G16.4427 Practical MRI 1
Pulse Design for Parallel MR Transmission
G16.4427 Practical MRI 1 – 12th March 2015
Outline
• K-space analysis of small tip-angle excitation
• RF shimming and Parallel MR Transmission
• B1 mapping
• Accelerated MR excitations
• Homogeneity and SAR minimization
G16.4427 Practical MRI 1 – 12th March 2015
K-Space Interpretation of Small Tip
Angle Excitation
• New point of view for analyzing selective
excitation
• Similar approach as that of data acquisition and
reconstruction
• Strictly valid only for small tip angles excitations
but holds also for flip angles ~ 90°
æ 0
Gix - B1, y ö æ M x ö
æ Mx ö
ç
֍
ç
÷
÷
0
B1,x ÷ ç M y ÷
ç M y ÷ = g ç -Gix
ç
֍
ç
÷
÷
çè M ÷ø
çè B
÷ø çè M ÷ø
B
0
1, y
1,x
z
z
G16.4427 Practical MRI 1 – 12th March 2015
Bloch equation in the
rotating frame
(ignoring T1 and T2)
Excitation K-Space
M z » M0
M xy = -ig GixM xy + ig B1 M 0
M xy » M x + iM y
T
M xy (x) = ig M 0 ò B1 (t)e
B1 » B1,x + iB1, y
0
T
k(t) = -g ò G(s) ds
t
(parametrically describes a
path through the spatial
frequency space)
S(k) = ò
T
0
{ d (k(t) - k) k(t) }dt
3
Parametric description of
the unit-weight trajectory
( sampling structure)
T
òt
- ig xi
G(s) ds
dt
T
M xy (x) = ig M 0 ò B1 (t)e- ig xik (t ) dt
0
M xy (x) = ig M 0 ò W (k)S(k)eik ix dk
K
W (k(t)) =
B1 (t)
g G(t)
Spatial-frequency weighting
of the k-space trajectory
G16.4427 Practical MRI 1 – 12th March 2015
Application to Slice-Selective Excitation
Conventional slice-selective excitation
pulse sequence
k-space interpretation
• K-space is scanned linearly as the RF field is applied
• The location in k-space at time t is the integral of the remaining
gradient waveform
– Origin reached when the remaining waveform integrates to zero
– RF weighting is symmetric with respect to the origin
– Slice profile (Fourier transform of RF weighting) is in phase
• The role of the refocusing lobe is to shift the k-space origin back in
the middle of the RF excitation
G16.4427 Practical MRI 1 – 12th March 2015
Multiple Coil Excitations
RF shimming
a2, φ2
• Distinct but time-constant
amplitudes and phases for each
element
a1, φ1
RF
a3, φ3
a4, φ4
a8, φ8
a7, φ7
• Common gradient and RF
waveform
a5, φ5
a6, φ6
RF
Parallel Transmission
• Distinct and time-varying
amplitudes and phases for each
element
• Common gradient waveform but
distinct RF waveform
RF
RF
RF
a2(t), φ2(t)
a6(t), φ6(t)
a3(t), φ3(t)
a7(t), φ7(t)
RF
G16.4427 Practical MRI 1 – 12th March 2015
RF
RF
RF
Parallel RF Transmission
• Parallel transmission may be used to correct RF
inhomogeneities, control SAR, tailor excitations
• Requires calibration of coil array excitation patterns,
and operates in close analogy to parallel reception
G16.4427 Practical MRI 1 – 12th March 2015
Small Flip Angle Excitation
• Homogeneous volume coil
excitation k-space sampling trajectory
(controlled by the switching gradients)
(Mxy is obtained by multiplying
the profile by iγM0)
m (x) = b(x) ò W (k)S(k)eik ix dk
K
B1 spatial
weighting
Question: what is the B1
spatial weighting?
