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L. S. N. I. Models and Equations for RF-Pulse Design*. Charles L. Epstein, PhD Departments of Mathematics and Radiology University of Pennsylvania. * This lecture is dedicated to the memory of NMR pioneer, and my former Penn colleague. Jack Leigh, 1939-2008.

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models and equations for rf pulse design





Models and Equations for RF-Pulse Design*

Charles L. Epstein, PhD

Departments of Mathematics and Radiology

University of Pennsylvania

*This lecture is dedicated to the memory of NMR pioneer, and my former Penn colleague

Jack Leigh, 1939-2008

declaration of conflict of interest or relationship

Speaker Name:Charles L. Epstein

I am the author of an “Introduction to the Mathematics of Medical Imaging,” published by SIAM Press, which bears some relationship to the topic of this talk.

Declaration of Conflict of Interest or Relationship

rf pulse design
RF-pulse design

In essentially every application of NMR, one needs to selectively excite spins, and this requires the design of an RF-pulse envelope.

We discuss the problem of designing an RF-envelope to attain a specified transverse magnetization, as a function of the offset frequency, for a single species of spins, assuming that there is no relaxation.


The message of this talk is that this problem, which has retained a certain mystique among MR-physicists, has an exact solution, with efficient numerical implementations, not much harder than the Fast Fourier Transform.


  • The Bloch Equation
  • Non-selective pulses
  • The problem of selective pulse design
  • Small flip angle pulses: the Fourier method
  • Large flip angle pulses
    • The Spin Domain Bloch Equation (SBDE)
    • Scattering and and Inverse Scattering for the SBDE
    • Selective pulse design and the Inverse Scattering Transform
  • The hard pulse approximation
  • SLR
  • DIST
  • Examples
the bloch equation
The Bloch equation

In MRI, the process of RF-pulse design begins with a single mathematical model, the Bloch Phenomenological Equation, without relaxation:

Here denotes the bulk magnetization produced by the nuclear spins, and denotes the applied magnetic induction field.

the b field
The B-Field

The B-field has three constituent parts:

We choose coordinates so that B0=(0,0,b0).

We write vectors in R3, as a complex number, paired with a real number, (a+ib,c). is the Larmor frequency defined by the background field.

the gradient fields
The gradient fields

The gradients are quasi-static fields, G(r,t), which produce a spatial variation in the Larmor frequency. We usef to denote the offset frequency, or local change in the Larmor frequency.

the rotating reference frame
The Rotating Reference Frame

In the sequel, we work with in the rotating reference defined byB0. The magnetization in this frame, m(t), is given by:

the bloch equation in the rotating frame
The Bloch Equation in the Rotating Frame

The vector m(t) satisfies the differential equation:

Recall that f is the offset frequency, and the RF-envelope, is the B1 field in the rotating frame.


The problem of selective pulse design reduces to that of understanding how solutions to the Bloch equation depends on the the B1-field. That is, how does m(f;t)depend on

While, this dependence is non-linear, it can still be understood in great detail.

non selective pulses
Non-selective pulses

The easiest case to analyze is when there is no gradient field, (so f=0) and the B1-field is aligned along a fixed axis:

In this situation the excitation is non-selective. Starting from equilibrium m(0)=(0,0,1), at time t the magnetization is rotated about the x-axis through an angle where:

selective pulse design
Selective pulse design

In the basic problem of selective RF-pulse design:

The data is: a target magnetization profile: mtar(f)=(mtarx(f),mtary(f), mtarz(f)).

The goal: To findan RF-envelope:

non-zero in [0,T] so that at time T:


As noted above, the map from the RF-envelope to m(f;T) is non-linear, so the problem of RF-pulse design is as well.

the small flip angle approximation
The small flip angle approximation

While the general pulse design problem is non-linear, so long as the maximum desired flip angle is “small”, a very simple linear approximation suffices:

Starting at equilibrium, the solution at time T is:


If we set then the solution at time T, is easily expressed in terms of the Fourier transform of 

Since we want m(f;T) to be , applying the inverse Fourier transform we find that:

small flip angle examples
Small flip angle examples

We illustrate the Fourier method by designing pulses intended to excite a window of width 2000Hz with transition regions of 200Hz on either side. Below are pulses with flip angles 30, 90, and 140, and the transverse components they produce starting from equilibrium.

2d 3d spatial spectral pulses
2D, 3D, Spatial-Spectral-Pulses

The small flip angle approximation can also be used to design 2D, 3D and spatial-spectral pulses. One combines varying gradients, and the formalism of excitation k-space to interpret the solution of the linearized Bloch equation as an approximation to a higher dimensional Fourier transform.

general properties of pulses
General properties of pulses
  • Sharp transitions in the pulse envelope produce “ringing” in the magnetization profile
  • A longer pulse is needed to produce a sharp transition in the magnetization profile.
  • Shifting an envelope in time leads to a linear phase change in the profile.
large flip angle pulses the spin domain bloch equation
“Large” flip angle pulsesThe Spin Domain Bloch Equation

The starting point for direct, large flip angle pulse design is the Spin Domain Bloch Equation (SDBE). The SDBEis related, in a simple way, to the Bloch equation for the magnetization, which is the quantum mechanical observable.

the spinor representation
The spinor representation

We represent the spin state as a pair of complex numbers

such that

It is related to the magnetization by

spin domain bloch equation
Spin Domain Bloch Equation

The vector is a function of time, and the spin domain offset frequency . It solves the Spin Domain Bloch Equation:


We call q(t) the “potential function”.

scattering theory for the sdbe
Scattering theory for the SDBE

In applications to NMR, the potential function is nonzero in a finite interval .

