Models and Models and Equations for Equations for

RF-Pulse RF-Pulse Design*Design*

Charles L. Epstein, PhDCharles L. Epstein, PhD

Departments of Mathematics and RadiologyDepartments of Mathematics and Radiology

University of PennsylvaniaUniversity of Pennsylvania

L SN I

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

Jack Leigh, 1939-2008

Declaration of Conflict of Declaration of Conflict of Interest or RelationshipInterest or Relationship

Speaker Name:Charles L. EpsteinSpeaker Name:Charles L. Epstein

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

RF-pulse designRF-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.

THERE IS AN ALGORITHMIC SOLUTION.

OutlineOutline

1. The Bloch Equation2. Non-selective pulses3. The problem of selective pulse design4. Small flip angle pulses: the Fourier method5. Large flip angle pulses

I. The Spin Domain Bloch Equation (SBDE)II. Scattering and and Inverse Scattering for the

SBDEIII. Selective pulse design and the Inverse

Scattering Transform6. The hard pulse approximation7. SLR8. DIST9. Examples

The Bloch equationThe Bloch equation

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

dt=γB×M

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

M =(Mx,My,Mz)

B =(Bx,By,Bz)

The B-FieldThe B-Field

The B-field has three constituent parts:

B =B0 + B1(t) +G(r,t).

We choose coordinates so that B0=(0,0,b0).B1(t) =(eiω0t(ω1(t) + iω2 (t)),0)

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.

ω0 = γ b0

The gradient fieldsThe gradient fields

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

The Rotating Reference FrameThe Rotating Reference Frame

In the sequel, we work with in the rotating reference defined by B0. The magnetization in this frame, m(t), is given by:

mx (t)

my (t)

mz (t)

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟=

cosω0t sinω0t 0−sinω0t cosω0t 0

0 0 1

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟

Mx(t)My(t)Mz(t)

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟.

The Bloch Equation in the Rotating FrameThe Bloch Equation in the Rotating Frame

d

dt

mx (t)

my (t)

mz (t)

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟=

0 −f γω2 (t)f 0 −γω1(t)

−γω2 (t) γω1(t) 0

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟

mx(t)my(t)mz(t)

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟.

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.

(ω1(t) + iω2 (t),0)

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

(ω1(t) + iω2 (t),0)?

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

Linear versus non-linear dependenceLinear versus non-linear dependence

Non-selective pulsesNon-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:

B1(t) =(ω1(t),0,0).

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: θ(t)

θ(t) = γω1

0

t

∫ (s)ds

The flip angle:

Selective pulse designSelective pulse design

In the basic problem of selective RF-pulse design: The data is: a target magnetization profile: mtar(f)=(mtar

x(f), mtary(f), mtar

z(f)).The goal: To find an RF-envelope: (ω1(t) + iω2 (t),0)

mx ( f ;T )

my ( f ;T )

mz f ;(T )

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟=

cosω0τ −sinω0τ 0sinω0τ cosω0τ 0

0 0 1

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟

mxtar( f )

mytar( f )

mztar( f )

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟.

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.

(ω1(t) + iω2 (t),0)

The small flip angle approximationThe small flip angle approximation

d(m1( f ;t)+ im2 ( f;t))dt

=if(m1( f;t) + im2 ( f;t))−iγ(ω1(t) + iω2 (t))

Starting at equilibrium, the solution at time T is:

m1( f ;T )+ im2 ( f;T ) =−ieifT γ(ω1(s) + iω2 (s))0

T

∫ e−isfds

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:

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

(t) = −iγ (ω1(t) + iω2 (t)),

mx ( f ;T )+ imy( f;T ) =eifT ∧( f )

(t) =1

2π(mx

tar

−∞

∞

∫ ( f ) + imytar ( f ))eiftdt

mxtar ( f )+ imy

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

Small flip angle examplesSmall 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.

30 Fourier pulse

90 Fourier pulse

140 Fourier pulse

2D, 3D, Spatial-Spectral-Pulses2D, 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 pulsesGeneral properties of pulses

1. Sharp transitions in the pulse envelope produce “ringing” in the magnetization profile

2. A longer pulse is needed to produce a sharp transition in the magnetization profile.

3. Shifting an envelope in time leads to a linear phase change in the profile.

““Large” flip angle pulsesLarge” flip angle pulsesThe Spin Domain Bloch EquationThe Spin Domain Bloch Equation

