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Transcript of Declaration of Conflict of Interest or Relationshipweb.yonsei.ac.kr/nipi/lectureNote/reciprocity by...
Declaration of Conflict of Interest or Relationship
Speaker Name: Greig Scott
Consultant for Boston Scientific. Our lab also receives funding from General Electric Healthcare.
I have no conflicts of interest to disclose with regard to the subject matter of this presentation
PMRIL Stanford Electrical Engineering
Topics
• Intro to Reciprocity• Faraday Law & Reciprocity• Circular Polarization Mathematics• Equivalent Sources• Lorentz Reciprocity• Applications
What is Reciprocity?
Left hand polarized
Response to a source is unchanged when source and measurer swapped.
What is Reciprocity?
Left hand polarized
Response to a source is unchanged when source and measurer swapped.
Series Resistor Capacitor
)(ωI
+- Cj
IVω
=AjIdVωε
2=ε)(ωI
E1=J/σσd1
d2
AIJ /=
E2=J/jωε
Assume ejωt so fields match electric circuit convention
A
)(ωIAj
IdVωε
2=ε )(ωI
E1=J1=0σd1
d2 E2=i/jωεA
+-Cj
IVω
=
Response is unchanged when source & observer are swapped
Time Domain Convolution
∫ −= τττ dthitv )()()(
)(ωI +
-
)(1
)( ωω
ω IRCj
RV+
=
)()( ttI δ= )(th
Frequency Domain Output: Product of input & filter response.
Time Domain Output: convolution of input and impulse response.
)(tI
Fourier transform
pair
)(ωH
BMdt
dM×= γ
zBoγω −=
1H, 31P, 19F: γ>0, negative (left hand) angular velocity
17O, electrons : γ< 0, positive (right hand) angular velocity
Spin Precession
Faraday Law & Reciprocity
∫ ⋅−=V
dvdtdemf MB
dtddlEemf Φ
−=⋅= ∫
I = 1
Hoult & Richards, J Magn. Reson, 24:71, 1976for vector potential proof: Haacke et al, Magnetic Resonance Imaging: Physical Principles and Sequence Design, pg 97-99, 1999
M
B
Quasistatic approximation of B. EMF imply loop contour integral.
Real Coils aren’t Simple Loops
Which loop do we use for contour integration?
How do I include matching capacitors, inductors etc?
Circular Polarizationyx jaaa −=+ yx jaaa +=−
yx
tjj
tteee
aaa
)sin()cos(][
φωφω
ωφ
+++=ℜ +
yx
tjj
tteee
aaa
)sin()cos(][
φωφω
ωφ
+−+=ℜ −
0=⋅ ∗+ −
aa 2=⋅−+ aa0=⋅
−− aa
y
x
y
φ
KEY PROPERTIES
xφ
Right hand Left hand
0=⋅ ++ aa
Quadrature Field
Chen, Hoult & Sank: Quadrature Detection Coils…J. Magn. Reson., 54:324,1983
To receive, a quadrature coil must create a field rotating opposite the direction of precession.
-90o
¼ cycle time delay
-90o
¼ cycle time delay
B M
-1
Transmit Receive
+
Constructing Polarized Fields
yyyxxxxy ththtΗ aa )cos()cos()( θωθω +++=
( ) ( ) −+ −++= aaΗ yxyx jy
jx
jy
jxxy ejhehejheh θθθθ
21
21
yx jaaa −=+yx jaaa +=−
yj
yxj
xxyyx eheh aaH θθ +=][)( tj
xyxy eetH ωHℜ= where
Time Domain:
Frequency Domain Phasor:
Circularly Polarized Phasor:
+ve/right hand -ve/ left hand
The Principle of Reciprocity in Signal Strength Calculations, D.I. Hoult, Concepts Magn. Reson. 12:173,2000.
