Supplementary Material for

Application of the limited-memory quasi-Newton algorithm for multi-dimensional,

large flip-angle RF pulses at 7T

Mads S. Vinding a,1,∗, Daniel Brennerb,1, Desmond H. Y. Tsec, Sebastian Vellmerd, Thomas Vosegaarda, Dieter Suterd,

Tony Stöckerb,e, Ivan I. Maximovd,∗

aCenter for Ultrahigh-Field NMR Spectroscopy, Interdisciplinary Nanoscience Center (iNANO) and Department of

Chemistry, Aarhus University, DK-8000 Aarhus, DenmarkbGerman Center for Neurodegenerative Diseases DZNE, Bonn, GermanycDepartment of Neuropsychology and Psychopharmacology, Faculty of Psychology and Neuroscience, Maastricht

University, The NetherlandsdExperimental Physics III, TU Dortmund University, 44221 Dortmund, Germany eDepartment of Physics and Astronomy, University of Bonn, Bonn, Germany

*Corresponding authors:

Email addresses: msv@inano.au.dk (Mads S. Vinding), ivan@e3.physik.tu-dortmund.de (Ivan I. Maximov)1These authors contributed equally to the work

Further details of the optimization algorithm

We consider the Bloch equation as the dynamical constraint on the magnetization vector, M (r , t )=¿ with r=¿ being

the spatial position and T is vector transposition. We assume that relaxation effects are negligible, and the Bloch

equation without relaxation reduces to

M (r ,t )=Ω (r , t )M (r ,t ) (1)

where the ∙ represents a time-derivative operation and

Ω (r , t )=[ 0 γ G ( t ) ∙ r+∆ω0(r) −ω y(r ,t )−γG ( t ) ∙ r−∆ω0(r ) 0 ωx (r , t)

ωy (r , t) −ωx (r , t) 0 ] (2)

with G(t)=¿ as the magnetic-field gradient, ∆ ω0(r ) as the B0 offset, and ωx (r , t) and ωx (r , t) as the net, resonant

RF field components. These RF field components can further be factored into their spatial and time-dependent parts as

ωx (r , t)=∑c=1

C

( Sc(R ) (r )uc ( t )−Sc( I )(r )vc (t))

(3)

and

ω y (r , t)=∑c=1

C

(Sc( I ) (r )uc (t )+Sc(R )(r)vc (t ))

(4)

whereSc(R)(r) and Sc

(I)(r) are the real and imaginary components of the complex RF field sensitivity, respectively, and

the sum signifies the superposition of C fields from a pTX coil of C channels. The time-dependent uc (t ) and vc ( t )

functions represent the controls we wish to optimize. The optimization objective is to bring the magnetization from an

equilibrium state M (r ,0)=¿ to a final state M (r ,T )at time T. The target state is described by a desired, spatial

dependent flip-angle, θ(r), and flip-phase φ (r ), i.e. M target (r )=M 0(r)¿, and our effort is to bring the

magnetization vector as close as possible to the target.

The optimization problem at each spatial position is written as a cost functional, J(r), to be maximized.

J (r )=Φ (M (r ,T ))−∫0

T

Ψ (uc (t ) , vc ( t ) )dt(5)

Here Φ (M (r ,T ) ) represent the excitation fidelity we want to maximize. It corresponds to the projection of the final

state onto the target state,

Φ (M (r ,T ) )=MT (r ,T )M target (r ) (6)

The functional, Ψ (uc (t ) , vc ( t ))represents a penalty on the controls on all channels. Typically, to avoid excessive

power demands the penalty is formulated to account for integrated RF power, thereby reducing the power dissipation

Ψ (uc ( t ) , vc (t ))=λ∑c=1

C

(uc2 (t )+vc2 (t ) )

(7)

The penalty is weighted by the λ parameter, and the term in Eq. (7) is a regularization term.

Later, J(r) will be integrated over the entire space to eliminate the spatial dependence and to provide us with a single

number of performance, but for clarity we keep it for now.

To solve the optimization problem in Eq. (5) constrained by Eq. (1) we follow the approach of Bryson and Ho [1]. We

introduce a Lagrange multiplier or an adjoined state, L (r ,t ) that yields

J (r )=Φ (M (r ,T ))+∫0

T

{LT (r , t )Ω (r ,t )M (r , t )−LT (r ,t ) M (r , t )−Ψ (uc ( t ) , v c ( t ))}dt .(8)

With M (r ,0)and T being fixed, it is convenient to also define a Hamiltonian of the problem in Eq. (8),

H=LT (r , t )Ω (r , t )M (r ,t )−Ψ (uc ( t ) , vc (t )), (9)

from which we can define the Euler-Lagrange equations that state the necessary conditions for a local optimum,

L (r ,T )=δΦ (M (r ,T ) )δ M (r ,T )

,

δ Hδ M (r ,t )

+ L (r , t )=0,

(10)

(11)

and

δ Hδ uc (t)

= δ Hδ vc (t)

=0.(12)

From Eq. (10) and (11) we get, respectively,

L (r ,T )=M target(r) (13)

and

L (r ,t )=−Ω (r , t )L (r , t ), (14)

where Eq. (13) and Eq. (14) define the fixed boundary and a dynamical equation, similar to Eq. (1), for the adjoined

state, respectively. This is fixed-time, two-point boundary-value optimization problem. The required condition in Eq.

