Method of designing and generating pulses for magnetic resonance imaging

Abstract

A method of designing a refocusing pulse or pulse train for Magnetic Resonance Imaging comprises the steps of: a) determining a phase-free performance criterion representative of a proximity between a rotation of nuclear spins induced by the pulse and a target operator, summed or averaged over one or more voxels of an imaging region of interest; and b) adjusting a plurality of control parameters of the pulse to maximize the phase-free performance criterion; wherein each target operator is chosen so the phase-free performance criterion takes a maximum value when the nuclear spins within all voxels undergo a rotation of a same angle around a rotation axis lying in a plane perpendicular to a magnetization field B.sub.0, called a transverse plane, with an arbitrary orientation; wherein the angle is different from M radians, with integer M, preferably with < radians and even preferably with 0.9.Math. radians.

Claims

1. A method of designing a refocusing pulse for Magnetic Resonance Imaging, said pulse being suitable to induce a rotation of nuclear spins around a rotation axis lying in a plane perpendicular to a magnetization field B.sub.0, called a transverse plane, with an arbitrary orientation, the method comprising adjusting a plurality of control parameters of said pulse to maximize a performance criterion , called phase-free performance criterion, given by: $\underset{n=1}{\overset{N}{.Math.}}{}_n$ N1 being the number of voxels, with:
.sub.n.sub.1,n+.sub.2,n wherein: ${}_{1,n}=.Math.\frac{.Math.U_{F1}|U_n\left(T\right).Math.}{2}.Math.,\mathrm{and}{}_{2,n}=.Math.\frac{.Math.U_{F2}|U_n\left(T\right).Math.}{2}.Math.;$ $\begin{array}{c}{U}_{F1}=\left[\begin{array}{cc}\cos \frac{}{2}& 0\\ 0& \cos \frac{}{2}\end{array}\right]\mathrm{and}\\ U_{F2}=\left[\begin{array}{cc}0& 2\sin \frac{}{2}\\ 0& 0\end{array}\right]\end{array}$ are two target operators in spin-domain representation; is a target value of the rotation angle of said nuclear spins induced by said pulse, said angle being different from M radians, with integer M; U.sub.n(T) is a rotation matrix expressing, also in spin-domain representation, the action of said pulse, of duration T, on nuclear spins of the n.sup.th voxel; and said method being carried out by a computer.

2. The method according to claim 1, wherein $U_n\left(T\right)=\left[\begin{array}{cc}& -{}^{*}\\ & {}^{*}\end{array}\right]$ is a solution of the spin-domain Bloch equation: $\left[\begin{array}{c}\stackrel{.}{}\\ \stackrel{.}{}\end{array}\right]=\frac{i}{2}\left[\begin{array}{cc}G.Math.r+B_0& B_1^{*}\\ B_1& -\left(G.Math.r+B_0\right)\end{array}\right]\left[\begin{array}{c}\\ \end{array}\right]$ where: i is the imaginary unit; is a nuclear spin gyromagnetic ratio G is a magnetic field gradient; r is a position vector of the n-th voxel; B.sub.0 expresses the non-uniformity of the magnetization field B.sub.0; and B.sub.1 expresses the spatial distribution of the pulse.

3. A method of designing a refocusing pulse train for Magnetic Resonance Imaging, said pulse train comprising a plurality of refocusing pulses, each of said refocusing pulses being adapted for inducing a rotation of nuclear spins of a respective angle .sub.i, said method comprising the steps of: A. using a computer for designing one said refocusing pulse, called reference pulse, by adjusting a plurality of control parameters of said pulse to maximize a performance criterion , called phase-free performance criterion, given by: $\underset{n=1}{\overset{N}{.Math.}}{}_n$ N1 being the number of voxels, with:
.sub.n.sub.1,n+.sub.2,n wherein: ${}_{1,n}=.Math.\frac{.Math.U_{F1}|U_n\left(T\right).Math.}{2}.Math.,\mathrm{and}{}_{2,n}=.Math.\frac{.Math.U_{F2}|U_n\left(T\right).Math.}{2}.Math.;$ $\begin{array}{c}{U}_{F1}=\left[\begin{array}{cc}\cos \frac{}{2}& 0\\ 0& \cos \frac{}{2}\end{array}\right]\mathrm{and}\\ U_{F2}=\left[\begin{array}{cc}0& 2\sin \frac{}{2}\\ 0& 0\end{array}\right]\end{array}$ are two target operators in spin-domain representation; is a target value of the rotation angle of said nuclear spins induced by said pulse, said angle being different from M radians, with integer M; and U.sub.n(T) is a rotation matrix expressing, also in spin-domain representation, the action of said pulse, of duration T, on nuclear spins of the n.sup.th voxel; B. for one or more voxels of said imaging region of interest, determining a voxel-dependent nuclear spin rotation axis, expressed by vector n.sub.T=(n.sub.xT, n.sub.yT, n.sub.zT=0) in a Cartesian frame having a z-axis parallel to said magnetization field B.sub.0, corresponding to said reference pulse; and C. for each of said refocusing pulses, other than said reference pulse, adjusting a plurality of control parameters of said pulse to maximize a performance criterion, called phase-constrained performance criterion, representative of a proximity between a rotation of nuclear spins induced by said pulse and a target operator, said phase-constrained performance criterion being summed or averaged over said voxels of an imaging region of interest; wherein, for each of said refocusing pulses other than said reference pulse, identified by an index i, said target operator is chosen such that said phase-constrained performance criterion takes a maximum value when the nuclear spins within each of said voxels undergo a rotation of a same angle .sub.i around a respective nuclear spin rotation axis n.sub.T determined at step B.