yjHHm yx ⇒∗−ℑ ]2/)[(xjHHe yx ⇒∗−ℜ ]2/)[(
( )
( ) yxxyy
xyyxx
yx
hh
hh
jHHe
a
a
a
θθ
θθ
sincos21
sincos21
]2/)([
−
++
⇒−ℜ −
In Electromagnetics :
In MRI (Hoult):
Imaginary part gives y component
Extract x, y components from real part of complex vector
Reciprocity Tool Kit
• Impressed Electric Current Source• Impressed Magnetic Current Source• Impressed Magnetic Dipole• Lorentz Reciprocity Theorem• Rumsey Reaction Integral
Electric Current Source Ji
iJEjH ++=×∇ )( ωεσHjE ωµ=×∇−
Ji is an impressed current element independent of field
δz zoi rrIJ a)( −= δ
Induces E, H J=σED=εE
B=µH
E
H
Ji
∫∫∫ −=⋅=⋅=⋅ IVdzEIdvIdzEdvJEVV
i )(rδ+
-
EV
Can compute a voltage across Ji from a field E!
N Port Impedance Matrix
aiJ
biJ
ciJ
diJ
Wherever we impress a current, we create current source port
aI
bI
cIdI
+
-
+
-
+-
+ -
ZIV =
ba
V
ai
b VIdvJE −=⋅∫Electric fields integrated over this source reduce to a port voltage.
Magnetic Current Source Ki
EjH )( ωεσ +=×∇
iKHjE +=×∇− ωµ
Ki = impressed magnetic current independent of field
δz zooi rrMjK a)( −= δωµ
Induces E, HJ=σE
D=εEB=µH
E
H
Ki
A magnetic current element can be physically created by a time varying magnetic dipole.
Ki
ioi MjK ωµ~
Voltage Source Concept
biK
ciK
diK
Magnetic current looping a wire creates a voltage source port
aV
bV
cVdV
YVI =
ab
V
ai
b VIdvKH −=⋅∫A loop of K induces an emf in the wire like a transformer.
+
-
+
-
+-
+ -
aiK
wire
Magnetic dipole toroid K
Lorentz Reciprocity Theorem
ai
aa JEjH ++=×∇ )( ωεσai
aa KHjE +=×∇− ωµ
bi
bb JEjH ++=×∇ )( ωεσbi
bb KHjE +=×∇− ωµ
aiK
aiJ
Exp. A: electric current source Jia,
magnet current source Kia
biK
biJ
Exp. B: electric current source Jib,
magnet current source Kib
0)(∫ =⋅×−×S
abba ndSHEHE
Reaction in Reciprocity
ai
aa JEjH ++=×∇ )( ωεσai
aa KHjE +=×∇− ωµ
bi
bb JEjH ++=×∇ )( ωεσbi
bb KHjE +=×∇− ωµ
aiK
aiJ
Exp. A: electric current source Jia,
magnet current source Kia
biK
biJ
Exp. B: electric current source Jib,
magnet current source Kib
∫ ∫ ⋅−⋅=⋅−⋅V V
ai
bai
bbi
abi
a dvKHJEdvKHJE )()(
Beware! σ, ε, µ are actually tensors and must be symmetric.
Reaction: <a,b> <b,a>
NMR Reciprocity CaseExp. A: electric current filament Ji
a Exp. B: magnetic current Kib=jωµoM
∫ ∫ ⋅−=⋅=−V V
bio
aai
bab dvMjHdvJEIV ωµcoil voltage
+ stuff0
aiJ
dtdMK
bi
obi µ=Unit current
I(ω) bV
+
- dtdj ↔ω
If symmetric σ, ε, µ
Vesselle et al, IEEE Trans. Biomed. Eng. 42, 497,1995
Ibrahim, T., Magn. Reson. Med. 54, 677, 2005
aaai EHJ ,→ bbb
io HEMj ,→ωµ
NMR Reciprocity CaseExp. A: electric current filament Ji
a Exp. B: magnetic current Kib=jωµoM
dvMjHIV bio
V
aab ωµ⋅−=− ∫
aaai EHJ ,→ bbb
io HEMj ,→ωµ
aiJ
dtdMK
bi
obi µ=Unit current
I(ω) bV
+
-( ) ++−−++ ⋅+=⋅ aaa mHHMH aab
ia
dvmHI
jVVa
ob+−∫=
ωµ2
Field rotating opposite precession determines sensitivity
( ) 0=⋅ ++++ aa mH
Time Domain Reciprocity
∫ ⋅−=t
io d
ddMtHtV
0
)()()( ττ
τµτ
)(tI δ=
Let a unit current impulse δ(t) generate the time varying field Hxy(t).
