Abstract-The proper design of RF pulses in magnetic resonanceimaging has a direct impact on the quality of acquired images.Several techniques have been proposed to obtain the RF pulseenvelope given the desired slice profile. Unfortunately, thesetechniques do not take into account the limitations of practicalimplementation such as limited amplitude resolution. Moreover,implementing constraints for special RF pulses on most techniquesis not possible. In this work, we propose an approach for designingoptimal RF under theoretically any constraints. The newtechnique poses the RF pulse design problem as a combinatorialoptimization problem and uses efficient techniques from this areato solve this problem. In particular, an objective function isproposed as the norm of the difference between the desired profileand the one obtained from solving the Bloch equations for thecurrent RF pulse design values. Two global optimizationtechniques were implemented using genetic algorithms andsimulated annealing. The results show a significant improvementover conventional design techniques and suggest the practicality ofusing of the new technique for clinical use.Keywords - MRI, RF pulse design, genetic algorithms, simulatedannealing.

I. INTRODUCTION

Magnetic resonance imaging (MRI) is one of the mostpromising imaging modalities. It offers true volumetricacquisition, ability to visualize and quantify flow, andspectroscopic imaging to image both anatomy and function.This technique relies on collecting the signal from an excitedslice or volume within the human body. This excitation isachieved using special RF pulses that are designed to providethe required localization within the imaged volume. The properdesign of RF pulses is important to avoid artifacts such ascross-talk in the acquired images.

The design of RF pulses is a rather difficult problemmathematically [1]. The basic goal of this design is to enablethe desired slice profile to be achieved using the available RFpulse generation hardware. The practical implementation of RFpulse systems consists of a computer that stores the array ofdigital values representing the RF pulse envelope amplitudes.These values are converted using a digital-to-analog converter(DAC) into actual voltage levels. This DAC has a limitedresolution in both amplitude and time. As a result, the generatedenvelope voltages appear like a piecewise constant curve with afixed time step and limited stepwise amplitudes. These levelsmodulate the amplitude of the output of an RF generator beforeapplying this output to the RF coils. The RF coils may be eitherlinear (i.e., allowing only the real component to be applied) orquadrature (i.e., allowing both the real and imaginarycomponents to be used). The actual frequency of the RFgenerator and the applied slice selection magnetic field gradient

determine the position of excited slice. On the other hand, theamplitudes of the RF pulse envelope points determine the shapeof the excitation profile as well as its flip angle.

For low flip angles, the problem of determining the RFpulse profiles can be approximated by the Fourier transform ofthe desired slice profile. An example of that is the use of Sinc-shaped RF pulse envelope to achieve a gate-like slice profile.As the flip angle gets higher, this approximation becomesinaccurate and more advanced pulse design techniques must beused [1,2]. Among the many design methodologies that havebeen proposed in the last decade, the Shinnar-Le Roux (SLR)method is probably the most widely used [2]. This methodworks by transforming the problem into one of designing afinite impulse response (FIR) filter. The solution of thisproblem is obtained as an FIR filter coefficients andsubsequently transformed back into the desired RF pulseenvelope.

Given the extensive literature on FIR filter design, thestrength of the SLR technique is that it taps into some of themost powerful FIR filter design techniques to solve the originalproblem of RF pulse design [2]. Nevertheless, it still has somelimitations that did not enable its performance to be optimal inpractice. In particular, this technique does not take intoconsideration the limited amplitude resolution in designing theRF filter. As it is well known in FIR filter design literature, thelimited precision (i.e., quantization or limited word lengtheffects) in implementing the digital filter may substantiallydeteriorate its performance [3]. Even if the FIR filter designtechnique is modified to takes care of this problem and provideoptimal filter coefficients for a given precision, there is noguarantee that the backward transformation to RF pulsecoefficients would preserve this property for RF pulsecoefficients. In other words, the characteristics of the SLRtransformation do not allow such constraints to be imposed. Infact, it is generally difficult to impose any type of constraints onthe solution (like for example adiabatic constraints). As a result,the obtained design may in fact be suboptimal in many casesthat are common in practical use. An example where difficultyto obtain accurate slice profiles is reported is the use ofunconventional spatial encoding techniques such as waveletencoding and pseudo-Fourier imaging [4]. The implementationof such techniques had to compromise between the need to uselow flip angles to obtain accurate slice profiles for correctencoding and the need for high flip angles for better signal-to-noise ratio. Therefore, a new RF pulse design technique thatcan incorporate practical constraints thus offering a true optimalperformance under the practical implementation constraintswould be rather helpful to solve these problems.

