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Time minimal saturation of a pair of spins and application in Magnetic Resonance Imaging
1. | Institut de Mathématiques de Bourgogne, Université de Bourgogne, 9 avenue Alain Savary, 21078 Dijon, France |
2. | INRIA, 2004 route des Lucioles F-06902, Sophia Antipolis, France |
3. | Toulouse Univ., INP-ENSEEIHT-IRIT, UMR CNRS 5505, 2 rue Camichel, 31071 Toulouse, France |
4. | EPF:École d'Ingénieur-e-s, 2 Rue F Sastre, 10430 Rosières-prés-Troyes, France |
5. | Institute for Algebra, Johannes Kepler University, 4040 Linz, Austria |
In this article, we analyze the time minimal control for the saturation of a pair of spins of the same species but with inhomogeneities of the applied RF-magnetic field, in relation with the contrast problem in Magnetic Resonance Imaging. We make a complete analysis based on geometric control to classify the optimal syntheses in the single spin case to pave the road to analyze the case of two spins. The ${\texttt {Bocop}}$ software is used to determine local minimizers for physical test cases and Linear Matrix Inequalities approach is applied to estimate the global optimal value and validate the previous computations. This is complemented by numerical computations combining shooting and continuation methods implemented in the ${\texttt {HamPath}}$ software to analyze the structure of the time minimal solution with respect to the set of parameters of the species. Symbolic computations techniques are used to handle the singularity analysis.
References:
[1] |
E. Allgower and K. Georg, Introduction to Numerical Continuation Methods, vol. 45 of Classics in Applied Mathematics, Soc. for Industrial and Applied Math., Philadelphia, PA, USA, 2003.
doi: 10.1137/1.9780898719154. |
[2] |
J. T. Betts, Practical Methods for Optimal Control Using Nonlinear Programming, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, 2001. |
[3] |
F. J. Bonnans, P. Martinon and V. Grélard, Bocop - A Collection of Examples, Technical report, INRIA (2012) RR-8053. |
[4] |
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), 207-236.
doi: 10.1051/cocv:2007012. |
[5] |
B. Bonnard and M. Chyba, Singular Trajectories and Their Role in Control Theory, vol 40 of Mathematics and Applications, Springer-Verlag, Berlin, 2003. |
[6] |
B. Bonnard, M. Chyba, A. Jacquemard and J. Marriott,
Algebraic geometric classification of the singular flow in the contrast imaging problem in nuclear magnetic resonance. Mathematical Control and Related Fields-AIMS, Special issue in the honor of Bernard Bonnard. Part II., 3 (2013), 397-432.
doi: 10.3934/mcrf.2013.3.397. |
[7] |
B. Bonnard, M. Chyba and J. Marriott,
Singular trajectories and the contrast imaging problem in nuclear magnetic resonance, SIAM J. Control Optim., 51 (2013), 1325-1349.
doi: 10.1137/110833427. |
[8] |
B. Bonnard, M. Claeys, O. Cots and P. Martinon,
Geometric and numerical methods in the contrast imaging problem in nuclear magnetic resonance, Acta Appl. Math., 135 (2015), 5-45.
doi: 10.1007/s10440-014-9947-3. |
[9] |
B. Bonnard and O. Cots,
Geometric numerical methods and results in the control imaging problem in nuclear magnetic resonance, Math. Models Methods Appl. Sci., 24 (2014), 187-212.
doi: 10.1142/S0218202513500504. |
[10] |
B. Bonnard, O. Cots, S. Glaser, M. Lapert, D. Sugny and Y. Zhang,
Geometric optimal control of the contrast imaging problem in nuclear magnetic resonance, IEEE Trans. Automat. Control, 57 (2012), 1957-1969.
