Mathematical Challenges In Magnetic Resonance Imaging (MRI)

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Mathematical challengesin magnetic resonance imaging (MRI)Jeffrey A. FesslerEECS DepartmentThe University of MichiganSIAM Conference on Imaging ScienceJuly 7, 2008Acknowledgements: Doug Noll, Brad Sutton,Valur Olafsson, Amanda Funai, Chunyu Yip, Will Grissom1

The EndsX-ray CTMRIwww.gehealthcare.comwww.cis.rit.eduMRI: excellent soft tissue contrast, and no ionizing radiation.(But, expensive, slow, big, small bone signal.)2

OverviewTwo inverse problems in MRI RF pulse design (spatially selective) Image reconstruction Nonuniform fast Fourier transform (NUFFT) Regularization issues (compressed sensing etc.)Image reconstruction toolbox:http://www.eecs.umich.edu/ fessler3

NMR / MRI History (Abbreviated) 1946. NMR phenomenon discovered independently by Felix Bloch (Stanford) Edward Purcell (Harvard) 1952. Nobel prize in physics to F. Bloch and E. Purcell 1966. Richard Ernst and W. Anderson develop Fourier transform spectroscopy NMR spectroscopy used in physics and chemistry 1971. Ray Damadian discriminates malignant tumors from normal tissueby NMR spectroscopy 1973. Paul Lauterbur and Peter Mansfield (independently) add magnetic field gradients,making images 1991. Nobel prize in chemistry to R. Ernst for NMR spectroscopy contributions 2002. Nobel prize in chemistry to Kurt Wüthrich for using NMR spectroscopyto determine 3D structure of biological macromolecules in solution 2003. Nobel prize in medicine to P. Lauterbur and Sir P. Mansfield! 2005. Lustig, Donoho, Pauly et al. apply compressed sensing ideas to MRI4

Physics5

MRI Scannerwww.magnet.fsu.edu6

Bloch Equation - OverviewNuclei with odd number of protons or neutrons (e.g., 1H) have nuclear spin angular momentum. These magnetic moments tend toalign with an applied magnetic field, and collectively the spins induce local magnetization.The (phenomenological) Bloch Equation describes thetime evolution of local magnetization M (rr ,t):Mx i My jdM(Mz M0)kk M γ B dtT2T1Precession Relaxation Equilibrium 7

Bloch Equation and ImagingMx i My j(Mz M0(rr ))kkd M (rr ,t)B(rr ,t) M (rr ,t) γBdtT2(rr )T1(rr )Image properties depend on: Steady-state magnetization M0(rr ) spin (Hydrogen) density Longitudinal (spin-lattice) relaxation T1(rr ) Transverse (spin-spin) relaxation T2(rr ) Chemical shift(resonant frequency of H is 3.5 ppm lower in fat than in water)Applied field B (rr ,t) includes three components we can control: Main field B0 (static) RF field B 1(t) Field gradients r · G (t) xGx(t) yGy(t) zGz(t)B (rr ,t) B0 B 1(t) r · G (t) k8

Systems view of MRIAppliedmagnetizationreceivedRF coil(s)field Patient pattern (Faraday signalM (rr ,t)sr (t)B (rr ,t)induction)basebandrecordeddemodulatesampledata signal (Larmor(A/D)s(t)yi, i 1, . . . , Mfrequency)displayedreconstruction imagealgorithmf ( r)Research areas: design of RF pulses / gradient waveforms (many possibilities!) coil design contrast agents reconstruction algorithm development / data processing9

Inverse Problem 1:RF Pulse Designfor “Excitation”10

RF Pulse Design: Forward ModelForward model:Applied fieldmagnetizationPatientB (rr ,t) {z}B0 B1(t)pattern} r · G{z(t) k} (Bloch {zM (rr ,t)equation)main RF gradientsRewriting Bloch equation:dB(rr ,t) T [MM (rr ,t) M (rr , 0)]M (rr ,t) M (rr ,t) γBdtwhere 1/T2(rr )00 .01/T2(rr )0T 001/T1(rr )RF pulse design goals: find RF waveform B1(t), 0 t t1 thatinduces some desired magnetization pattern M d (rr ,t1) at pulse end.This is a “noiseless” inverse problem.11

