely solved. 
     In order to remedy this ambiguity in the determination of the spectral components, it is to take that various variables into account in the spectral model are assumed to be only slightly dependent on position. For example, a variable that describes the field inhomogeneities of a basic magnetic field of an MR system that is used can be assumed to be relatively slightly dependent on position. A relatively slight dependency on position can in particular mean: only a small change on the length scale of an image point of the MR data. An image point of the MR data can have a size of 1 mm×1 mm×1 mm, for example. This size of the image point typically determines the spatial resolution of the final MR image, i.e. in particular the resolution with which the first and second spectral components are determined. Another factor, however, is that the basic magnetic field can exhibit inhomogeneities that vary only on an order of 1 cm or more, thus do not or do not significantly vary between two adjacent image points of the MR data. 
     In such a case, after the numerical discovery of the multiple solution candidates, one of these solution candidates can be chosen as the physically correct solution, and in fact depending on a solution already found beforehand for a neighboring image point. Such techniques are known to those skilled in the art, for instance under the name “region growing techniques”. 
     Such techniques have limitations or disadvantages. The implementation of the numerical optimization thus can be computationally intensive. 
     SUMMARY OF THE INVENTION 
     A need therefore exists for techniques which enable an improved determination of spectral components from MR data. In particular, there is a need for such techniques that enable a particularly simple and less computationally intensive determination of the spectral components. A need also exists for such techniques that determine the spectral components with a relatively high precision. 
     According to one aspect of the invention, a method is provided for magnetic resonance measurement of a first spectral component and a second spectral component of an examination subject by using a two-point Dixon technique at a first echo time and a second echo time. The predetermined spectral model of the two-point Dixon technique includes the first spectral component, the second spectral component, a phase at the first echo time and a phase evolution due to field inhomogeneities and/or eddy current effects between the first echo time and the second echo time. The method includes the acquisition of MR data for multiple image points, respectively at the first echo time and at the second echo time. Furthermore, the method includes the determination of a computation grid of low resolution in comparison to the MR data, wherein each grid point of the computation grid encompasses a predetermined number of adjacent image points of the MR data. For each image point of the MR data, the method furthermore includes the implementation of a numerical optimization that determines an optimized phase at the first echo time and/or an optimized phase evolution. The optimization is based on an equation that takes into account that the phases at the first echo time and/or the phase evolution for all image points that are encompassed by a grid point of the computation grid is constant. The method furthermore includes the analytical calculation of the first spectral component and of the second spectral component based on the phase determined by the optimization at the first echo time and/or the phase evolution. 
     The resolution of the MR data can be determined by the variable of an image point of the MR data, for example as a number of image points per area. The MR data can be composed of an MR signal at the first echo time and an MR signal at the second echo time. The first and second echo time can typically be determined relative to a time period between an MR signal and a radiated radio-frequency (RF) excitation pulse. For example, the acquisition of the MR data can take place by means of a spin echo MR measurement sequence and/or by means of a gradient echo MR measurement sequence. Within the scope of the MR measurement sequence, two echoes can then be formed that correspond to the MR signals at the first and second echo time. The gradient echo MR measurement sequence can be bipolar or monopolar. 
     In other words, a grid point of the computation grid can designate that region within which the phases at the first echo time and/or the phase evolution are assumed to be constant in the numerical optimization, i.e. have a fixed value. The scale of a grid point of the computation grid can correlate with that length scale at which it is assumed that the phase at the first echo time and/or the phase evolution exhibit no significant change. For example, a grid point can be quadratic or rectangular, i.e. encompasses a different number of image points of the MR data along different spatial directions; more complex influences of varying unknowns that have spatial dependencies of different strengths for different spatial directions can therefore be taken into account. Merely as an example, a grid point of the computation grid can encompass 2×2 or 2×4 or 6×6 or 20×20 or 100×50 image points of the MR data. It would also be possible that the size of a grid point of the computation grid is different at various locations. For instance, more complex spatial dependencies of the field inhomogeneities and/or of the eddy current effects can therefore be considered. For example, the analytical calculation of the first and second spectral components can take place for each image point, but can also take place jointly for multiple image points. 
     In principle, the numerical optimization can be delimited compared to analytical techniques. For example, within the scope of the numerical optimization iterative techniques can be implemented, for example with regard to solutions of the equation. For example, it is possible that the numerical optimization yields multiple solutions of the equation as result candidates. Within the scope of the optimization, it can then be required that, for each image point, one solution is chosen from the multiple result candidates as the optimized phase at the first echo time, and/or the optimized phase evolution. However, it is also possible for the numerical optimization also include analytical calculation steps, for example in addition to the numerical techniques noted above. 
     As described above, within the scope of the discovery of solutions of the equation within the scope of the numerical optimization it can already be taken into account that the phase at the first echo time and/or the phase evolution is constant within a grid point of the computation grid. Stated differently, at the point in time of implementing the numerical optimization, it can thus already be taken into account that the phase at the first echo time and/or the phase evolution has a smaller dependency on position than the MR data themselves. In spite of that, the phase at the first echo time and/or the phase evolution can thus be assumed to be constant in parts in the numerical optimization. 
     An efficient and less computationally intensive numerical optimization is thereby achieved, particularly in comparison to reference implementations in which, only after the numerical optimization (in particular after the discovery of result candidates), is it taken into account that the phase at the first echo time and/or the phase evolution are constant in parts. 
     A precise determination of the first spectral component and of the second spectral component is thereby also achieved. This is because a higher signal-to-noise ratio of the phase images is achieved through the assumption that the phase at the first echo time and the phase evolution are constant in subsets within a grid point, and as a result of this a higher signal-to-noise ratio of the spectral components subsequently calculated based on these in the individual image points is also achieved. Within the scope of the numerical optimization, the signal-to-noise ratio can be increased since the data foundation on which the numerical optimization is implemented is increased via averaging over multiple MR signals. 
     In general, the numerical optimization can be implemented based on any optimization technique known in principle to those skilled in the art. For example, it is possible for the optimization to be a chi-square optimization or an Lp-norm optimization. For example, the optimization problems can be solved by the Marquardt-Levenberg method. 
     For example, the optimization can be implemented with regard to the phase at the first echo time or with regard to the phase evolution, or both with regard to the optimized phase and with regard to the phase evolution. In general, numerical optimization techniques are known that provide result candidates for one, two or more unknowns of the equation on which the numerical optimization is based. In general, the precision is greater (lesser) the fewer (more) unknowns that are determined within the scope of the numerical optimization. 
