mr imaging: k-space formalism

Colóquio IFSC - Março/2011

MR Imaging:k-Space formalism

A. Tannús – 11/2006IFSC - USP


1952 (1946): Physics : F. Bloch & E. Purcell:

Precision measurement of Nuclear Magnetism

1942 (1930): Physics: I. Rabbi:Resonant method for measuringmagnetic properties of atomic nuclei.

Nobel prizes:NMR as a source of insight


Nobel prizes in MR 1992 (1966): Chemistry: R. Ernst: High Resolution Pulsed Magnetic Resonance - Spectroscopy.

2003 (1973): Medicine:

P. Mansfield & P. C. Lauterbur

Magnetic Resonance Imaging.


MRI temporal and spatial resolution

Improve

Improve


NMR Phenomena

Quantum Mechanical approach:Easy for spin ½;Gets complex when dealing with

different nuclear species in a system. Classical Approach.

Explain almost completely the development of Imaging methodologies.

To QM..


Classical Approach

μBγ μdtd

L

Fundamental properties of nuclei

Evolution described by an equation of a precessing rotor

LB


Spinning Top in a gravitational field:a very bad example…

“Spinning Top”

Reaction from base

L=angular momentum

“Spinning nucleus”

= magnetic momentL=angular momentum

t = magnetically induced torque = - x B0

t = torque produced by the binary forces:Weight and reaction at contact point

Weight force

B0


Relaxation

2

,,

1

0

)( )(

dtd

))(( )(dtd

TtM

tM

TMtMtM

yxyx

zz

T1 and T2 are determined based on experimental results!

(Phenomenology)

M Macroscopic Magnetization


Bloch Phenomenological Equations:

1

0

2

2

))(()()( )(dtd

)()()()(

dtd

)()()()(dtd

TMtMtBtMtM

TtM

tBtMtM

TtMtBtMtM

zzz

yyy

xxx


Excitation/Detection Scheme

z

x y M

e.m.f

V(t)

B o a)

B1

e.m.f


Detected signal

t

FIDInduced e.m.f.


x

x

y

y

z

z

x

x

x x

y

y

y y

z

z

z z

M L

M L

MT

ML

ML=Mo

MT=Mo

Exc ita tion

MT = M0 e-t/T2

ML = M0 ( 1 - e-t/T1)

tT2 T1<


Imaging Scanner Overview:HardwareFully digital, multichannel now!

Work in progress at our group


Magnet: superconducting,axial access.


Other Magnet TypesPermanent magnets, e.g. light weight rare earth magnets, <0.3T

“H” type, transverse access

“C” type, transverse access (open systems)


Other Magnet Types

“H” and “C” mixed type, transverse

access(open systems)


Other Magnet Types

Electromagnet <0.3T


Field is aligned to subject;Other designs than solenoidal must be used.

Saddle coil allows axial access. Efficiency is low, and homogeneity is poor

RF CoilsRemember:

Brf (B1) must be orthogonal to B0 !!


By mapping the spins according to their position. How?Using their frequency/position correspondence

(r) =B0(r)

Now that we have an NMR signal, how to get an image?

x

y

z

Gy

x

y

z

Gx

x

y

z

Gz

Imaging basic principles: encoding


A bit of history…First 2D NMR image:

came from an annoyance for spectroscopists!!!

P. C. Lauterbur - (1973) State University - New York

z

B0

Gz

x y

G

Projection/Reconstructionmethod(same as in CT)


10 years later…


Encoding inmore than one dimension:

solving the projection paradox.Magnetic field gradients add as vectors, giving a newly oriented

gradient!!


Magnetic Field Gradients

t

zo

z

t

zo

y

t

zo

x

zB(t)= G

, y

B(t)= G

, x

B(t)= G

Now,

gradients

are

time

dependent!!