RF Pulse Design: Inverse ProblemProblem is typically over-determined, so apply LS approach:arg min M (rr ,t1) M d (rr ,t1)2{B1(n t )} rsubject to constraints: RF amplitude, bandwidth (hardware) RF power deposition (patient safety)Challenge: no general solution to Bloch equation forward model requires numerical methods inverse problem slow (fine grid sampling in r )12

RF Excitation: Applications(Exciting all spins is relatively easy, cf. NMR spectroscopy) slice selection (1D) spatially selective excitation (2D and 3D) imaging small regions compensating for undesired spin phase evolution (fMRI) compensating for nonuniform coil sensitivity (high field)13

RF Excitation: Slice SelectionBefore Excitation (Equilibrium)50 5 8 6 4 202468468After Ideal Slab Excitationy50 5 8 6 4 20z214

RF Excitation: Slice SelectionoBefore ExcitationAfter 90 ExcitationMzMzzM xyzM xyzzHere, forward model simplifies to (roughly speaking) a Fourier relationship between RF pulse B1(t) and slice profile.Practical RF design methods exist.(Pauly et al., IEEE T-MI, Mar. 1991)15

RF Excitation: Spatially SelectiveExcite only spins within some region of interestChallenges: Computation Magnetic field inhomogeneity Coil field pattern nonuniformity Multiple coils Joint design of RF pulse B1(t) and gradient waveforms G (t)This is an active research area.16

Multiple-coil RF Pulse Design ExampleGrissom et al., MRM, Sep. 2006Approach: linearization,nonuniform FFT, iterative CG17

Example: Iterative RF Pulse DesignTailored RF pulses for through-plane dephasing compensationYip et al., MRM, Nov. 2006Challenge: patient specific, requiring “on line” computation18

Inverse Problem 2:MR Image Reconstruction19

Example: Iterative Reconstruction under B020

Standard MR Image ReconstructionMR k space data Reconstructed ImageCartesian sampling in k-space. An inverse FFT. End of story.Commercial MR system quotes 400 FFTs (2562) per second.21

Non-Cartesian MR Image Reconstruction“k-space” datay (y1, . . . , yM )imagef ( r)kykx k-space trajectory: κ(t) (kx(t), ky(t))spatial coordinates:d r R22

Textbook MRI Measurement ModelIgnoring lots of things, the standard measurement model is:yi s(ti) noisei,i 1, . . . , Ms(t) Zf ( r) e ı2π κ(t) · r d r F( κ(t)) . r: spatial coordinates κ(t): k-space trajectory of the MR pulse sequencef ( r): object’s unknown transverse magnetizationF( κ): Fourier transform of f ( r). We get noisy samples of this!e ı2π κ(t) · r provides spatial information Nobel PrizeGoal of image reconstruction: find f ( r) from measurements {yi}Mi 1.The unknown object f ( r) is a continuous-space function,but the recorded measurements y (y1, . . . , yM ) are finite.Under-determined (ill posed) problem no canonical solution.All MR scans provide only “partial” k-space data.23

Image Reconstruction Strategies Continuous-continuous formulationPretend that a continuum of measurementsare available:ZF( κ) f ( r) e ı2π κ · r d r .The “solution” is an inverse Fouriertransform:Zf ( r) F( κ) eı2π κ · r d κ .Now discretize the integral solution:MMi 1i 1fˆ( r) F( κi) eı2π κi · r wi yiwi eı2π κi · r ,where wi values are “sampling density compensation factors.”Numerous methods for choosing wi values in the literature.For Cartesian sampling, using wi 1/N suffices,and the summation is an inverse FFT.For non-Cartesian sampling, replace summation with gridding.24