OPTIMAL DESIGN OF RF PULSES WITH ARBITRARY PROFILES INMAGNETIC RESONANCE IMAGING

Ayman M. Khalifa1,2, Abou-Bakr M. Youssef1 and Yasser M. Kadah11Biomedical Engineering Department, Cairo University, Giza, Egypt, E-mail: ymk@internetegypt.com

2Biomedical Engineering Department, Helwan University, Cairo, Egypt

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Title and Subtitle Optimal Design of RF Pulses With Arbitrary Profiles inMagnetic Resonance Imaging

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Supplementary Notes Papers from 23rd Annual International Conference of the IEEE Engineering in Medicine and Biology Society, October25-28, 2001, held in Istanbul Turkey. See also ADM001351 for entire Conference on cd-rom., The original documentcontains color images.

Abstract

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In this work, we formulate the problem of RF pulse designas a combinatorial optimization problem with an arbitrarynumber of constraints. This formulation takes into account thelimited precision of RF pulse generation and provides theoptimal results at any given precision. Unlike SLR technique,the objective function used is based on the computed sliceprofile, which offers a feedback loop to improve the results.Two optimization techniques are applied to solve this problem,namely, genetic algorithms and simulated annealing. Weprovide the detailed implementation details for each and presenttheir results compared to those of the SLR technique.

II. THEORY

Given the definition of the RF pulse, it is possible tocompute the expected slice profile using the solution to theBloch equations. This solution relies on using the analyticalform for the slice profile from a single rectangular pulse ofarbitrary magnitude given in [1]. The Bloch equations relatesthe derivative of the magnetization M to the currentmagnetization value in the form,

(1)

Here, Gz is the slice selection gradient, Bx and By are the twoquadrature components of the RF pulse, γ is the gyromagneticratio. For a rectangular pulse, the solution can be simplycomputed as,

, (2)where t is the duration of the rectangular RF pulse and thematrix exponent can be computed analytically for this problemas in [1].

Keeping in mind the practical implementation of RF pulsesin the form of piecewise constant envelope pulses (i.e., asequence of rectangular pulses of arbitrary amplitudes), theoutput magnetization from one piece serves as the initialcondition for the next. Hence, given any design for the RFpulse, the slice profile can be computed this method. Given thatthe amplitudes of the RF pulses must be represented within acertain number of bits, the problem now becomes the one offinding the optimal combination of amplitudes that would givea slice profile closest to the desired. This problem descriptionshows that this problem is indeed a combinatorial optimizationproblem. Using the rich literature of this area, the solution canbe obtained efficiently and accurately. In this work, we exploretwo of the most prominent techniques in this area, namely,genetic algorithms and simulated annealing.

III. METHODS

A. Optimization Using Genetic Algorithms (GA)

The GA method employs mechanisms analogous to thoseinvolved in natural selection to conduct a search through agiven parameter space for the global optimum of someobjective function [5]. The main features of this approach are:

• A point in the search space is encoded as a chromosome. • A population of N chromosomes/search points ismaintained.

• New points are generated by probabilistically combiningexisting solutions.

• Optimal solutions are evolved by iteratively producing newgenerations of chromosomes using a selective breedingstrategy based on relative values of the objective function forthe different members of the population.

A solution is encoded as a string of genes to form achromosome representing an individual. In many applicationsthe gene values are [0,1] and the chromosomes are simply bitstrings. An objective function, f, is supplied which can decodethe chromosome and assign a fitness value to the individual thechromosome represents.

Given a population of chromosomes the genetic operatorscrossover and mutation can be applied in order to propagatevariation within the population. Crossover takes two parentchromosomes, cuts them at some random gene/bit position andrecombines the opposing sections to create two children. Forexample, for one-point crossover crossing the chromosomes010-11010 and 100-00101 at randomly selected position 3-4gives 010-00101 and 100-11010. Another example is for two-point crossover, where crossing the chromosomes 01-011-010and 10-000-101 at randomly selected two positions 2-3 and 5-6gives 01-000-010 and 10-011-101. Mutation is a backgroundoperator, which selects a gene at random on a given individualand mutates the value for that gene (for bit strings the bit iscomplemented).