spatial-frequency weighting
(proportional to the coil driving current)
N
• Transmit coil array
b̂l (x) = å cn,l bn (x)
n=1
L
m (x) = å b̂l (x) ò Wl (k)S(k)e
l=1
K
ik ix
dk
G16.4427 Practical MRI 1 – 12th March 2015
Effective spatial
weighting, to account for
coupling-induced intercoil
correlations
Illustration of Parallel Transmission
L
 (x)   bˆl (x)  Wl (k )S (k )ei 2 k x dk
l 1
k
f l ( x)
1 RF pulse
+ the gradient pulse
G16.4427 Practical MRI 1 – 12th March 2015
Illustration of Parallel Transmission
L
 (x)   bˆl (x)  Wl (k )S (k )ei 2 k x dk
l 1
k
f l ( x)
1 RF pulse
+ the gradient pulse
+ B1 weighting
G16.4427 Practical MRI 1 – 12th March 2015
Illustration of Parallel Transmission
L
 (x)   bˆl (x)  Wl (k )S (k )ei 2 k x dk
l 1
k
f l ( x)
L (coils) RF pulses
+ the gradient pulse
+ B1 weighting
G16.4427 Practical MRI 1 – 12th March 2015
Example: 2D Selective Excitation
G16.4427 Practical MRI 1 – 12th March 2015
Example: 3D Selective Excitation
G16.4427 Practical MRI 1 – 12th March 2015
Outline
• K-space analysis of small tip-angle excitation
• RF shimming and Parallel MR Transmission
• B1 mapping
• Accelerated MR excitations
• Homogeneity and SAR minimization
G16.4427 Practical MRI 1 – 12th March 2015
B1 Mapping
• Accurate transmit RF field (B1+) or flip angle maps are
needed for many MR applications. Examples?
– Correct the results of quantitative methods
– Validate theoretical models for EM calculations
– Testing MR compatibility of implanted objects
– Compensate for B1 inhomogeneities
• Image-based RF field measurements are needed for
in-vivo applications
• Several B1 mapping techniques exists, but further
improvements (time efficiency, anatomical coverage,
accuracy) are needed to use them in the routine
practice and for parallel transmission
G16.4427 Practical MRI 1 – 12th March 2015
Multi-Point Intensity Method
• Non-selective RF pulse (long TR) and FID
– Signal is largely independent from T1 and T2
– S ∝sin(α)
• Step through transmit voltage until the first
signal maximum is found (i.e. α = 90°)
• Other pulse amplitudes would then be set
relative to this calibration pulse
– For GRE:
S
Smax
=
B1+
+
B1,max
sin(g B1+t )
G16.4427 Practical MRI 1 – 12th March 2015
Double Angle Method (DAM)
• Collect two scans, one of which uses twice the RF amplitude of
the other.
sin(2 j )
image value at unknown actual
“Double angle formula”

2cos(

)
j
pixel j
flip angle
sin( j )
S j1 = m j sin(a j ) + e1 j
S j 2 = m j sin(2a j ) + e 2 j
object magnetization
at voxel j
error
G16.4427 Practical MRI 1 – 12th March 2015
æ 1 S j2 ö
â j = arccos ç
÷
è 2 S j1 ø
Double Angle Method (DAM)
• Collect two scans, one of which uses twice the RF amplitude of
the other.
sin(2 j )
image value at unknown actual
“Double angle formula”

2cos(

)
j
pixel j
flip angle
sin( j )
S j1 = m j sin(a j ) + e1 j
S j 2 = m j sin(2a j ) + e 2 j
object magnetization
at voxel j
æ 1 S j2 ö
â j = arccos ç
÷
è 2 S j1 ø
error
• Inefficient method (TR ≥ 5T1 required)
• Performs poorly in regions of low signal
• 2 ambiguities if  j is too large, sensitive to noise if
 j is too small
G16.4427 Practical MRI 1 – 12th March 2015
Phase-Based Method
• Exploits the fact that rotations do not commute
• Final Mxy differs by a phase that depends on the magnitude of
the flip angle α
αx  αy  α-x  α-y  αx  αy
αy  αx  α-y  α-x  αy  αx
G16.4427 Practical MRI 1 – 12th March 2015
3
Df @ a 2
2
Question:
What are pros and cons of the phasebased method?