     In particular, the equation can have no explicit dependency on the first spectral component and the second spectral component. No explicit dependency means a partial derivation of the equation according to one of the first spectral component and the second spectral component yields zero. 
     The equation on which the optimization is based can be derived from the predetermined spectral model. In this regard, a number of techniques are known that allow the spectral model to be reformulated such that the equation has no explicit dependency on the first and second spectral components. 
     For the case that the equation has no explicit dependency on the first and second spectral components, the effect of a particularly simplified numerical optimization can be achieved. In such a case it can be unnecessary for the numerical optimization to yield direct result candidates for the first spectral component and/or the second spectral component. In other words: in such a case the numerical optimization only provides result candidates for the phase at the first echo time and/or the phase evolution. The optimization thus can inherently take into account the first and second spectral portion but without itself providing a direct solution for the first and spectral portions. In general, the computing resources necessary to implement the numerical optimization are greater (lesser) for a larger (smaller) number of variables to be optimized. 
     The predetermined spectral model can assume real-value weightings for the first and second spectral components. Alternatively, the predetermined spectral model can assume complex-valued weightings, i.e. in other words additional weightings associated with a phase for the two spectral components. For the latter case, it is possible that the phase at the first echo time to be expressed by the complex-valued weightings of the first and second spectral components. 
     For example, the equation can be described by a variable projection of real-value weightings of the two spectral components on the basis of the spectral model. If the spectral model is based on the real-value weightings for the two spectral components, a particularly simple elimination of explicit dependencies on the phase at the first echo time and/or the phase evolution can take place. 
     The determination of the computation grid can furthermore encompass the establishment of the predetermined number of adjacent image points of the MR data that are encompassed by a grid point, depending on a user input and/depending on a machine parameter of a magnetic resonance system. 
     In other words: the determination of the computation grid can furthermore encompass the establishment of a dimension of a grid point of the computation grid depending on the user input and/or the machine parameter. If a larger (smaller) number of adjacent image points of the MR data that are encompassed by a grid point of the computation grid is established, in general the implementation of the numerical optimization can require lesser (greater) computing capacities. In general, defined computation operations within the scope of the numerical optimization can be implemented simultaneously and without differentiation for all of those adjacent image points of the MR data that are encompassed by a grid point of the computation grid. Therefore, given a larger (smaller) number of image points of the MR data that are encompassed by a grid point of the computation grid, the number of computation operations within the scope of the numerical optimization can be reduced (increased). At the same time, a maximized precision can be achieved for a defined number of image points that are encompassed by a grid point. Namely, the signal-to-noise ratio can be increased by the averaging (described in the preceding) of the MR signals of multiple image points within the scope of the numerical optimization. At the same time, given grid points of the computation grid that are chosen to be too large, the spatial dependency of the phase at the first echo time and/or the spatial dependency of the phase evolution can no longer be sufficiently precisely described. Therefore, an optimum of the size of the grid points can be provided which can be determined depending on machine parameters, for example. The optimum can take into account precision and computing capacities. 
     For example, the machine parameters can be selected from a group that includes the following values: quality of a basic magnetic field of the magnetic resonance system; quality of a shielding from eddy current effects of the MR system; computing capacity of a computer of the MR system; size of an image point of the MR data; and/or strength of gradient fields of the MR system etc. All such machine parameters can have an influence on a change of field inhomogeneities and/or of eddy current effects as a function of the location. 
     For the case that only the phase evolution is numerically optimized, the equation can take into account that the phase at the first echo time varies for image points that are encompassed by a grid point of the computation grid. For the case that only the phase is numerically optimized at the first echo time, the equation can take into account that the phase evolution varies for image points that are encompassed by a grid point of the computation grid. 
     In other words, if the numerical optimization is implemented only with regard to either the phase at the first echo time or with regard to the phase evolution, the respective variable that is not considered is assumed to be variable within a grid point of the computation grid. 
     In such a technique, a particularly high precision can be achieved in the determination of the optimized phase at the first echo time and the determination of the optimized phase evolution. At the same time, however, it can be necessary to occupy increased computing capacity to implement the numerical optimization. 
     The calculation of the first and second components can furthermore include the interpolation of the phase at the first echo time and/or of the phase evolution between adjacent grid points of the computation grid. The determination of the first and second spectral components can be based on the interpolated phase at the first echo time and/or the interpolated phase evolution. 
     By means of such described techniques, the phase at the first echo time and/or the phase evolution can be assumed to be constant within a grid point of the computation grid. In contrast to this, the analytical calculation of the first spectral component and of the second spectral component for each image point of the MR data can be implemented separately within the scope of the method. The first and second spectral components therefore can be provided with a resolution that corresponds to that of the acquired MR data. For example, if a water MR image and a fat MR image are created on the basis of the defined first and second components, this high resolution of the water MR image and of the fat MR image can be worthwhile for a subsequent clinical or medical application. Therefore, it can be worthwhile to also allow a certain variation of the phases at the first echo time and/or of the phase evolution within a grid point of the computation grid by the interpolation after the implementation of the optimization. The first spectral component and the second spectral component thus can be calculated with a higher precision. At the same time, the interpolation can be a less computationally intensive operation, such that the necessary computing capacities are not significantly increased by this interpolation. Only physical changes of the phase at the first echo time and/or of the phase evolution that are of subordinate relevance, are sudden or that occur in stages can therefore be reduced. In other words, conditional artifacts can be reduced in the technique to determine the first and second spectral components. 
     According to a further aspect, the invention concerns a method for MR measurement of a first spectral component and a second spectral component of an examination subject by means of a two-point Dixon technique at a first echo time and a second echo time. A predetermined spectral model of the two-point Dixon technique includes the first spectral component, the second spectral component, a phase at the first echo time and a phase evolution due to field inhomogeneities and/or eddy current effects between the first echo time and the second echo time. The method includes the acquisition of MR data for multiple image points, respectively at the first echo time and at the second echo time. For each image point of the MR data, the method furthermore includes the implementation of a numerical optimization that determines an optimized phase at the first echo time and/or an optimized phase evolution. The optimization can be based on an equation that has no explicit dependency on the first and second spectral components. The method also includes the analytical calculation of the first spectral component and of the second spectral component based on the phase determined by the optimization at the first echo time and/or on the determined phase evolution. 
