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Approximate Representations of Shaped Pulses Using the Homotopy Analysis Method

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Approximate Representations of Shaped Pulses Using the Homotopy Analysis Method Magn. Reson., 2, 175–186, 2021 Open Access https://doi.org/10.5194/mr-2-175-2021 © Author(s) 2021. This work is distributed under the Creative Commons Attribution 4.0 License. Approximate representations of shaped pulses using the homotopy analysis method Timothy Crawley and Arthur G. Palmer III Department of Biochemistry and Molecular Biophysics, Columbia University, 630 West 168th Street, New York, NY 10032, United States Correspondence: Arthur G. Palmer III ([email protected]) Received: 11 January 2021 – Discussion started: 15 January 2021 Revised: 19 March 2021 – Accepted: 23 March 2021 – Published: 16 April 2021 Abstract. The evolution of nuclear spin magnetization during a radiofrequency pulse in the absence of relax- ation or coupling interactions can be described by three Euler angles. The Euler angles, in turn, can be obtained from the solution of a Riccati differential equation; however, analytic solutions exist only for rectangular and hyperbolic-secant pulses. The homotopy analysis method is used to obtain new approximate solutions to the Riccati equation for shaped radiofrequency pulses in nuclear magnetic resonance (NMR) spectroscopy. The re- sults of even relatively low orders of approximation are highly accurate and can be calculated very efficiently. The results are extended in a second application of the homotopy analysis method to represent relaxation as a perturbation of the magnetization trajectory calculated in the absence of relaxation. The homotopy analysis method is powerful and flexible and is likely to have other applications in magnetic resonance. 1 Introduction ical calculation in which the shaped pulse is represented by a series of K short rectangular pulses with appropriate am- Numerous aspects of nuclear magnetic resonance (NMR) plitudes and phases. Thus, the propagator for a shaped pulse spectroscopy are formulated in terms of differential equa- expressed in the Cartesian basis is given by the following tions, few of which have closed-form analytical solutions. (Siminovitch, 1995): In an era characterized by ever-increasing computational ca- − − − pabilities, numerical solutions to such differential equations U D e iγ (τp)Iz e iβ(τp)Ix e iα(τp)Iz are always possible and are frequently the preferred approach " C β(τ ) − β(τ ) # −iχ (τp) p − iχ (τp) p e cos( 2 ) ie sin( 2 ) for applications such as data analysis. However, approximate D − β(τ ) C β(τ ) − −iχ (τp) p iχ (τp) p solutions can provide useful formulas and insights that are ie sin( 2 ) e cos( 2 ) difficult to discern from purely numerical results. K Y As one example, the net evolution of the magnetization D Uk; (1) of an isolated spin during a radiofrequency pulse, i.e. in the kD1 absence of relaxation and scalar or other coupling interac- in which χ ±(τ ) is given as follows: tions, can be described by three rotations with Euler an- p gles α(τ ), β(τ ), and γ (τ ), in which τ is the pulse length ± p p p p χ (τ ) DTα(τ ) ± γ (τ )U=2; (2) (Zhou et al., 1994; Siminovitch, 1997a,b). Shaped pulses, p p p in which the amplitude (Rabi frequency), phase, or radiofre- Ik are the Cartesian spin operators, and the product is time- quency are time dependent, are widely applied in modern ordered from right to left. The propagator for the kth rectan- NMR spectroscopy and other magnetic resonance techniques gular pulse segment is given as follows: (Geen and Freeman, 1991; Emsley and Bodenhausen, 1992; ∗ ∗ Kupce˘ et al., 1995; Cavanagh et al., 2007). The Euler angles κ (τp) −iη (τp) Uk D ; (3) for an arbitrary shaped pulse can be extracted from a numer- −iη(τp) κ(τp) Published by Copernicus Publications on behalf of the Groupement AMPERE. 