G16.4427 Practical MRI 1 – 12th March 2015
Phase-Based Method
• Exploits the fact that rotations do not commute
• Final Mxy differs by a phase that depends on the magnitude of
the flip angle α
αx  αy  α-x  α-y  αx  αy
αy  αx  α-y  α-x  αy  αx
3
Df @ a 2
2
• Works better for larger α and shorter pulses ( SAR limitation)
• Only for 3D and sensitive to motion/flow
G16.4427 Practical MRI 1 – 12th March 2015
Actual Flip Angle Imaging (AFI)
• Two identical RF pulses followed by two delays of different
duration (TR1 < TR2 < T1)
Assumption: at the end of both TR1 and
TR2 the transverse magnetization is
completely spoiled (need RF spoiling
with dummy repetition to reach steady
state)
Before each excitation pulse:
M z1 = M 0
M z 2 = M0
E1,2 = e
-
1- E2 + (1- E1 )E2 cos a
1- E1 E2 cos 2 a
The observed signals are:
S1,2 = M z1,2 e
-
TE
T2*
sin a
1- E1 + (1- E2 )E1 cos a
TR1,2
T1
1- E1 E2 cos 2 a
Their ratio is:
S2 1- E1 + (1- E2 )E1 cos a
=
S1 1- E2 + (1- E1 )E2 cos a
»
æ
ç
a » arccos ç
ç
çè
ö
S2 TR2
- 1÷
S1 TR1
÷
TR2 S2 ÷
TR1 S1 ÷ø
G16.4427 Practical MRI 1 – 12th March 2015
1+ (TR2 / TR1 )cos a
(TR2 / TR1 ) + cos a
Any questions?
G16.4427 Practical MRI 1 – 12th March 2015
Outline
• K-space analysis of small tip-angle excitation
• RF shimming and Parallel MR Transmission
• B1 mapping
• Accelerated MR excitations
• Homogeneity and SAR minimization
G16.4427 Practical MRI 1 – 12th March 2015
Parallel Transmit For 2D EPI Excitation
L
m (x) = å b̂l (x) ò Wl (k)S(k)eik ix dk
K
l=1
Complex-valued excitation profile
In the case of a 2D EPI excitation trajectory, let’s define:
fl (x, y) =
òò Wl (kx ,k y )e
i( k x x+k y y )
dk x dk y
k x ,k y
L
+¥
l=1
m=-¥
Periodic excitation pattern associated
with the RF pulse of the lth transmit coil
m (x, y) = å b̂l (x, y) å fl (x - mD x , y)
If the sampling interval in excitation
k-space is sufficiently small, then Δx = 1/Δkx
is big enough that all the aliasing lobes are
outside the FOV:
L
m (x, y) = å b̂l (x, y) f l (x, y)
l=1
If we undersample (i.e. use a larger
sampling interval) in excitation k-space,
then M lobes will alias inside the FOV:
L
M -1
l=1
m=0
m (x, y) = å b̂l (x, y) å f l (x - mD x , y)
G16.4427 Practical MRI 1 – 12th March 2015
Accelerated Parallel MR Excitations
To design our parallel transmit pulse design we need to find the periodic
excitation patterns for each transmit coil such that:
(
æ f (x , y ) ö
1
n
n
ç
÷
...
ç
÷
b̂1 (xn , yn ) ... b̂l (xn , yn ) ... b̂L (xn , yn ) ç f l (xn , yn ) ÷ = m (xn , yn )
ç
÷
...
ç
÷
ç f (x , y ) ÷
è L n n ø
)
(in the case of the EPI
excitation trajectory
we can treat each
position separately)
Question: how would the equation above change for an accelerated RF excitation?
G16.4427 Practical MRI 1 – 12th March 2015
Accelerated Parallel MR Excitations
To design our parallel transmit pulse design we need to find the periodic
excitation patterns for each transmit coil such that:
(
æ f (x , y ) ö
1
n
n
ç
÷
...
ç
÷
b̂1 (xn , yn ) ... b̂l (xn , yn ) ... b̂L (xn , yn ) ç f l (xn , yn ) ÷ = m (xn , yn )
ç
÷
...
ç
÷
ç f (x , y ) ÷
è L n n ø
)
(in the case of the EPI
excitation trajectory
we can treat each
position separately)
We can exploit the extra degrees of freedom to under sample the excitation by a factor M:
æ
b̂1 (xn , yn )
ç
...
ç
ç b̂ (x + mD , y )
x
n
ç 1 n
ç
...
ç
çè b̂1 (xn + M D x , yn )
...
b̂l (xn , yn )
...
...
...
...
... b̂l (xn + mD x , yn ) ...
...
...
...
... b̂l (xn + M D x , yn ) ...
ö æ f (x , y ) ö æ
m (xn , yn )
÷ç 1 n n ÷ ç
...
֍
...
...
