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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 t 1 2 y H = woIz H|+> = (1/2)wo|+> H|-> = -(1/2)wo|-> = exp(-iHt)exp(-iIy)Izexp(iIy)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(-iIy)Izexp(iIy)exp(iHt)|+> = exp(iwot/2) <-|exp(-iHt)exp(-iIy)Izexp(iIy)|+> = ? 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(-iIy)Izexp(iIy)|+> = exp(iwot) <-|exp(-iIy) 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? 0.8 0.6 0.4 Signal 0.2 0 -0.2 -0.4 -0.6 -0.8 -100 -80 -60 -40 -20 0 20 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 “ “ 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 4