4. The method according to claim 3, wherein: said phase-constrained performance criterion is given by $\underset{n=1}{\overset{N}{.Math.}}{}_n,N1$ being the number of voxels, with: .sub.n|U.sub.F|U.sub.n(T)|.sup.2 where: $U_F=\left[\begin{array}{cc}{}_T& -{}_T^{*}\\ {}_T& {}_T^{*}\end{array}\right],\mathrm{with}\text{:}$ $\{\begin{array}{c}{}_T=\cos \frac{{}_i}{2}\\ {}_T-i\left(n_{\mathrm{xT}}+{\mathrm{in}}_{\mathrm{yT}}\right)\sin \frac{{}_i}{2}\end{array}$ is the target operator for each voxel and U.sub.n(T) is a rotation matrix expressing, also in spin-domain representation, the action of said pulse or pulse sequence on nuclear spins of the n.sup.th voxel.

5. The method according to claim 4, wherein $U_n\left(T\right)=\left[\begin{array}{cc}& -{}^{*}\\ & {}^{*}\end{array}\right]$ is a solution of the spin-domain Bloch equation: $\left[\begin{array}{c}\stackrel{.}{}\\ \stackrel{.}{}\end{array}\right]=\frac{i}{2}\left[\begin{array}{cc}G.Math.r+B_0& B_1^{*}\\ B_1& -\left(G.Math.r+B_0\right)\end{array}\right]\left[\begin{array}{c}\\ \end{array}\right]$ wherein: i is the imaginary unit; is a nuclear spin gyromagnetic ratio G is a magnetic field gradient; r is a position vector of the n-th voxel; B.sub.0 expresses the non-uniformity of the magnetization field B.sub.0; and B.sub.1 expresses the spatial distribution of the pulse.

6. The method according to claim 3, wherein when carried out step C., said parameters are adjusted iteratively by taking, as initial values, those of corresponding parameters of an adjacent pulse within the pulse train for which said performance criterion has previously been maximized.

7. A method of designing a Magnetic Resonance Imaging pulse sequence comprising: i. designing a refocusing pulse train by: A. using a computer for designing one said refocusing pulse, called reference pulse, by adjusting a plurality of control parameters of said pulse to maximize a performance criterion , called phase-free performance criterion, given by: $\underset{n=1}{\overset{N}{.Math.}}{}_n$ N1 being the number of voxels, with:
.sub.n.sub.1,n+.sub.2,n wherein: ${}_{1,n}=.Math.\frac{.Math.U_{F1}|U_n\left(T\right).Math.}{2}.Math.,\mathrm{and}{}_{2,n}=.Math.\frac{.Math.U_{F2}|U_n\left(T\right).Math.}{2}.Math.;$ $\begin{array}{c}{U}_{F1}=\left[\begin{array}{cc}\cos \frac{}{2}& 0\\ 0& \cos \frac{}{2}\end{array}\right]\mathrm{and}\\ U_{F2}=\left[\begin{array}{cc}0& 2\sin \frac{}{2}\\ 0& 0\end{array}\right]\end{array}$ are two target operators in spin-domain representation; is a target value of the rotation angle of said nuclear spins induced by said pulse, said angle being different from M radians, with integer M; and U.sub.n(T) is a rotation matrix expressing, also in spin-domain representation, the action of said pulse, of duration T, on nuclear spins of the n.sup.th voxel; B. for one or more voxels of said imaging region of interest, determining a voxel-dependent nuclear spin rotation axis, expressed by vector n.sub.T=(n.sub.xT, n.sub.yT, n.sub.zT=0) in a Cartesian frame having a z-axis parallel to said magnetization field B.sub.0, corresponding to said reference pulse; C. for each of said refocusing pulses, other than said reference pulse, adjusting a plurality of control parameters of said pulse to maximize a performance criterion, called phase-constrained performance criterion, representative of a proximity between a rotation of nuclear spins induced by said pulse and a target operator, said phase-constrained performance criterion being summed or averaged over said voxels of an imaging region of interest; wherein, for each of said refocusing pulses other than said reference pulse, identified by an index i, said target operator is chosen such that said phase-constrained performance criterion takes a maximum value when the nuclear spins within each of said voxels undergo a rotation of a same angle .sub.i around a respective nuclear spin rotation axis n.sub.T determined at step B; and ii. designing a nuclear spin excitation pulse, to be generated before said refocusing pulse or pulse train, by adjusting a plurality of control parameters of said pulse to maximize a performance criterion, called phase-constrained performance criterion, representative of a proximity between a rotation of nuclear spins induced by said excitation pulse and a target operator, said phase-constrained performance criterion being summed or averaged over said voxels of an imaging region of interest; wherein said target operator is chosen such that said phase-constrained performance criterion takes a maximum value when the nuclear spins within each of said voxels undergo a rotation of a same flipping angle FA around a respective nuclear spin rotation axis lying in said transverse plain and orthogonal to the nuclear spin rotation axis corresponding to said refocusing pulse train.