÷ ç
b̂L (xn + mD x , yn ) ÷ ç fl (xn , yn ) ÷ = ç m (xn + mD x , yn )
֍
÷ ç
÷
...
...
ç
÷ ç
...
֍
÷ ç
b̂L (xn + M D x , yn ) ÷ø è f L (xn , yn ) ø è m (xn + M D x , yn )
b̂L (xn , yn )
G16.4427 Practical MRI 1 – 12th March 2015
ö
÷
÷
÷
÷
÷
÷
ø
Outline
• K-space analysis of small tip-angle excitation
• RF shimming and Parallel MR Transmission
• B1 mapping
• Accelerated MR excitations
• Homogeneity and SAR minimization
G16.4427 Practical MRI 1 – 12th March 2015
SAR and RF Homogeneity
• SAR management and RF homogeneity are
critical issues at high magnetic field
strengths
– SAR is a potentially elevated
safety concern
– B1 focusing compromises
the underlying SNR increase
G16.4427 Practical MRI 1 – 12th March 2015
RF Power Deposition
in Multiple Coil Excitations
unit current electric field
L
Net Electric Field
E(r, pDt) = å I l ( pDt)e l (r)
l=1
RF energy dissipated in Dt
xp =
2
1
H
s
(r)
E
(r,
pDt)
dtdv
=
I
F Ip
p
òò
2 V Dt
electric
noise
fieldcovariance
covariancematrix
matrix
1 P H
x = åI p F I p
P p=1
Global SAR
EPI excitation trajectory
Small tip angle
{
}
I l ( pDt) µF -1 fl (rn )
RF excitation patterns
Image-domain global SAR
1 N H
x = å fn F fn
N n=1
G16.4427 Practical MRI 1 – 12th March 2015
f n « f (rn )
Pulse Design for SAR Reduction
target excitation
profile at rn
Homogeneous excitation
with minimum SAR
weighting
Optimal excitation patterns
for Parallel Transmission
Minimum global SAR
Time-independent RF shimming
ij
I l ( pDt) = A l e l I( pDt) = a l I p
optimal modulation
Optimal modulation coefficients
shared excitation profile
G16.4427 Practical MRI 1 – 12th March 2015
Minimum SAR with Parallel Tx
8 coils
12 coils
20 coils
SAR = 7.9
SAR = 5.5
SAR = 3.3
Parallel Transmission - 7 Tesla - No Acceleration
G16.4427 Practical MRI 1 – 12th March 2015
Ultimate
Basis Set
SAR = 1
SAR vs. Profile Homogeneity
Bo = 7T
G16.4427 Practical MRI 1 – 12th March 2015
Calibrating the Phi Matrix
1
2
3
x1global
é F11 F12
êF F
= [ 10 0 0 ] ê 21 22
F F
ê 31 32
êë F41 F 42
F13 F14
F 23 F24
F33 F34
F 43 F44
ù
úé 1 ù
ú = F11
úê0
0
ú êë 0 úû
úû
x2global
é F11 F12
êF F
= [ 0 10 0 ] ê 21 22
F F
ê 31 32
êë F41 F 42
F13 F14
F 23 F24
F33 F34
F 43 F44
ù
úé0 ù
ú ê 01 ú = F 22
ú êë 0 úû
úû
4
1
2
3
4
G16.4427 Practical MRI 1 – 20th March 2012
Calibrating the Phi Matrix
1
2
3
x3global
é F11 F12
êF F
= [ 0 0 10 ] ê 21 22
F F
ê 31 32
êë F41 F 42
F13 F14
F 23 F24
F33 F34
F 43 F44
ù
úé0 ù
ú ê 01 ú = F33
ú êë 0 úû
úû
x4global
é F11 F12
êF F
= [ 0 0 0 1 ] ê 21 22
F F
ê 31 32
êë F41 F 42
F13 F14
F 23 F24
F33 F34
F 43 F44
ù
úé0 ù
ú = F 44
úê0
0
ú êë 1 úû
úû
4
1
2
3
4
G16.4427 Practical MRI 1 – 20th March 2012
Calibrating the Phi Matrix
1
2
x5global
3
é F11 F12
êF F
= [ 110 0 ] ê 21 22
F F
ê 31 32
êë F 41 F 42
F13 F14
F 23 F 24
F33 F34
F 43 F 44
ù
úé 1 ù
ú ê 01 ú
ú êë 0 úû
úû
= F11 + F12 + F 21 + F 22
4
F12 = (F21 )*
1
2
3
4
x5global
é F11 F12
êF F
= [ 1 i 0 0 ] ê 21 22
F F
ê 31 32
êë F 41 F 42
F13 F14
F 23 F 24
F33 F34
F 43 F 44
ù
úé 1 ù
ú ê 0i ú
ú êë 0 úû
úû
= F11 + iF12 + iF 21 - F 22
iF12 = (iF21 )*
G16.4427 Practical MRI 1 – 20th March 2012
How Many Measurements?