     For the method for MR measurement according to this aspect, corresponding techniques can be used as described above with regard to the method for MR measurement according to a further aspect of the present invention. For such techniques in which the numerical optimization is based on an equation without explicit dependency on the first and second spectral components, results can be achieved as explained above. 
     As described above, it is possible to implement the numerical optimization both with regard to the phase at the first echo time and with regard to the phase evolution, but it is also possible to implement the numerical optimization only with regard to either the phases at the first echo time or the phase evolution. 
     If only the phase evolution is numerically optimized, the equation can have no explicit dependency on the phase at the first echo time. Accordingly, if only the phase at the first echo time is numerically optimized, the equation can have no explicit dependency on the phase evolution. 
     In other words, it is possible for the equation to be based on the spectral model and, in addition to the elimination of the explicit dependency on the first and second spectral components for the explicit dependency on either the phase evolution or the phase at the first echo time to be eliminated. 
     A particularly simple and less computationally intensive numerical optimization is thereby achieved. The numerical optimization can be implemented only with regard to one of the phase evolution and the phase at the first echo time, and he other can be obtained by analytical calculation based on the optimized variable, for example. The precision can also be increased because, in the numerical optimization, the respective explicitly eliminated phase or phase evolution is inherently and precisely taken into account, and is not adulterated by the possibly limited precision of the numerical optimization. 
     The analytical calculation of the first spectral component and the second spectral component can occur based on a formula that is based on a variable back-projection of real-value weightings of the first and second spectral components. 
     If the explicit dependency at the first and second spectral components is eliminated from the spectral model by the variable projection, the analytical calculation of this elimination can be taken into account and in general represent an inverse calculation operation. The analytical calculation of the first and second spectral components can be less computationally intensive and be implemented quickly and particularly precisely. 
     In general, techniques of variable projection are known to those skilled in the art, for example from the article by G. H. Golub and V. Pereyra, “The differentiation of pseudoinverses and nonlinear least squares problems whose variables separate” in SIAM J. Numer. Anal. 10 (1973), 413-432. Therefore, there is no need to present additional details at this point with regard to the variable projection need not be presented herein. 
     As noted above, the implementation of the numerical optimization can provide multiple result candidates for an image point. The optimization can furthermore include the implementation of a region growing technique for the multiple image points of the MR data and, for each image point, with value then being selected from the multiple result candidates a value as the optimized phase at the first echo time and/or the optimized phase evolution. In general, the region growing technique can take into account results of the numerical optimization for adjacent image points of the MR data, i.e. for respective adjacent image points, the respective optimized phase at the first echo time and/or the optimized phase evolution is selected based on an initial image point. Suitable techniques are known in principle to the those skilled in the art, for example from H. Yu et al. “Field map estimation with a region growing scheme for iterative 3-point water-fat-decomposition” in Mag. Reson. Met. 54 (2005), 1032-1039. Therefore, further explanation of the region growing technique is not necessary herein. 
     According to a further aspect, the invention concerns an MR system that is designed for MR measurement of a first spectral component and a second spectral component of an examination subject, using a two-point Dixon technique, at a first echo time and a second echo time. A predetermined spectral model of the two-point Dixon technique includes the first spectral component, the second spectral component, a phase at the first echo time, and a phase evolution due to field inhomogeneities and/or eddy current effects between the first echo time and the second echo time. The MR system has an acquisition unit and a computer. The acquisition unit is designed in order to acquire MR data for multiple image points, respectively at the first echo time and at the second echo time. The computer is designed in order to determine a computation grid that is of low resolution in comparison to the MR data, wherein each grid point of the computation grid encompasses a predetermined number of adjacent image points. The computer is furthermore configured to implement a numerical optimization for each image point of the MR data, the numerical optimization determining an optimized phase at the first echo time and/or an optimized phase evolution. The optimization is based on an equation that takes into account that the phase at the first echo time and/or the phase evolution is constant for all image points that are encompassed by a grid point of the computation grid. The computer is furthermore configured to analytically calculate the first spectral component and the second spectral component based on the phase at the first echo time as determined by the optimization and/or based on the phase evolution determined by the optimization. 
     According to a further aspect, the invention concerns an MR system that is designed for MR measurement of a first spectral component and a second spectral component of an examination subject, with a two-point Dixon technique, at a first echo time and a second echo time. A predetermined spectral model of the two-point Dixon technique includes the first spectral component, the second spectral component, a phase at the first echo time, and a phase evolution due to field inhomogeneities and/or eddy current effects between the first echo time and the second echo time. The MR system has an acquisition unit and a computer. The acquisition unit is designed in order to acquire MR data for multiple image points, respectively at the first echo time and at the second echo time. The computer is designed in order to implement a numerical optimization for each image point of the MR data, which numerical optimization determines an optimized phase at the first echo time and/or an optimized phase evolution. The optimization is based on an equation that has no explicit dependency on the first and second spectral components. The computer is furthermore configured to analytically calculate the first spectral component and the second spectral component based on the phase at the first echo time as determined by the optimization and/or based on the phase evolution determined by the optimization. 
     The MR system is designed in order to implement the method according to the present invention. 
     MR systems according to the invention achieve results and advantages that are comparable to those achieved with methods according to the invention. 
     The features presented above and features that are described in the following can be used not only in the corresponding explicitly presented combinations, but also in additional combinations or in isolation, without departing from the scope of the invention. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  schematically illustrates an MR system. 
         FIG. 2  shows a gradient echo MR measurement sequence in which two MR signals are acquired in a bipolar manner as MR data, respectively at a first echo time and at a second echo time. 
         FIG. 3  illustrates a phase at the first echo time. 
         FIG. 4  illustrates a phase evolution between the first and second echo times. 
         FIG. 5  schematically shows image points of the MR data, grid points of a computation grid, and first and second spectral components for the different image points. 
         FIG. 6  illustrates numerical optimization for two image points. 
         FIG. 7  is a flowchart of the method of the invention according to various embodiments. 
     
