Nonlinear Reconstruction Methodsfor Magnetic Resonance Imaging

Martin Uecker

Biomedizinische NMR Forschungs GmbHam Max-Planck-Institut fur biophysikalische Chemie, Gottingen

Overview

I Image reconstruction as (nonlinear) inverse problem

I Autocalibrated parallel imaging

I Undersampled radial MRI with total-variation penalty

I Model-based reconstruction for fast spin-echo acquisitions

I Real-time imaging

Direct Image Reconstruction

I Assumption: Signal is Fourier transform of the image:

s(t) =

∫d~x ρ(~x)e−i

~k(t)~x

I Image reconstruction with an FFT algorithm

kx

ky

~k(t ′) = γ∫ t′

0 dt ~G (t)

⇒

k-space

⇒iDFT

image

Requirements:I Short readout (signal equation holds for small t only)I Sampling on a Cartesian grid⇒ Line-by-line scanning

Image Reconstruction as Inverse Problem

Forward problem:

y = Fx + n

x image (and more), F (nonlinear) operator, n noise, y data

Regularized solution:

x? = argmin ‖Fx − y‖22︸ ︷︷ ︸

data consistency

+ αR(x)︸ ︷︷ ︸regularization

Advantages:

I Simple extension to non-Cartesian trajectories

I Modelling of physical effects

I Prior knowledge via suitable regularization terms

Extension to Non-Cartesian Trajectories

Cartesian radial spiral

Practical implementation issues:

I Imperfect gradient waveforms (e.g. delays)

I Efficient implementation of the reconstruction algorithm

Extension to Non-Cartesian Trajectories

Cartesian radial spiral

Practical implementation issues:

I Imperfect gradient waveforms (e.g. delays)

I Efficient implementation of the reconstruction algorithm

Modelling of Physical Effects

Examples:

I Coil sensitivities (parallel imaging)

sj(t) =

∫d~x cj(~x)ρ(~x)e−i

~k(t)~x

I T2 Relaxation

sj(t) =

∫d~x cj(~x)e−R(~x)tρ(~x)e−i

~k(t)~x

I Field inhomogeneities

sj(t) =

∫d~x cj(~x)e−i∆B0(~x)tρ(~x)e−i

~k(t)~x

I Diffusion, flow, motion, ...

⇒ Nonlinear equations with additional unknowns (cj ,R,∆B, ...)

Regularization

I Introduces additional information about the solution

I In case of ill-conditioning: needed for stabilization

Common choices:

I Tikhonov (small norm)

R(x) = ‖W (x − xR)‖22 (often: W = I and xR = 0)

I Total variation (piece-wise constant images)

R(x) =

∫d~r

√|∂1x(~r)|2 + |∂2x(~r)|2

I L1 regularization (sparsity)

R(x) = ‖W (x − xR)‖1

Parallel MRI as Inverse Problem

I Signal from multiple coils (image ρ, sensitivities cj):

sj(t) =

∫Vd~x ρ(~x)cj(~x)e−i~x ·

~k(t)

I Assumption: known sensitivities cj⇒ linear relation between image ρ and data y

I Image reconstruction is a linear inverse problem:

Aρ = y

Ra and Rim, Magn Reson Med 30:142–145 (1993)

Pruessmann et al., Magn Reson Med 4:952–962 (1999)

Auto-calibrated Parallel MRI

Estimation of the coil sensitivities during the measurement

I Object influences sensitivities (dielectric)

I Problems with consistency due to movement (e.g. breathing)

Complete acquisition of the k-space center

I Reconstruction of the fully sampled center

I Removal of the image content

I Postprocessing: smoothing, extrapolation, ...

kx

ky

⇒ ⇒

Nonlinear Inversion

The signal equation for unknown image ρ and unknown coilsensitivities cj is a nonlinear equation Fx = y .

Forward operator:

F : H l([0, 1]3 ,CN)× L2([0, 1]3 ,C)→ L2(range(~k),CN)

(cj , ρ) 7→ yj =

∫d~x cj(~x)ρ(~x) e−i

~k(t)·~x

Reconstruction:I Iteratively regularized Gauss-Newton method (IRGNM)I Smoothness penalty for the coil sensitivities:

‖ρ‖22 + ‖(1 + s|~k |2)lFTcj‖2

2

Advantages:I Better estimation of the coil sensitivities

⇒ Improved image quality

Uecker et al., Magn Reson Med 60:674–682, 2008

Nonlinear Inversion

Algorithm:

I Initialization: ρ = 1 and cj = 0

I Update rule (IRGNM):

(DF (xn)HDF (xn) + αnI )δx = DF (xn)H(y − F (xn)) + αn(x0 − xn)

(solved with the conjugate gradient algorithm)

Regularization:

I αn = qnα0, e.g. q = 1/2

I αn‖ρ‖22 + αn‖(1 + s|~k|2)lFTcj‖2

2

(smoothness of the sensitivities)

I Implementation: multiplication with aweighting matrix: F ′ = F ◦W

Nonlinear Inversion

Experiment:

I Siemens Tim Trio 3 T, 12-channel head coil

I 3D-FLASH, acceleration R = 2× 2

Results:

iterative reconstruction of image and coil sensitivities

Undersampled Radial MRI with Total-Variation Constraint

A motivating example:

I Fourier data from 24 spokes (Shepp-Logan phantom)

I With total variation: exact reconstruction!

