Method and apparatus for determination of a magnetic resonance system control sequence

Abstract

In a method and a control sequence determination device for determining a magnetic resonance system control sequence includes at least one radio-frequency pulse train to be emitted by a magnetic resonance system, a target magnetization is acquired and a k-space trajectory is determined. A radio-frequency pulse train for the k-space trajectory is then determined in an RF pulse optimization method using a target function, wherein the target function includes a combination of different trajectory curve functions, of which at least one trajectory curve function is based on a trajectory error model. A method for operating a magnetic resonance system uses such a control sequence and a magnetic resonance system has such a control sequence determination device.

Claims

1. A computerized method to determine a magnetic resonance system control sequence for operating a magnetic resonance apparatus in an acquisition procedure to acquire magnetic resonance data from a subject situated in the magnetic resonance apparatus, said method comprising: providing a computer with a target magnetization of nuclear spins in the subject, to be produced a radio-frequency pulse train of said magnetic resonance system control sequence; in said computer, determining, or receiving a designation of, a k-space trajectory for entering raw data from the subject into k-space in said acquisition procedure, the entry of said raw data into k-space during said acquisition procedure being subject to deviation from said k-space trajectory due to an error source that occurs in operation of said magnetic resonance apparatus in said acquisition procedure; in said computer, automatically optimizing said radio-frequency pulse train for said k-space trajectory by executing an RF pulse optimization algorithm in said computer using a target function comprising a combination of a plurality of different trajectory curve functions, with at least one of said trajectory curve functions being based on a trajectory error model that models the error produced by said at least one error source; and making said magnetic resonance system control sequence with the optimized radio-frequency pulse train therein, available in electronic form at an output of said computer, in a format configured to operate said magnetic resonance apparatus.

2. A method as claimed in claim 1 comprising, employing as said target function in said computer, a target function wherein said different trajectory curve functions are based on different trajectory error models.

3. A method as claimed in claim 2 comprising generating said different trajectory error models from a same trajectory error model type, but using different error parameters.

4. A method as claimed in claim 2 comprising generating said different trajectory error models as different trajectory error model types.

5. A method as claimed in claim 1 comprising, in said computer, entering said different trajectory models into said target function with predetermined respective weightings.

6. A method as claimed in claim 1 comprising generating said trajectory error models independently of said raw magnetic resonance data acquired in said acquisition procedure.

7. A method as claimed in claim 1 comprising generating said error model to model an error produced by at least one error source selected from the group consisting of amplitude scaling errors, eddy current errors, basic magnetic field errors, gradient mixed term effects, and coupling effects among radio-frequency antennas.

8. A method as claimed in claim 1 comprising using and optimizing, as said radio-frequency pulse train, a multi-channel pulse train comprising multiple, individual radio-frequency pulse trains emitted in parallel in said acquisition procedure by said magnetic resonance apparatus, via multiple, different, independent radio-frequency transmission channels of said magnetic resonance apparatus.

9. A method as claimed in claim 1 comprising employing, as said target function, a target function formulated as an A-matrix that comprised multiple, different sub-matrices, the respective sub-matrices that are respectively based on said different trajectory curve functions.

10. A method as claimed in claim 9 wherein one of said sub-matrices is based on an ideal trajectory curve function.

11. A computerized method to operate a magnetic resonance apparatus in an acquisition procedure to acquire magnetic resonance data from a subject situated in the magnetic resonance apparatus according to a magnetic resonance system control sequence, said method comprising: providing a computer with a target magnetization of nuclear spins in the subject, to be produced a radio-frequency pulse train of a magnetic resonance system control sequence; in said computer, determining, or receiving a designation of, a k-space trajectory for entering raw data from the subject into k-space in said acquisition procedure, the entry of said raw data into k-space during said acquisition procedure being subject to deviation from said k-space trajectory due to an error source that occurs in operation of said magnetic resonance apparatus in said acquisition procedure; in said computer, automatically optimizing said radio-frequency pulse train for said k-space trajectory by executing an RF pulse optimization algorithm in said computer using a target function comprising a combination of a plurality of different trajectory curve functions, with at least one of said trajectory curve functions being based on a trajectory error model that models the error produced by said at least one error source; and making said magnetic resonance system control sequence with the optimized radio-frequency pulse train therein, available in electronic form at an output of said computer, and operating said magnetic resonance apparatus according to said magnetic resonance system control sequence with the optimized radio-frequency pulse train therein.