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

The spinor representationThe spinor representation

We represent the spin state as a pair of complex numbers Ψ =(ψ 1,ψ 2 )

such that |ψ 1 |2 + |ψ 2 |

2=1

It is related to the magnetization by

m =(mx + imy,mz) =(2ψ 1*ψ 2 ,|ψ 1 |

2 −|ψ 2 |2 )

Spin Domain Bloch EquationSpin Domain Bloch Equation

Ψf =2ξ.

dΨdt

(ξ;t) =−iξ q(t)

−q* (t) iξ⎛

⎝⎜⎞

⎠⎟Ψ(ξ;t)

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

q(t) =γ2(ω2 (t)−iω1(t)).Where

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

Scattering theory for the SDBEScattering theory for the SDBE

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

q(t)[t0 , t1]

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

Ψ1− (ξ;t) = (e−iξ t ,0)

(a(ξ),b(ξ))t > t1

Ψ1− (ξ;t) = (a(ξ )e−iξ t ,b(ξ )eiξ t ).

Scattering by an RF-Scattering by an RF-envelopeenvelope

Scattering data and the target profileScattering data and the target profile

r(ξ) =b(ξ)a(ξ)

=mx(2ξ;t) + imy(2ξ;t)

1+ mz(2ξ;t)e−2iξt

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.

e−2iξt

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:

r(ξ)

rf

2⎛⎝⎜

⎞⎠⎟=

mxtar( f ) + imy

tar( f )1+ mz

tar( f ).

Parseval relationParseval relation

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

|ω1∫ (t) + iω2 (t) |2 dt≥

2πγ 2 log 1+ |r(ξ) |2( )∫ dξ

Power ∝Wlog1

180 −θ⎛⎝⎜

⎞⎠⎟

Pulse design and Inverse ScatteringPulse design and Inverse Scattering

q(t)r(ξ)

mtar(f)

IST

(ω1(t) + iω2 (t),0)

Stereographic projection

Truth in advertisingTruth 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 ISTThe classical IST

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

kt (s)+ F(t,x)ktt

∞

∫ (x)dx=g(t+ s)

The potential is found from:

q(t) =−2kt(t).

The hard pulse approximationThe hard pulse approximation

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

qh (t) = μ j∑ δ(t− jΔ)

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

Hard pulse approximationHard pulse approximation

Hard Pulse Recursion EquationHard Pulse Recursion Equation

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

A j+1(w)Bj+1(w)

⎛

⎝⎜⎞

⎠⎟=

α j −β j*

wβ j wα j

⎛

⎝⎜⎞

⎠⎟Aj (w)Bj (w)

⎛

⎝⎜⎞

⎠⎟

Where:β j =

μ j*

| μ j |

1 + cos | μ j |

2α j = 1− | β j |2w =e2iξΔ

Scattering theory for the Hard Pulse Scattering theory for the Hard Pulse Recursion EquationRecursion 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:

(A j−(w),Bj−(w))

j → −∞

R(w) =lim

j → ∞w−jBj−(w)

Aj−(w)

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

R(w) =rlogw2iΔ

⎛⎝⎜

⎞⎠⎟

Inverse scattering for the HPRE and pulse Inverse scattering for the HPRE and pulse designdesign

The pulse design problem is now reduced to solving the inverse scattering problem for the HPRE: Find a sequence of coefficients μjso 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 algorithmSLR 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:

| R(w) |2

1+ |R(w) |2

Note that the flip angle is:

θ(w) = 2sin−1(| B(w) |)

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

| A(w) |2 + |B(w) |2=1 for |w|=1.

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

SLR schematicSLR schematic

R(w) |B(w)| (A(w),B(w))

{μj}

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 for B(w). On the other hand, the duration of the pulse is specified, in advance by the choice of Δ and the degree of the polynomial B(w).

The Discrete Inverse Scattering Transform The Discrete Inverse Scattering Transform (DIST)(DIST)

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

RD (w) = rjj=−M

N1

∑ wj

The upper limit N1, specifies the rephasing time to be N1Δ.

(AN1 +(w),BN1 +

(w)) =(1,0)

R(w) Rapp(w)

(Aj(w),Bj(w); {μj} )

“Polynomial” design

DIST transform

DIST SchematicDIST Schematic

•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 examplesDIST 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.

140 pulses

SLR DIST

Magnetization profilesMagnetization profilesSLR DIST

Multi-band pulsesMulti-band pulses

SLR DIST

Start band End band Flip angle Phase -1 kHz 1 kHz 140 degrees 0 degrees 1 kHz 3 kHz 90 degrees 60 degrees

SLR DIST

AcknowledgementsAcknowledgements• 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.