The receive signal is convolution of the field impulse response with the time varying magnetization
)(tV dtdM
oµ)(tH xy+
-
Reciprocal Media
HB µ=
=
z
y
x
zzyzxz
yzyyxy
xzxyxx
z
y
x
HHH
BBB
µµµµµµµµµ
=
z
y
x
z
y
x
z
y
x
HHH
BBB
µµ
µ
000000
scalar
x
y
z
x
y
z
x
y
zPrincipal axes align x,y,z Principal axes arbitrary
Reciprocal Media have symmetric material property tensors. Reciprocity is satisfied if material is reciprocal.
T][][ µµ = 0][][ =⋅−⋅ abba HHHH µµIf then
Circuit Reciprocity
)(1 ωi
+- )(2 ωv
+=
2
1
2
1
11
11
ii
CjCj
CjCjR
vv
ωω
ωω
2221
1211
zzzz
1i 2i
2v1v
2112 zz =In reciprocal circuits, Zmn = Znm for n!=m
When Can Reciprocity Fail?
Left hand polarized Left hand
polarized
ionosphereq-
Earth magnetic field
Cyclotron motion
Hmm? What if I’ve got charged ions moving in a magnetic field? Or electron spin acting like a gyroscope, or NMR?
Gyrotropic Media
−=
z
y
x
oz
y
x
HHH
jj
BBB
µµκκµ
0000
Material tensors not symmetric so reciprocity is not satisfied.
T][][ µµ ≠ abba HHHH ][][ µµ ⋅≠⋅If then
−=
z
y
x
z
tt
tt
z
y
x
EEE
jj
DDD
εεηηε
0000
Magnetized plasma (electrons undergo cyclotron motion
Magnetized ferrite
Foundations for Microwave Engineering, Robert E Collin
Bloch equation describes electron motion in ferrites!
Phenomena & Applications
• Surface coil sensitivity asymmetry• Guide wire artifact patterns• Reversed Polarization• RF Current Density Imaging• Electric Properties Tomography
Transmit & Receive Asymmetry
Reciprocity & Gyrotropism in magnetic resonance transduction, James Tropp, Phys. Rev. A, 74, 062103, fig 3, 2006
Different Excitation & Reception Distributions with a Single Loop Transmit-Receive Surface Coil near a head-sized spherical phantom at 300 MHz, C.M. Collins et al, MRM, 47, 1026, fig 2, 2002
3T7T
Guidewire Artifacts
15° flip Ross Venook
Transmit and Receive coupling of a guidewire to a body coil creates two distinct null locations
experiment simulation
Reversed RF PolarizationForward Polarization Reversed Polarization
• MR signal is created only by one circular polarization• This has been exploited for wireless catheter tracking
Haydar Celik et al., MRM 58:1224, 2007.
Pacemaker Lead
Forward Polarization Reversed Polarization
Linear polarized fields generated by conducting structures
Electric Properties Tomography
U Katscher et al, Proc ISMRM 14, 3035, 2006
U Katscher et al, Proc ISMRM 15, 1774, 2007
σ
raw
ε ε
qoq HjjH )(2 ωεσωµ +=∇++ +=∇ HjjH o )(2 ωεσωµ
−− +=∇ HjjH o )(2 ωεσωµ
CHALLENGE: Independently measure H+ and H-
q=x,y,z
Summary
• Lorentz Reciprocity Theorem central to deriving NMR signal detection.