8. The method of designing a Magnetic Resonance Imaging pulse sequence according to claim 7, further comprising: iii. designing a nuclear spin inversion pulse, to be generated before said nuclear spin excitation pulse, by adjusting a plurality of control parameters of said pulse to maximize a phase-free performance criterion, representative of a proximity between a rotation of nuclear spins induced by said nuclear spin inversion pulse and a target operator, said operator being summed or averaged over one or more voxels of an imaging region of interest; and iv. wherein said target operator is chosen such that said phase-free performance criterion takes a maximum value when the nuclear spins within all said voxels undergo a rotation of a same angle of radians around a rotation axis lying in said transverse plane, with an arbitrary orientation.

9. The method according to claim 1, wherein said step of adjusting a plurality of control parameters of said or each said pulse to maximize a said performance criterion is performed using a gradient ascent algorithm with analytically-computed gradients.

10. The method according to claim 1, wherein at least some of said control parameters represent samples of a complex envelope of said or one said pulse.

11. A method of generating a Magnetic Resonance Imaging pulse, pulse train, or pulse sequence comprising: a step of designing said pulse, pulse train, or pulse sequence by using a computer for designing one said refocusing pulse, called reference pulse, by adjusting a plurality of control parameters of said pulse to maximize a performance criterion , called phase-free performance criterion, given by: $\underset{n=1}{\overset{N}{.Math.}}{}_n$ N1 being the number of voxels, with:
.sub.n.sub.1,n+.sub.2,n wherein: ${}_{1,n}=.Math.\frac{.Math.U_{F1}|U_n\left(T\right).Math.}{2}.Math.,\mathrm{and}{}_{2,n}=.Math.\frac{.Math.U_{F2}|U_n\left(T\right).Math.}{2}.Math.;$ $\begin{array}{c}{U}_{F1}=\left[\begin{array}{cc}\cos \frac{}{2}& 0\\ 0& \cos \frac{}{2}\end{array}\right]\mathrm{and}\\ U_{F2}=\left[\begin{array}{cc}0& 2\sin \frac{}{2}\\ 0& 0\end{array}\right]\end{array}$ are two target operators in spin-domain representation; is a target value of the rotation angle of said nuclear spins induced by said pulse, said angle being different from M radians, with integer M; and U.sub.n(T) is a rotation matrix expressing, also in spin-domain representation, the action of said pulse, of duration T, on nuclear spins of the n.sup.th voxel; and a step of actually generating said pulse, pulse train, or pulse sequence.

12. A method of performing Magnetic Resonance Imaging comprising a step of generating a pulse, pulse train or pulse sequence using: a step of designing said pulse, pulse train, or pulse sequence by using a computer for designing one said refocusing pulse, called reference pulse, by adjusting a plurality of control parameters of said pulse to maximize a performance criterion , called phase-free performance criterion, given by: $\underset{n=1}{\overset{N}{.Math.}}{}_n$ N1 being the number of voxels, with:
.sub.n.sub.1,n+.sub.2,n wherein: ${}_{1,n}=.Math.\frac{.Math.U_{F1}|U_n\left(T\right).Math.}{2}.Math.,\mathrm{and}{}_{2,n}=.Math.\frac{.Math.U_{F2}|U_n\left(T\right).Math.}{2}.Math.;$ $\begin{array}{c}{U}_{F1}=\left[\begin{array}{cc}\cos \frac{}{2}& 0\\ 0& \cos \frac{}{2}\end{array}\right]\mathrm{and}\\ U_{F2}=\left[\begin{array}{cc}0& 2\sin \frac{}{2}\\ 0& 0\end{array}\right]\end{array}$ are two target operators in spin-domain representation; is a target value of the rotation angle of said nuclear spins induced by said pulse, said angle being different from M radians, with integer M; and U.sub.n(T) is a rotation matrix expressing, also in spin-domain representation, the action of said pulse, of duration T, on nuclear spins of the n.sup.th voxel; and a step of actually generating said pulse, pulse train, or pulse sequence.

13. A non-transitory computer readable medium comprising instructions that, when executed on a computer, cause the computer to automatically perform a method of designing a refocusing pulse for Magnetic Resonance Imaging, said pulse being suitable to induce a rotation of nuclear spins around a rotation axis lying in a plane perpendicular to a magnetization field B.sub.0, called a transverse plane, with an arbitrary orientation, the method comprising adjusting a plurality of control parameters of said pulse to maximize a performance criterion , called phase-free performance criterion, given by: $\underset{n=1}{\overset{N}{.Math.}}{}_n$ N1 being the number of voxels, with:
.sub.n.sub.1,n+.sub.2,n wherein: ${}_{1,n}=.Math.\frac{.Math.U_{F1}|U_n\left(T\right).Math.}{2}.Math.,\mathrm{and}{}_{2,n}=.Math.\frac{.Math.U_{F2}|U_n\left(T\right).Math.}{2}.Math.;$ $\begin{array}{c}{U}_{F1}=\left[\begin{array}{cc}\cos \frac{}{2}& 0\\ 0& \cos \frac{}{2}\end{array}\right]\mathrm{and}\\ U_{F2}=\left[\begin{array}{cc}0& 2\sin \frac{}{2}\\ 0& 0\end{array}\right]\end{array}$ are two target operators in spin-domain representation; is a target value of the rotation angle of said nuclear spins induced by said pulse, said angle being different from M radians, with integer M; and U.sub.n(T) is a rotation matrix expressing, also in spin-domain representation, the action of said pulse, of duration T, on nuclear spins of the n.sup.th voxel.