12. A control sequence determination device to determine a magnetic resonance system control sequence for operating a magnetic resonance apparatus in an acquisition procedure to acquire magnetic resonance data from a subject situated in the magnetic resonance apparatus, said device comprising: a computer having an input, provided with a target magnetization of nuclear spins in the subject, to be produced a radio-frequency pulse train of said magnetic resonance system control sequence; said computer being configured to determine, or to receive via said input a designation of, a k-space trajectory for entering raw data from the subject into k-space in said acquisition procedure, the entry of said raw data into k-space during said acquisition procedure being subject to deviation from said k-space trajectory due to an error source that occurs in operation of said magnetic resonance apparatus in said acquisition procedure; said computer being configured to automatically optimize said radio-frequency pulse train for said k-space trajectory by executing an RF pulse optimization algorithm in said computer using a target function comprising a combination of a plurality of different trajectory curve functions, with at least one of said trajectory curve functions being based on a trajectory error model that models the error produced by said at least one error source; and said computer being configured to make said magnetic resonance system control sequence with the optimized radio-frequency pulse train therein, available in electronic form at an output of said computer, in a format configured to operate said magnetic resonance apparatus.

13. A magnetic resonance apparatus comprising: a magnetic resonance data acquisition unit operable with a magnetic resonance system control sequence in an acquisition procedure to acquire magnetic resonance data from a subject situated in the magnetic resonance data acquisition unit; a computer provided with a target magnetization of nuclear spins in the subject, to be produced a radio-frequency pulse train of said magnetic resonance system control sequence; said computer being configured to determine, or to receive a designation of, a k-space trajectory for entering raw data from the subject into k-space in said acquisition procedure, the entry of said raw data into k-space during said acquisition procedure being subject to deviation from said k-space trajectory due to an error source that occurs in operation of said magnetic resonance apparatus in said acquisition procedure; said computer being configured to automatically optimize said radio-frequency pulse train for said k-space trajectory by executing an RF pulse optimization algorithm in said computer using a target function comprising a combination of a plurality of different trajectory curve functions, with at least one of said trajectory curve functions being based on a trajectory error model that models the error produced by said at least one error source; and said computer being configured to make said magnetic resonance system control sequence with the optimized radio-frequency pulse train therein, available in electronic form at an output of said computer, and to operate said magnetic resonance data acquisition unit according to said magnetic resonance system control sequence with the optimized radio-frequency pulse train therein.

14. A non-transitory, computer-readable data storage medium encoded with programming instructions to determine a magnetic resonance system control sequence for operating a magnetic resonance apparatus in an acquisition procedure to acquire magnetic resonance data from a subject situated in the magnetic resonance apparatus, said storage medium being loaded into a computer and said programming instructions causing said computer to: receive a target magnetization of nuclear spins in the subject, to be produced a radio-frequency pulse train of said magnetic resonance system control sequence; determine, or receive a designation of, a k-space trajectory for entering raw data from the subject into k-space in said acquisition procedure, the entry of said raw data into k-space during said acquisition procedure being subject to deviation from said k-space trajectory due to an error source that occurs in operation of said magnetic resonance apparatus in said acquisition procedure; optimize said radio-frequency pulse train for said k-space trajectory by executing an RF pulse optimization algorithm in said computer using a target function comprising a combination of a plurality of different trajectory curve functions, with at least one of said trajectory curve functions being based on a trajectory error model that models the error produced by said at least one error source; and make said magnetic resonance system control sequence with the optimized radio-frequency pulse train therein, available in electronic form at an output of said computer, in a format configured to operate said magnetic resonance apparatus.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) FIG. 1 is a schematic depiction of an exemplary embodiment of a magnetic resonance system according to the invention.

(2) FIG. 2 is a flowchart of an exemplary embodiment of the method according to the invention for determination of a control sequence.

(3) FIG. 3 shows a simulation of the achieved magnetization (given specification of an L-shaped target magnetization) under assumption of different gradient delay and eddy current errors given execution of the gradient pulses in a method without application of the invention (upper images) in comparison to a method with application of the invention (lower images).