    
    
     DESCRIPTION OF THE PREFERRED EMBODIMENTS 
     In the following, the present invention is explained in detail using preferred embodiments with reference to the drawings. In the figures, identical reference characters designate identical or similar elements. The subsequent description of embodiments with reference to the figures should not be construed as limiting. The figures are only illustrative. 
     The figures are schematic representations of different embodiments of the invention. Components shown in the figures are not necessarily true to scale. Rather, the different components presented in the figures are rendered such that their function and general purpose are comprehensible to those skilled in the art. Connections and couplings between functional units and elements that are depicted in the figures can also be implemented as indirect connections or couplings. A connection or coupling can be implemented via wires or wirelessly. Functional units can be implemented as hardware, software or a combination of hardware and software. 
     In the following, techniques are presented by which a first spectral component and a second spectral component can be determined from MR data. For example, the first spectral component can indicate a fat content (shortened to fat in the following) and the second spectral component can indicate a water content (shortened to water in the following). In general, however, any spin species—thus also silicone, for instance—can be considered as a first and second spectral component. 
     The MR data are acquired with a two-point Dixon technique, and thus include a first MR signal and a second MR signal, respectively at first and second echo times. A spectral model is also used that, in addition to the fat component and water component, also takes into account a phase at the first echo time and a phase evolution between the first and second echo times. Weightings of the fat component and of the water component that are used in the spectral model are typically assumed to have real values. 
     Within the scope of the spectral model, the first MR signal and the second MR signal S 0 (x), S 1 (x) can be expressed as
 