14. A Magnetic Resonance Imaging apparatus comprising: at least a magnet for generating a magnetization field; a set of gradient coils for generating magnetic field gradients; at least one radio-frequency coil for transmitting and receiving radio-frequency pulses; at least a radio-frequency emitter and a radio-frequency receiver connected to said radio-frequency coil or coils; and a data processor configured or programmed for designing a Magnetic Resonance Imaging pulse, pulse train, or pulse sequence and for driving at least said radio-frequency emitter or emitters to actually generate said pulse, pulse train, or pulse sequence, wherein the data processor is configured or programmed for designing said Magnetic Resonance Imaging Pulse, pulse train, or pulse sequence by adjusting a plurality of control parameters of said pulse to maximize a performance criterion , called phase-free performance criterion, given by: $\underset{n=1}{\overset{N}{.Math.}}{}_n$ N1 being the number of voxels, with:
.sub.n.sub.1,n+.sub.2,n wherein: ${}_{1,n}=.Math.\frac{.Math.U_{F1}|U_n\left(T\right).Math.}{2}.Math.,\mathrm{and}{}_{2,n}=.Math.\frac{.Math.U_{F2}|U_n\left(T\right).Math.}{2}.Math.;$ $\begin{array}{c}{U}_{F1}=\left[\begin{array}{cc}\cos \frac{}{2}& 0\\ 0& \cos \frac{}{2}\end{array}\right]\mathrm{and}\\ U_{F2}=\left[\begin{array}{cc}0& 2\sin \frac{}{2}\\ 0& 0\end{array}\right]\end{array}$ are two target operators in spin-domain representation; is a target value of the rotation angle of said nuclear spins induced by said pulse, said angle being different from M radians, with integer M; and U.sub.n(T) is a rotation matrix expressing, also in spin-domain representation, the action of said pulse, of duration T, on nuclear spins of the n.sup.th voxel.

15. The method according to claim 1, wherein < radians.

16. The method according to claim 1, wherein <0.9 radians.

17. The method according to claim 3, wherein < radians.

18. The method according to claim 3, wherein <0.9 radians.

19. The method according to claim 7, wherein < radians.

20. The method according to claim 7, wherein <0.9 radians.

21. The method according to claim 11, wherein < radians.

22. The method according to claim 11, wherein <0.9 radians.

23. The non-transitory computer readable medium according to claim 13, wherein < radians.

24. The non-transitory computer readable medium according to claim 13, wherein <0.9 radians.

25. The Magnetic Resonance Imaging apparatus according to claim 14, wherein < radians.

26. The Magnetic Resonance Imaging apparatus according to claim 14, wherein <0.9 radians.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) Additional features and advantages of the present invention will become apparent from the subsequent description, taken in conjunction with the accompanying drawings, which show:

(2) FIG. 1, the flow-chart of a method according to a particular embodiment of the invention;

(3) FIGS. 2A-2E, the implementation of a method according to another particular embodiment of the invention;

(4) FIG. 3, T.sub.2-weighted MRI images obtained using prior art techniques and a method according to the embodiment of the invention whose implementation is illustrated on FIGS. 2A-2E; and

(5) FIG. 4, a MRI apparatus (scanner) suitable for carrying out a method according to the invention.

DETAILED DESCRIPTION

(6) The GRAPE method applied to the design of a refocusing pulse will now be described using the SU(2) group formalism. If relaxation effects are neglected, the Bloch dynamics of the magnetization is simply expressed by a 22 unitary matrix, the so-called spin-domain representation. In this domain, a rotation of the nuclear spins within the n.sup.th voxel of a MRI region of interest by an angle about a vector n (n.sub.y, n.sub.y, n.sub.z) can be described by the complex-valued Cayley-Klein parameters (, ) [20]

(7) $\begin{array}{cc}{U}_n\left(T\right)=\left[\begin{array}{cc}& -{}^{*}\\ & {}^{*}\end{array}\right]& \left(1\right)\end{array}$
with ||.sup.2+||.sup.2=1 and, in the specific case of a vector n lying in the transverse plane (xy planeB.sub.0)

(8) $\begin{array}{cc}\{\begin{array}{c}=\cos \frac{}{2}-{\mathrm{in}}_z\sin \frac{}{2}\\ =-i\left(n_x+{\mathrm{in}}_y\right)\sin \frac{}{2}\end{array}& \left(2\right)\end{array}$

(9) For a given RF pulse B.sub.1(r, t), gradient waveform G(t) and static magnetization field offset of B.sub.0, the and parameters representing the rotation they induce at a spatial location r are obtained by solving the spin-domain Bloch equation:

(10) $\begin{array}{cc}\left[\begin{array}{c}\stackrel{.}{}\\ \stackrel{.}{}\end{array}\right]=\frac{i_{}}{2}\left[\begin{array}{cc}G.Math.r+B_0& B_1^{*}\\ B_1& -\left(G.Math.r+B_0\right)\end{array}\right]\left[\begin{array}{c}\\ \end{array}\right]& \left(3a\right)\end{array}$