DESCRIPTION OF THE PREFERRED EMBODIMENTS

(4) A magnetic resonance system 1 according to the invention is schematically depicted in FIG. 1. The system includes the actual magnetic resonance scanner 2 with an examination space 8 or patient tunnel therein. A bed 7 can be driven into this patient tunnel 8 so that, during an examination, an examination subject 0 (patient/test subject) lying on said bed 7 can be supported at a defined position within the magnetic resonance scanner 2 relative to the magnet system and radio-frequency system arranged therein, or can be driven between different positions during a measurement.

(5) Basic components of the magnetic resonance scanner 2 are a basic field magnet 3, a gradient system 4 with magnetic field gradient coils in order to apply arbitrary magnetic field gradients in the x-, y- and z-directions, and a whole-body radio-frequency coil 5. The reception of magnetic resonance signals induced in the examination subject O can take place via the whole-body coil 5 with which the radio-frequency signals are normally also emitted to induce said magnetic resonance signals. However, these signals are typically received with local coils 6 placed on or below the examination subject O. All of these components are known in principle to those skilled in the art, and therefore are only schematically shown in FIG. 1.

(6) Here the whole-body radio-frequency coil 5 is designed in the form of a birdcage antenna and has a number N of individual antenna rods that proceeds parallel to the patient tunnel 8 and are arranged distributed uniformly on a periphery around the patient tunnel 8. At the ends, the individual antenna rods are each connected capacitively in the form of a ring. The individual antenna rods are separately controllable by a control device 10 as individual transmission channels S.sub.1, . . . , S.sub.N, which means that the magnetic resonance tomography system is a pTX-capable system. However, the method according to the invention is also applicable to classical magnetic resonance tomography apparatuses with only one transmission channel. Since the method according to the invention offers particular advantages given pTX sequences, in the following such an example is assumed (insofar as it is not stated otherwise), without limitation of the generality.

(7) The control device 10 can be a control computer that can be composed of a number of individual computers (which may be spatially separated and connected among one another via suitable cables or the like). This control device 10 is connected via a terminal interface 17 with a terminal 20 via which an operator can control the entire magnetic resonance system 1. In the present case, this terminal 20 is equipped as a computer with keyboard, one or more screens 28 as well as additional input devices (for example a mouse or the like) so that a graphical user interface is provided to the operator.

(8) Among other things, the control device 10 has a gradient control unit 11 that can in turn be composed of multiple sub-components. The individual gradient coils are supplied via this gradient control unit 11 with control signals SG.sub.x, SG.sub.y, SG.sub.z. These represent gradient pulses that, during a measurement, are activated at precisely set time positions and with a precisely predetermined time curve.

(9) Moreover, the control device 10 has a radio-frequency transmission/reception unit 12. This RF transmission/reception unit 12 likewise has multiple sub-components in order to feed radio-frequency pulses separately and in parallel to the individual transmission channels S.sub.1, . . . , S.sub.N, i.e. to the individually controllable antenna rods of the body coil. Magnetic resonance signals can also be received via the transmission/reception unit 12. However, this typically occurs with the aid of the local coils 6. The raw data RD acquired with these local coils 6 are read out and processed by an RF reception unit 13. The magnetic resonance signals received by these (or by the whole body coil) by means of the RF transmission/reception unit 12 are passed as raw data RD to a reconstruction unit 14, which reconstructs the image data BD from these and stores these in a memory 16 and/or passes them to the terminal 20 via the interface 17 so that the operator can view them. The image data BD can also be stored and/or displayed and evaluated at other locations via a network NW. Insofar as the local coils have a suitable switching unit, these can also be connected to an RF transmission/reception unit in order to also use the local coils to transmit.