doi: 10.1109/TAC.2012.2195859. |
[11] |
B. Bonnard, O. Cots, J. Rouot and T. Verron, Working Notes on the Time Minimal Saturation of a Pair of Spins and Application in Magnetic Resonance Imaging, http://hal.archives-ouvertes.fr/hal-01721845/ |
[12] |
B. Bonnard, J.-C. Faugére, A. Jacquemard, M. Safey El Din and T. Verron, Determinantal sets, singularities and application to optimal control in medical imagery, Proceedings of the 2016 ACM International Symposium on Symbolic and Algebraic Computation, 103–110, ACM, New York, 2016. |
[13] |
B. Bonnard and I. Kupka,
Théorie des singularités de l'application entrée/sortie et optimalité des trajectoires singulières dans le problème du temps minimal, Forum Math., 5 (1993), 111-159.
doi: 10.1515/form.1993.5.111. |
[14] |
U. Boscain and B. Piccoli, Optimal Syntheses for Control Systems on 2-D Manifolds, Springer SMAI, 2004. |
[15] |
R. Bulirsch and J. Stoer, Introduction to Numerical Analysis, vol.12 of Texts in Applied Mathematics, Springer-Verlag, New York, 2$^{nd}$ edition, 1993.
doi: 10.1007/978-1-4757-2272-7. |
[16] |
J.-B. Caillau, O. Cots and J. Gergaud,
Differential continuation for regular optimal control problems, Optimization Methods and Software, 27 (2011), 177-196.
doi: 10.1080/10556788.2011.593625. |
[17] |
M. Claeys, J. Daafouz and D. Henrion,
Modal occupation measures and {LMI} relaxations for nonlinear switched systems control, Automatica, 64 (2016), 143-154.
doi: 10.1016/j.automatica.2015.11.003. |
[18] |
S. Conolly, D. Nishimura and A. Macovski,
Optimal control solutions to the magnetic resonance selective excitation problem, IEEE Trans. Med. Imaging, 5 (1986), 106-115.
|
[19] |
M. Gerdts, Optimal Control of ODEs and DAEs, ed. De Gruyter, Berlin, 2012.
doi: 10.1515/9783110249996. |
[20] |
D. Henrion, J. B. Lasserre and J. Löfberg,
GloptiPoly 3: Moments, optimization and semidefinite programming, Optim. Methods and Software, 24 (2009), 761-779.
doi: 10.1080/10556780802699201. |
[21] |
A. J. Krener,
The high order maximal principle and its application to singular extremals, SIAM J. Control Optim., 15 (1977), 256-293.
doi: 10.1137/0315019. |
[22] |
I. Kupka,
Geometric theory of extremals in optimal control problems. Ⅰ. The fold and Maxwell case, Trans. Amer. Math. Soc., 299 (1987), 225-243.
doi: 10.2307/2000491. |
[23] |
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), 083001.
|
[24] |
M. Lapert, Y. Zhang, M. A. Janich, S. J. Glaser and D. Sugny,
Exploring the physical limits of saturation contrast in magnetic resonance imaging, Sci. Rep., 589 (2012).
|
[25] |
J.-B. Lasserre, Moments, Positive Polynomials and Their Applications, Imperial College Press, London, 2010.
![]() ![]() |
[26] |
J.-B. Lasserre, D. Henrion, C. Prieur and E. Trélat,
Nonlinear optimal control via occupation measures and LMI-relaxations, SIAM J. Control Optim., 47 (2008), 1643-1666.
doi: 10.1137/070685051. |
[27] |
M. H. Levitt, Spin Dynamics: Basics of Nuclear Magnetic Resonance, John Wiley and Sons, New York-London-Sydney, 2008. |
[28] |
L. Markus,
Quadratic differential equations and non-associative algebras, Princeton Univ. Press, Princeton, N.J., 5 (1960), 185-213.
|
[29] |
H. Maurer,
Numerical solution of singular control problems using multiple shooting techniques, J. Optim. Theory Appl., 18 (1976), 235-257.