(11) Equation 3a can be solved exactly if the pulse is rectangular or, more generally, is decomposed into a sequence of elementary rectangular (constant amplitude and phase) sub-pulses B.sub.1,j of duration t, each inducing a spin rotation of an angle .sub.j about an axis n.sub.j:

(12) $\begin{array}{cc}{}_j=-t\sqrt{{.Math.B_{1,j}.Math.}^2+{\left(G.Math.r+B_0\right)}^2}& \left(3b\right)\\ n_j=\frac{t}{{}_j}\left(B_{1,j}x,B_{1,j}y,G.Math.r+B_0\right)& \left(3c\right)\end{array}$
(in equation (3c), B.sub.1,jx and B.sub.1,jy are the components of B.sub.1,j along the axis x and y, B.sub.1,jz being supposed to be zero). Explicit expressions for and are straightforwardly obtained by combining equations (2), (3b) and (3c).

(13) When an array of m transmit coils is used, the total effective B.sub.1 field is a function of both space and time. S.sup.R and S.sup.I are the real and imaginary parts of the transmit sensitivities, while u.sub.k and v.sub.k are the control parameters, which represent real and imaginary parts of the RF shape of the k.sup.th transmitter:

(14) $\begin{array}{cc}{B}_1\left(r,t\right)=\underset{k=1}{\overset{m}{.Math.}}\left[S_k^R\left(r\right)+{\mathrm{iS}}_k^I\left(r\right)\right]\left[u_k\left(t\right)+{\mathrm{iv}}_k\left(t\right)\right]& \left(4\right)\end{array}$

(15) The problem to be solved consists in determining the control parameters u.sub.k and v.sub.k (and, optionally, the gradients G) to approximate as closely as possible a target transformation.

(16) In the case of refocusing, the target is a rotation matrix of a given angle , with a purely transverse axis, whose in-plane direction (phase) is left free. Once a proper target is established, a metric to optimize its distance with the candidate pulse needs to be found. For a general value of , two virtual target rotation matrices decomposing the desired operation can be used:

(17) $\begin{array}{cc}{U}_{F1}=\left[\begin{array}{cc}\cos \frac{}{2}& 0\\ 0& \cos \frac{}{2}\end{array}\right]\mathrm{and}& \left(5a\right)\\ U_{F2}=\left[\begin{array}{cc}0& 2\sin \frac{}{2}\\ 0& 0\end{array}\right]& \left(5b\right)\end{array}$

(18) To tailor the RF pulse rotation matrix in the n.sup.th voxel, the proposed performance criterion .sub.n (which can be called phase-free performance criterion, as the phase of the rotation is left free) is:

(19) $\begin{array}{cc}{}_n={}_{1,n}+{}_{2,n}=.Math..Math.U_{F1}.Math.U_n\left(T\right).Math./2.Math.+.Math..Math.U_{F2}.Math.U_n\left(T\right).Math./2.Math.& \left(6\right)\end{array}$

(20) Indeed, rewriting this expression knowing that U.sub.F1|U.sub.n(T)=Tr(U.sub.F1.sup.U.sub.n(T)) gives:

(21) $\begin{array}{cc}{}_n=.Math.\cos \frac{}{2}\cos \frac{}{2}.Math.+\sqrt{n_x^2+n_y^2}.Math.\sin \frac{}{2}\sin \frac{}{2}.Math..& \left(7\right)\end{array}$

(22) Evaluation of .sub.n shows that this criterion is equal to one, its maximum value, if and only if:

(23) $\begin{array}{cc}\{\begin{array}{c}=\\ \sqrt{n_x^2+n_y^2}=1\end{array}\mathrm{or}\{\begin{array}{c}=2k-\\ \sqrt{n_x^2+n_y^2}=1\end{array}& \left(8\right)\end{array}$ where k is an integer.

(24) In practice, the second case is not encountered, as the required energy would be considerably higher for the same pulse duration. After summing this performance criterion for all voxels

(25) 0$\left(=\frac{1}{N}\underset{n=1}{\overset{N}{.Math.}}{}_n\right),$
a cost function equal to 1 is minimal, if and only if the rotation angle is and the rotation axis is purely transverse everywhere in the ROI. It is important to note that U.sub.F1 and U.sub.F2 do not correspond to physical rotation matrices, as they are not unitary. Their use is simply a mathematical convenience that removes the phase constraint on the transverse rotation axis. After time discretization, the derivatives of this performance function with respect to all control parameters u.sub.k, v.sub.k (which are the real and imaginary parts of the RF pulse on each coil channel) are taken to compute its gradient, knowing that for each voxel:

(26) $\begin{array}{cc}\frac{{}_{1,2}}{u_k}=\frac{1}{2}\frac{1}{{}_{1,2}}\frac{{}_{1,2}^2}{u_k}& \left(9a\right)\\ \frac{{}_{1,2}}{v_k}=\frac{1}{2}\frac{1}{{}_{1,2}}\frac{{}_{1,2}^2}{v_k}& \left(9b\right)\end{array}$

(27) In this way, the original analytical formulation of the derivative proper to the GRAPE algorithm could be retrieved, acknowledging the fact that this function is not strictly speaking differentiable at .sub.1,2=0 (see references [18-19] for the whole expression of the derivative with 22 rotations matrices). Control parameters are then updated using the conjugate gradient method. As in [18], gradient blips intensities can also be considered as additional degrees of freedom and optimized with the RF in order to modulate the k-space trajectory.

