Transcript Slide 1

Problems in MR that really need quantum
mechanics: The density matrix approach
Robert V. Mulkern, PhD
Department of Radiology
Children’s Hospital
Boston, MA
Nuclear Spin: An inherently Quantum
Mechanical (QM) Phenomenon
Angular momentum operators represent spin I
But problems in MR that need QM?
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Proton imaging? Not really…
Relaxation? Not really…
Radiological interpretations? Sometimes…
Spectroscopy? Absolutely…
Spectroscopic imaging? Yes indeed…
X-nuclei? Why not!
Proton Imaging: Our Bread and Butter
T2 Contrast
T1 Contrast
Tissue relaxation rates and pulse sequence specifics determine
Tissue contrast – all understood via the classical Bloch equations
BPP Theory: Used QM to calculate T1, T2 –
1950’s – rarely used in practice
Fluctuations of Dipolar Hamiltonian
QM in Radiological Interpretations?
• Magic angle effect (3cos2 – 1) = 0
• Bright fat effect (quenching of J-coupling with
multiple 180’s)
“When molecules lie at 54.74° there is
lengthening of T2 times (don't understand why, but it involves 'bipolar coupling')”
“Dipolar Coupling” - Magnetic energy
between two dipoles
The Dipolar Hamiltonian
Bright Fat Phenomenon
Where QM Really Rules: Coupled Spin
Systems and Spectroscopy
“Shut up and Calculate”
Richard Feynman
The real beauty of the Density
Matrix Formalism – no thinking…
Spin ½ Rules of the Road
Iz|+> = ½ |+>
Iz|-> = -1/2 |->
Ix = (I+ + I-)/2
Iy = (I+ - I-)/2i
I+|+> = 0
I+|-> = |+>
I-|-> = 0
I-|+> = |->
h = 1, let’s be friends
Commutation Relations
[I,S] = 0 (two spins)
[Ii,Ij] = ijkIk
Typical Hamiltonians of Interest
1) H = woIz
2) H = (wo + /2)Iz + (wo – /2)Sz + JIzSz
3) H = (wo + /2)Iz + (wo – /2)Sz + JIxSx + J IySy + JIzSz
4) H = w1Iy or w1Ix
RF pulses
Weak vs strong and “secular” terms: J
<<  means weak and no secular terms
Density Matrix Example: Free
Precession
t
1
2
y
H = woIz
H|+> = (1/2)wo|+>
H|-> = -(1/2)wo|->
 = exp(-iHt)exp(-iIy)Izexp(iIy)exp(iHt)
Calculate the Signal as Tr{(Ix+iIy)} = Tr{I+}
The Matrix and its Trace
Tr{(Ix+iIy)} = Tr{I+}
<+|I+|+>
<-|I+|+>
<+|I+|->
<-|I+|->
<-|I+|-> = only nonvanishing diagonal element
<-|exp(-iHt)exp(-iIy)Izexp(iIy)exp(iHt)|+> =
exp(iwot/2) <-|exp(-iHt)exp(-iIy)Izexp(iIy)|+> = ?
How to handle the RF pulses?
The Pauli Spin Matrices
Wolfgang Pauli
Matrix Representations of Angular
Momentum Operators
A2 =1 0
0 1
= The Identity Matrix
So…keep on trucking to get the
classical FID result
exp(iwot/2) <-|exp(-iHt)exp(-iIy)Izexp(iIy)|+> =
exp(iwot) <-|exp(-iIy) Iz (cos/2 + sin/2 (I+-I-))|+>
=…
exp(iwot) cos/2 sin/2 = (1/2) exp(iwot) sin
t
1
y
2
The general approach
• Identify pulse sequence, Hamiltonian(s)
• Construct density matrix operator 
• Calculate Tr({ I+} to get time domain signal –
the diagonal elements
• Multiply by exp(-R2t) and Fourier transform for
spectrum
The citrate molecule
AB System
Citrate quantitation and prostate
cancer
Projection Operator: Sum over States
(when you get stuck)
Two Spin Hard Pulse RF Operators
Fy = Iy + Sy
[I,S] = 0, I and S commute
So…shut up and calculate!
Localization with PRESS sequence
The Best Day of My Life?
Theory
Experiment
Joining the Greats!
Inverted lactate at TE = 140 ms
AX3 system
The lactate molecule
Lactate (AX3) Calculation
Why is lactate inverted at TE = 140
ms and up again at 240 ms?
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Signal
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-0.2
-0.4
-0.6
-0.8
-100
-80
-60
-40
-20
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frequency
40
60
80
100
Ethanol Detection with brain MRS
270 ms TE
An A2X3 Calculation…Optimize Ethanol detection
in the Brain
6 minute scans
18 minute scan
31P
MRI of ATP
RARE Sequence and Density Matrix
With J = J = J and J = 0
J-Coupled modulation of k-space lines
Hey you great guys and girl - Thanks
for the QM!
…and we still have a lot
to calculate…
Magn Reson Med 1993;29:38-33
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Be careful what you say in print…
Every Pulse Sequence has a Density
Matrix Operator
t
Gradient Echo
1
2
y

t
Spin Echo
1
90y
2 3
180x
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