December  2013, 3(4): 375-396. doi: 10.3934/mcrf.2013.3.375

On the application of geometric optimal control theory to Nuclear Magnetic Resonance

1. 

Laboratoire Interdisciplinaire Carnot de Bourgogne (ICB), UMR 5209 CNRS-Université de Bourgogne, 9 Av. A. Savary, BP 47 870, F-21078 DIJON Cedex, France, France, France

2. 

Department of Chemistry, Technische Universität München, Lichtenbergstrasse 4, D-85747 Garching, Germany

Received  October 2012 Revised  April 2013 Published  September 2013

We present some applications of geometric optimal control theory to control problems in Nuclear Magnetic Resonance (NMR) and Magnetic Resonance Imaging (MRI). Using the Pontryagin Maximum Principle (PMP), the optimal trajectories are found as solutions of a pseudo-Hamiltonian system. This computation can be completed by second-order optimality conditions based on the concept of conjugate points. After a brief physical introduction to NMR, this approach is applied to analyze two relevant optimal control issues in NMR and MRI: the control of a spin 1/2 particle in presence of radiation damping effect and the maximization of the contrast in MRI. The theoretical analysis is completed by numerical computations. This work has been made possible by the central and essential role of B. Bonnard, who has been at the heart of this project since 2009.
Citation: Elie Assémat, Marc Lapert, Dominique Sugny, Steffen J. Glaser. On the application of geometric optimal control theory to Nuclear Magnetic Resonance. Mathematical Control & Related Fields, 2013, 3 (4) : 375-396. doi: 10.3934/mcrf.2013.3.375
References:
[1]

K. Beauchard, J.-M. Coron and P. Rouchon, Controllability issues for continuous-spectrum systems and ensemble controllability of Bloch equations,, Comm. Math. Phys., 296 (2010), 525. doi: 10.1007/s00220-010-1008-9. Google Scholar

[2]

M. A. Bernstein, K. F. King and X. J. Zhou, Handbook of MRI pulse sequences,, Medical Physics, 32 (2005). doi: 10.1118/1.1904597. Google Scholar

[3]

A. Bhattacharya, Chemistry: Breaking the billion-hertz barrier,, Nature, 463 (2010), 605. doi: 10.1038/463605a. Google Scholar

[4]

N. Bloembergen and R. V. Pound, Radiation damping in magnetic resonance experiments,, Phys. Rev., 95 (1954), 8. doi: 10.1103/PhysRev.95.8. Google Scholar

[5]

B. Bonnard, J.-B. Caillau and E. Trélat, Second order optimality conditions in the smooth case and applications in optimal control,, ESAIM COCV, 13 (2007), 207. doi: 10.1051/cocv:2007012. Google Scholar

[6]

B. Bonnard, J.-B. Caillau, R. Sinclair and M. Tanaka, Conjugate and cut loci of a two-sphere of revolution with application to optimal control,, Ann. Inst. H. Poincaré Anal. Non linéaire, 26 (2009), 1081. doi: 10.1016/j.anihpc.2008.03.010. Google Scholar

[7]

B. Bonnard and M. Chyba, "Singular Trajectories and their Role in Control Theory,", Mathématiques & Applications (Berlin) [Mathematics & Applications], 40 (2003). Google Scholar

[8]

B. Bonnard, M. Chyba and J. Marriott, Singular trajectories and the contrast imaging problem in nuclear magnetic resonance,, SIAM, 51 (2013), 1325. doi: 10.1137/110833427. Google Scholar

[9]

B. Bonnard, M. Chyba and D. Sugny, Time-minimal control of dissipative two-level quantum systems: The generic case,, IEEE Trans. Auto. Control, 54 (2009), 2598. doi: 10.1109/TAC.2009.2031212. Google Scholar

[10]

B. Bonnard, M. Claeys, O. Cots and P. Martinon, Comparison of numerical methods in the contrast imaging problem in NMR,, submitted to IEEE Trans. Auto. Control, (2013). Google Scholar

[11]

