Time minimal saturation of a pair of spins and application in Magnetic Resonance Imaging

Abstract

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 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 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.

Connection between singular arcs in optimal control using bridges. Physical occurence and Mathematical model

“…IN MAGNETIC RESONANCE IMAGING [2] We consider the problem of transferring the system from the North pole N = (0, 1) to the center O = (0, 0) of the Bloch ball so that the system takes the form…”

“…. This phenomenon was obtained in many cases in the problem of multisaturation of spin particles see [2] or [10] for a single spin, that is to drive the magnetization vector from the North pole to the center of the Bloch ball, where the system is of the form:…”

“…∂H ∂p (x(t), p(t), u(t)),ṗ(t) = − ∂H ∂x (x(t), p(t), u(t)), (1) H(x(t), p(t), u(t)) = M (x(t), p(t)) (2) where H(x, p, u) = p · (F (x) + u G(x)) is the pseudo Hamiltonian, · stands for the scalar product and M (x, p) = max |u|≤1 H(x, p, u) is the maximized Hamiltonian. Moreover M (x(t), p(t)) is a non-negative constant along an extremal solution and one has p(t f ) ⊥ T x(t f ) N (transversality equation).…”

“…In section II, we present a recap of the classification of extremals near the switching surface in the so-called fold case [8] which is the basis of the geometric construction of bridges. In section III, the saturation of the spin case is described in details based on [2]. This leads to construct in section IV a planar mathematical model.…”

Towards Geometric Time Minimal Control without Legendre Condition and with Multiple Singular Extremals for Chemical Networks

“…The key point for this extension is the analysis of singular trajectories and their role in the synthesis. This is connected with an important question and the need to extend the standard synthesis related to the turnpike phenomenon [29] to deal with cases, where the strict Legendre-Clebsch condition is not satisfied, a situation encountered in a recent application in MRI [6].…”