Department of Radiology 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? 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 1 2 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 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 y t 1 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

The lactate molecule AX3 system

Lactate (AX3) Calculation

Why is lactate inverted at TE = 140 ms and up again at 240 ms?

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…

Be careful what you say in print… Magn Reson Med 1993;29:38-33 “ “ Be careful what you say in print…

Every Pulse Sequence has a Density Matrix Operator 1 2 Gradient Echo 90y t 180x  1 2 3 4 Spin Echo