Spatially encoded frequency and phase:more than one dimension?

t t

oo

o

(t')dt'Gγrt + ω,t')dt' = rω(,t)=rΘ(

tr(t).Gr + γ,t) = ωrω(

(t),Gr,t) = BrB(

thenis and at phase The

then0


Generalizing the definition of k(t)

(t)krΘ(r,t)=

so

(t')dt'G(t) = γkt

o


The 3D Image Equation!!

vo

(t)kri dv)r(M(t))kS( e

3D Signal 3D Image


K-Space properties

velocity(t)Gγ

trajectory(t)k

(t')dt'G(t) = γkt

o

:


Only spins inside this band are excited

Steps to NMR ImagingSelective excitation Absorption line broadening Narrow bandwidth RF pulses

Gz

B0

z

x

y e.m.f.

Gz RF


Selective excitation Absorption line broadening Narrow bandwidth RF pulses

Gz

B0

z

x

y e.m.f

Gy

Phase encoding

Encoding in this dimension is done through the initial phase.

Gy

Principles of NMR Imaging



Gz

Gx

Frequency encoding

B0 z

x

y e.m.f

Gx

Phase encoding

Encoding in this dimension is done through the spatially dependent frequency.

Gy




Gz

Gx Gy

Frequency encodingPhase encoding

dxdyeyxM

dxdyeyxMkkSyx

yxx

iykixk

yGitxGiyx

),(

),(),( t



Acquisition sequences and image formation :

• Spin Echo ( SE )• Echo Planar Imaging ( EPI )• Gradient Recalled Echo ( GRE )

Spin Echo ( SE )

FID

Signal

tA

Gy A

ECO

tC 2t tB

Gx

Preparation

C’

A’

B’

A’’

C’’ B’’ Acquisition

C

Gz

Gx Gy

0

RF p/2

kX

ky

0

Gz

p

t

p

B



k-space

k-space is the raw data space before Fourier transformation into the image

2D image will be represented by a 2D array of data points spread throughout k-space(it could be 3D!!)

Changing the k-space trajectory will alter image contrast


k-space

k-space must be sampled in equally spaced intervals in order to allow 2D FFT.

As a consequence the image is also presented in equally spaced sampled values.

All concepts of discrete Fourier formalism applies.


Image vs. k-space data

(r) S(k)k(t)=

/2pG(t)dt


(r) S(k)k(t)= /2pG(t)dt



(r) S(k)k(t)=

/2pG(t)dt



(r) S(k)k(t)=

/2pG(t)dt

FFT



GE k-space trajectory

(r) S(k)k(t)=

/2pG(t)dt

RF

G S

G R

G P

S(t)


(r) S(k)k(t)=

/2pG(t)dt

RF

G S

G R

G P

S(t)

-kr +kr



(r) S(k)k(t)=

/2pG(t)dt

RF

G S

G R

G P

S(t)

-kr +kr

-kp

+kp



Why does MRI take so long Answer

Only one phase encode line acquired per excitation

Spin Echo: 256*3s for T2, 256*0.6s for T1 Gradient Echo: 256*35ms (but have to do 3D

Solution get more phase encode lines per excitation


Presenting Echo Planar Imaging (EPI)Main tool for Neurosciences

Fastest imaging method Typical Acquisition times: 30-100ms Lower RF deposition Very fast gradient switching Highly demanding on MRI hardware

B0 homogeneitygradient switching

P. Mansfield


(r) S(k)k(t)=

/2pG(t)dt

RF

G S

G R

G P

S(t)

-kr +kr

-kp

+kp

GE-EPI k-space trajectory


Localized Spectroscopy (d)

Following metabolites and biochemical process inside the body

Spectralinformation

Spatialinformation

ppm020406080100120140160180200

ppm020406080100120140160180200

ppm020406080100120140160180200

13C, 31P


Angiography (V)Transverse: Head Coronal: Neck-Chest

Aneurisms


functional MRI (T2*)

Mapping of a finger tapping experiment

mr imaging: k-space formalism

Documents

colquio ifsc maro20111952

colquio ifsc maro2011how

aquisies e

u s s c

w i t h g

aquisio e um proce

magnetic momentl

nas tcnicas tridimensionais