B. Bonnard, O. Cots, S. J. Glaser, M. Lapert, D. Sugny and Y. Zhang, Geometric optimal control of the contrast imaging problem in nuclear magnetic resonance,, IEEE Transactions on Automatic and Control, 57 (2012), 1957. doi: 10.1109/TAC.2012.2195859. Google Scholar

[12]

B. Bonnard, O. Cots, N. Shcherbakova and D. Sugny, The energy minimization problem for two-level dissipative quantum systems,, J. Math. Phys., 51 (2010). doi: 10.1063/1.3479390. Google Scholar

[13]

B. Bonnard and D. Sugny, Time minimal control of dissipative two-level quantum systems: The integrable case,, SIAM, 48 (2009), 1289. doi: 10.1137/080717043. Google Scholar

[14]

U. Boscain and P. Mason, Time minimal trajectories for a spin 1/2 particle in a magnetic field,, J. Math. Phys., 47 (2006). doi: 10.1063/1.2203236. Google Scholar

[15]

D. O. Brunner, N. De Zanche, J. Fröhlich, J. Paska and K. P. Pruessmann, Travelling-wave nuclear magnetic resonance,, Nature, 457 (2009), 994. doi: 10.1038/nature07752. Google Scholar

[16]

G. M. Bydder, J. V. Hajnal and I. R. Young, MRI: Use of the inversion recovery pulse sequence,, Clinical Radiology, 53 (1998), 159. doi: 10.1016/S0009-9260(98)80096-2. Google Scholar

[17]

G. M. Bydder and I. R. Yound, MR imaging: Clinical use of the inversion recovery sequence,, Journal of Computed Assisted Tomography, 9 (1985). doi: 10.1097/00004728-198507010-00002. Google Scholar

[18]

S. Conolly, D. Nishimura and A. Macovski, Optimal control solutions to the magnetic resonance selective excitation problem,, IEEE Trans. Med. Imaging, 5 (1986), 106. doi: 10.1109/TMI.1986.4307754. Google Scholar

[19]

O. Cots, "Contrôle Optimal Géométrique: Méthodes Homotopiques et Applications,", Ph.D thesis, (2012). Google Scholar

[20]

D. D'Alessandro and M. Dahleh, Optimal control of two-level quantum systems,, IEEE Trans. Auto. Control, 46 (2001), 866. doi: 10.1109/9.928587. Google Scholar

[21]

R. R. Ernst, G. Bodenhausen and A. Wokaun, "Principles of Nuclear Magnetic Resonance in One and Two Dimensions,", International Series of Monographs on Chemistry, (1990). doi: 10.1063/1.2811094. Google Scholar

[22]

L. Frydman and D. Blazina, Ultrafast two-dimensional nuclear magnetic resonance spectroscopy of hyperpolarized solutions,, Nature Physics, 3 (2007), 415. doi: 10.1038/nphys597. Google Scholar

[23]

N. I. Gershenzon, K. Kobzar, B. Luy, S. J. Glaser and T. E. Skinner, Optimal control design of excitation pulses that accomodate relaxation,, J. Magn. Reson., 188 (2007), 330. doi: 10.1016/j.jmr.2007.08.007. Google Scholar

[24]

V. Jurdjevic, "Geometric Control Theory,", Cambridge Studies in Advanced Mathematics, 52 (1997). Google Scholar

[25]

N. Khaneja, R. Brockett and S. J. Glaser, Time optimal control in spin systems,, Phys. Rev. A, 63 (2001). doi: 10.1103/PhysRevA.63.032308. Google Scholar

[26]

N. Khaneja, S. J. Glaser and R. Brockett, Sub-Riemannian geometry and time optimal control of three spin systems: Quantum gates and coherence transfer,, Phys. Rev. A (3), 65 (2002). doi: 10.1103/PhysRevA.65.032301. Google Scholar

[27]

N. Khaneja, F. Kramer and S. J. Glaser, Optimal experiments for maximizing coherence transfer between coupled spins,, J. Magn. Reson., 173 (2005), 116. doi: 10.1016/j.jmr.2004.11.023. Google Scholar