Candes E, Romberg J, Tao T, Robust uncertainty principles: Exact signal reconstruction from highlyincomplete frequency information. IEEE Transactions on Information Theory, 52:489–509, 2006

Undersampled Radial MRI with Total-Variation Constraint

Forward operator:

A : ρ 7→∫

d~x cj(~x)ρ(~x)e−i~k(t)~x

~k(t)Reconstruction

I Use of a total-variation regularization term:

x? = argmin‖Ax − y‖22 + αTV (~x)

I Total variation (anisotropic version):

TV (x) =

∫d~r |∂1x(~r)|+ |∂2x(~r)|

I Nonlinear conjugate gradient algorithm

Block KT, Uecker M, Frahm J. Magn Reson Med 57:1086-1098, 2007

Undersampled Radial MRI with Total-Variation Constraint

Experiments:

I Siemens Tim Trio 3 T, 12-channel head coil

I Radial spin-echo sequence with 94, 48, 24 spokes

Results:

Image Reconstruction for Fast Spin-echo Acquisitions

90◦ 180◦

TE1

180◦

TE2

180◦

TE3

e−rt

I Repeated acquisition of echo trains

I T2 relaxation during each trainsampling scheme

(R = 3)

Forward operator:

F : (ρ,R) 7→∫

d~x cj(~x)ρ(~x)e−R(~x)te−i~k(t)~x

Reconstruction:

I Minimization of ‖F (ρ,R)− y‖22

I Nonlinear conjugate gradient algorithm

Image Reconstruction for Fast Spin-echo Acquisitions

Experiments:

I Siemens Tim Trio 3 T, 32-channel coil

I 16 echos, ∆TE = 12.2ms,TR = 3000ms, accl. R = 5

Results:

spin density T2 map synthetic images

Sumpf T, Uecker M, Boretius S, Frahm J, Model-based Nonlinear Inverse Reconstruction for T2 Mappingof Highly Undersampled Spin-Echo MRI Data, submitted 2010.

For radial trajectories: Block KT, Uecker M, Frahm J, IEEE Trans Med Imag, 28, 2009

Real-time MRI

Echo

RF

α

Gz

Gx

Gy

TR

Sequence for fast low-angle shot (FLASH) MRI and interleaved radial k-space scheme.

Advantages of radial sampling:

I Robustness to motion

I Tolerance to undersampling

I Continuous updating of image data

I Self calibration for parallel imaging

Zhang et al., J Magn Reson Imaging, 31:101-109, 2010.

Real-time MRI

Objective:I Robust real-time imaging with high temporal resolution

Reconstruction:I Autocalibrated parallel imaging based on nonlinear inversionI Algorithm extended to non-Cartesian (radial) samplingI Further improvements for real-time imaging

gridding nonlinear inversion real-time version

Figure: short-axis view of a human heart, 15 spokes (30 ms)

Real-time MRI

β = 0 β = 0.8

Figure: short-axis view of a human heart, 15 spokes (30 ms)

Improved regularization:

I Previous frame as prior knowledge: ‖x − βxprev‖22

I Damping factor β to avoid accumulation of errors

⇒ Enhanced recovery of high frequencies

Real-time MRI

unfiltered filtered

Figure: short-axis view of a human heart, 15 spokes (30 ms)

Median Filter:I Applied in the temporal domainI Removes streaking artefactsI Preserves sharp transitions t

1

0

0 5 10 15 20 25

Real-time MRI

unfiltered filtered

Figure: short-axis view of a human heart, 15 spokes (30 ms)

Median Filter:I Applied in the temporal domainI Removes streaking artefactsI Preserves sharp transitions t

1

0

0 5 10 15 20 25

Real-time MRI

Why does the median filter remove streaking artefacts?

I Interleaved k-space sampling scheme

median

I Median: invariant to reordering

⇒ Removes flickering artefacts for static image content

Median as L1 minimization:

x3x1

x?

x2x5x4

x? = argminx

{N∑

k=1

‖xk − x‖1

}

Real-time MRI

before after

Figure: short-axis view of a human heart, 15 spokes (30 ms)

Image filter:

I Edge enhancement

I Denoising

Real-time MRI

Figure: short-axis view of a human heart, 15 spokes (30 ms)

Reconstruction steps:

1. Parallel imaging with nonlinear inverse reconstruction

2. Improved regularization

3. Median filter

4. Further image enhancement

Real-time MRI

Experiments:

I Siemens Tim Trio 2.9 T

I 32 channel cardiac coil(array compression to 12 virtual channels)

I RF-spoiled radial FLASH

I Healthy volunteers

I Free breathing, no synchronisation to ECG

I Image reconstruction with four Tesla C1060 GPUs (Nvidia)

Real-time MRI: Movies of the Human Heart

Acquisition time 50 ms (25 spokes)Spatial resolution 2.0x2.0x8 mm3

Real-time MRI: Movies of the Human Heart

Short-axis view 2-Chamber view

Acquisition time 18 ms (9 spokes)Spatial resolution 2.0x2.0x8 mm3

Acquisition time 22 ms (11 spokes)Spatial resolution 2.0x2.0x8 mm3

Real-time MRI: Movies of the Human Heart

Vessels + coronary artery 2-Chamber view

Acquisition time 30 ms (15 spokes)Spatial resolution 1.5x1.5x8 mm3

Acquisition time 30 ms (15 spokes)Spatial resolution 1.5x1.5x8 mm3

Real-time MRI: Speaking

Acquisition time 50 ms (25 spokes)Spatial resolution 1.7x1.7x10 mm3

Real-time MRI: Flow Dynamics

Acquisition time 30 ms (21 spokes)Spatial resolution 1.5x1.5x5 mm3

Summary

Image reconstruction as inverse problem:

I Simple extension to non-Cartesian acquisitions

I Modelling of physical effects

I Prior knowledge with regularization

Examples:

I Autocalibrated parallel imaging

I Undersampled radial MRI with total-variation penalty

I Model-based reconstruction for fast spin-echo acquisitions

I Real-time imaging