doi: 10.1007/BF00935706. |
[30] |
A. Mosek, The Mosek Optimization Toolbox for Matlab Manual, Version 7.1 Revision, 2015. |
[31] |
J. Nocedal and S. J. Wright, Numerical Optimization, Springer-Verlag, New York, 1999.
doi: 10.1007/b98874. |
[32] |
L. S. Pontryagin, V. G. Boltyanskiĭ, R. V. Gamkrelidze and E. F. Mishchenko, The Mathematical Theory of Optimal Processes, Translated from the Russian by K. N. Trirogoff, edited by L. W. Neustadt, Interscience Publishers John Wiley and Sons, Inc., New York-London, 1962. |
[33] |
M. Putinar,
Positive polynomials on compact semi-algebraic sets, Indiana Univ. Math. J., 42 (1993), 969-984.
doi: 10.1512/iumj.1993.42.42045. |
[34] |
H. Schättler,
The local structure of time-optimal trajectories in dimension three under generic conditions, SIAM J. Contr. Opt., 26 (1988), 899-918.
doi: 10.1137/0326050. |
[35] |
H. J. Sussmann, Time-optimal control in the plane,, in Feedback Control of Linear and Nonlinear Systems, (Bielefeld/Rome, 1981), vol. 39 of Lecture Notes in Control and Inform. Sci., Springer, Berlin, (1982), 244–260.
doi: 10.1007/BFb0006833. |
[36] |
H. J. Sussmann,
Regular synthesis for time-optimal control of single-input real-analytic systems in the plane, SIAM J. Control and Opt., 25 (1987), 1145-1162.
doi: 10.1137/0325062. |
[37] |
E. Van Reeth, H. Ratiney, M. Tesch, D. Grenier, O. Beuf, S. J. Glaser and D. Sugny,
Optimal control design of preparation pulses for contrast optimization in MRI, J. Magn. Reson., 279 (2017), 39-50.
|
show all references
References:
[1] |
E. Allgower and K. Georg, Introduction to Numerical Continuation Methods, vol. 45 of Classics in Applied Mathematics, Soc. for Industrial and Applied Math., Philadelphia, PA, USA, 2003.
doi: 10.1137/1.9780898719154. |
[2] |
J. T. Betts, Practical Methods for Optimal Control Using Nonlinear Programming, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, 2001. |
[3] |
F. J. Bonnans, P. Martinon and V. Grélard, Bocop - A Collection of Examples, Technical report, INRIA (2012) RR-8053. |
[4] |
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), 207-236.
doi: 10.1051/cocv:2007012. |
[5] |
B. Bonnard and M. Chyba, Singular Trajectories and Their Role in Control Theory, vol 40 of Mathematics and Applications, Springer-Verlag, Berlin, 2003. |
[6] |
B. Bonnard, M. Chyba, A. Jacquemard and J. Marriott,
Algebraic geometric classification of the singular flow in the contrast imaging problem in nuclear magnetic resonance. Mathematical Control and Related Fields-AIMS, Special issue in the honor of Bernard Bonnard. Part II., 3 (2013), 397-432.
doi: 10.3934/mcrf.2013.3.397. |
[7] |
B. Bonnard, M. Chyba and J. Marriott,
Singular trajectories and the contrast imaging problem in nuclear magnetic resonance, SIAM J. Control Optim., 51 (2013), 1325-1349.
doi: 10.1137/110833427. |
[8] |
B. Bonnard, M. Claeys, O. Cots and P. Martinon,
Geometric and numerical methods in the contrast imaging problem in nuclear magnetic resonance, Acta Appl. Math., 135 (2015), 5-45.
doi: 10.1007/s10440-014-9947-3. |
[9] |
B. Bonnard and O. Cots,
Geometric numerical methods and results in the control imaging problem in nuclear magnetic resonance, Math. Models Methods Appl. Sci., 24 (2014), 187-212.