[28]

N. Khaneja, B. Luy and S. J. Glaser, Boundary of quantum evolution under decoherence,, Proc. Natl. Acad. Sci. USA, 100 (2003), 13162. doi: 10.1073/pnas.2134111100. Google Scholar

[29]

N. Khaneja, T. Reiss, C. Kehlet, T. Schulte-Herbrüggen and S. J. Glaser, Optimal control of coupled spin dynamics: Design of NMR pulse sequences by gradient ascent algorithms,, J. Magn. Reson., 172 (2005), 296. doi: 10.1016/j.jmr.2004.11.004. Google Scholar

[30]

N. Khaneja, T.Reiss, B. Luy and S. J. Glaser, Optimal control of spin dynamics in the presence of relaxation,, J. Magn. Reson., 162 (2003), 311. doi: 10.1016/S1090-7807(03)00003-X. Google Scholar

[31]

K. Kobzar, B. Luy, N. Khaneja and S. J. Glaser, Pattern pulses: Design of arbitrary excitation profiles as a function of pulse amplitude and offset,, J. Magn. Reson., 173 (2005), 229. doi: 10.1016/j.jmr.2004.12.005. Google Scholar

[32]

K. Kobzar, T. E. Skinner, N. Khaneja, S. J. Glaser and B. Luy, Exploring the limits of broadband excitation and inversion pulses,, J. Magn. Reson., 170 (2004), 236. doi: 10.1016/j.jmr.2004.06.017. Google Scholar

[33]

K. Kobzar, T. E. Skinner, N. Khaneja, S. J. Glaser and B. Luy, Exploring the limits of broadband excitation and inversion: II. Rf-power optimized pulses,, J. Magn. Reson., 194 (2008), 58. doi: 10.1016/j.jmr.2008.05.023. Google Scholar

[34]

M. Lapert, Y. Zhang, M. Braun, S. J. Glaser and D. Sugny, Singular extremals for the time-optimal control of dissipative spin 1/2 particles,, Phys. Rev. Lett., 104 (2010). doi: 10.1103/PhysRevLett.104.083001. Google Scholar

[35]

M. Lapert, Y. Zhang, M. Janich, S. J. Glaser and D. Sugny, Exploring the physical limits of contrast in magnetic resonance imaging,, Sci. Rep., 2 (2012). doi: 10.1038/srep00589. Google Scholar

[36]

M. H. Levitt, "Spin Dynamics: Basics of Nuclear Magnetic Resonance,", John Wiley & sons, (2008). doi: 10.1118/1.3273534. Google Scholar

[37]

J.-S. Li and N. Khaneja, Ensemble controllability of the Bloch equations,, in, (2006), 2483. Google Scholar

[38]

J. Mao, T. H. Mareci, K. N. Scott and E. R. Andrew, Selective inversion radiofrequency pulses by optimal control,, J. Magn. Reson., 70 (1986), 310. doi: 10.1016/0022-2364(86)90016-8. Google Scholar

[39]

N. C. Nielsen, C. Kehlet, S. J. Glaser and N. Khaneja, "Optimal Control Methods in NMR Spectroscopy,", Encyclopedia of Nuclear Magnetic Resonance, (2010). Google Scholar

[40]

G. Pileio, M. Carravetta and M. H. Levitt, Extremely low-frequency spectroscopy in low-field nuclear magnetic resonance,, Phys. Rev. Lett., 103 (2009). doi: 10.1103/PhysRevLett.103.083002. Google Scholar

[41]

L. Pontryagin, et al., "Mathematical Theory of Optimal Processes,", Mir, (1974). Google Scholar

[42]

T. O. Reiss, N. Khaneja and S. J. Glaser, Broadband geodesic pulses for three spin systems: Time-optimal realization of effective trilinear coupling terms and indirect SWAP gates,, J. Magn. Reson., 165 (2003), 95. doi: 10.1016/S1090-7807(03)00245-3. Google Scholar

[43]