 S   0 ( x )=( W ( x )+ c   0   F ( x )) e   iφ(x)  
 
 S   1 ( x )=( W ( x )+ c   1   F ( x )) e   iφ(x)+φ(x) ,  (1)
 
where W(x) and F(x) are real-value water and fat components in an image point x of the MR data; c 0  and c 1  model the time dependency of fat and are known in principle and physically dependent on a spectral composition of the fat and the echo times themselves; φ(x) is the phase at the first echo time and φ(x) is the phase evolution between the first and second echo time. The phase evolution φ(x) has its physical cause in field inhomogeneities of a basic magnetic field of the MR system and in eddy currents.
 
     As noted above, a spectral model corresponding to Equation (1) can also be directly set up for other species than fat and water, but for simplicity only water and fat are referred to for the purposes of better illustration. 
     Equation (1) can also be depicted schematically:
 
 S ( x )=Φ( x ) Av ( x ),  (2)
 
wherein
 
     
       
         
           
             
               
                 
                   
                     
                       
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     Due to the underlying physical effects, the phase at the first echo time φ(x) and the phase evolution φ(x) vary continuously. In particular, they exhibit a smaller variation as a function of location than the MR data themselves. 
     In the following, techniques are explained that enable a determination of the first and second spectral components W(x), F(x) on the basis of the spectral model, i.e. on the basis of Equations (1)-(3). However, the fundamentals of the MR system that can be used for the MR measurement are initially explained with reference to  FIG. 1 . 
     In  FIG. 1 , an MR system  100  is shown that is designed to implement techniques, methods and steps according to the invention. The MR system  100  has a basic field magnet  110  that defines a tube  111 . The magnet  110  generates the basic magnetic field parallel to its longitudinal axis. The basic magnetic field can exhibit inhomogeneities, thus local deviations from a desired value. An examination subject (here an examined person  101 ) can be slid on a bed table  102  into the magnet  110 . Furthermore, the MR system  100  has a gradient system  140  to generate gradient fields that are used for MR imaging and for spatial coding of acquired raw data. The gradient system  140  typically has at least three gradient coils  141  that are separately controllable and positioned with good definition relative to one another. The gradient coils  141  enable gradient fields to be applied and switched along defined spatial directions (gradient axes). Switching the gradient fields can cause eddy current effects that, in turn, produce local magnetic fields. The gradient fields can be used for slice selection, for frequency coding (in the readout direction) and for phase coding, for example. A spatial coding of the raw data can thereby be achieved. The spatial directions that are respectively parallel to slice selection gradient fields, phase coding gradient fields and readout gradient fields do not necessarily need to be coincident with the machine coordinate system. Rather, they can be defined in relation to a k-space trajectory (for example), which can in turn be established on the basis of specific requirements of the respective MR measurement sequence and/or can be established based on anatomical properties of the examined person  101 . 
     To deflect the nuclear spins from the polarization or alignment of their magnetization in the longitudinal direction that results in the basic magnetic field, an RF coil arrangement  121  is provided that radiates an amplitude-modulated RF excitation pulse in the examined person  101 . A transverse magnetization of the nuclear spins can thereby be generated. To generate such RF excitation pulses, an RF transmission unit  131  is connected via an RF switch  130  with the RF coil arrangement  121 . The RF transmission unit  131  can include an RF generator and an RF amplitude modulation unit. The RF excitation pulses deflect (flip) the transverse magnetization in 1D (slice-selectively) or 2D/3D (spatially selectively or globally) out of the steady state. 
     Furthermore, an RF acquisition unit  132  is coupled via the RF switch  130  with the RF coil arrangement  121 . Via the RF acquisition unit  132 , MR signals of the relaxing transverse magnetization (for example due to inductive injection into the RF coil arrangement  121 ) can be acquired as MR data. 
     In general, it is possible to use separate RF coil arrangements  121  for the radiation of the RF excitation pulses by means of the RF transmission unit  131  and for the acquisition of the MR data by means of the RF acquisition unit  132 . For example, a volume coil  121  can be used for the radiation of RF pulses and a surface coil (not shown) which can be an array of RF coils can be used for the acquisition of raw data. For example, the surface coil can have 32 individual RF coils for the acquisition of the raw data, and therefore can be particularly suitable for partially parallel imaging (PPA, partially parallel acquisition). Suitable techniques are known to those skilled in the art and thus need not be explained herein. 
     The MR system  100  furthermore has an operating unit  150  that, for example, can include a monitor, a keyboard, a mouse, etc. User entries can be detected, and outputs to the user can be implemented by the operating unit  150 . For example, via the operating unit  150  it is possible for individual operating modes or operating parameters of the MR system to be set by the user and/or automatically and/or by remote control. 
     Furthermore, the MR system  100  has a computer  160 . For example, the computer  160  can be configured to implement diverse computation operations within the scope of the determination of the fat component and the water component. For example, the computer  160  can be configured in order to implement a numerical optimization and/or to process MR data with a Fourier transformation. 
     In  FIG. 2 , a two-point Dixon gradient echo MR measurement sequence  5  is shown. A radio-frequency axis  10 , a gradient field component  11  and a readout channel  12  are shown. An RF excitation pulse  15  is initially radiated. Readout gradient fields  16  are subsequently switched that form two gradient echoes at the first echo time  21  and the second echo time  22 . The MR data  25 —namely one MR signal at each echo times  21 ,  22 —are received by the analog/digital converter, graphically indicated by the measurement blocks on the readout channel  12 . The first and second echo times  21 ,  22  are defined in relation to a point in time known as the isodelay point in time of the RF excitation pulse  15  which, for example, lies approximately in the middle of the RF excitation pulse with a sinc amplitude envelope. Other definitions of the first and second echo times  21 ,  22  are possible and do not need to be discussed in detail in this context. 
       FIG. 2  is a simplified presentation since at least one slice selection gradient field and one phase coding gradient field (which are typically required for complete spatial coding of an image point of the MR data  25 ) are not shown. However, the MR data  25  are obtained with resolution for different image points (illustrated by the grid in  FIG. 2 ), such that the additional gradient fields are also typically used for spatial coding. 
     Although a gradient echo MR measurement sequence is shown in  FIG. 2 , other two-point Dixon MR measurement sequences can also be used. For example, a spin echo MR measurement sequence could be used, or a monopolar gradient echo MR measurement sequence 
     The RF excitation pulse  15  deflects the magnetization out of its steady state along the longitudinal direction, such that what is known as a transverse component is created. The transversal component is typically depicted in the x-y plane (see  FIGS. 3 and 4 ). In  FIG. 3 , the phase position of the water component  35  and of the fat component  36  at the first echo time  21  is shown. In particular, in  FIG. 3  a situation is shown in which the MR measurement sequence  5  is adjusted to the water component  35 . As can be seen from  FIG. 3 , the water component  35  has a phase φ relative to a zero degree position (defined as a reference) along the x-axis. Due to the frequency shift between the water component  35  and the fat component  36 , the fat component  36  has a different phase position than the water component  35 . 
     In  FIG. 4 , the phase position of the water component  35  and of the fat component  36  at the second echo time  22  is shown. Now the water component  35  has a phase shift relative to the zero degree position (defined as a reference phase) along the x-axis of φ+φ. The phase evolution φ thus designates an additionally acquired phase between the first and second echo times  21 ,  22  that, for example, is due to the field inhomogeneities and/or the eddy current effects. 
     As is explained above with regard to Equations (1)-(3), the spectral model can take into account this phase φ at the first echo time  21  and the phase evolution φ between the first and second echo times  21 ,  22 . It is now possible for an equation on the basis of which a numerical optimization to be implemented to determine the phase φ at the first echo time  21  and/or the phase evolution φ has no explicit dependency on the water component  35  and the fat component  36 . 
     Alternatively or additionally, phase φ at the first echo time  21  and/or the phase evolution φ can be assumed to be constant within a defined region. This is shown in  FIG. 5 . In  FIG. 5 , grid points  40  of a computation grid are represented with dashed lines. The image points  30  of the MR data  25  are also represented with solid lines in  FIG. 5 . As can be seen from  FIG. 5 , the computation grid is defined such that it is of lower resolution in comparison to the MR data  25 , meaning that a grid point  40  is larger than an image point  30 . Each grid point  40  encompasses a predetermined number of adjacent image points  30  of the MR data  25 . In the case of  FIG. 5 , sixteen image points  30  are respectively encompassed given quadratic grid points  40 . For example, within the scope of the determination of the computation grid, the number of adjacent image points  30  of the MR data  25  that are encompassed by a grid point  40  are established depending on a user input and/or depending on a machine parameter of the MR system  100 . 
     In the following, techniques are presented in which it is assumed, within the scope of a numerical optimization, that the phase φ at the first echo time  21  and/or the phase evolution φ between the first and second echo times  21 ,  22  are respectively constant within a grid point  40  of the computation grid. The phase φ at the first echo time and/or the phase evolution φ can thus also be designated as constant in parts. The water component  35  and the fat component  36  thus can be determined in a simple manner. 
     The water component  35  and the fat component  36  are illustrated in  FIG. 5  in a schematic manner for only a few image points  30  of the MR data  25 . In general, however, it is possible to determine the water component  35  and the fat component  36  for all image points  30  of the MR data, for example to determine them individually. 
     For example, the numerical optimization can be a chi-square optimization, such that this equation (based on Equation (3)) has the following form: 
                       x   2     ⁡     (       {     φ   ⁡     (   x   )       }     ,     {     ϕ   ⁡     (   x   )       }     ,     {     v   ⁡     (   x   )       }       )       =           Σ     χ   ∈   U       ⁡     (       D   ⁡     (   x   )       -       Φ   ⁡     (   x   )       ⁢     Av   ⁡     (   x   )           )       T     ⁢       (       D   ⁡     (   x   )       -       Φ   ⁡     (   x   )       ⁢     Av   ⁡     (   x   )           )     .               (   4   )               
wherein U designates a grid point  4  of the computation grid. The equation 4 is bilinear with regard to the water component  35  and fat component  36 . Therefore, these can be eliminated by means of what is known as variable projection.
 
