The evolution of nuclear spin magnetization during a radiofrequency pulse in the absence of relaxation 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 results 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.

Numerous aspects of nuclear magnetic resonance (NMR) spectroscopy are formulated in terms of differential equations, few of which have closed-form analytical solutions. In an era characterized by ever-increasing computational capabilities, numerical solutions to such differential equations are always possible and are frequently the preferred approach for applications such as data analysis. However, approximate solutions can provide useful formulas and insights that are difficult to discern from purely numerical results.

As one example, the net evolution of the magnetization of an isolated spin
during a radiofrequency pulse, i.e. in the absence of relaxation and
scalar or other coupling interactions, can be described by three
rotations with Euler angles

Alternatively, the Euler angles can be determined from the solution of the following Riccati equation

The homotopy analysis method (HAM) is a fairly recent development, first
reported in 1992

In topology, a pair of functions defining different topological spaces
are said to be homotopic if the shape defined by one function can be
continuously transformed (deformed in the lexicon of topology) into the
shape defined by the other. Analogously, the essence of HAM is to map a
function of interest, here

This relationship is established by constructing the homotopy as follows

The iterative algorithm in HAM is illustrated by application to the
second-order differential form of the Riccati equation. In the first
example, the nonlinear operator is obtained from Eq. (

From the relationships of Eqs. (

The derivative of Eq. (

The higher-order approximations

The above choices of

The

Numerical integration was performed using the trapezoid method implemented in Python 3.6. Pulse shapes were discretized in 1000 increments. Rectangular pulses were simulated using

Equation (

In the present applications, HAM converts the second-order Riccati
differential equation, Eq. (

A first example of the results of the above analysis is given for a
rectangular 90

In contrast to the results of method 1, the power series for

HAM approximations for on-resonance 90

A more challenging example is given by the 90

HAM approximations for 90

The Gaussian Q5 90

HAM approximations for 90

The application of HAM is not limited to 90

HAM approximations for

Method 2 yields a power series for

in which the second equality is the expansion to first order
in

The above explications have focused on solutions to the transformed
Riccati equation in Eq. (

For many applications, the Euler angles for a shaped pulse are easily
obtained from Eqs. (

The Euler angle representation is particularly convenient because, once
calculated, the Euler angles can be used to determine the outcome of a
shaped pulse applied to arbitrary initial magnetization. The Riccati
equation can be extended to incorporate radiation damping, but not
relaxation, as discussed by

The initial zeroth-order approximations for HAM are

The

For a rectangular pulse applied to equilibrium magnetization (with
magnitude set to unity for convenience), the initial approximations are as follows:

Figure

HAM approximations for rectangular

Fast, accurate methods for solving differential equations have
widespread application in NMR spectroscopy. The present work has
illustrated the homotopy analysis method

RMarkdown and bibtex files are provided in the Supplement. The RMarkdown file contains the Python code for all calculations described in the paper.

The supplement related to this article is available online at:

AGP conceived the project. The theoretical derivations, numerical calculations, and writing of the paper were performed by TC and AGP.

The authors declare that they have no conflict of interest.

This article is part of the special issue “Robert Kaptein Festschrift”. It is not associated with a conference.

Some of the work presented here was conducted at the Center on Macromolecular Dynamics by NMR Spectroscopy located at the New York Structural Biology Center. Arthur G. Palmer is a member of the New York Structural Biology Center. This paper is dedicated to Robert Kaptein on the occasion of his 80th birthday.

This research has been supported by the National Institutes of Health (NIH; grant nos. R35GM130398 and P41GM118302).

This paper was edited by Malcolm Levitt and reviewed by Fabien Ferrage and one anonymous referee.