Pulse shaping, inverse scattering theory, and solitons

David Rourke

The nmr spin system is often described by

$${\frac{\partial}{\partial t}} \begin{pmatrix}m_x\\ m_y\\ m_z\end{pmatrix} \begin{pmatrix}-\frac{1}{T_2}&-\omega_3&\omega_2(t)\\ \omega_3&-\frac{1}{T_2}&-\omega_1(t)\\ -\omega_2(t)&\omega_1(t)&-\frac{1}{T_1}\end{pmatrix} \begin{pmatrix}m_x\\ m_y\\ m_z\end{pmatrix} \begin{pmatrix}0\\ 0\\ \frac{1}{T_1}\end{pmatrix}$$ (1)

These three simultaneous linear first order ordinary differential equations are known as the Bloch equations [1]. Hence, spins with a free-precession frequency (in a rotating frame) $\omega_{3}^{}$, are described by a magnetization vector m = (m_x, m_y, m_z) which evolves under a radiofrequency pulse $\omega$(t) = $\omega_{1}^{}$(t) + i$\omega_{2}^{}$(t) according to Eq. (1). The magnetization also experiences longitudinal and transverse relaxation with characteristic times T₁ and T₂.

The pulse design problem is to find a function $\omega$(t) such that, for a given initial m, the magnetization vector at some later time is a given function of $\omega_{3}^{}$. That is, the pulse design problem requires the Bloch equations to be ``inverted''.

In the presence of relaxation, this is still an open problem. It is not even known what constraints there are on allowable final functions m($\omega_{3}^{}$). Its solution is important, for example, in the design of ``slice-selective'' pulses that only excite spins that have a given range of free-precession frequencies.

Without relaxation, the inversion is a solved problem. The best method of solution appears to be to rewrite the Bloch equations as an equivalent spinor equation of motion (which is a linear second order differential equation), and then note that this is a scattering problem--the Zakharov-Shabat eigenvalue problem [2,3,4]. Hence, the inversion of the Bloch equations is an inverse scattering problem. Furthermore, the design of pulses that do something interesting (e.g., that result in slice selection) can be reduced to finding pulses that do nothing (such pulses correspond to ``solitons'' in scattering theory). Since solitons can be calculated very efficiently, this provides a fast way of solving the original problem.

In addition to trying to extend these ideas to the Bloch equations with relaxation, we are interested in the ``multidimensional'' case, where now $\omega_{3}^{}$ = r ^. g(t). That is, the free-precession frequency is given by a time-varying magnetic field gradient, g(t), and the spatial position, r. Now both initial and final magnetizations are specified as functions of r, and the design problem is to find $\omega$(t) for a given g(t) (assuming such an $\omega$(t) exists).

Finally, even though the system with no relaxation, and time-constant $\omega_{3}^{}$, is ``solved'', there remain important open questions. For example, can a pulse shape be found that results in a desired magnetization response, and that is also ``shape-invariant''? That is, the response remains constant (or approximately so) even if the amplitude of the pulse is changed, i.e., $\omega$(t) is replaced by k$\omega$(t), for any k in the range [1, 1 + $\delta$). Powerful approaches to this problem include the ``adiabatic approximation'' [5,6,7], and the ``recursive expansion'' method [8]. This is an important problem, as the rf coils used to produce the pulse do not produce a uniform strength of pulse throughout the sample. Different rf coils would result in a different value of $\delta$ above.

Click here to see a list of papers that are relevant in the history of pulse design.

Bibliography

1

F. Bloch.
Nuclear induction.
Phys. Rev., 70(7,8):460-474, 1946.

2

V. E. Zakharov and A. B. Shabat.
Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media.
Zh. Éksp. Teor. Fiz., 61:118-134, 1971.
[Sov. Phys. JETP, Vol. 34, pp. 62-69 (1972)].

3

David E. Rourke and Peter G. Morris.
Half-solitons as solutions to the Zakharov-Shabat eigenvalue problem for rational reflection coefficient, with application in the design of selective pulses in NMR.
Phys. Rev. A, 46(7):3631-3636, 1992.

4

David E. Rourke and John K. Saunders.
Half-solitons as solutions to the Zakharov-Shabat eigenvalue problem for rational reflection coefficient. II. Potentials on infinite support.
J. Math. Phys., 35(2):848-872, 1994.

5

J. Baum, R. Tycko, and A. Pines.
Broadband and adiabatic inversion of a two-level system by phase-modulated pulses.
Phys. Rev. A, 32(6):3435-3447, 1985.

6

M. S. Silver, R. I. Joseph, and D. I. Hoult.
Selective spin inversion in nuclear magnetic resonance and coherent optics through an exact solution of the Bloch-Ricatti equation.
Phys. Rev. A, 31(4):2753-2755, 1985.

7

Michael Garwood and Yong Ke.
Symmetric pulses to induce arbitrary flip angles with compensation for RF inhomogeneity and resonance offsets.
J. Magn. Reson., 94:511-525, 1991.

8

Malcolm H. Levitt and R. R. Ernst.
Composite pulses constructed by a recursive expansion procedure.
J. Magn. Reson., 55:247-254, 1983.