     In contrast to known reference implementations in which both the weighting W of the water component  35  and the weighting F of the fat component  36  are assumed to have complex values, the present approach has the advantage that it reduces the number of variables taken into account within the scope of the numerical optimization given use for two-point Dixon techniques. In the following, the difference relative to the aforementioned reference implementations is illustrated in detail. 
                       If   ⁢           ⁢     A   R       =       ⁢     (   A   )     ⁢           ⁢   and   ⁢           ⁢     A   I       =     ⁢     (   A   )     ⁢           ⁢   are   ⁢           ⁢   provided         ,     
     ⁢             v   ⁡     (   x   )       =         (         A   R   T     ⁢     A   R       +       A   I   T     ⁢     A   I         )       -   1       ⁢   ⁢     (       A   T     ⁢       ϕ   ⁡     (   x   )       T     ⁢     D   ⁡     (   x   )         )                   =         (         A   R   T     ⁢     A   R       +       A   I   T     ⁢     A   I         )       -   1       ⁢     (       A   R   T     ⁢     A   I   T       )     ⁢     (           ⁢     (         Φ   ⁡     (   x   )       T     ⁢     D   ⁡     (   x   )         )                 ⁢     (         Φ   ⁡     (   x   )       T     ⁢     D   ⁡     (   x   )         )             )                       (   5   )               
is obtained and
 
     
       
         
           
             
               
                 
                   
                     
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     The matrix B R  has a real value, is symmetrical and 2nd-order, with eigenvalues of 1, and represents a projection onto the space that is spanned by the columns (A R ,A I ) T . 
     Therefore B R =Σ j=1,2  {tilde over (w)} j {tilde over (w)} j   T  applies, wherein the vectors {tilde over (w)} j =(w R,j +w+w I,j ) T  are real-valued and orthonormal. If u j =w R,j +iw I,j  is defined, 
                       x   2     ⁡     (       {     φ   ⁡     (   x   )       }     ,     {     ϕ   ⁡     (   x   )       }       )       =       Σ     χ   ∈   U       ⁡     (             D   ⁡     (   x   )       T     ⁢     D   ⁡     (   x   )         -     Σ       j   =   1     ,   2         |     ⁢     (       u   j   T     ⁢       Φ   ⁡     (   x   )       T     ⁢     D   ⁡     (   x   )         )       ⁢     |   2       )               (   7   )               
is obtained.
 