(12) provides us with functions or gradients for the control updates, i.e., how to change the controls until an optimum is

reached. Inserting Eq. (7) in Eq. (9) we get from Eq. (12)

2 λuc (t )=LT (r , t ) δ Ω (r , t )δ uc (t )

M (r , t ) , c=1 ,…,C,(15)

and

2 λvc ( t )=LT (r , t ) δΩ (r , t )δ vc ( t )

M (r , t ) , c=1 ,…,C .(16)

In gradient-ascent approaches, Eq. (15) (or Eq. (16)) is, e.g., used in the control update as [2]

uc( k+1 ) ( t )=uc

( k ) (t )+ε (−2 λuc( k ) (t )+LT (r , t ) δΩ (r , t )

δ uc ( t )M (r , t ))

¿uc( k ) ( t )+ε∇J u , c=1 ,…,C ,

(17)

(18)

where k is the iteration counter, and ε is the step size. We refer to the term in the parentheses, or ∇ J u, as the gradient

with respect to u (∇ being the gradient operator), which the step is taken along. The expressions are similar for vc(k) (t ).

It is well-known that the use of second-order gradient information significantly improves the optimization process. Thus

in Newton optimization methods the second-order derivative manifested in the Hessian matrix is also part of the update.

However, the Hessian matrix itself can be difficult to express and tedious to compute [3] and quasi-Newton methods

exploit an estimate of the Hessian matrix instead.

The BFGS scheme is a particularly popular quasi-Newton method. Still, the derived limited-memory version (L-BFGS)

that emphasizes the use of vectors instead of dense matrices is computationally advantageous. The update scheme from

one iteration k to 𝑘+1 with the approximate Hessian matrix, H u, gradient, ∇ J u, and step-size, ε , reads

uc( k+1 ) ( t )=uc

( k ) (t )+εH u∇ J u , c=1 ,…,C. (19)

We have implemented the optimization problem and update schemes in a MATLAB (MathWorks, Inc. Natick, MA,

USA) software package named “blOCh”. We calculate the gradients through an explicit script; however, the Hessian-

matrix is estimated through MATLABs optimization function fmincon. The gradients can be calculated or estimated

numerically through multiple methods, ideally with exact analytical expressions to facilitate the best Hessian estimation

and optimization performance, but in practice also through first, second or multiple order approximations depending on

the required precision [4]. In this study we use gradients precise to first order. Instead of regularizing the RF power

(with the term in Eq. (7)), we further included a strict limit on the peak RF amplitude through a built-in facility of the

fmincon optimizer. Hence, we kept the λ-value in Eq. (7) equal to 0.

Details of the DREAM settings

DREAM[5–8] acquires quasi simultaneously free induction decay (FID) and stimulated echo (STE) signals with a

centric 2D readout. Initial rectangular preparation pulses (here 500 µs with 33.3 V per coil element) with dephasing

gradients create the necessary magnetization that can then be related to the actual local flip angle via the STE and FID

signals. The whole process was repeated for a total of 16 TX configurations with varying coil phases as described in

Ref. [9]

References

1. Bryson AE, Ho Y-C (1975) Applied optimal control: optimization, estimation, and control. Washington, D. C.: Hemisphere publishing corporation2. Vinding MS, Maximov II, Tošner Z, Nielsen NC (2012) Fast numerical design of spatial-selective rf pulses in MRI using Krotov and quasi-Newton based optimal control methods. J Chem Phys 137:054203.3. Majewski K, Ritter D (2015) First and second order derivatives for optimizing parallel RF excitation waveforms. J Magn Reson 258:65–80.4. De Fouquieres P, Schirmer SG, Glaser SJ, Kuprov I (2011) Second order gradient ascent pulse engineering. J Magn Reson 212:412–417.

5. Nehrke K, Börnert P (2012) DREAM—a novel approach for robust, ultrafast, multislice B1 mapping. Magn Reson Med 68:1517–1526.6. Brenner D, Tse DHY, Pracht ED, Feiweier T, Stirnberg R, Stöcker T (2014) 3DREAM – A Three-Dimensional Variant of the DREAM Sequence. Proc Intl Soc Mag Reson Med. p 14557. Brenner D, Tse DHY, Ledden PJ, Neumann C, Stöcker T (2015) Rapid and Accurate PTX B1 Mapping Using 3DREAM with Dual Interferometry. Proc Intl Soc Mag Reson Med. p 01098. Nehrke K, Versluis MJ, Webb A, Börnert P (2014) Volumetric B1+ Mapping of the Brain at 7T using DREAM. Magn Reson Med 71:246–256.9. Tse D, Poole MS, Magill AW, Felder J, Brenner D, Jon SN (2014) Encoding methods for B1 (+) mapping in parallel transmit systems at ultra high field. J Magn Reson 245:125–132.