doi: 10.1142/S0218202513500504. |
[10] |
B. Bonnard, O. Cots, S. Glaser, M. Lapert, D. Sugny and Y. Zhang,
Geometric optimal control of the contrast imaging problem in nuclear magnetic resonance, IEEE Trans. Automat. Control, 57 (2012), 1957-1969.
doi: 10.1109/TAC.2012.2195859. |
[11] |
B. Bonnard, O. Cots, J. Rouot and T. Verron, Working Notes on the Time Minimal Saturation of a Pair of Spins and Application in Magnetic Resonance Imaging, http://hal.archives-ouvertes.fr/hal-01721845/ |
[12] |
B. Bonnard, J.-C. Faugére, A. Jacquemard, M. Safey El Din and T. Verron, Determinantal sets, singularities and application to optimal control in medical imagery, Proceedings of the 2016 ACM International Symposium on Symbolic and Algebraic Computation, 103–110, ACM, New York, 2016. |
[13] |
B. Bonnard and I. Kupka,
Théorie des singularités de l'application entrée/sortie et optimalité des trajectoires singulières dans le problème du temps minimal, Forum Math., 5 (1993), 111-159.
doi: 10.1515/form.1993.5.111. |
[14] |
U. Boscain and B. Piccoli, Optimal Syntheses for Control Systems on 2-D Manifolds, Springer SMAI, 2004. |
[15] |
R. Bulirsch and J. Stoer, Introduction to Numerical Analysis, vol.12 of Texts in Applied Mathematics, Springer-Verlag, New York, 2$^{nd}$ edition, 1993.
doi: 10.1007/978-1-4757-2272-7. |
[16] |
J.-B. Caillau, O. Cots and J. Gergaud,
Differential continuation for regular optimal control problems, Optimization Methods and Software, 27 (2011), 177-196.
doi: 10.1080/10556788.2011.593625. |
[17] |
M. Claeys, J. Daafouz and D. Henrion,
Modal occupation measures and {LMI} relaxations for nonlinear switched systems control, Automatica, 64 (2016), 143-154.
doi: 10.1016/j.automatica.2015.11.003. |
[18] |
S. Conolly, D. Nishimura and A. Macovski,
Optimal control solutions to the magnetic resonance selective excitation problem, IEEE Trans. Med. Imaging, 5 (1986), 106-115.
|
[19] |
M. Gerdts, Optimal Control of ODEs and DAEs, ed. De Gruyter, Berlin, 2012.
doi: 10.1515/9783110249996. |
[20] |
D. Henrion, J. B. Lasserre and J. Löfberg,
GloptiPoly 3: Moments, optimization and semidefinite programming, Optim. Methods and Software, 24 (2009), 761-779.
doi: 10.1080/10556780802699201. |
[21] |
A. J. Krener,
The high order maximal principle and its application to singular extremals, SIAM J. Control Optim., 15 (1977), 256-293.
doi: 10.1137/0315019. |
[22] |
I. Kupka,
Geometric theory of extremals in optimal control problems. Ⅰ. The fold and Maxwell case, Trans. Amer. Math. Soc., 299 (1987), 225-243.
doi: 10.2307/2000491. |
[23] |
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), 083001.
|
[24] |
M. Lapert, Y. Zhang, M. A. Janich, S. J. Glaser and D. Sugny,
Exploring the physical limits of saturation contrast in magnetic resonance imaging, Sci. Rep., 589 (2012).
|
[25] |
J.-B. Lasserre, Moments, Positive Polynomials and Their Applications, Imperial College Press, London, 2010.