(28) In the case of the SPACE sequence, the formalism described above is used to tailor a reference refocusing pulse of the RF echo train. This pulse then dictates a phase pattern (i.e. a spatial distribution of the orientation of the rotation axis in the transverse plane) for the whole sequence. For all subsequent refocusing pulses, as well as the initial excitation pulse, care has to be taken as the CPMG condition fulfillment is important to obtain the desired refocusing of magnetization. Indeed, as several types of echoes (primary echo, stimulated echo, etc.) arise from multiple refocusing pathways, it is required that simultaneously acquired echoes are kept phase-coherent. This can be obtained by targeting a specific phase pattern, precisely the one of the reference refocusing pulse, and change only the rotation angle. Hence, the target rotation matrix is:

(29) $\begin{array}{cc}{U}_F=\left[\begin{array}{cc}{}_T& -{}_T^{*}\\ {}_T& {}_T^{*}\end{array}\right],\mathrm{with}\text{:}\{\begin{array}{c}{}_T=\cos \frac{}{2}\\ {}_T=-i\left(n_{\mathrm{xT}}+{\mathrm{in}}_{\mathrm{yT}}\right)\sin \frac{}{2}\end{array}& \left(11\right)\end{array}$
where is the new desired target rotation angle, n.sub.xT and n.sub.yT the normalized components of the first refocusing pulse axis of rotation (voxel-dependent). The phase-constrained performance criterion, called is then:

(30) $\begin{array}{cc}=\frac{1}{N}\underset{n=1}{\overset{N}{.Math.}}{}_n=\frac{1}{N}\underset{n=1}{\overset{N}{.Math.}}{.Math..Math.U_F.Math.U_n\left(T\right).Math..Math.}^2.& \left(12\right)\end{array}$

(31) The optimization procedure and the updating of the control parameters then is the same as above, the only difference being smaller latitude in the optimization, as the target operator is more specific. Last, thanks to the smooth variation of the angle values along the RF train, a previously designed pulse can be used as an initial guess for the next one, thus greatly speeding up the algorithm convergence.

(32) A flowchart of the whole procedure in order to tailor every single RF pulse of a SPACE-like sequence, and more precisely a SPACE/FLAIR sequence, is provided in FIG. 1.

(33) Step 1: The initial guess waveform IGP is the solution returned by the small tip angle approximation, using the k.sub.T-points method [12]. It is a composite pulse comprising e.g. three rectangular sub-pulses and gradient blips (not represented).

(34) Step 2: The reference refocusing pulse RRP is designed with a phase-free rotation axis. The reference pulse is not necessarily the first pulse of the train; more advantageously, it is chosen such that the corresponding rotation angle has an intermediate value (it is neither the largest nor the smallest rotation angle, and preferably is approximately halfway between them). It can be seen on the figure that the reference refocusing pulse is a composite pulse comprising the same number of sub-pulses than the initial guess waveform; but now the sub-pulses are shaped. Again, gradient blips are not represented.

(35) Step 3: The phase pattern PHP of the rotation axis is stored and used as a target for the design of all subsequent refocusing pulses (step 4).

(36) Step 4: The other refocusing pulses of the refocusing pulse train RPT are designed, with the phase pattern determined at step 3. Pulses are designed one by one, taking an adjacent, already designed pulse as an initial guess. For the sake of simplicity, pulses are represented as rectangular waveforms, even if in reality they are composite and, after optimization, shaped.

(37) Step 5: The excitation pulse EXP is designed in a similar way, targeting 90 and the same phase pattern dephased by +/2 to fulfill CPMG condition.

(38) Step 6: The GRAPE algorithm is used to design an additional inversion pulse IP if needed (e.g. for suppressing the signal coming from the Cerebrospinal fluid, according to a method known as FLAIR).

(39) Experimental validation was performed on a 7 Tesla Magnetom scanner (Siemens, Erlangen, Germany), equipped with parallel transmission capabilities and an AC84 head gradient system allowing amplitudes up to of 50 mT/m and a slew rate of 333 T/m/s, on four different healthy volunteers. The study was approved by the applicant's review board and informed consent was obtained from all volunteers. For both RF transmission and reception, a home-made transceiver-array head coil was used [21]. The array consists of 8 stripline dipoles distributed every 42.5 on a cylindrical surface of 27.6-cm diameter, leaving a small open space in front of the subject's eyes.