The search for an optimal solution starts with a randomlygenerated population of chromosomes and an iterativeprocedure is used to conduct the search. For each iteration, aprocess of selection from the current generation ofchromosomes is followed by applying the genetic operators andre-evaluation of the resulting chromosomes. Selection allocatesa number of trials to each individual according to its relativefitness value. The fitter an individual the more trials it will beallocated and vice versa. Average individuals are allocated onlyone trial. Trials are conducted by applying the geneticoperators (in particular crossover) to selected individuals, thusproducing a new generation of chromosomes. The algorithmprogresses by allocating, at each iteration, ever more trials tothe high performance areas of the search space under theassumption that these areas are associated with shortsubsections of chromosomes which can be recombined usingthe random cut-and-mix of crossover to generate even bettersolutions [5].

The use of GA is robust in that they are not affected byspurious local optima in the solution space. Nevertheless, Theparameters that control the GA can significantly affect itsperformance, and there is no guidance in theory as to howproperly select them. The most important parameters are thepopulation size, the crossover rate, the mutation probability andthe fitness function. The following are the proposed algorithmdetails.

.0

00

MM

BB

BG

BG

M

xy

xz

yz rr&r ⋅=⋅

−−

−⋅= Aγ

)Exp()()( 0 tzMzM ⋅−⋅= Arr

1) Population of the chromosomes: Population represents thesize of the solutions that we are working with. The biggeris the population size, the faster the convergence, but themore the computational burden at the initial iterations. Itcan be fixed or dynamically changed allover the run of thealgorithm. In our problem, a fixed population size of 1000individual has been used.

2) Population Initialization: The chromosomes are initializedrandomly. A chromosome that represents the SLR solutionis added to the initial population.

3) Chromosome structure: Binary chromosome is used. TheRF pulse is encoded into the chromosome as follows. Thereal RF pulse values are converted into discrete onesaccording to the resolution of the D/A of the MRI machine(12 or 16 bit for example). Every bit represents a gene. Themost significant bits of all values are placed adjacent toeach other, then the second most bits and so on untilplacing the least significant bits together at the end. Hence,the chromosome size equals the number of the RF pulseenvelope values times the bit resolution (usually 12 or 16).

4) Fitness criterion: The chromosome is decode to obtain thecorresponding RF pulse envelope amplitudes and its sliceprofile is computed by solving the Bloch equations. Then,an error measure is calculated for the difference betweenthe response of this RF pulse and the desired response.This measure is usually taken as either the 1-norm or 2-norm of the difference vector. The results of this paperwere obtained using the 1-norm.

5) Selection scheme: A biased roulette-wheel selection isused. In this method, the fitness of all the chromosomes inthe population is evaluated. Divide each of these fitnessvalues by the total summation of these fitness values andthen multiply it by 100 to get the percentage that thischromosome can be evolved in the next generation.

6) Crossover: One point crossover is used the probability ofcrossover is taken as 90%.

7) Mutation: The probability of mutation is taken as small as1%.

B. Optimization Using Simulated Annealing (SA)

The concepts of simulated annealing are based on a stronganalogy between the physical annealing process of solids andthe problem of solving large combinatorial optimizationproblems. Annealing is a thermal process for obtaining lowenergy states of a solid in a heat bath. It contains two steps.First, the temperature of the heat bath is increased to amaximum value at which the solid melts. Second, it is decreasecarefully until the particles arrange themselves in the groundstate of the solid. In the ground state the particles are arrangedin a highly structured lattice and the energy of the system isminimal. The ground state of the solid is obtained only if themaximum temperature is sufficiently high and the cooling isdone sufficiently slow. Otherwise the solid will be frozen into ameta-stable state rather than ground state.

The Metropolis algorithm is a numerical technique thatsimulates the annealing process [6]. In this algorithm, given a

current state i of the solid with energy Ei, then a subsequentstate j is generated by applying a perturbation mechanismwhich transforms the current state into a next state by a smalldistortion (for instance by displacement of a particle). Theenergy of the next state is Ej. If the energy difference is lessthan or equal to 0, the state j is accepted as the current state. Ifthe energy difference is greater than 0, the state j is acceptedwith a certain probability which is given by Exp((Ei–Ej)/(KB.T)), where T denotes the temperature of the heat bath andKB is a physical constant known as the Boltzmann constant. Theacceptance rule described above is known as the Metropoliscriterion and the algorithm that goes with it is known as theMetropolis algorithm.

We can apply the Metropolis algorithm to generate asequence of solutions of a combinatorial optimization problem.For this purpose, we assume an analogy between a physicalmany-particle systems and a combinatorial optimizationproblem. This is based on the equivalencies that solutions in thecombinatorial optimization problem are equivalent to states of aphysical system, and that the cost of a solution is equivalent tothe energy of a state.