Continuous-discrete formulationUse many-to-one linear model: y A f εε, where A : L 2(Rd ) CM .Minimum norm solution (cf. “natural pixels”):min fˆ 2 subject to y A fˆfˆ ı2π κi · rc y.ce,whereAAfˆ A (A A ) 1 y Mii 1 Discrete-discrete formulationAssume parametric model for object:Nf ( r) f j p j( r) .j 1Estimate parameter vector f ( f1, . . . , fN ) from data vector y .25

Why Iterative Image Reconstruction? “Non-Fourier” physical effects such as field inhomogeneity Incorporate prior information, e.g.: support constraints (piecewise) smoothness phase constraints No density compensation needed Statistical modeling may reduce noisePrimary drawbacks of Iterative Methods Algorithm speed Complexity, e.g., choosing regularization parameter(s)26

Model-Based Image Reconstruction: Overview27

Model-Based Image ReconstructionMR signal equation with more complete physics:s(t) Zcoilf ( r) s ı ω( r)t R 2( r)t ı2π κ(t) · red r( r) eeyi s(ti) noisei,i 1, . . . , M scoil( r) Receive-coil sensitivity pattern(s) (for SENSE) ω( r) Off-resonance frequency map(due to field inhomogeneity / magnetic susceptibility) R 2( r) Relaxation mapOther physical factors (?) Eddy current effects; in κ(t) Concomitant gradient terms Chemical shift MotionGoal?(it depends)28

Field Inhomogeneity-Corrected Reconstructions(t) Zcoilf ( r) s ıω( r)t R 2( r)t ı2π κ(t) · red re( r) eGoal: reconstruct f ( r) given field map ω( r).(Assume all other terms are known or unimportant.)MotivationEssential for functional MRI of brain regions near sinus cavities!(Sutton et al., ISMRM 2001; T-MI 2003)29

Sensitivity-Encoded (SENSE) Reconstructions(t) Zcoilf ( r)s ı ω( r)t R 2( r)t ı2π κ(t) · red r( r) eeGoal: reconstruct f ( r) given sensitivity maps scoil( r).(Assume all other terms are known or unimportant.)Can combine SENSE with field inhomogeneity correction “easily.”(Sutton et al., ISMRM 2001, Olafsson et al., ISBI 2006)30

Joint Estimation of Image and Field-Maps(t) Zcoilf ( r) s ıω( r)t R 2( r)t ı2π κ(t) · red re( r) eGoal: estimate both the image f ( r) and the field map ω( r)(Assume all other terms are known or unimportant.)Analogy:joint estimation of emission image and attenuation map in PET.(Sutton et al., ISMRM Workshop, 2001; ISBI 2002; ISMRM 2002;ISMRM 2003; MRM 2004)31

The Kitchen Sinks(t) Zcoilf ( r) s ıω( r)t R 2( r)t ı2π κ(t) · re( r) eed rGoal: estimate image f ( r), field map ω( r), and relaxation map R 2( r)Requires “suitable” k-space trajectory.(Sutton et al., ISMRM 2002; Twieg, MRM, 2003)32

Estimation of Dynamic Rate Mapss(t) Zcoilf ( r) s ıω( r)t R 2( r)t ı2π κ(t) · reed r( r) eGoal: estimate dynamic field map ω( r) and “BOLD effect” R 2( r)given baseline image f ( r) in fMRI.Motion.(Olafsson et al., IEEE T-MI 2008)33

Model-Based Image Reconstruction: Details34

Basic Signal Modelyi s(ti) εi,E[yi] s(ti) Zf ( r) e ı2π κi · r d rGoal: reconstruct f ( r) from y (y1, . . . , yM ).Series expansion of unknown object:Nf ( r) f j p( r r j) usually 2D rect functions.j 1Substituting into signal model yields"#ZN E[yi] f j p( r r j ) e ı2π κi · r d r j 1N ai j f j,N Z p( r r j ) e ı2π κi · r d r f jj 1ai j P( κi) e ı2π κi · r j ,FTp( r) P( κ).j 1Discrete-discrete measurement model with system matrix A {ai j }:y A f ε.Goal: estimate coefficients (pixel values) f ( f1, . . . , fN ) from y .35