(10) The gradient coil controller 11, the RF transmission/reception unit 12 and the reception unit 13 for the local coils 6 are respectively controlled as coordinated by a measurement control unit 15. Via corresponding commands, this ensures that a desired gradient pulse train GP is emitted via suitable gradient coil signals SG.sub.x, SG.sub.y, SG.sub.z and controls the RF transmission/reception unit 12 in parallel so that a multichannel pulse train MP is emitted, meaning that the radio-frequency pulses matching the individual transmission channels S.sub.1, . . . , S.sub.N are provided in parallel to the individual transmission rods of the whole-body coil 5. Moreover, it must be ensured that the magnetic resonance signals are read out at the local coils 6 via the RF reception unit 13 or, respectively, that possible signals are read out at the whole-body coil 5 via the RF transmission/reception unit 12 at the matching point in time and are processed further. The measurement control unit 15 provides the corresponding signals—in particular the multichannel pulse train MP—to the radio-frequency transmission/reception unit 12 and the gradient pulse train GP to the gradient control unit 11 according to a predetermined control protocol P. In this control protocol P, all control data are stored that must be set during a measurement according to a predetermined control sequence AS.

(11) A number of control protocols P for different measurements are typically stored in a memory 16. These could be selected by the operator via the terminal 20 and possibly be varied in order to then provide for the currently desired measurement a matching control protocol P with which the measurement control unit 15 can operate. Moreover, the operator can also retrieve control protocols P (for example from a manufacturer of the magnetic resonance system) via a network NW and can then modify and use these as necessary.

(12) The basic workflow of such a magnetic resonance measurement and the cited components for control are known to those skilled in the art, and thus need not be described in further detail herein. Moreover, such a magnetic resonance scanner 2 as well as the associated control device 10 can still have a number of additional components that are likewise not explained in detail herein. It is noted that the magnetic resonance scanner 2 can be of a different design, for example with a laterally open patient space, and that in principle the radio-frequency whole-body coil does not need to be designed as a birdcage antenna.

(13) Moreover, a control sequence determination device 22 according to the invention that serves to determine a magnetic resonance system control sequence AS is schematically depicted here in FIG. 1. Among other things, for a specific measurement this magnetic resonance system control sequence AS includes a pulse sequence with a gradient pulse train GP in order to traverse a specific trajectory in k-space, as well as a radio-frequency pulse train (here a multichannel pulse train MP) coordinated with this to control the individual transmission channels S.sub.1, . . . , S.sub.N. In the present case, the magnetic resonance system control sequence AS is created as part of the control protocol P.

(14) Here the control sequence determination device 22 is depicted as part of the terminal 20 and can be realized in the form of software components at the computer of this terminal 20. In principle, however, the control sequence determination device 22 can also be part of the control device 10 itself, or be realized at a separate computer system, and the finished control sequences AS are transmitted via a network NW to the magnetic resonance system 1 (possibly also within the framework of a complete control protocol P).

(15) Here the control sequence determination device 22 has an input interface arrangement 23, 24 which can be composed of multiple sub-interfaces, where here is symbolized by the 2 reference characters 23, 24. Via this input interface arrangement 23, 24, the control sequence determination device 22 receives a target magnetization m that predetermines what the flip angle distribution should be in the desired measurement. The control sequence device also receives a transmission k-space trajectory k(t), and may receive a B.sub.0 map ΔB.sub.0; multiple B.sub.1 maps ΔB.sub.1, and possible additional input parameters that are explained again in detail below in connection with FIG. 2.

(16) With an error model determination unit 25 (shown separately here), a trajectory error model TFM.sub.1, TFM.sub.2, . . . , TFM.sub.m or multiple trajectory error models TFM.sub.1, TFM.sub.2, . . . , TFM.sub.m or a combination of trajectory error models TFM.sub.1, TFM.sub.2, . . . , TFM.sub.m that is/are to be considered is/are selected. This error module determination unit 25 can also be an interface, for example a part of the input interface arrangement. However, it can also be a memory in which finished trajectory error model types or the like are stored and into which, for example, only parameters for trajectory error models TFM.sub.1, TFM.sub.2, . . . , TFM.sub.m that are defined via a user interface are to be input, or these parameters are selected depending on the desired control sequence (for example the type of sequence and the type of trajectory), for example. It is normally the case that specific error types can occur with specific types of trajectories.

(17) All of these data are then passed to a target function determination unit 26 which determines a suitable target function f.sub.z that then includes the desired combination of different k-space trajectory curve functions that are based on the defined trajectory error models TFM.sub.1, TFM.sub.2, . . . , TFM.sub.m. The determined target function f.sub.z is then passed to an RF pulse optimization unit 27 in which an optimal radio-frequency pulse train MP is then determined for the desired trajectory using the predetermined target function f.sub.z.