(40) The variable flip angle series of the SPACE sequence that yields prescribed signal evolution was calculated for specified T.sub.1=1400 ms and T.sub.2=40 ms relaxation times (min. angle: 10, max angle: 100). Conventional square pulse durations were set to 600 s and 900 s for the CP-mode and RF-shimming respectively. An initial candidate waveform fed to the GRAPE algorithm (i.e. the inventive method) consisted of solving the Magnitude Least Squares problem [22] with a 3 k.sub.T-point self-refocused trajectory (i.e. k(0)=0) surrounding the center of k-space. The location of the k.sub.T-points was determined empirically off-line for an initial case study [12] and was kept the same for all the subjects, keeping in mind that these locations were then free to move thanks to the GRAPE algorithm. The reference refocusing pulse design was achieved after about 100 iterations of the first adaptation of the GRAPE algorithm presented in the theory section. For the other pulses, a quasi-linear scaling of their duration with respect to the prescribed angle was followed by 2 to 10 iterations of the GRAPE algorithm presented above. With this setup, about 50 different pulses needed to be designed, since several similar pulses are used in the echo train. The peak amplitudes of the designed waveforms were constrained to the maximum voltage available per channel (180 V). Replacing the original hard pulses of the SPACE sequence with sets of sub-pulses and gradients blips inevitably increases their durations and Specific Absorption Rate (SAR) contributions. This inherently affects the TR of the sequence, the echo spacing ES and the shape of the RF echo train. The following sequence protocol was implemented: TR: 6 s, ES: 9 ms, effective TE: 315 ms, Echo Train Length: 96 pulses, resolution: 1 mm isotropic, matrix size: 256224160, GRAPPA factor: 2, Partial Fourier: 6/8, TA: 12 min.

(41) FIG. 2A shows the measured transmit sensitivity magnitudes of the eight transmit elements (central axial slices). FIG. 2B is a B.sub.0 map measured in Hz, central sagittal slice. FIG. 2C defines the region of interest, extracted with conventional brain volume extraction software (central slices). FIG. 2D illustrates the flip-angle train designed for the SPACE sequence used in the present study. FIG. 2E shows an example of a tailored RF pulse on one of the Tx-channels (amplitude, phase and gradient blips).

(42) T.sub.2-weighted SPACE images for three configurations (CP-mode, RF-shim configuration and GRAPE design) are displayed on FIG. 3 for one of the subjects.

(43) The first line of the figure, labeled a, corresponds to a conventional CP-mode method with hard pulses, implemented at 7 Tesla resulting in strong signal voids in the cerebellum and in temporal lobes.

(44) The second line of the figure, labeled b, corresponds to the application of a subject-specific static RF-shim with hard pulses. Image quality is improved to some extent, thus allowing cerebellar GM/WM to be distinguished. Even so, residual B.sub.1.sup.+ non-uniformities introduce significant signal variations in other regions of the brain, complicating distinction between GM and WM in these regions with confidence.

(45) The third line of the figure, labeled c, corresponds to a method according to an embodiment of the invention, using GRAPE-tailored k.sub.T-points. Use of this method improves every refocusing profile and leads to higher signal homogeneity for a given tissue across the brain. Considerable improvements occur in the cerebellum, in the upper brain region and in the temporal lobes. In addition, intricate structures of the cortex can now be resolved in greater detail. This improvement comes at the cost of less than 10 preliminary minutes of subject-specific data acquisition and pulse design (4 minutes for B.sub.1 mapping, 20 sec. for B.sub.0 mapping, 1 min. for brain masking and about three minutes for GRAPE pulse designtimes are indicated on FIGS. 2A-2E).

(46) FIG. 4 representsvery schematicallya Magnetic Resonance Imaging (MRI) scanner comprising:

(47) a magnet M for generating magnetization field B.sub.0, which is a static and substantially uniform magnetic field oriented along direction z (longitudinal), in which is immersed a patient body (or a part of it) PB to be imaged;

(48) a radio-frequency coil RFC for exposing said body to transverse radio-frequency pulses (as discussed above, transverse here means having a polarization which is perpendicular to B.sub.0, and therefore lying in a x-y plane) and for detecting signal emitted by flipped nuclear spins within said body; in parallel transmission, there are as many radio-frequency coils as there are transmit and/or receive channels;

(49) three sets of gradient coils GC for generating gradient fields, i.e. magnetic fields directed along the z direction which vary linearly along a respective spatial direction across the region of interest (ROI);

(50) a transmitter Tx for generating radio-frequency pulses, which feed the radio-frequency coil (in parallel transmission, there are as many transmitters as there are transmit channels);

(51) a receiver R for amplifying said spin resonance signal before demodulating and digitizing it (in parallel imaging, the receivers are duplicated as many times as there are receive channels) and

(52) a data processor DP driving the transmitter(s) and the gradient coils, and processing the signal received by the receiver(s).

(53) The data processor DP can be a computeror a set of electronic programmable computerscomprising at least a memory for storing a computer program (i.e. a piece of executable code) and a least one processor for executing said program. The hardware part of the scanner can be conventional, while the software is adapted for carrying out the method of the invention. Therefore software meanse.g. code stored on a computer-readable storage medium such as a CD-ROMcan turn a standard scanner into a device according to the invention, without any need for hardware modifications.

(54) The invention can be applied to any MRI sequence based on the use of a refocusing pulse (rotating nuclear spins by a fixed angle, different from 180), or a train of interacting refocusing pulse pulses (e.g. the SPACE method, discussed above). This sequence can be either 3D (e.g. SPACE) or 2D. The sequence can be T.sub.2- or T.sub.1-weighted, or used a mixed weighting (T.sub.2-weighted SPACE has been discussed, but T.sub.1-weighting is also possible).