S. Rice and M. Zhao, "Optimal Control of Molecular Dynamics,", Wiley, (2000). Google Scholar

[44]

D. Rosenfeld and Y. Zun, Design of adiabatic selective pulses using optimal control theory,, Magn. Reson. Med., 36 (1996), 401. doi: 10.1002/mrm.1910360311. Google Scholar

[45]

J. N. Rydberg, S. J. Riederer, C. H. Rydberg and C. R. Jack, Contrast optimization of fluid-attenuated inversion recovery (flair) imaging,, Magnetic Resonance in Medicine, 34 (1995), 868. doi: 10.1002/mrm.1910340612. Google Scholar

[46]

T. E. Skinner, T. O. Reiss, B. Luy, N. Khaneja and S. J. Glaser, Application of optimal control theory to the design of broadband excitation pulses for high resolution NMR,, J. Magn. Reson., 163 (2003), 8. doi: 10.1016/S1090-7807(03)00153-8. Google Scholar

[47]

D. Stefanatos, N. Khaneja and S. J. Glaser, Optimal control of coupled spins in presence of longitudinal and transverse relaxation,, Phys. Rev. A, 69 (2004). Google Scholar

[48]

D. Stefanatos, N. Khaneja and S. J. Glaser, Relaxation optimized transfer of spin order in ising chains,, Phys. Rev. A, 72 (2005). doi: 10.1103/PhysRevA.72.062320. Google Scholar

[49]

D. J. Tannor, "Introduction to Quantum Mechanics: A Time-Dependent Perspective,", University Science Books, (2007). Google Scholar

[50]

Z. Tosner, T. Vosegaard, C. Kehlet, N. Khaneja, S. J. Glaser and N. C. Nielsen, Optimal control in NMR spectroscopy: Numerical implementation in SIMPSON,, J. Magn. Reson., 197 (2009), 120. Google Scholar

[51]

L. M. K. Vandersypen and I. L. Chuang, NMR techniques for quantum control and computation,, Rev. Mod. Phys., 76 (2004), 1037. Google Scholar

[52]

M. S. Vinding, I. I. Maximov, Z. Tosner and N. C. Nielsen, Fast numerical design of spatil-selective rf pulses in MRI using Krotov and quasi-Newton based optimal control methods,, J. Chem. Phys., 137 (2012). doi: 10.1063/1.4739755. Google Scholar

[53]

W. S. Warren, S. L. Hammes and J. L. Bates, Dynamics of radiation damping in nuclear magnetic resonance,, J. Chem. Phys., 91 (1989). doi: 10.1063/1.457458. Google Scholar

[54]

Y. Zhang, M. Lapert, M. Braun, D. Sugny and S. J. Glaser, Time-optimal control of spin 1/2 particles in presence of relaxation and radiation damping effects,, J. Chem. Phys., 134 (2011). Google Scholar

show all references

References:
[1]

K. Beauchard, J.-M. Coron and P. Rouchon, Controllability issues for continuous-spectrum systems and ensemble controllability of Bloch equations,, Comm. Math. Phys., 296 (2010), 525. doi: 10.1007/s00220-010-1008-9. Google Scholar

[2]

M. A. Bernstein, K. F. King and X. J. Zhou, Handbook of MRI pulse sequences,, Medical Physics, 32 (2005). doi: 10.1118/1.1904597. Google Scholar

[3]

A. Bhattacharya, Chemistry: Breaking the billion-hertz barrier,, Nature, 463 (2010), 605. doi: 10.1038/463605a. Google Scholar

[4]

N. Bloembergen and R. V. Pound, Radiation damping in magnetic resonance experiments,, Phys. Rev., 95 (1954), 8. doi: 10.1103/PhysRev.95.8. Google Scholar

[5]

B. Bonnard, J.-B. Caillau and E. Trélat, Second order optimality conditions in the smooth case and applications in optimal control,, ESAIM COCV, 13 (2007), 207. doi: 10.1051/cocv:2007012. Google Scholar

[6]