For t<t0, the function is a solution to the SDBE, representing the equilibrium state. There are functions of the frequency, so that, for

scattering data and the target profile
Scattering data and the target profile

The functions, a and b are called the scattering coefficients.If m(f;t) is the corresponding solution of the Bloch Equation, then, for t>t1, we have the fundamental relation:

The exponential is connected to rephasing.


The right hand side does not depend on time! The function is called the reflection coefficient. To define a selective excitation we specify a target magnetization profile. This is equivalent to specifying a reflection coefficient:

parseval relation
Parseval relation

The potential q(t) and the scattering coefficient r() are like a Fourier transform pair. They satisfy a non-linear Parseval relation:

pulse design and inverse scattering
Pulse design and Inverse Scattering


Stereographic projection


truth in advertising
Truth in advertising

The inverse scattering problem has optional auxiliary parameters, called bound states. This means that there are infinitely many different solutions to any pulse design problem. If no auxiliary parameters are specified, then one obtains the minimum energy solution. No more will be said about this topic today.

the classical ist
The classical IST

The inverse scattering transform finds q(t) given r(). To find q(t), for each t, we can solve an integral equation of the form:

The potential is found from:

the hard pulse approximation
The hard pulse approximation

We model the RF-pulse envelope as a sum of equally spaced Dirac delta functions:

The Shinnar-Leigh-Le Roux (SLR) method of pulse design makes essential use of the SPDE and the hard pulse approximation.

hard pulse recursion equation
Hard Pulse Recursion Equation

A limiting solution to the SDBE has jumps at the times {j}, and freely precesses in the gradient field between the jumps. At the jumps we have a simple recursion relation (HPRE):


scattering theory for the hard pulse recursion equation
Scattering theory for the Hard Pulse Recursion Equation

Let denote the solution to the recursion that tends to (1,0) as the index then the reflection coefficient , R(w), is the limit:


If we choose the spacing sufficiently small, then this function is related to the target magnetization profile by:

inverse scattering for the hpre and pulse design
Inverse scattering for the HPRE and pulse design

The pulse design problem is now reduced to solving the inverse scattering problem for the HPRE:

Find a sequence of coefficients jso that the reflection coefficient is a good approximation to that defined by the target magnetization profile.

SLR and DIST can be used to solve this problem

slr as an inverse scattering algorithm
SLR as an inverse scattering algorithm

First we find polynomials, (A(w),B(w)),so that the ratio B(w)/A(w) is, in some sense, an approximation to R(w). In most implementations of SLR, one first chooses a polynomial B(w), so that |B(w)|2 is a good approximation to:


Note that the flip angle is:

A polynomial A(w) is then determined using the relation:

for |w|=1.

The phase of B(w) is then selected using standard filter design tools.

slr schematic
SLR schematic

Polynomial design

Hilbert transform




Inverse SLR



A limitation of this approach is that the phase of the magnetization profile is not specified, but is “recovered” in the process of finding the polynomial A(w)and the choice of phase forB(w). On the other hand, the duration of the pulse is specified, in advance by the choice of  and the degree of the polynomial B(w).

the discrete inverse scattering transform dist
The Discrete Inverse Scattering Transform (DIST)

The DIST is another approach to solving the inverse scattering problem for the (HPRE). With DIST we directly approximate R(w):

The upper limit N1, specifies the rephasing time to beN1.

dist schematic
DIST Schematic

“Polynomial” design



DIST transform

(Aj(w),Bj(w); {mj} )


The DIST algorithm provides direct control on the phase, flip angle and rephasing time.

  • It sacrifices direct control on the duration of the pulse.
  • Both algorithms have an approx-imation step and a recursion step.
  • The recursion steps have a computa-tional complexity similar to that of the Fast Fourier Transform.
dist and slr examples
DIST and SLR examples
  • These pulses are designed with the indicated algorithms to produce flip angle 140 in a 2kHz window, with a .2 kHz transition band on either side. The nominal rephasing time is 5ms.



  • Thanks to my collaborator Jeremy Magland for his help understanding this subject and for creating MR pulsetool.
  • Thanks to Felix Wehrli and LSNI.
  • Research partially supported by*
      • NIH R01-AR050068, R01-AR053156
      • DARPA: HR00110510057
      • NSF: DMS06-03973

*Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the NIH, NSF, or DARPA.