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Problems in MR that really need quantum mechanics: The density matrix approach

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.

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Problems in MR that really need quantum mechanics: The density matrix approach

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  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

  2. Nuclear Spin: An inherently Quantum Mechanical (QM) Phenomenon Angular momentum operators represent spin I

  3. 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!

  4. 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

  5. BPP Theory: Used QM to calculate T1, T2 – 1950’s – rarely used in practice Fluctuations of Dipolar Hamiltonian

  6. QM in Radiological Interpretations? • Magic angle effect (3cos2 – 1) = 0 • Bright fat effect (quenching of J-coupling with multiple 180’s)

  7. “When molecules lie at 54.74° there is lengthening of T2 times (don't understand why, but it involves 'bipolar coupling')”

  8. “Dipolar Coupling” - Magnetic energy between two dipoles

  9. The Dipolar Hamiltonian

  10. Bright Fat Phenomenon

  11. Where QM Really Rules: Coupled Spin Systems and Spectroscopy

  12. “Shut up and Calculate” Richard Feynman The real beauty of the Density Matrix Formalism – no thinking…

  13. Spin ½ Rules of the Road h = 1, let’s be friends Iz|+> = ½ |+> Iz|-> = -1/2 |-> Ix = (I+ + I-)/2 Iy = (I+ - I-)/2i I+|+> = 0 I+|-> = |+> I-|-> = 0 I-|+> = |-> Commutation Relations [I,S] = 0 (two spins) [Ii,Ij] = ijkIk

  14. 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

  15. t 1 2 y Density Matrix Example: Free Precession 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+}

  16. 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?

  17. The Pauli Spin Matrices Wolfgang Pauli

  18. 0 0 1 A2 = = The Identity Matrix Matrix Representations of Angular Momentum Operators

  19. t 1 2 y 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

  20. 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

  21. The citrate molecule AB System

  22. Citrate quantitation and prostate cancer

  23. Projection Operator: Sum over States(when you get stuck)

  24. Two Spin Hard Pulse RF Operators Fy = Iy + Sy [I,S] = 0, I and S commute

  25. So…shut up and calculate!

  26. Localization with PRESS sequence

  27. Theory Experiment The Best Day of My Life?

  28. Joining the Greats!

  29. Inverted lactate at TE = 140 ms

  30. The lactate molecule AX3 system

  31. Lactate (AX3) Calculation

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

  33. Ethanol Detection with brain MRS 270 ms TE

  34. An A2X3 Calculation…Optimize Ethanol detection in the Brain

  35. 6 minute scans 18 minute scan

  36. 31P MRI of ATP

  37. RARE Sequence and Density Matrix

  38. With J = J = J and J = 0

  39. J-Coupled modulation of k-space lines

  40. Hey you great guys and girl - Thanks for the QM! …and we still have a lot to calculate…

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

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

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