• Impressed current dipole creates H.• Magnetic dipole induces V at current
dipole location.• Rotating phasor a+ “reacts” with a-• Time domain convolution requires
time reversal of one field.
Problems
• I need final freq domain formula showing H+ and m-, to go with time domain convolution, and show H- dot m- gives 0.
• Want time domain to show as picture the applied impulse and then the voltage response.
• See if can make picture of B- and B+ (just as in my paper!)
• Photocopy the Lorentz proof from a textbook.• Can I get a dyadic green’s function plot?• Add impedance definitions, power definitions &
computation
NMR Reciprocity Case
ai
aaa J
dtdDJH ++=×∇
dtdBE
aa =×∇−
dtdDJH
bbb +=×∇
dtdM
dtdBE
bi
o
bb µ+=×∇−
Exp. A: electric current filament Jia Exp. B: magnetic current Ki
b=jωµoM
∫ ∫ ⋅−=⋅V V
bi
oaa
ib dv
dtdMHdvJE µ)(
coil voltage
+ stuff0
Warning: Not time domain YET! Assume ωjdtd
=
aiJ
dtdMK
bi
obi µ=Unit current
I(ω) bV
+
-
Reciprocity Theorems
• Response to a source is unchanged when source and measurer swapped
• Relate response at one source due to a second source to the response at the second source due to the first source.
PMRIL Stanford Electrical Engineering
What is Reciprocity?
Left hand polarized Left hand
polarized
ionosphereq-
Earth magnetic field
Cyclotron motion
Response to a source is unchanged when source and measurer swapped.
Hmm? What if I’ve got charged ions moving in a magnetic field? Or electron spin acting like a gyroscope, or NMR?
Time Domain Interpretation
yx tttm aa )sin()cos()( ωω −=−
yx tttH aa )sin()cos()( ωω −=−
yx tttH aa )sin()cos()( ωω +=+
0=⋅−−−− aa Hm 0)()()(
0
→−⋅= −−∫ τττ dtHmtyt
0≠⋅ ++−− aa Hm )cos()()()(0
tdtHmtyt
ωτττ →−⋅= +−∫
Time convolution is zero for same sense of rotation.
Time convolution yields cos(ωt) for opposing rotation.
Ignored “Stuff”
[ ] dvdt
dBHdt
dBHdvJEJEdvdt
dDEdt
dDEa
bb
aabbaa
bb
a ∫∫∫
⋅−⋅+⋅−⋅+
⋅−⋅
( )abba EEEEj ][][ εεω ⋅−⋅ ( )abba EEEE ][][ σσ ⋅−⋅ ( )abba HHHHj ][][ µµω ⋅−⋅
ED ][ε= EJ ][σ= HB ][µ=
If [ε], [σ], [µ] are symmetric tensors, all terms are zero!
0 0 0
Reciprocal Media have symmetric material property tensors.
NMR Reciprocity CaseExp. A: electric current filament Ji
a Exp. B: magnetic current Kib=jωµoM
∫ ∫ ⋅−=⋅=−V V
bio
aai
bab dvMjHdvJEIV ωµcoil voltage
+ stuff0
ai
aa JEjH ++=×∇ )( ωεσaa HjE ωµ=×∇−
bb EjH )( ωεσ +=×∇bio
bb MjHjE ωµωµ +=×∇−
aiJ
dtdMK
bi
obi µ=Unit current
I(ω) bV
+
-
dtdj ↔ω If symmetric
σ, ε, µ
Vesselle et al, IEEE Trans. Biomed. Eng. 42, 497,1995
Ibrahim, T., Magn. Reson. Med. 54, 677, 2005