B. Bonnard, J.-B. Caillau, R. Sinclair and M. Tanaka, Conjugate and cut loci of a two-sphere of revolution with application to optimal control,, Ann. Inst. H. Poincaré Anal. Non linéaire, 26 (2009), 1081. doi: 10.1016/j.anihpc.2008.03.010. Google Scholar

[7]

B. Bonnard and M. Chyba, "Singular Trajectories and their Role in Control Theory,", Mathématiques & Applications (Berlin) [Mathematics & Applications], 40 (2003). Google Scholar

[8]

B. Bonnard, M. Chyba and J. Marriott, Singular trajectories and the contrast imaging problem in nuclear magnetic resonance,, SIAM, 51 (2013), 1325. doi: 10.1137/110833427. Google Scholar

[9]

B. Bonnard, M. Chyba and D. Sugny, Time-minimal control of dissipative two-level quantum systems: The generic case,, IEEE Trans. Auto. Control, 54 (2009), 2598. doi: 10.1109/TAC.2009.2031212. Google Scholar

[10]

B. Bonnard, M. Claeys, O. Cots and P. Martinon, Comparison of numerical methods in the contrast imaging problem in NMR,, submitted to IEEE Trans. Auto. Control, (2013). Google Scholar

[11]

B. Bonnard, O. Cots, S. J. Glaser, M. Lapert, D. Sugny and Y. Zhang, Geometric optimal control of the contrast imaging problem in nuclear magnetic resonance,, IEEE Transactions on Automatic and Control, 57 (2012), 1957. doi: 10.1109/TAC.2012.2195859. Google Scholar

[12]

B. Bonnard, O. Cots, N. Shcherbakova and D. Sugny, The energy minimization problem for two-level dissipative quantum systems,, J. Math. Phys., 51 (2010). doi: 10.1063/1.3479390. Google Scholar

[13]

B. Bonnard and D. Sugny, Time minimal control of dissipative two-level quantum systems: The integrable case,, SIAM, 48 (2009), 1289. doi: 10.1137/080717043. Google Scholar

[14]

U. Boscain and P. Mason, Time minimal trajectories for a spin 1/2 particle in a magnetic field,, J. Math. Phys., 47 (2006). doi: 10.1063/1.2203236. Google Scholar

[15]

D. O. Brunner, N. De Zanche, J. Fröhlich, J. Paska and K. P. Pruessmann, Travelling-wave nuclear magnetic resonance,, Nature, 457 (2009), 994. doi: 10.1038/nature07752. Google Scholar

[16]

G. M. Bydder, J. V. Hajnal and I. R. Young, MRI: Use of the inversion recovery pulse sequence,, Clinical Radiology, 53 (1998), 159. doi: 10.1016/S0009-9260(98)80096-2. Google Scholar

[17]

G. M. Bydder and I. R. Yound, MR imaging: Clinical use of the inversion recovery sequence,, Journal of Computed Assisted Tomography, 9 (1985). doi: 10.1097/00004728-198507010-00002. Google Scholar

[18]

S. Conolly, D. Nishimura and A. Macovski, Optimal control solutions to the magnetic resonance selective excitation problem,, IEEE Trans. Med. Imaging, 5 (1986), 106. doi: 10.1109/TMI.1986.4307754. Google Scholar

[19]

O. Cots, "Contrôle Optimal Géométrique: Méthodes Homotopiques et Applications,", Ph.D thesis, (2012). Google Scholar

[20]

D. D'Alessandro and M. Dahleh, Optimal control of two-level quantum systems,, IEEE Trans. Auto. Control, 46 (2001), 866. doi: 10.1109/9.928587. Google Scholar

[21]

R. R. Ernst, G. Bodenhausen and A. Wokaun, "Principles of Nuclear Magnetic Resonance in One and Two Dimensions,", International Series of Monographs on Chemistry, (1990). doi: 10.1063/1.2811094. Google Scholar

[22]

L. Frydman and D. Blazina, Ultrafast two-dimensional nuclear magnetic resonance spectroscopy of hyperpolarized solutions,, Nature Physics, 3 (2007), 415. doi: 10.1038/nphys597. Google Scholar