176 T. Crawley and A. G. Palmer III: HAM for shaped pulses in which κ(τp) and η(τp) are given by the following: (Silver et al., 1985; Zhou et al., 1994; Rourke, 2002). Ap- proximate solutions for arbitrary shaped pulses have been de- κ(τp) D cos(!ek1tk=2) C i cosθk sin(!ek1tk=2) rived by the perturbation theory for the limits of small, using iφk Eq. (11), and large, using Eq. (5), resonance offsets (Li et al., η(τp) D e sinθk sin(!ek1tk=2): (4) 2014); however, perturbation theory is unwieldy to apply to In Eq. (4), !1k, φk, and 1tk are the radiofrequency field a high order and, obviously, depends on the perturbation pa- strength, phase angle, and duration of the kth pulse segment; rameters being small in some respects. k is the resonance offset in the rotating frame of refer- The homotopy analysis method (HAM) is a fairly recent ence during the kth pulse segment (and is constant if the off- development, first reported in 1992 (Liao, 1992), for ap- D 2 C 2 1=2 proximating solutions to differential equations, particularly set is fixed); !ek (!1k k) is the effective field; and −1 θk D tan (!1k/k) is the tilt angle. Values of α(τp), β(τp), nonlinear ones. HAM does not depend on small parame- and γ (τp) are then obtained from the matrix elements of U. ters, unlike perturbation theory, and has proven powerful in a Alternatively, the Euler angles can be determined from the number of applications outside of NMR spectroscopy (Liao, solution of the following Riccati equation (Zhou et al., 1994): 2012). The present paper illustrates HAM by application to the solutions of Eqs. (5) and (11) and, subsequently, by ex- df (t) 1 1 D !C(t)f 2(t) C i(t)f (t) C !−(t); (5) tension to the Bloch equations, including relaxation. dt 2 2 in which the following applies: 2 Theory β(t) f (t) D tan eiγ (t); (6) In topology, a pair of functions defining different topologi- 2 cal spaces are said to be homotopic if the shape defined by ± one function can be continuously transformed (deformed in where ! (t) D !x(t) ± i!y(t) and !x(t) and !y(t) are the the lexicon of topology) into the shape defined by the other. Cartesian amplitude components of the radiofrequency field Analogously, the essence of HAM is to map a function of in- in the rotating frame of reference. After solution of the Ric- terest, here y(t), to a second function, 8(tIq), which has a cati equation, β(τp) and γ (τp) are obtained from the mag- known solution and is a function of both t and an embedding nitude and argument of f (τp), and the value of α(τp) is ob- parameter q 2 T0;1U. tained by integration as follows: This relationship is established by constructing the homo- τp topy as follows (Liao, 2012): Z α(τp) D dtf!x(t)sinTγ (t)U − !y(t)cosTγ (t)Ug=sinTβ(t)U: (7) H 8(tIq) V q D (1 − q)L 8(tIq) − y0(t) 0 − qc0H(t)N 8(tIq) ; (12) The Riccati equation can be transformed into a second-order differential equation as follows: in which LTU is a linear (differential) operator, and N TU is an (nonlinear differential) operator satisfying the following: " # d2y(t) dln!−(t) dy(t) 1 − C i(t) C j!(t)j2y(t) D 0; (8) dt2 dt dt 4 L[0] D 0 (13) N y(t) D 0; (14) by using the following definition: where y0(t) is an initial approximation for the desired so- dln y(t) 1 − 6D D − ! (t)f (t): (9) lution y(t), c0 0 is a convergence control parameter, and dt 2 H (t) 6D 0 is an auxiliary function (vide infra). When q D 0, A more compact form is obtained by defining the following: the homotopy becomes as follows: 2 t 3 H[8(tI0) V 0] D L 8(tI0) − y (t) : (15) Z 0 !O −(t) D exp i (t0)dt0 !−(t); (10) 4 5 Therefore, when H[8(tI0) V 0] D 0, Eq. (13) requires 0 8(t;0) D y0(t). Similarly, when q D 1, the homotopy be- to yield the following: comes as follows: − d2y(t) dln !O (t) dy(t) 1 H[8(tI1) V 1] D −c0H(t)N [8(tI1)]: (16) − C j! O (t)j2y(t) D 0: (11) dt2 dt dt 4 Therefore, when H[8(tI1) V 1] D 0, Eq. (14) requires The Riccati differential equation only can be solved ana- 8(t;1) D y(t). Stated more succinctly, as q increases from lytically for a single rectangular or hyperbolic-secant pulse 0 ! 1, 8(tIq) deforms from the initial approximation y0(t) Magn. Reson., 2, 175–186, 2021 https://doi.org/10.5194/mr-2-175-2021 T. Crawley and A. G. Palmer III: HAM for shaped pulses 177 to the exact solution y(t). To proceed, the Maclaurin series the following: for 8(tIq) is assumed to exist as follows: d2y (t) dln!O −(t) dy (t) 1 1 − 1 X n 2 8(tIq) D yn(t)q ; (17) dt dt dt − ! nD0 d2y (t) dln !O (t) dy (t) 1 D c 0 − 0 C j! O t j2y t 0 2 ( ) 0( ) in which yn(t) is given by the following: dt dt dt 4 c 1 dn8(tIq) D 0 j! O (t)j2; (25) y (t) D : (18) 4 n W n n dq qD0 in which the final line is obtained using dy0(t)=dt D 0. This Conditions concerning convergence of the series are dis- differential equation does not contain a term proportional to cussed by Liao(2012). Equation (17) has the desired prop- y1(t).
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