Channel
Experiment 1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
1
2
3
4
é F11 F12 F13 F14
êF F F F
ê F21 F22 F23 F24
ê 31 32 33 34
ë F 41 F 42 F 43 F 44
4 real diagonal elements
6 complex off-diagonal elements
ù Question:
ú Why 16 measurements?
ú
ú
û
16 variables to estimate
16 (# of channels × # of channels) measurements required
G16.4427 Practical MRI 1 – 20th March 2012
Simulation Results
Simulation set up
Measured and predicted power
Measured and Predicted Power
0.3
1
0.25
Power(Watts)
2
3
4
0.2
0.15
0.1
0.05
1
2
3
4
5
Experiment Number
6
7
8
1
2
3
4
5
Experiment Number
6
7
8
% Error
3
2
1
0
Φ calibrated using
simulated E fields
Φ calibrated using
power measurements
Identical !
G16.4427 Practical MRI 1 – 20th March 2012
Experimental Set Up (7 Tesla)
Directional couplers
Knee setup,
8-channel
parallel Tx
stripline coil
Power meter
Rhodes & Schwarz
NRP-Z11
Computer that
automates the
measurement
RF switch
National Instrument Dual 16x1 MUX
G16.4427 Practical MRI 1 – 20th March 2012
In-Vivo Results
Φ calibrated at 60V
40-miliseconds
- measured
power
• predicted
power
Φ calibrated at 120V
40-miliseconds
G16.4427 Practical MRI 1 – 20th March 2012
Uses of Phi Matrix Calibration
• Prediction and real-time monitoring of
global SAR
• Prediction and real-time monitoring of
individual channel FWD and RFL power
• Real-time detection of Tx chain hardware
failures
• Optimization of RF pulse design for RF
shimming and parallel transmission
G16.4427 Practical MRI 1 – 20th March 2012
Maximum Efficiency RF Shimming
Array transmit efficiency metric:
Average
B1+
strength squared:
Total power deposition
w H w
 H
w w
B1 strength squared

dissipated power
 B
1 
B1
M
H

1
 w H M 1C H Cw  w H w


2 
1
wwHl w
bl,x  ibl ,y  Cw
    r E (r) Bdv
1
# of spatial
l
2 V locations
Maximum Efficiency RF Shimming
 Find w that maximize η
It can be treated as a generalized eigenvalue problem:
• Largest eigenvalue = maximum transmit efficiency ηmax
• Corresponding eigenvector = wmax for maximum efficiency RF shimming
G16.4427 Practical MRI 1 – 20th March 2012
Experiment: Hip Imaging at 7 T
No Shimming
Excitation: 4 ch Tx/Rx loop coils
Receive: 10 ch Tx/Rx
(5 loop/stripline modules)
Conservative parallel transmit SAR
limits were used
Φ matrix computed from forward
and reflected power measures
Flip Angle Maps
90
Γ - matrix
Φ - matrix
0
π
-π
RF Shim weights
G16.4427 Practical MRI 1 – 20th March 2012
RF Shimming Weights Calculation
B1+ map acquisition and
field extraction: ~1 min
Maximum efficiency
weights calculation: < 1 s
Φ matrix calibration:
< 5 seconds
G16.4427 Practical MRI 1 – 20th March 2012
Results: RF Shimming at 7 T
No RF Shimming
Maximum Efficiency RF
Shimming
Transmit Efficiency (η)
117
286
Measured Total Average
Energy Deposition
(Watts)
155
58.8
28.8° ± 10.7°
25.1° ± 10.9°
Mean Flip angle in ROI
G16.4427 Practical MRI 1 – 20th March 2012
Any questions?
G16.4427 Practical MRI 1 – 12th March 2015
See you after Spring Break!
G16.4427 Practical MRI 1 – 12th March 2015