(18) The data that define this radio-frequency pulse train MP and the selected trajectory k(t) are then emitted as an output via a control sequence output interface 28 and can then be provided to the control device 10, for example within the scope of a control protocol P in which additional specifications for controlling the magnetic resonance system 1 are indicated (for example parameters for reconstruction of the images from the raw data etc.).

(19) In the following, the workflow of a method according to the invention for determining a magnetic resonance system control sequence AS is explained in a simplified example using the workflow diagram according to FIG. 2.

(20) In step I, different parameters used within the further method are initially predetermined or, respectively, adopted. For example, in step Ia system-specific parameters SP (such as the number of transmission channels, a maximum slew rate, a maximum gradient amplitude etc.) are adopted; in step Ib, different examination-specific parameters UP (such as the positioning of the slices etc. to be acquired) are adopted; and in step Ic, the B.sub.1 maps ΔB.sub.1 for the individual transmission channels are adopted. Moreover, in step Id a currently measured B.sub.0 map ΔB.sub.0 can be provided. In step II, a desired target magnetization m is provided. Finally, in step III a k-space trajectory k(t) is defined, for example by specification of a fixed k-space trajectory k(t) or via specification of a trajectory type, and subsequent determination of an optimized k-space trajectory k(t) of this type. The specification of the trajectory or of the type can take place via the selected control protocol since the trajectory often depends on the type of measurement. The method steps Ia through Id, II and III can also be executed in a different order.

(21) The design of the radio-frequency pulse train (here a multichannel pulse train MP) then takes place automatically in step V after the determination of the target function f.sub.z in step IV. The individual RF pulse series for the different transmission channels are hereby developed, which means that which RF pulse train must be sent at which channel is calculated precisely. An iterative optimization method is applied since this has proven to be particularly suitable. Specifically, the known conjugate gradient method (CG method, or method of conjugated gradients) is used. In principle, however, other optimization methods (also non-iterative methods) are also usable.

(22) This can take place with arbitrary methods. In many known methods, the optimization method takes place so that (for example) the quadratic mean deviation (least mean square) between the target magnetization and the real magnetization is minimized. This means that a solution is sought for the following target function f.sub.z:
b=arg.sub.bmin{∥|m.sub.ist|−m∥.sub.w.sup.2+R(b)}  (1)

(23) The target magnetization is thereby m, and m.sub.real=A.Math.b is the (theoretical) real magnetization achieve via an RF pulse train p, wherein A is what is known as the A-matrix, comprised of a system of linear complex equations into which the spatial coil profiles and the present B.sub.0 maps and B.sub.1 maps and the k-space trajectory that is used are entered. For example, an A-matrix (also called system matrix or design matrix) that is typically used is described by W. Grissom et al.: “Spatial Domain Method for the Design of RF Pulses in Multicoil Parallel Excitation”, Mag. Res. Med. 56, 620-629, 2006. b is a (time-dependent) vector that includes the (for example) N functions b.sub.c (a time-dependent function of the RF amplitude for each transmission channel c=1 through N). R(b) is an (optional) regularization term, preferably a Thikonov regularization. W defines the volume of interest (region of interest), i.e. the volume that is to be achieved via the RF pulse series.

(24) Equation (1) or the target function f.sub.z (the part to be minimized within the curly brackets) that is used therein or its A-matrix here is set up beforehand in a manner according to the invention in step IV so that the target function f.sub.z includes a combination of different trajectory curve functions, of which at least one trajectory curve function is based on a trajectory error model TFM.sub.1, TFM.sub.2, . . . , TFM.sub.M. In this way it can be taken into account that the actual gradient pulse shapes that are executed later do not need to coincide with the previously calculated theoretical gradient pulse shapes, and therefore the k-space trajectory can differ from the theoretically calculated desired k-space trajectory.