     The difference relative to the aforementioned complex-valued approach according to various reference implementations can be displayed using Equation (7). In the various reference implementations that are based on complex-valued weightings W, F it is not necessary to establish the real part as above in Equation (7). The eigenvalues u j  can differ depending on the imaginary part A I . 
     As can be seen from Equation 7, this has no explicit dependency on the water component  35  and the fat component  36 . These were eliminated by means of the variable projection. 
     Equation (7) can be used as a basis for a numerical optimization in which the optimized phase φ(x) at the first echo time  21  and the optimized phase evolution φ(x) are determined. In other words: an optimization with regard to these two variables φ(x), φ(x) can thus take place. 
     However, it is possible to simplify further, such that the phase φ(x) at the first echo time  21  and/or the phase evolution φ(x) are eliminated. The optimization can then respectively be implemented only with regard to the variable φ(x), φ(x) that has not been eliminated. 
     This elimination is shown as an example for the phase φ(x) at the first echo time  21 , but corresponding techniques can also be directly applied for the eliminate of the phase evolution φ(x). In other words, the equation taken into account within the numerical optimization can have no explicit dependency on the phase φ at the first echo time  21  if only the phase evolution φ is numerically optimized. Accordingly, the equation can have no explicit dependency on the phase evolution φ if only the phase φ at the first echo time  21  is numerically optimized. If the equation taken into account in the numerical optimization has no explicit dependency on the phase evolution φ, either it can be assumed that the phase φ at the first echo time  21  varies within a grid point  40  of the computation grid or it can be assumed that the phase φ at the first echo time  21  is constant within a grid point  40  of the computation grid. Accordingly, it analogously applies for the case that the equation has no explicit dependency on the phase  9  at the first echo time  21 . 
     The latter case is illustrated in the following, wherein corresponding techniques can be directly applied to the elimination of the phase evolution φ. It can thus be selected whether the phase φ at the first echo time  21  has a lower resolution than the MR data  25  (i.e. is constant within a grid point  40 ) or has a high resolution (i.e. has the same resolution as the MR data  25 ). For the two options it is respectively defined: x φ   2 ({φ(x)}) and x φ(x)   2 ({φ(x)}). 
     In both cases, the optimization problem is of the form: 
                             min   α     ⁢     (     -         Σ   j     ⁡     (     ⁢     (       c   j     ⁢     e     i   ⁢           ⁢   α         )       )       2       )       =         -     1   2       ⁢     Σ   j       |     c   j     ⁢     |   2     ⁢     -       max   α     ⁢     (       1   4     ⁢       Σ   j     ⁡     (         c   j   2     ⁢     e     2   ⁢   i   ⁢           ⁢   α         +       c   j     *   2       ⁢     e       -   2     ⁢   i   ⁢           ⁢   α           )         )                         =         -     1   2       ⁢     Σ   j       |     c   j     ⁢     |   2     ⁢     -     1   2       |       Σ   j     ⁢     c   j   2       |       ,                 (   8   )               
wherein e iα =|Σ j c j   2 | 1/2 /(Σ j c j   2 ) 1/2 . Due to the extraction of the root, there is an ambiguity in the algebraic sign that, for example, can be made unambiguous via the selection of the weighting W(x) of the fat component  36  in a positive value. This selection of the algebraic sign can then be limited: φε[0, π)
 
                       x   φ   2     ⁡     (     {     ϕ   ⁡     (   x   )       }     )       =         Σ     χ   ∈   U       ⁢       D   ⁡     (   x   )       T     ⁢     D   ⁡     (   x   )         -       1   2     ⁢     Σ     x   ∈   U       ⁢     Σ       j   =   1     ,   2       ⁢             |       u   j   T     ⁢       Ψ   ⁡     (   x   )       T     ⁢     D   ⁡     (   x   )         ⁢     |   2     ⁢     -     |       Σ     x   ∈   U       ⁢         Σ       j   =   1     ,   2       ⁡     (       u   j   T     ⁢       Ψ   ⁡     (   x   )       T     ⁢     D   ⁡     (   x   )         )       2       |     
     ⁢       x     φ   ⁡     (   x   )       2     ⁡     (     {     ϕ   ⁡     (   x   )       }     )             =       Σ     x   ∈   U       (           D   ⁡     (   x   )       T     ⁢     D   ⁡     (   x   )         -       1   2     ⁢     Σ       j   =   1     ,   2       ⁢           |       u   j   T     ⁢       Ψ   ⁡     (   x   )       T     ⁢     D   ⁡     (   x   )         ⁢     |   2     ⁢     -     |         Σ       j   =   1     ,   2       ⁡     (       u   j   T     ⁢       Ψ   ⁡     (   x   )       T     ⁢     D   ⁡     (   x   )         )       2     |         )                           (   9   )               
is then obtained, where Ψ(x)=e −iφ(x) Φ(x) has been defined. From a comparison of the two terms in Equation 9 it is apparent that the fact of whether φ is assumed to be constant within a grid point  40  or not affects the position of the sum over the different image points  30  of the MR data  25 .
 