Least-Squares EstimationEstimate object by minimizing a simple cost function:f̂f arg min Ψ( f ),Ψ( f ) kyy A f k2f CN data fit term kyy A f k2corresponds to negative log-likelihood of Gaussian distribution Equivalent to maximum-likelihood (ML) estimationIssues: computing minimizer rapidly stopping iteration (?) image quality36

Iterative Minimization by Conjugate GradientsChoose initial guess f (0) (e.g., fast conjugate phase / gridding).Iteration (unregularized): (n)(n)A f (n) y ) gradientg Ψ f A ′(Ap (n) P g (n)preconditionn 0 0,hgg(n), p(n)iγn (n 1) (n 1) , n 0hgg,pid (n) pp(n) γnd (n 1)search directionv (n) A d (n)2αn hdd (n), gg(n)i/kvv(n)kstep sizef (n 1) f (n) αnd (n)updateBottlenecks: computing A f (n) and A ′ r . A is too large to store explicitly (not sparse) Even if A were stored, directly computing A f is O(MN)per iteration, whereas FFT is only O(M log M).37

Computing A f RapidlyNA f ]i [AN ı2π κi · r j af P(κ)ef j,i ij jj 1i 1, . . . , Mj 1 Pixel locations { r j } are uniformly spaced k-space locations { κi} are unequally spaced needs nonuniform fast Fourier transform (NUFFT)38

NUFFT (Type 2) Compute over-sampled FFT of equally-spaced signal samplesInterpolate onto desired unequally-spaced frequency locationsDutt & Rokhlin, SIAM JSC, 1993, Gaussian bell interpolatorFessler & Sutton, IEEE T-SP, 2003, min-max interpolatorand min-max optimized Kaiser-Bessel interpolator.NUFFT toolbox: http://www.eecs.umich.edu/ fessler/codeX(ω)100500 π π/2ω?π/2π39

Worst-Case NUFFT Interpolation ErrorMaximum error for K/N 2 210 4Emax10 610Min Max (uniform)Gaussian (best)Min Max (best L 2)Kaiser Bessel (best)Min Max (L 13, β 1 fit) 810 10102468J1040

NUFFT InterpolationIdeal interpolator would be (impractical) sinc-like (Dirichlet kernel)In practice, we use finite-support frequency-domain interpolators;these have nonuniform spatial response.Spatial “scaling” of the signal before FFT is necessaryto compensate for imperfect interpolation.Open problem: determining optimal scaling function.(Reciprocal of Fourier transform of Kaiser-Bessel function worksreasonably well.)41

Further Acceleration using Toeplitz MatricesCost-function gradient:g(n) A′(AA f (n) y) T f (n) b ,whereT , A ′A ,b , A ′y .In the absence of field inhomogeneity, the Gram matrix T is Toeplitz:M ′ 2A A jk P( κi) e ı2π κi ·( r j rk) .i 1Computing T f (n) requires an ordinary (2 over-sampled) FFT.(Chan & Ng, SIAM Review, 1996)In 2D: block Toeplitz with Toeplitz blocks (BTTB).Precomputing the first column of T and b requires a couple NUFFTs.(Wajer, ISMRM 2001, Eggers ISMRM 2002, Liu ISMRM 2005)This formulation seems ideal for “hardware” FFT systems.42

Unregularized Example: Simulated Data4 x under sampled radial: 6760Phantom Objectπ/40ωY20 π/4π/2ωXπ4 under-sampled radial k-space dataAnalytical k-space data generation43

Unregularized Example: ImagesUnregularized CG, 1:4:60, SNR 401212801128Iterations 1:4:60 of unregularized CG reconstruction44