(25) The way that the trajectory curve functions enter into the A-matrix is shown in the mathematical definition of their individual matrix elements a.sub.ij, which here is as follows:
a.sub.ij=iγm.sub.0Δte.sup.ir.sup.i.sup.k(t.sup.j.sup.)e.sup.iγΔB.sup.0.sup.(r.sup.i.sup.)(t.sup.j.sup.−T)  (2)
In this equation, i and j are rows/columns of the A-matrix or, respectively, the i-th spatial and j-th temporal sample point; γ is the gyromagnetic ratio; m.sub.0 is the steady state magnetization, i.e. likewise a material constant; r is the spatial coordinates within k-space; ΔB.sub.0 is the value of the B.sub.0 map at the location r.sub.i; Δt is the discrete sampling time interval; and T is the pulse length, i.e. the duration to traverse the trajectory k(t) (all partial pulses thereby form a single common “RF pulse” along a k-space trajectory in the sense of this equation). The k-space trajectory or, respectively, the k-space trajectory curve function k(t) is provided as follows:

(26) $\begin{array}{cc}k\left(t\right)=-\gamma {\int }_t^{T}G\left(\tau \right)d\tau & \left(3\right)\end{array}$
wherein G is the gradient amplitude (gradient waveform) at the point in time t. t is hereby simply the integration index, and T is again the pulse length (for example in [s]). The gradient amplitude G is written as a vector since it relates to the gradient amplitude shape in all three spatial directions, i.e. G.sub.x,y,z(t) (for example in [mT/m]). k(t) is the position in k-space, indicated as a vector k.sub.x,y,z(t) (for example in [1/mm]).

(27) In order to ensure that the target function according to Equation (1) includes a combination of different k-space trajectory curve functions that are based on different trajectory error models, according to the preferred variant of the invention a typical A-matrix is now not used, but rather an A-matrix is used that is comprised of multiple sub-matrices. For this, in Equation (1) the real magnetization m.sub.real achieved via an RF pulse train b is defined as follows:

(28) $\begin{array}{cc}\left[\begin{array}{c}m\\ m\\ \mathrm{.Math.}\\ m\end{array}\right]=\left[\begin{array}{c}{A}_{\mathrm{ideal}}\\ A_{\mathrm{em}1}\\ \mathrm{.Math.}\\ A_{\mathrm{emn}}\end{array}\right]b& \left(4\right)\end{array}$
A.sub.ideal is the conventional A-matrix as used in the article by Setsompop, for example. This A-matrix describes the case that the k-space trajectory is traversed in an ideal manner, meaning that it assumes an ideal k-space trajectory curve function.

(29) This matrix was expanded by additional sub-matrices A.sub.em1, . . . , A.sub.emn which are respectively constructed in the same manner, meaning that each of these sub-matrices A.sub.em1, . . . , A.sub.emn is comprised of elements a.sub.ij as they are defined in Equation (2). The individual sub-matrices A.sub.em1, . . . , A.sub.emn differ only by, instead of the definition of the k-space trajectory curve functions according to Equation (3), the sub-matrix A.sub.emi for the i-th error model now being constructed according to

(30) $\begin{array}{cc}{k}_{\mathrm{emi}}\left(t\right)=-\gamma {\int }_t^T{G}_{\mathrm{emi}}\left(\tau \right)d\tau & \left(5\right)\end{array}$

(31) The gradient amplitudes (gradient waveforms) G.sub.emi(t) that are based on errors can thereby be modeled with the use of anisotropic hardware delay times T.sub.d and linear eddy currents G.sub.EC(T) according to
G.sub.emi(t)=T.sub.d(G.sub.ideal(t)+G.sub.EC(t))  (6)
The operator T.sub.d defining the delay time is thereby defined as follows:

(32) $\begin{array}{cc}{G}_d=T_d\left(G\left(t\right)\right)=\left[\begin{array}{c}{G}_x\left(t-{\tau }_x\right)\\ G_y\left(t-{\tau }_y\right)\\ G_z\left(t-{\tau }_z\right)\end{array}\right]& \left(7\right)\end{array}$
wherein t.sub.x, t.sub.y, t.sub.z are respectively the delay times in the direction of the x-, y- and z-axis with regard to the RF pulse emission. The operator T.sub.d can be constructed so that it has the same delay times in all spatial directions, but also so that different delay times are provided for each spatial direction.