![]() ![]() |
[26] |
J.-B. Lasserre, D. Henrion, C. Prieur and E. Trélat,
Nonlinear optimal control via occupation measures and LMI-relaxations, SIAM J. Control Optim., 47 (2008), 1643-1666.
doi: 10.1137/070685051. |
[27] |
M. H. Levitt, Spin Dynamics: Basics of Nuclear Magnetic Resonance, John Wiley and Sons, New York-London-Sydney, 2008. |
[28] |
L. Markus,
Quadratic differential equations and non-associative algebras, Princeton Univ. Press, Princeton, N.J., 5 (1960), 185-213.
|
[29] |
H. Maurer,
Numerical solution of singular control problems using multiple shooting techniques, J. Optim. Theory Appl., 18 (1976), 235-257.
doi: 10.1007/BF00935706. |
[30] |
A. Mosek, The Mosek Optimization Toolbox for Matlab Manual, Version 7.1 Revision, 2015. |
[31] |
J. Nocedal and S. J. Wright, Numerical Optimization, Springer-Verlag, New York, 1999.
doi: 10.1007/b98874. |
[32] |
L. S. Pontryagin, V. G. Boltyanskiĭ, R. V. Gamkrelidze and E. F. Mishchenko, The Mathematical Theory of Optimal Processes, Translated from the Russian by K. N. Trirogoff, edited by L. W. Neustadt, Interscience Publishers John Wiley and Sons, Inc., New York-London, 1962. |
[33] |
M. Putinar,
Positive polynomials on compact semi-algebraic sets, Indiana Univ. Math. J., 42 (1993), 969-984.
doi: 10.1512/iumj.1993.42.42045. |
[34] |
H. Schättler,
The local structure of time-optimal trajectories in dimension three under generic conditions, SIAM J. Contr. Opt., 26 (1988), 899-918.
doi: 10.1137/0326050. |
[35] |
H. J. Sussmann, Time-optimal control in the plane,, in Feedback Control of Linear and Nonlinear Systems, (Bielefeld/Rome, 1981), vol. 39 of Lecture Notes in Control and Inform. Sci., Springer, Berlin, (1982), 244–260.
doi: 10.1007/BFb0006833. |
[36] |
H. J. Sussmann,
Regular synthesis for time-optimal control of single-input real-analytic systems in the plane, SIAM J. Control and Opt., 25 (1987), 1145-1162.
doi: 10.1137/0325062. |
[37] |
E. Van Reeth, H. Ratiney, M. Tesch, D. Grenier, O. Beuf, S. J. Glaser and D. Sugny,
Optimal control design of preparation pulses for contrast optimization in MRI, J. Magn. Reson., 279 (2017), 39-50.
|
















Name | ||||
Water | 2.5 | 2.5 | 1.0 | 0.7854 |
Fat | 0.2 | 0.1 | 0.5 | 0.4636 |
Cerebrospinal Fluid | 2.0 | 0.3 | 0.15 | 0.1489 |
Oxygenated blood | 1.35 | 0.2 | 0.1481 | 0.1471 |
White cerebral matter | 0.78 | 0.09 | 0.1154 | 0.1148 |
Gray cerebral matter | 0.92 | 0.1 | 0.1087 | 0.1083 |