     In principle, an algebraic simplification of the aforementioned Equation 9 is not possible. If the phase evolution φ is assumed to be constant within a grid point  40 , 
                               x   φ   2     ⁡     (   ϕ   )       ⁢       =         Σ     x   ∈   U       ⁡     (             D   ⁡     (   x   )       T     ⁢     D   ⁡     (   x   )         -       1   2     ⁢         Σ       j   =   1     ,   2       ⁡     (     |       u     j   ,   o     T     ⁢       D   o     ⁡     (   x   )         |     )       2       +     |       u     j   ,   1     T     ⁢       D   1     ⁡     (   x   )         ⁢     |   2       )       -                     ⁢         1   2     ⁢     (       Σ     x   ∈   U       ⁢       Σ       j   =   1     ,   2       ⁡     (       u     j   ,   0     T     ⁢       D   0     ⁡     (   x   )         )       ⁢       (       u     j   ,   1     T     ⁢       D   1     ⁡     (   x   )         )     *       )     ⁢     e       -   i     ⁢           ⁢   ϕ         -                     ⁢       1   2     ⁢     (       Σ     x   ∈   U       ⁢         Σ       j   =   1     ,   2       ⁡     (       u     j   ,   0     T     ⁢       D   0     ⁡     (   x   )         )       *     ⁢     (       u     j   ,   1     T     ⁢       D   1     ⁡     (   x   )         )       )     ⁢     e     i   ⁢           ⁢   ϕ                         ⁢     |       Σ     x   ∈   U       ⁢       Σ       j   =   1     ,   2       (         (       u     j   ,   0     T     ⁢       D   0     ⁡     (   x   )         )     2     +         (       u     j   ,   1     T     ⁢       D   1     ⁡     (   x   )         )     2     ⁢     e     2   ⁢   i   ⁢           ⁢   ϕ         +                             ⁢     2   ⁢     (       u     j   ,   0     T     ⁢       D   0     ⁡     (   x   )         )     ⁢     (       u     j   ,   1     T     ⁢       D   1     ⁡     (   x   )         )     ⁢     e     i   ⁢           ⁢   ϕ         )     |           ⁢     
     ⁢           ⁢   and   ⁢     
     ⁢               x     φ   ⁡     (   x   )       2     ⁡     (   ϕ   )       =       ⁢         Σ     x   ∈   U       ⁡     (             D   ⁡     (   x   )       T     ⁢     D   ⁡     (   x   )         -       1   2     ⁢         Σ       j   =   1     ,   2       ⁡     (     |       u     j   ,   o     T     ⁢       D   o     ⁡     (   x   )         |     )       2       +     |       u     j   ,   1     T     ⁢       D   1     ⁡     (   x   )         ⁢     |   2       )       -                     ⁢       1   2     ⁢     (       Σ     x   ∈   U       ⁢         Σ       j   =   1     ,   2       ⁡     (       u     j   ,   0     T     ⁢       D   0     ⁡     (   x   )         )       *     ⁢     (       u     j   ,   1     T     ⁢       D   1     ⁡     (   x   )         )       )     ⁢     e     i   ⁢           ⁢   ϕ                         ⁢       Σ     x   ∈   U       |       Σ       j   =   1     ,   2       (         (       u     j   ,   0     T     ⁢       D   0     ⁡     (   x   )         )     2     +         (       u     j   ,   1     T     ⁢       D   1     ⁡     (   x   )         )     2     ⁢     e     2   ⁢   i   ⁢           ⁢   ϕ         +                           ⁢     2   ⁢     (       u     j   ,   0     ′     ⁢       D   0     ⁡     (   x   )         )     ⁢     (       u     j   ,   1     T     ⁢       D   1     ⁡     (   x   )         )     ⁢     e     i   ⁢           ⁢   ϕ         )     |                   (   10   )               
are obtained.
 
     It can be shown that Equations (10) and (11) have at most two minima for φε[0,2π) but must be solved numerically. Equations (10) and (11) can thus be taken into account on the basis of the numerical optimization to determine the phase evolution φ. After obtaining the optimized phase evolution φ, the phase φ at the first echo time  21  can then be determined analytically and the analytical calculation of the water component  35  and of the fat component  36  can take place. This analytical calculation can be based on a variable back-projection of the real-value weightings W, F of the water component  35  and the fat component  36 . 
     In  FIG. 6 , a situation is shown in which the implementation of the first numerical optimization (here with regard to the phase evolution φ) provides two respective result candidates (labeled with stars in  FIG. 6 ) for the image points  30 - 1 ,  30 - 2 . In the scenario of  FIG. 6 , these image points  30 - 1 ,  30 - 2  are directly adjacent. The optimization can furthermore include the implementation of a region growing technique for the multiple image points  30 - 1 ,  30 - 2  of the MR data  25 . For each image point  30 - 1 ,  30 - 2 , a value can therefore be selected from the multiple result candidates as the optimized phase evolution φ. For example, after the smaller value of φ has been identified as the actual value for the physically relevant solution for the image point  30 - 2 , the smaller value of φ could also be identified as the actual physically relevant solution for the image point  30 - 1  (respectively illustrated by an arrow and the vertical dashed lines in  FIG. 6 ). In principle, region growing techniques are known to those skilled in the art in connection with the discovery of the relevant solution from multiple result candidates in connection with the optimization in Dixon techniques, such that no additional details need be explained herein. 
     A flowchart of a method according to various aspects of the present invention is shown in  FIG. 7 . The method begins in step S 1 . In step S 2 , the MR data  25  are initially acquired at the first echo time  21  and at the second echo time  22 . 
     The determination unit of the computation grid then takes place in step S 3 . For example, within the scope of step S 3  the number of image points  30  per grid point  40  can be established. 
     In step S 4  a check is made as to whether the optimization should occur only in one variable, i.e. with regard to either the phase φ at the first echo time  21  or with regard to the phase evolution φ, for example. If this is the case, in step S 5  a check is made as to whether the numerical optimization should occur with regard to the phase φ at the first echo time  21 . If this is the case, in step S 6  the chi-square optimization is implemented to determine the optimized phase φ at the first echo time  21 . The phase evolution φ is subsequently determined analytically in step S 7 . 
     If it is established in step S 5  that the optimization should not occur with regard to the phase φ at the first echo time  21 , in step S 8  the implementation takes place in the numerical optimization to determine the optimized phase. The analytical determination of the phase φ at the first echo time  21  subsequently takes place in step S 9 . 
     In contrast to this, if it is established in step S 4  that the optimization should occur not only in one of the phase φ at the first echo time  21  and the phase evolution φ, in step S 10  the implementation of the chi-square optimization takes place both to determine the optimized phase φ at the first echo time  21  and to determine the optimized phase evolution φ. 
     Independent of the output of the checks in step S 4  and step S 5 , a value for both the phase φ at the first echo time  21  and for the phase evolution φ thus respectively exists after the steps S 7 , S 9  or S 10 . 
     The determination of the fat component  35  and the water component  36  can then take place in step S 11 . The method ends in step S 12 . 
     Although modifications and changes may be suggested by those skilled in the art, it is the intention of the inventor to embody within the patent warranted hereon all changes and modifications as reasonably and properly come within the scope of his contribution to the art.