(33) The operator G.sub.EC for definition of the eddy current errors can, for example, be defined as depicted in the following as an example for the x-axis:

(34) $\begin{array}{cc}{G}_{{\mathrm{EC}}_x}\left(t\right)=-\frac{d{G}_x}{dt}.Math.e_{\mathrm{xx}}\left(t\right)-\frac{d{G}_y}{dt}.Math.e_{\mathrm{xy}}\left(t\right)-\frac{d{G}_z}{dt}.Math.e_{\mathrm{xz}}\left(t\right)& \left(8\right)\end{array}$
e.sub.xx(t), e.sub.xy(t) and e.sub.xz(t) are the exponential functions that respectively describe the decay response of the eddy current terms. The last two cross products could optionally also be ignored because the eddy current effects due to cross terms are relatively small in relation to the first term. The first term can then be developed as follows in a linear Taylor series:

(35) $\begin{array}{cc}\begin{array}{c}{G}_{{\mathrm{EC}}_x}\left(t\right)\approx -\frac{d{G}_x}{dt}.Math.e_{\mathrm{xx}}\left(t\right)\\ =-\frac{d{G}_x}{dt}.Math.\underset{n}{.Math.}{a}_{n}h\left(t\right)\exp \left(-t/{\tau }_n\right)\\ \approx -\frac{d{G}_x}{dt}.Math.\underset{n}{.Math.}{a}_{n}h\left(t\right)\left(1-t/{\tau }_n\right)\end{array}\hspace{1em}& \left(9\right)\end{array}$
wherein τ.sub.n represents the decay time of the eddy current (for example typically 20-150 μs), a.sub.n is the amplitude of the eddy current (for example typically 1%-2% of the gradient amplitude G) and

(36) $\begin{array}{cc}h\left(t\right)=\{\begin{array}{cc}1& t\ge 0\\ 0& t<0\end{array}& \left(10\right)\end{array}$
Equation (9) can then also be written as follows:

(37) $\begin{array}{cc}{G}_{\mathrm{ECx}}\left(t\right)={\mathrm{AG}}_x\left(t\right)+{\mathrm{BG}}_x\left(t\right)t-B{\int }_0^t\frac{d{G}_x}{d{t}^{\prime }}{t}^{\prime }d{t}^{\prime }\mathrm{with}& \left(11\right)\\ A=-\underset{n}{.Math.}{a}_n\mathrm{and}& \left(12\right)\\ B=-\underset{n}{.Math.}\left(a_n/{\tau }_n\right)& \left(13\right)\end{array}$

(38) The gradient error model G.sub.emi(t) defined according to Equation (6) is an example of a combined error model that takes into account both different gradient delays and eddy currents. Alternatively, it is also possible to construct two different trajectory error model types (gradient error model type), wherein one model type takes into account only the delay times and the other model type takes into account only the delay times, and then define one or more individual sub-matrices for each trajectory error model type, which sub-matrices then enter into the complete A-matrix according to Equation (4).

(39) More elaborate error models or error model types can similarly be used that, for example, also include nonlinear eddy currents, cross products, mixed products between gradient fields of the different coils due to the Maxwell terms, overcoupling effects, or also static or, respectively, dynamic B0 field effects, as well as incorrect amplitude scalings of the gradient coils etc.

(40) For each error model type, multiple error models or, respectively, multiple sub-matrices can thereby also be used according to Equation (4), wherein different error parameters are used in the same error model type. For example, in Equation (4) a first error model for a first sub-matrix A.sub.em1 could be constructed according to Equation (6), wherein t.sub.d1=+10 μs is assumed as a delay time in all gradient directions. An additional sub-matrix A.sub.em2 is likewise constructed according to Equation (6), i.e. with the same error model type, wherein then the delay times are, however, t.sub.d2=−10 μs etc. For each error model it is thereby also possible to provide different delay times for different spatial directions etc.

(41) In order to keep the calculation times as low as possible, it is in particular also possible to limit the error models that are used to “worst case” error values known a priori, for example (as described above) maximum delay times of +/−10 μs in each gradient direction.

