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A Genetic Algorithm for Optimal Design A Genetic Algorithm for Optimal Design of Spectrally Selective k-Spaceof Spectrally Selective k-Space

Douglas C. Noll, Ph.D.Douglas C. Noll, Ph.D.

Depts. of Biomedical Engineering and RadiologyDepts. of Biomedical Engineering and Radiology

University of Michigan, Ann ArborUniversity of Michigan, Ann Arbor

Supported by NIH Grant NS32756Supported by NIH Grant NS32756

Acknowledge the assistance of Sangwoo LeeAcknowledge the assistance of Sangwoo Lee

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OutlineOutline

• Background on Spectral-Spatial ImagingBackground on Spectral-Spatial Imaging• Optimization using Genetic AlgorithmsOptimization using Genetic Algorithms• Optimization ResultsOptimization Results• Experimental FindingsExperimental Findings• SummarySummary

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Stochastic AcquisitionsStochastic Acquisitions

• Sheffler and Hennig Sheffler and Hennig (MRM, 35:569-576, 1996)(MRM, 35:569-576, 1996)

• Recognition that particular acquisitions could Recognition that particular acquisitions could be spectrally and spatially selectivebe spectrally and spatially selective

• Spectral bandwidth ~ 1/TSpectral bandwidth ~ 1/Treadread

StochasticStochasticK-SpaceK-Space

Water OilWater Oil

(From Sheffler & Hennig, MRM, 35:569-576, 1996)(From Sheffler & Hennig, MRM, 35:569-576, 1996)

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Rosette AcquisitionsRosette Acquisitions

• Spectral properties similar to stochastic Spectral properties similar to stochastic imaging, but:imaging, but:– Extra suppression of low spatial frequenciesExtra suppression of low spatial frequencies– Simple parameterizationSimple parameterization– No sharp corners in k-space (reduced slew req.)No sharp corners in k-space (reduced slew req.)

WaterWater FatFat

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SMART ImagingSMART Imaging

• Simultaneous Multislice Acquisition using Simultaneous Multislice Acquisition using Rosette Trajectories (SMART)Rosette Trajectories (SMART)

• Excitation of several (e.g. 3) slicesExcitation of several (e.g. 3) slices• Use of slice gradient to modulate slices to Use of slice gradient to modulate slices to

different frequencies different frequencies • Use of spectral properties of acquisition to Use of spectral properties of acquisition to

differentiate slicesdifferentiate slices• Demodulation of raw data shifts from one Demodulation of raw data shifts from one

slice to anotherslice to another

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SMART ImagingSMART Imaging

3 Runs - Single-slice3 Runs - Single-sliceRosette ImagingRosette Imaging

1 Run - Triple-slice1 Run - Triple-sliceSMART ImagingSMART Imaging

Slice 1Slice 1 Slice 2Slice 2 Slice 3Slice 3

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The Rosette k-space TrajectoryThe Rosette k-space Trajectory

• K-space can be described by:K-space can be described by:

kk(t) = A sin((t) = A sin(11 t)exp(i t)exp(i 22 t) t)

11 = oscillation frequency = oscillation frequency

22 = rotation frequency = rotation frequency

• Peak gradient and slew rate Peak gradient and slew rate constraints:constraints:

ggmaxmax = (2 = (2//) A ) A 11

ssmaxmax = (2 = (2//) A () A (1122 + + 22

22))

11

22

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Stochastic RosettesStochastic Rosettes

• Rosette acquisitions can be randomized by Rosette acquisitions can be randomized by treating each petal as a separate unittreating each petal as a separate unit

• Each petal can be characterized by two Each petal can be characterized by two random numbersrandom numbers

• Method:Method:1.1. Randomly select A from [0.9, 1.1]xARandomly select A from [0.9, 1.1]xA00

2.2. Determine Determine 11 from g from gmaxmax equation equation

3.3. Determine Determine 2,max2,max from s from smaxmax equation equation

4.4. Randomly select Randomly select 22 from [0.5, 1.0]x from [0.5, 1.0]x 2,max2,max

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Stochastic RosettesStochastic Rosettes

• Petals are spliced together so that there are Petals are spliced together so that there are no discontinuities in the gradient waveformsno discontinuities in the gradient waveforms