REFERENCES

(55) [1] de Graaf W L, Zwanenburg J J, Visser F, Wattjes M P, Pouwels P J, Geurts J J, Polman C H, Barkhof F, Luijten P R, Castelijns J A. Lesion detection at seven Tesla in multiple sclerosis using magnetization prepared 3D-FLAIR and 3D-DIR. Eur Radiol 2012; 22:221-231. [2] van der Kolk A G, Zwanenburg J J, Brundel M, Biessels G J, Visser F, Luijten P R, Hendrikse J. Intracranial vessel wall imaging at 7.0-T MRI. Stroke 2011; 42:2478-2484. [3] Hennig J, Nauerth A, Friedburg H. RARE imaging: a fast imaging method for clinical M R. Magn Reson Med 1986; 3:823-833. [4] Hennig J, Scheffler K. Hyperechoes. Magn Reson Med 2001; 46:6-12. [5] Busse R F, Hariharan H, Vu A, Brittain J H. Fast Spin Echo sequences with very long echo trains: Design of variable refocusing flip angle schedules and generation of clinical T2 contrast. Magn Reson Med 2006; 55:1030-1037. [6] Mugler J P. Three-dimensional T2-weighted imaging of the brain using very long spin-echo trains. In Proceedings of the 8th Annual Meeting of ISMRM, Denver, Colo., USA, 2000. p. 687. [7] Visser F, Zwanenburg J J, Hoogduin J M, Luijten P R. High-resolution magnetization-prepared 3D-FLAIR imaging at 7.0 Tesla. Magn Reson Med 2010; 64:194-202. [8] U. Katscher, P. Bornert, C. Leussler, J. S. Van Den Brink, Transmit SENSE, Magn. Reson. Med. 49 (2003) 144-150. [9] Y. Zhu, Parallel excitation with an array of transmit coils, Magn. Reson. Med. 51 (2004) 775-784. [10] P. F. Van de Moortele, C. Akgun, G. Adriany, S. Moeller, J. Ritter, C. M. Collins, M. B. Smith, T. Vaughan, K. Ugurbil, B1 destructive interferences and spatialphase patterns at 7 T with a head transceiver array coil, Magn. Reson. Med. 54 (2005) 1503-1518. [11] Malik S J, Padormo F, Price A N, Hajnal J V. Spatially resolved extended phase graphs: modeling and design of multipulse sequences with parallel transmission. Magn Reson Med 2012; 68:1481-1494. [12] M. A. Cloos, N. Boulant, M. Luong, G. Ferrand, E. Giacomini, D. Le Bihan, A. Amadon, kT-Points: short three-dimensional tailored RF pulses for flip-angle homogenization over an extended volume, Magn. Reson. Med. 67 (2012) 72-80. [13] D. Xu, K. F. King, Y. Zhu, G. C. McKinnon, Z. P. Liang, Designing multichannel multidimensional, arbitrary flip angle RF pulses using an optimal control approach, Magn. Reson. Med. 59 (2008) 547-560. [14] M. A. Cloos, N. Boulant, M. Luong, G. Ferrand, E. Giacomini, M.-F. Hang, C. J. Wiggins, D. Le Bihan, A. Amadon, Parallel-transmission-enabled magnetization-prepared rapid gradient-echo T1-weighted imaging of the human brain at 7 T, Neuroimage 62 (2012) 2140-2150. [15] J. Pauly, D. Nishimura, A. Macovski, A linear class of large-tip-angle selective excitation pulses, J. Magn. Reson. 82 (1989) 571-587. [16] Setsompop K, Alagappan V, Zelinski A C, Potthast A, Fontius U, Hebrank F, Schmitt F, Wald L L, Adalsteinsson E. High-flip-angle slice-selective parallel RF transmission with 8 channels at 7 T. J Magn Reson 2008; 195:76-84. [17] Eggenschwiler F, O'Brien K R, Gruetter R, Marques J P. Improving T2-weighted imaging at High Field through the use of kT-points. Magn Reson Med 2014. doi: 10.1002/mrm.24805. [18] Khaneja N, Reiss T, Kehlet C, Schulte-Herbrggen T, Glaser S J. Optimal control of coupled spin dynamics: design of NMR pulse sequences by gradient ascent algorithms. J Magn Reson 2005; 172: 296-305. [19] Massire A, Cloos M A, Vignaud A, Le Bihan D, Amadon A, Boulant N. Design of non-selective refocusing pulses with phase-free rotation axis by gradient ascent pulse engineering algorithm in parallel transmission at 7 T. J Magn Reson 2013; 230:76-83. [20] Pauly J, Le Roux P, Nishimura D, Macovski A. Parameter relations for the Shinnar-Le Roux selective excitation pulse design algorithm. IEEE Trans. Med. Imaging 1991; 10:53-64. [21] Ferrand G, Luong M, France A. Rsonateur Linaire d'une Antenne Haute Frquence pour Appareil d'lmagerie par Rsonance Magntique Nuclaire, WO/2011/098713, PCT/FR2011/098713. [22] Setsompop K, Wald L L, Alagappan V, Gagoski B A, Adalsteinsson E. Magnitude least squares optimization for parallel radio frequency excitation design demonstrated at 7 T with eight channels. Magn Reson Med 2008; 59:908-915.