[23]

N. I. Gershenzon, K. Kobzar, B. Luy, S. J. Glaser and T. E. Skinner, Optimal control design of excitation pulses that accomodate relaxation,, J. Magn. Reson., 188 (2007), 330. doi: 10.1016/j.jmr.2007.08.007. Google Scholar

[24]

V. Jurdjevic, "Geometric Control Theory,", Cambridge Studies in Advanced Mathematics, 52 (1997). Google Scholar

[25]

N. Khaneja, R. Brockett and S. J. Glaser, Time optimal control in spin systems,, Phys. Rev. A, 63 (2001). doi: 10.1103/PhysRevA.63.032308. Google Scholar

[26]

N. Khaneja, S. J. Glaser and R. Brockett, Sub-Riemannian geometry and time optimal control of three spin systems: Quantum gates and coherence transfer,, Phys. Rev. A (3), 65 (2002). doi: 10.1103/PhysRevA.65.032301. Google Scholar

[27]

N. Khaneja, F. Kramer and S. J. Glaser, Optimal experiments for maximizing coherence transfer between coupled spins,, J. Magn. Reson., 173 (2005), 116. doi: 10.1016/j.jmr.2004.11.023. Google Scholar

[28]

N. Khaneja, B. Luy and S. J. Glaser, Boundary of quantum evolution under decoherence,, Proc. Natl. Acad. Sci. USA, 100 (2003), 13162. doi: 10.1073/pnas.2134111100. Google Scholar

[29]

N. Khaneja, T. Reiss, C. Kehlet, T. Schulte-Herbrüggen and S. J. Glaser, Optimal control of coupled spin dynamics: Design of NMR pulse sequences by gradient ascent algorithms,, J. Magn. Reson., 172 (2005), 296. doi: 10.1016/j.jmr.2004.11.004. Google Scholar

[30]

N. Khaneja, T.Reiss, B. Luy and S. J. Glaser, Optimal control of spin dynamics in the presence of relaxation,, J. Magn. Reson., 162 (2003), 311. doi: 10.1016/S1090-7807(03)00003-X. Google Scholar

[31]

K. Kobzar, B. Luy, N. Khaneja and S. J. Glaser, Pattern pulses: Design of arbitrary excitation profiles as a function of pulse amplitude and offset,, J. Magn. Reson., 173 (2005), 229. doi: 10.1016/j.jmr.2004.12.005. Google Scholar

[32]

K. Kobzar, T. E. Skinner, N. Khaneja, S. J. Glaser and B. Luy, Exploring the limits of broadband excitation and inversion pulses,, J. Magn. Reson., 170 (2004), 236. doi: 10.1016/j.jmr.2004.06.017. Google Scholar

[33]

K. Kobzar, T. E. Skinner, N. Khaneja, S. J. Glaser and B. Luy, Exploring the limits of broadband excitation and inversion: II. Rf-power optimized pulses,, J. Magn. Reson., 194 (2008), 58. doi: 10.1016/j.jmr.2008.05.023. Google Scholar

[34]

M. Lapert, Y. Zhang, M. Braun, S. J. Glaser and D. Sugny, Singular extremals for the time-optimal control of dissipative spin 1/2 particles,, Phys. Rev. Lett., 104 (2010). doi: 10.1103/PhysRevLett.104.083001. Google Scholar

[35]

M. Lapert, Y. Zhang, M. Janich, S. J. Glaser and D. Sugny, Exploring the physical limits of contrast in magnetic resonance imaging,, Sci. Rep., 2 (2012). doi: 10.1038/srep00589. Google Scholar

[36]

M. H. Levitt, "Spin Dynamics: Basics of Nuclear Magnetic Resonance,", John Wiley & sons, (2008). doi: 10.1118/1.3273534. Google Scholar

[37]

J.-S. Li and N. Khaneja, Ensemble controllability of the Bloch equations,, in, (2006), 2483. Google Scholar

[38]