(42) The possibility to construct multiple trajectory error models TFM.sub.1, TFM.sub.2, . . . , TFM.sub.M on the basis of different trajectory error model types FMT.sub.a, FMT.sub.b, and to have these enter into the target function f.sub.z in the described manner, is symbolized in FIG. (2). Here it is shown how the first trajectory error models TFM.sub.1, TFM.sub.2 are based on a first trajectory error model type FMT.sub.b, in contrast to which additional trajectory error models TFM.sub.m are based on other trajectory error model types (here the trajectory error model type FMT.sub.b).

(43) If the matching target function f.sub.z was set up according to Equations (1) through (13) in step IV, and the solution for this was found in Step V, a function of the amplitude depending on the time for all present transmission channels exists as a result.

(44) The multichannel pulse series can thereby initially be obtained for what is known as the “low flip range” with flip angles below 5°, since in this range the magnetization response still runs linearly. The obtained values are then scaled up in order to achieve the actual desired target magnetization that, for example, goes up to a 90° flip angle. This takes place simply via multiplication of the amplitudes of the individual pulses with the desired scaling factor. The error that can occur upon scaling up can be corrected via a (partial) Bloch simulation.

(45) To conclude, in step VI the control sequence AS is then passed on for caching or immediate execution.

(46) In order to test the method according to the invention, simulations have been calculated in which an L-shaped target magnetization distribution (target flip angle distribution) has respectively been predetermined for a data set of a human torso. In the calculations, a numerical optimization takes place according to the above method, wherein 8 independent transmission channels and an excitation field of 400×600 mm has been assumed. The excitation resolution has been assumed with 10 mm, and what is known as a tick-tock optimizer has been assumed. 15° has been assumed as a target flip angle.

(47) FIG. 3 shows the result of these simulations. The images respectively show the spatial distribution of the achieved flip angle within an x/y-plane in positional space, wherein the coordinates on the x-axis and y-axis are indicated in pixels. The achieved flip angle is indicated by the nearby greyscaling. The quadratic error (or root mean square error) relative to the target magnetization (i.e. the desired “L” pattern) is indicated in the flip angle (α.sub.rms) via the image.

(48) Thereby shown in the upper line are the achieved target magnetizations without the method according to the invention, i.e. without any correction with regard to possible trajectory errors, wherein in the first, left image a gradient delay time d.sub.1 of −10 μs is assumed in each spatial direction, and additional eddy current errors have been simulated; in the middle image an ideal trajectory is assumed; and in the last, right image a gradient delay time d.sub.2 of +10 μs in each spatial direction and an eddy current error are assumed. These images unambiguously show that (simulated) gradient errors have a significant effect on the achieved target magnetization.

(49) Shown in the line below this are the achieved target magnetizations given respective identical starting conditions as in the upper image line, but given application of a method according to the invention. Here it is shown that the achieved target magnetization is easily slightly disrupted by the consideration according to the invention only in the (improbable) event that absolutely no gradient errors occur.

(50) However, insofar as an error actually occurs due to eddy currents or, respectively, due to a gradient delay of −10 μs to +10 μs (left and right images in the lower line), the desired target magnetization is achieved significantly better than without the method according to the invention. In particular, the simulations show that it is not absolutely necessary to determine concrete gradient errors due to previously implemented measurements of the gradient trajectory, and then to use these in the additional measurements. In these simulations, the trajectory error model that was defined according to the above equations (4) through (13) has been used under consideration of “worst case” delay times of +/−10 μs in each spatial direction, as well as under consideration of eddy currents.

(51) As described above, the method is applicable for arbitrary k-space trajectories, and also for an arbitrary number of transmission channels. The method likewise shows that it is sufficient to consider only the worst case scenarios, and that the numerical complexity can thereby be reduced.

(52) The method can also be used advantageously if the traversal through k-space simultaneously with the RF pulses (i.e. the k-space trajectories themselves) is also optimized, which would not be possible without further measures given methods that are based on a pre-measurement of the actually achieved k-space trajectory. Therefore, the method according to the invention can be used not only to achieve ideal RF pulses for predetermined k-space trajectories, but also in methods in which optimized k-space trajectories and radio-frequency pulses that have simultaneously been optimized for these are determined.

(53) Although modifications and changes may be suggested by those skilled in the art, it is the intention of the inventors to embody within the patent warranted hereon all changes and modifications as reasonably and properly come within the scope of their contribution to the art.