EPI Bildgebung

German Chapter of ISMRM

Doktorantentraining

Freiburg 30 Mai - 1 Juni 2001

Review of k-space and FTThe EPI sequenceTechnical requirementsChoice of the readout waveform

ramp sampling regriddingtrajectory measurement

The N/2 ghostphase correction with referenceimage-based correctionregridding effects

Off-resonance effects (distortions)field inhomogeneity and Maxwell shiftsunwarping methods

Slowing downinterleavingghosting and phase correctionecho-time shifting

Speeding uphalf-FTparalell methods

Application examples

MRI: FOURIER ENCODING

FT

RAW DATA(k - space)

IMAGE(r - space)

ACQUI-SITION

MRI: SCANNING STRATEGIES

RF

Gx

Gy

sample ky

ky

kx

ky

RF

Gx

Gy

ky

kx

STANDARD MRI: LOW DUTY CYCLE

TR

ACQ

RF

GS

GR

GP

Echo-Planar Imaging (EPI)

kx

ky

RF

Gz

Gx

Gy

TRACQ

High duty cycleInstantaneous Mz sampling

Some useful maths

CONVOLUTION

∫+∞

∞−

−=⊗ dzzxgzfxgf )()())((

TAPE RECORD

HEAD PROFILE

f

g

gf ⊗

[ ] [ ] [ ]gfgf FTFTFT =⊗CONVOLUTION THEOREM

SAMPLING DISTRIBUTIONS

∫+∞

∞−

=− )()()( afdxxfaxδa x

f(x)Single sample - Dirac's delta:

Equidistant sampling – Bracewell's "shah"

∑+∞

−∞=−=

nnxx )()III( δ

∑∫+∞

−∞==

nnfdxxfx )()()III(

Correct notation: (x)

0 1 2 …

Properties of (x)

)III( k

∑ −=

)(III

1nak

ak

aδ

k

k

x

x

)III(x

)III(ax

k x

)III( 21+x ( ) ( )[ ]2

122

1 IIIIII −− xx

FT

a 1/a

DISCRETE FT

dkeksldk Ndkni

l

/2)()( πδ∫ ∑

−=

Π=

Ndn

ksNdk

dk

)()(IIIFT

"sampled signal"

∑−

−==

12/

2/

/2~N

Nl

Nnliln ess π

)(ldssl =

x

x

)(ks

Π

( )dkIII

=

DISCRETE FT(2)

)(~)sinc()(III

)()(IIIFT

xsxNdxd

ksNdk

dk

⊗⋅⊗⋅

=

Π

FOV = 1/d RESOLUTION = 1/Nd

TECHNICAL REQUIREMENTS

Goal: EPI 128x128, 2x2mm resolution total time 82ms, without ramp sampling.

Example of a whole-body gradient coil:*efficiency: 0.075 mT/m/AL = 170 µHR = 40 mΩ

* Bruker BGA55 (55 cm)

signal bandwidth: (0.32ms/128) -1 = 400 kHzgradient: 1/(2mm * 0.32ms) = 15.6 kHz/cm = 36 mT/mcurrent: 36/0.075 = 480 Avoltage: 170uH*480A/0.16ms = 510Vpeak heating power: (480A) 2 * 40mΩ = 9.2 kW

0.32 0.16 ms

READOUT SHAPE

Gamplitude

duration t

t = 2G/slew + 1/(G*resol)

FASTEST WAVEFORM: plateau = 2*ramp

RAMP SAMPLING

+50% k-spacerange

r 2r r

G

-30% time2G

-25% timeG

EFFECT of NON-EQUIDISTANT SAMPLING

k

G(t)

FFT: ALL POINTS WITHOUT RAMPS

REGRIDDING

CONVOLUTIONKERNEL W

~ ( )S S W k k kk l ll

l= −∑ ∆

Sl

Needed: kl distribution (trajectory)

REGRIDDING - A CLOSER LOOK

non-equidistant distribution: )(/)()( kDkkkW ll

−= ∑δ

)()( kskWsampled signal:

( )[ ])()()()/(III 0 kskWkKkk ⊗regridded signal:

convolution kernel

sampling density

( )[ ])(~)(~

)(~

)(III 0 xsxWxKxk ⊗⋅⊗image:

FOV depends on initial sampling density

image intensity needs correction

REGRIDDING - NYQUIST CONDITION

k-SPACEGRADIENT

n samples

G

IMAGE DOMAINFT

FIELD OF VIEWsampling rate >= Gmax / FOV

)(kW )(~ xW

Gπ/2

nπ /2 samples

Gπ/2

MEASUREMENT OF k-SPACE TRAJECTORY

Onodera et al. J. Phys. E 20, 416 (1987)

k0

k

Echo centre: k = -k0

k0

time

k(t) plot

REGRIDDING WITH A MEASURED TRAJECTORY

RAW REGRID (TRAPEZE) REGRID (MEASURED)

0

10

20

30

40

50

1 6 11 16 21 26 31 36 41 46 51 56 611 6 11 16 21 26 31 36 41 46 51 56 61

g(t)

k(t)

THE GHOST N/2 GHOST

DATA

EVEN ODD

2DFT 2DFT

IMAGE

Θ(x,y)

π + Θ(x,y+N/2)Θ(x,y) =

A +

Bx +

f(x) +

Cy +

g(x,y)

Gx(t)

δB(x,y)

CAN BECORRECTED

CAN BEADJUSTED

?