J. Mao, T. H. Mareci, K. N. Scott and E. R. Andrew, Selective inversion radiofrequency pulses by optimal control,, J. Magn. Reson., 70 (1986), 310. doi: 10.1016/0022-2364(86)90016-8. Google Scholar

[39]

N. C. Nielsen, C. Kehlet, S. J. Glaser and N. Khaneja, "Optimal Control Methods in NMR Spectroscopy,", Encyclopedia of Nuclear Magnetic Resonance, (2010). Google Scholar

[40]

G. Pileio, M. Carravetta and M. H. Levitt, Extremely low-frequency spectroscopy in low-field nuclear magnetic resonance,, Phys. Rev. Lett., 103 (2009). doi: 10.1103/PhysRevLett.103.083002. Google Scholar

[41]

L. Pontryagin, et al., "Mathematical Theory of Optimal Processes,", Mir, (1974). Google Scholar

[42]

T. O. Reiss, N. Khaneja and S. J. Glaser, Broadband geodesic pulses for three spin systems: Time-optimal realization of effective trilinear coupling terms and indirect SWAP gates,, J. Magn. Reson., 165 (2003), 95. doi: 10.1016/S1090-7807(03)00245-3. Google Scholar

[43]

S. Rice and M. Zhao, "Optimal Control of Molecular Dynamics,", Wiley, (2000). Google Scholar

[44]

D. Rosenfeld and Y. Zun, Design of adiabatic selective pulses using optimal control theory,, Magn. Reson. Med., 36 (1996), 401. doi: 10.1002/mrm.1910360311. Google Scholar

[45]

J. N. Rydberg, S. J. Riederer, C. H. Rydberg and C. R. Jack, Contrast optimization of fluid-attenuated inversion recovery (flair) imaging,, Magnetic Resonance in Medicine, 34 (1995), 868. doi: 10.1002/mrm.1910340612. Google Scholar

[46]

T. E. Skinner, T. O. Reiss, B. Luy, N. Khaneja and S. J. Glaser, Application of optimal control theory to the design of broadband excitation pulses for high resolution NMR,, J. Magn. Reson., 163 (2003), 8. doi: 10.1016/S1090-7807(03)00153-8. Google Scholar

[47]

D. Stefanatos, N. Khaneja and S. J. Glaser, Optimal control of coupled spins in presence of longitudinal and transverse relaxation,, Phys. Rev. A, 69 (2004). Google Scholar

[48]

D. Stefanatos, N. Khaneja and S. J. Glaser, Relaxation optimized transfer of spin order in ising chains,, Phys. Rev. A, 72 (2005). doi: 10.1103/PhysRevA.72.062320. Google Scholar

[49]

D. J. Tannor, "Introduction to Quantum Mechanics: A Time-Dependent Perspective,", University Science Books, (2007). Google Scholar

[50]

Z. Tosner, T. Vosegaard, C. Kehlet, N. Khaneja, S. J. Glaser and N. C. Nielsen, Optimal control in NMR spectroscopy: Numerical implementation in SIMPSON,, J. Magn. Reson., 197 (2009), 120. Google Scholar

[51]

L. M. K. Vandersypen and I. L. Chuang, NMR techniques for quantum control and computation,, Rev. Mod. Phys., 76 (2004), 1037. Google Scholar

[52]

M. S. Vinding, I. I. Maximov, Z. Tosner and N. C. Nielsen, Fast numerical design of spatil-selective rf pulses in MRI using Krotov and quasi-Newton based optimal control methods,, J. Chem. Phys., 137 (2012). doi: 10.1063/1.4739755. Google Scholar

[53]

W. S. Warren, S. L. Hammes and J. L. Bates, Dynamics of radiation damping in nuclear magnetic resonance,, J. Chem. Phys., 91 (1989). doi: 10.1063/1.457458. Google Scholar

[54]

Y. Zhang, M. Lapert, M. Braun, D. Sugny and S. J. Glaser, Time-optimal control of spin 1/2 particles in presence of relaxation and radiation damping effects,, J. Chem. Phys., 134 (2011). Google Scholar

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