The control parameter here is equivalent to the temperature.That is, if the energy difference is greater than 0, the state j isaccepted with a certain probability which is given by theMetropolis criterion where KB.T is replaced by a single controlparameter c. Initially the control parameter contains a largevalue. This means that most changes even those with largedeterioration to the objective function will be accepted. Thevalue of c is gradually decreased with each step until itapproaches a very small value near 0. Then no deterioration tothe objective function will be possible. This feature allows thesimulated annealing algorithm to escape from local minima [6].

In order to apply SA to our RF pulse problem, we start withthe RF pulse designed by SLR Algorithm. A new solution isgenerated by adding a vector of random numbers to the vectorcontaining the RF pulse values. The random vector should bewith small values to generate a new solution with energy (costvalue) with small distortion of the old solution. The costfunction used is the same as the evaluation function used in theGA. The control parameter c is decreased linearly with eachstep. The constraints on the solution are applied to the newvalues in every step to make sure that the solution is acceptable(e.g., the constraints of limited amplitude resolution). The newvalues are accepted or rejected according to the outcomes fromthe objective function and the Metropolis criterion. The processin repeated until the changes in the solution become very small.

VI. RESULTS AND DISCUSSION

The proposed techniques were applied to design two RFpulses with rectangular spatial profiles at π/2 and π flip angles.The outcome of the SLR technique is compared to theoutcomes from optimized RF pulses using the GA and SAmethods. The results are shown in Figs.1 and 2. The 1-norm ofthe error between the desired and obtained is shown in Fig. 3.As can be observed, both techniques show a significantimprovement over the design computed using the SLR

Figure 2: Comparison between RF pulses for flip angle of 180 degrees.

Figure 1: Comparison between slice profiles for flip angle of 90 degrees.

Figure 3: Comparison between error in slice profiles versus flip angle.

technique. Also, the design using the SA method appears betterthan that from the GA technique. Even though both techniquesshould theoretically reach the global optimum, we have torealize that both techniques are iterative. This means that theuser decides when the iteration stops both in time and accuracy.Hence, the above results from comparing GA and SAtechniques indicate that the SA technique reaches the solutionfaster than GA in the number of experiments we performed.

One of the most important applications for the proposedtechnique is in the area of unconventional spatial encodingwhere complex RF pulse are to be generate at high accuracy.The proposed technique is expected to enable betterreconstruction accuracy, less image artifacts, and higher signal-to-noise ratio. The study of this area requires the investigationof several of these techniques and the assessment of the resultsand is left to future work.

The addition of constraints on the solution using theproposed approaches is straightforward. In each of thealgorithms for GA or SA, the obtained new solution afterperturbation is passed through a ‘filter’ that checks the validityof the new solution from the point of view of the requiredconstraints. In very much the same way this was used forpractical amplitude resolution, other conditions can also beincluded for specialized RF pulse such as adiabatic pulses. Theinvestigation of this application should be addressed in futurework.

V. CONCLUSIONS

A new optimized RF pulse design approach is proposed.The new technique relies on posing the problem as acombinatorial optimization problem and uses geneticalgorithms and simulated annealing to compute the solutionunder any type of constraints. The results demonstrate thesuccess of the new approach and suggest its potential forpractical use in clinical magnetic resonance imaging.

ACKNOWLEDGMENT

This work was supported in part by IBE Technologies,Egypt.

REFERENCES[1] E.T. Jaynes, “Matrix treatment of nuclear induction,”Phys. Rev., vol. 98, no. 4, pp. 1099-1105, May 1955.

[2] J. Pauly, P. Le Roux, D. Nishimura, and A. Macovski,“Parameter relations for the Shinnar-Le Roux selectiveexcitation pulse design algorithm,” IEEE Trans. Med. Imag.,vol. 10, no. 1, pp. 53-65, March 1991.

[3] C.S. Burrus, et al, Computer-based Exercises for SignalProcessing, Prentice Hall, New Jersey, 1994.

[4] Y.M. Kadah and X. Hu, "Pseudo-Fourier imaging: Atechnique for spatial encoding in MRI," IEEE Trans. Med.Imag., vol. 16, no. 6, pp.893-902, Dec. 1997.

[5] D.E. Goldberg, Genetic Algorithms in Search,Optimization, and Machine Learning, Addison-Wesley,Berkeley, 1989.

[6] E. Aarts and J. Korst, Simulated Annealing and BoltzmannMachines, John Wiley & Sons, New York, 1989.

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