The N/2 ghost - linear case

READOUTGRADIENT

TRAJECTORY IMAGE

REFERENCE SCAN METHOD

G read

S (k) S (k)

R (x)e

oe

R (x)o

FT

CALIBRATION PROCESSING

Phase Correction: exp(iφ(x)) = arg( R (x)e R (x) )o

-1

1. FT in readread2. Apply exp(iφ(x)) to odd lines 3. FT phase direction

IMAGE-BASED METHOD

1. FT in read direction2. Split even and odd lines3. FT in phase direction (even and odd):

4. Select a ghost-only line

[ ][ ]

R x y R x y R x y

R x y R x y R x y e

e

oi x

( , ) ( , ) ( , / )

( , ) ( , ) ( , / ) ( )

= + ±

= − ±

FOV

FOV

2

2 φ

y R x y R x yg g g: ( , ) & ( , / )= ± ≠0 2 0FOV

5. Phase correction:

[ ]1),(),(arg)( −−= gego yxRyxRxφ

GHOST CORRECTION - RESULTS

ORIGINAL NON-LINEAR PHASECORRECTION

LINEAR PHASECORRECTION

PERSISTENT GHOSTS

LINEAR 1D PHASE CORRECTION

PHASE-SHIFT MAP

-40

-30

-20

-10

0

10

20

30

1 2 3 4 5 6

φ (rad)

FIT:A + Bx + Cy + D(x2-y2) + E 2xy

Θ (deg)

LINEAR 1D 2nd ORDER 2DCORRECTION:

CONTOUR GHOST EFFECT OF GRIDDING ERRORS

simple regrid even-odd regrid

OFF-RESONANCE EFFECT- SPIN WARP

kx

ky

tim

e

xy

freq

uen

cy

SPIN DENSITY:FATWATER

IMAGE

OFF-RESONANCE EFFECT - EPI

kx

ky

tim

e

xy

freq

uen

cy

SPIN DENSITY:FATWATER

IMAGE

EPI: SUSCEPTIBILITY EFFECTS

5 mT/m 10 mT/m 16 mT/m 80 ms130 ms250 ms

MAXWELL SHIFTS(concomitant gradients)

zB

xB xz

∂∂

=∂

∂ because,without el. fieldsor currents:

0=×∇ B

field generated by "x-grad coil": xz ˆˆ zGxG +

xGB +0

zG B

0

22

022

0 2)()(

BGz

xGBzGxGB ++≈++=B

Maxwell shift at 1T, 20mT/m, z=10cm: 85 Hz

UNWARPING

field map methods

reference scan methods

UNWARPINGO

RIG

INA

LC

OR

RE

CT

ION

EPI 64x64 100kHz 3T

UNWARPING

ORIGINAL CORRECTION REFERENCE

EPI 256x256, 200kHz, 3T

EPI: SEQUENCE

multi-shot (interleaved)

single shot

INTERLEAVING

t

ky

kx SINGLE SHOT

MULTI SHOT

NUMBER OF INTERLEAVES:

1 2 4

8 16

EVEN-ODD ECHO SHIFT

SINGLE SHOT INTERLEAVED

READOUTGRADIENT

TRAJECTORIES

ECHO-SHIFT ARTEFACT

1 SHOT 9 SHOTS

GHOSTING in Multi-Shot EPI

image from even echoes:

image from odd echoes:

R x y R x y A R x y md B R x y mde mm

mm

( , ) ( , ) ( , ) ( , )= + + + +∑ ∑even odd

R x y R x y A R x y md B R x y md ee mm

mm

i x( , ) ( , ) ( , ) ( , ) ( )= + + − +

∑ ∑

even odd

φ

d = FOV/(2*n_shots)

Deriving φ(x) from the image?There should be no even ghosts - stability required!Only high order (week) ghosts can be used - high SNR required!

IMAGE BASED DEGHOSTING -ATTEMPT

STABLESIGNAL:only oddghosts

UNSTABLESIGNAL:even and oddghosts

SOLUTION:ALTERNATING TRAJECTORIES

IMAGE-BASED DEGHOSTING

A

D

STA

ND

AR

DIN

TER

LEA

VIN

GA

LTE

RN

ATI

NG

INTE

RLE

AV

ING

raw non-linear correction linear correction