Petal 1Petal 1Petal 3Petal 3

Petal 2Petal 2

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Challenge: OptimizationChallenge: Optimization

• Stochastic rosette acquisitions:Stochastic rosette acquisitions:– Easy to designEasy to design– Large number of parametersLarge number of parameters– No obvious relationship between parameters and No obvious relationship between parameters and

acquisition performanceacquisition performance

• There are an infinite choice of parameters for There are an infinite choice of parameters for stochastic rosette acquisitionsstochastic rosette acquisitions

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Parameterization of Each TrajectoryParameterization of Each Trajectory• Each petal is characterized by two random Each petal is characterized by two random

numbers, which we will call “genes”numbers, which we will call “genes”• For a trajectory with K=56 petals, there are K For a trajectory with K=56 petals, there are K

genes that make up a “chromosome”genes that make up a “chromosome”

• Each candidate trajectory is characterized by Each candidate trajectory is characterized by a chromosome a chromosome

PetalsPetals 11 22 33 44 …… KK

GenesGenes

AA11 AA22 AA33 AA44 AAKK

2,12,1 2,22,2 2,32,3 2,42,4 2,K2,K

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Genetic AlgorithmGenetic Algorithm

Create InitialCreate InitialPopulationPopulation(e.g. N=64)(e.g. N=64)

Random MutationsRandom Mutations(e.g. 2%)(e.g. 2%)

Evaluate CostEvaluate CostFunctionFunction

Select “Mates”Select “Mates”

Mate by SwappingMate by SwappingRandom SegmentsRandom Segmentsof Chromosomesof Chromosomes

Done?Done?

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Cost FunctionCost Function

• Each trajectory was evaluated by creating k-Each trajectory was evaluated by creating k-space data and reconstructing a simulation space data and reconstructing a simulation object:object:

• The cost function that had two components:The cost function that had two components:– Fidelity of on-resonant reconstruction Fidelity of on-resonant reconstruction

(squared error vs. an artifact-free image)(squared error vs. an artifact-free image)– Suppression of off-resonant data Suppression of off-resonant data

(average image energy for a range of off-resonant (average image energy for a range of off-resonant reconstruction frequencies)reconstruction frequencies)

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Genetic Algorithm ResultsGenetic Algorithm Results

• Average and best cost functions over 200 Average and best cost functions over 200 generations:generations:

Rapid early reduction Rapid early reduction from elimination of “unfit” from elimination of “unfit” members from the members from the breeding poolbreeding pool

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Off-Resonance BehaviorOff-Resonance Behavior

0 Hz0 Hz 50 10050 100 150 150 200 Hz 200 Hz

Stochastic RosettesStochastic Rosettes

Standard RosettesStandard Rosettes

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Off-Resonance BehaviorOff-Resonance Behavior

• Periodic structure in Periodic structure in regular rosettes gives regular rosettes gives uneven spectral uneven spectral behaviorbehavior

• Stochastic rosettes Stochastic rosettes have a more uniform have a more uniform response, though at response, though at times largertimes larger

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Experimental Results -Experimental Results -

• Water/Oil Images in Phantom – each pair of Water/Oil Images in Phantom – each pair of images is reconstructed for a single data setimages is reconstructed for a single data set

Residual Residual waterwater

Water OilWater Oil

StochasticStochasticRosettesRosettes

StandardStandardRosettesRosettes

28 ms Readout28 ms ReadoutRes: 3.3 x 3.3 mm Res: 3.3 x 3.3 mm

ggmaxmax = 22 mT/m = 22 mT/m

ssmaxmax = 175 T/m/s = 175 T/m/s

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SummarySummary

• Stochastic rosette acquisitions are both Stochastic rosette acquisitions are both spatially and spectrally selectivespatially and spectrally selective

• Optimization of acquisition parameters is a Optimization of acquisition parameters is a daunting task:daunting task:– Approximately 100 parametersApproximately 100 parameters– No obvious relationship between parameters and No obvious relationship between parameters and

performanceperformance– Gradient-based optimization methods do not work Gradient-based optimization methods do not work

because the cost function space is too roughbecause the cost function space is too rough

• Genetic algorithms are appropriate for this Genetic algorithms are appropriate for this kind of problemkind of problem