Brain | 1.062 | 0.052 | 0.0490 | 0.0489 |
Deoxygenated blood | 1.35 | 0.05 | 0.0370 | 0.0370 |
Parietal muscle | 1.2 | 0.029 | 0.0242 | 0.0242 |
Name | ||||
Water | 2.5 | 2.5 | 1.0 | 0.7854 |
Fat | 0.2 | 0.1 | 0.5 | 0.4636 |
Cerebrospinal Fluid | 2.0 | 0.3 | 0.15 | 0.1489 |
Oxygenated blood | 1.35 | 0.2 | 0.1481 | 0.1471 |
White cerebral matter | 0.78 | 0.09 | 0.1154 | 0.1148 |
Gray cerebral matter | 0.92 | 0.1 | 0.1087 | 0.1083 |
Brain | 1.062 | 0.052 | 0.0490 | 0.0489 |
Deoxygenated blood | 1.35 | 0.05 | 0.0370 | 0.0370 |
Parietal muscle | 1.2 | 0.029 | 0.0242 | 0.0242 |
Case | ||||
|
9.855 |
3.65 |
44.769 | 42.685 |
2.464 |
3.65 |
113.86 | 110.44 | |
1.642 |
2.464 |
168.32 | 164.46 | |
9.855 |
9.855 |
15.0237 | 8.7445 |
Case | ||||
|
9.855 |
3.65 |
44.769 | 42.685 |
2.464 |
3.65 |
113.86 | 110.44 | |
1.642 |
2.464 |
168.32 | 164.46 | |
9.855 |
9.855 |
15.0237 | 8.7445 |
1st Formulation | 2nd Formulation | |||
r | ||||
1 | 25 | 0.8143 | 30 | 0.818 |
2 | 105 | 0.5164 | 105 | 0.5958 |
3 | 294 | 0.2611 | 252 | 0.4355 |
4 | 660 | 0.1491 | 495 | 0.1842 |
5 | 1287 | 0.0932 | 858 | 0.1284 |
6 | 2275 | 0.0643 | 1365 | 0.096 |
7 | 3740 | 0.0517 | 2040 | 0.0797 |
8 | 5814 | 0.0461 | 2907 | 0.0716 |
1st Formulation | 2nd Formulation | |||
r | ||||
1 | 25 | 0.8143 | 30 | 0.818 |
2 | 105 | 0.5164 | 105 | 0.5958 |
3 | 294 | 0.2611 | 252 | 0.4355 |
4 | 660 | 0.1491 | 495 | 0.1842 |
5 | 1287 | 0.0932 | 858 | 0.1284 |
6 | 2275 | 0.0643 | 1365 | 0.096 |
7 | 3740 | 0.0517 | 2040 | 0.0797 |
8 | 5814 | 0.0461 | 2907 | 0.0716 |
Name | Init | Transition | End |
H1a | θmax ≈ 1.1071 | θ1a, 1 ≈ 0.5069 | θ1a, 2 ≈ 0.2722 > θmin = 0.02 |
σ-σsσ+σ0 | σ-σsσ-σsσ+σ0 | σ-σsσ-σ+σ0 | |
H1b | θmin | θ1b, 1 ≈ 0.2752 | θ1b, 2 ≈ 0.3018 < θmax |
σ-σsσ+σsσ+σ0 | σ-σsσ+σ0 | σ-σs -σ+σ0 | |
H2 | θ = 0.2 | θ2, 1 ≈ 0.3618 | θ = atan(1.5) ≈ 0.9828 |
σ+σsσ+σsσ+σ0 | σ+σsσ+σ0 | σ+σsσ+σ0 | |
H3 | θ = 0.25 | θ3, 1 ≈ 0.4778 | θ = θmax |
σ+σsσ+σsσ+σ0 | σ+σsσ+σ0 | σ+σsσ+σ0 |
Name | Init | Transition | End |
H1a | θmax ≈ 1.1071 | θ1a, 1 ≈ 0.5069 | θ1a, 2 ≈ 0.2722 > θmin = 0.02 |
σ-σsσ+σ0 | σ-σsσ-σsσ+σ0 | σ-σsσ-σ+σ0 | |
H1b | θmin | θ1b, 1 ≈ 0.2752 | θ1b, 2 ≈ 0.3018 < θmax |
σ-σsσ+σsσ+σ0 | σ-σsσ+σ0 | σ-σs -σ+σ0 | |
H2 | θ = 0.2 | θ2, 1 ≈ 0.3618 | θ = atan(1.5) ≈ 0.9828 |
σ+σsσ+σsσ+σ0 | σ+σsσ+σ0 | σ+σsσ+σ0 | |
H3 | θ = 0.25 | θ3, 1 ≈ 0.4778 | θ = θmax |
σ+σsσ+σsσ+σ0 | σ+σsσ+σ0 | σ+σsσ+σ0 |
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