Unregularized Example: Movie(movie in pdf)45

Unregularized Example: RMS ErrorUnregularized CG100Zero image9080NRMS Error (%)7060504030Noisy LS image2010"Best" image?0010Complexity: when to stop?2030Iteration405060A solution: regularization.46

Unregularized EigenspectrumEigenvalues of A’A for 4x under sampled radial, 32x325100eigenvalue10 510 1010 151001024index47

Regularized Example: Movie(movie in pdf)48

Regularized Example: Image ComparisonTrue Unregularized Edge preserving regularization49

Regularized Example: RMS Error100UnregularizedRegularizedNRMS Error (%)806040200010203040CG Iteration506050

Regularized Least-Squares EstimationEstimate object by minimizing a regularized cost function:Ψ( f ) kyy A f k2 αR( f )f̂f arg min Ψ( f ),f CN data fit term kyy A f k2corresponds to negative log-likelihood of Gaussian distribution regularizing term R( f ) controls noise by penalizing roughness,e.g. :R( f ) Zk f k2 d r regularization parameter α 0controls tradeoff between spatial resolution and noise Equivalent to Bayesian MAP estimation with prior e α R( f )Complexities: choosing R( f ) choosing α computing minimizer rapidly.51

Quadratic regularization1D example: squared differences between neighboring pixel values:N1R( f ) f j f j 1 2 .j 2 2C f k2 whereIn matrix-vector notation, R( f ) 12 kC 1 1 0 0 . . . 0f2 f1 0 1 1 0 . . . 0 . , so C f .C . . fN fN 10 . . . 0 0 1 1For 2D and higher-order differences, modify differencing matrix C .Leads to closed-form solution:C f k2f̂f arg min kyy A f k2 αkCf ′ 1 ′′C C A y. A A αC(a formula of limited practical use for computing f̂f )52

Choosing the Regularization ParameterSpatial resolution analysis (Fessler & Rogers, IEEE T-IP, 1996): ′ 1 ′′CC Ayf̂f A A αCh i 1 ′′′C C A E[yy]E f̂f A A αCh i 1 ′′′CC AA fE f̂f A A αC{z} blurA ′A and C ′C are Toeplitz blur is approximately shift-invariant.Frequency response of blur:H(ω)L(ω) H(ω) αR(ω)A′A e j ) (lowpass) H(ωk ) FFT(AC ′C e j ) (highpass) R(ωk ) FFT(CAdjust α to achieve desired spatial resolution.53

Spatial Resolution Exampleα C’C ejA’A ej101010555000 5 5 5 10 100 10 10100 10 1010R(ω)π0000ωXπ π πωYπ π π0ωX010L H/(H R)πωYYH(ω)ωPSFπ π π0ωXπRadial k-space trajectory, FWHM of PSF is 1.2 pixels54

Spatial Resolution Example: Profiles5H(ω)10x 10500800R(ω)60040020000L(ω)10.80.6 π0ωπ55

Tabulating Spatial Resolution vs Regularization42nd order1st orderFWHM [pixels]3.532.521.51 6 4 20log (β)24682Trajectory specific, but easily computed using a few FFTsWorks only for quadratic regularization56

Resolution/noise tradeoffsNoise analysis:n o 1 ′ 1 ′′′′C C A Cov{yy} A A A αCCCCov f̂f A A αCUsing circulant approximations to A′ A and C ′C yields: H(ωk )2ˆVar f j σε 2k (H(ωk ) αR(ωk ))A′A e j ) (lowpass) H(ωk ) FFT(AC ′C e j ) (highpass) R(ωk ) FFT(C Predicting reconstructed image noise requires just 2 FFTs.(cf. gridding approach?)Adjust α to achieve desired spatial resolution / noise tradeoff.57

