MRI: from Physics to Signal Processing

Matthieu Guerquin-Kern

Biomedical Imaging Group

Institute of Micro-engineeringEPFL, Lausanne, Switzerland

Internal presentation, June 2008

TABLE OF CONTENT

2

Physics Excitation & Relaxation Detection

Imaging Principle in 1D Imaging with gradients of field

Parallel MRI Signal Processing point of view

Scalar products Linear problem formalism First step to reconstruction

Physics

3

Resulting magnetization

Spinʼs motion

B0

M

dMdt

= !M!B

Bloch equation

!L = "B0

B1

B0

! ! !L

Physics: Excitation

4

Resulting magnetization

dMdt

= !M!B

Bloch equation

Transverse RF pulse

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

X−component of the magnetization

Y−co

mpo

nent

of t

he m

agne

tizat

ion

100 200 300 400 500 600 700−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

time in ps

X−co

mpo

nent

of t

he m

agne

tizat

ion

T2*

63%

0 200 400 600 800 1000 1200 1400 1600 1800 20000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

time in ps

Z−co

mpo

nent

of t

he m

agne

tizat

ion

T1

63%

Free induction decay

Physics: Relaxation

5

B0

MR detection

6

Some formulae from electro-magnetism:Equivalent current distribution: JM(r, t) =!!!"M(r, t).

Magnetic vector potential A(r, t) = µ04!

!R3

JM(r!,t)!r"r!! d3r#.

Magnetic field B(r, t) =!!!"A(r, t)

Flux !"(t) =!C!

B(r, t) · d2S ="

#C!A(r, t) · dr,

!"(t) =#

R3M(r#, t) · Bu

"(r#) d3r#,

where we define

Bu"(r#) =

µ0

4!

$

#C!

dr" (r# r#)$r# r#$3

,

7

MR detection By

Bx

B

I

!!(t) =!

R3M(r!, t) · Bu

!(r!) d3r!Flux

Synchronous demodulation

m!(t) ! !0

!

R3M(r, 0)e!j

R t0 !"(r,#) d#S!(r) d3r

Complex notations:M(r, t) = Mx(r, t) + jMy(r, t) = M(r, 0)e!j

R t0 !(r,") d"

S!(r) = Bu!,x(r)! jBu

!,y(r)

e!(t) = ! d!!

dt(t)

" ! Im!

!0

"

R3M(r, 0)e!j

R t0 "(r,#) d#S!(r) d3r

#Emf

Imaging: principle in 1D

8Space

B = B0 + Gx

x

! = "BV (t) =

!!(x)ej!t dx

= ej"B0t

!!(x)ej"Gtx dx

" #$ %m(t)

Frequency

t

m(t) = !!(!"Gt)

!(x)

!(x1) < !(x2) < !(x3)

Imaging: apply gradients!

9

z

Gz

!exc

!!

!z

zexc

•Slice selection

•Phase encoding m!kx(t), ky(t)

"=

##!(x, y)ej2!(kx(t)x+ky(t)y) dx dy

with kx(t) =!

2"

! t

0Gx(#) d# and ky(t) =

!

2"

! t

0Gy(#) d#

−8 −6 −4 −2 0 2 4 6 8

−8

−6

−4

−2

0

2

4

6

8 k−space trajectory

kx

k y

10

Array of non-homogeneous coils

Non-cartesian and undersampled trajectory

SENSitivity Encoding[Pruessmann, 1999]

Phase encoding

−60 −40 −20 0 20 40 60−60

−40

−20

0

20

40

60

SenseSpiral k−space trajectory

kx

k y

500 1000 1500 2000 2500 3000 3500 4000 1

2

3

4

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

x 105

index of the coil

modulus of the measurements

index of the k−space sample

Parallel MRI: generalization

SP point of view: projection

11

with !!,k(r) = S!!(r)e"2j"k·rm!(k) = !!!,k , ""

Linear problem formalism

12

Continuous forward model:

Discretization: k(t) !" k[n]

S!(r)!(r) !!

p!Cs

s! [p]c[p]"(r" p)

Discretized model:

m!(k) =!

R2S!(r)!(r)e2j"!k , r" dr + b!(k)

m! [n] = !!(!2"kn)"

p!Cs

s! [p]c[p]e2j""kn , p# + b! [n]

Vector representation: m = Ec + b

measurements original signal

error to the model(noise)

First step to reconstruction

13

Reconstruction often requires the adjoint operator to be computed...

MRI adjoint: MRIH {m!}(r) = S!!(r)

!

Rm!(t)e2j""k(t) , r# dt

MRI measurement followed by adjoint:MRIH MRI{!}(r) = S!

!(r) (g ! S!!) (r)

with g(r) =!

Re2j!!k(t) , r" dt

and "g(!!!) =!

R"(!!! + 2#k(t)) dt

Questions?

14

Thanks for your attention!

Any question?