1Under sampled radialNyquist sampled radialCartesian0 αRelative standard deviationResolution/Noise Tradeoff Example0.80.60.40.2α 011.21.41.6PSF FWHM [pixels]1.8In short: one can choose α rapidly and predictably for quadratic regularization.258

NUFFT with Field Inhomogeneity?Combine NUFFT with min-max temporal interpolator(Sutton et al., IEEE T-MI, 2003)(forward version of “time segmentation”, Noll, T-MI, 1991)Recall signal model including field inhomogeneity:s(t) Zf ( r) e ıω( r)t e ı2π κ(t) · r d r .Temporal interpolation approximation (aka “time segmentation”):Le ı ω( r)t al (t) e ı ω( r) τll 1for min-max optimized temporal interpolation functions {al (·)}Ll 1.Z hiLf ( r) e ı ω( r) τl e ı2π κ(t) · r d rs(t) al (t)l 1Linear combination of L NUFFT calls.59

Field Corrected Reconstruction ExampleSimulation using known field map ω( r).60

Simulation Quantitative Comparison Computation time? NRMSE between f̂f and f true?Reconstruction Method Time (s) NRMSE NRMSEcomplex magnitudeNo Correction0.061.350.22Full Conjugate Phase4.070.310.19Fast Conjugate Phase0.330.320.19Fast Iterative (10 iters)2.200.040.04Exact Iterative (10 iters)128.160.040.0461

Human Data: Field Correction62

Joint Field-Map / Image ReconstructionSignal model:yi s(ti) εi,After discretization:ω) f ε ,y A (ωs(t) Zf ( r) e ıω( r)t e ı2π κ(t) · r d r .ω) P( κi) e ıω jti e ı2π κi · r j .ai j (ωJoint estimation via regularized (nonlinear) least-squares:ω) arg min kyy A (ωω) f k2 β1R1( f ) β2R2(ωω).( f̂f , ω̂ω RNf CN ,ωAlternating minimization:ω, Using current estimate of fieldmap ω̂update f̂f using CG algorithm. Using current estimate f̂f of image,ω using gradient descent.update fieldmap ω̂Use spiral-in / spiral-out sequence or “racetrack” EPI.(Sutton et al., MRM, 2004)63

Joint Estimation Example(a) uncorr., (b) std. map, (c) joint map, (d) T1 ref, (e) using std, (f) using joint.64

Activation Results: Static vs Dynamic Field MapsFunctional results for the two reconstructions for 3 human subjects.65

Reconstruction using the standard field mapfor (a) subject 1, (b) subject 2, and (c) subject 3.Reconstruction using the jointly estimated field mapfor (d) subject 1, (e) subject 2, and (f) subject 3.Number of pixels with correlation coefficients higher than thresholdsfor (g) subject 1, (h) subject 2, and (i) subject 3.Take home message: dynamic field mapping is possible, using iterative reconstruction as an essential tool.(Standard field maps based on echo-time differences work poorlyfor spiral-in / spiral-out sequences due to phase discrepancies.)

Tracking Respiration-Induced Field Changes67

Nonquadratic RegularizationQuadratic regularization is simple and reduces noise but impairsspatial resolution.Nonquadratic regularization attempts to circumvent this tradeoffEdge-preserving regularization has been investigated some for MRI:N1R( f ) ψ( f j f j 1),j 2 2where ψ rises less rapidly than a parabola, e.g., a hyperbola:pψ(t) 1 (t/δ)2.Challenges choosing regularization parameter(s) characterizing nonlinear reconstruction results68

Edge-Preserving Regularization ExampleTrueQuadraticEdge preservingNRMS 12.6%NRMS 11.0%TV-like convex regularization (see next plenary by Dr. Leonid

Mathematical challenges in magnetic resonance imaging (MRI) Jeffrey A. Fessler EECS Department The University of Michigan SIAM Conference on Imaging Science July 7, 2008 Acknowledgements: Doug Noll, Brad Sutton, Valur Olafsson, Amanda Funai, Chunyu Yip, Will Grissom

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