1. Introduction to Magnetic Resonance Imaging Howard Halpern

2. Basic Interaction Magnetic Moment  let bold indicate vectors) Magnetic Field B Energy of interaction: – E = -µ∙B = -µB cos(θ)

3. Magnetic Moment μ ~ Classical Orbital Dipole Moment –Charge q in orbit with diameter r, area A = πr 2 –Charge moves with velocity v; Current is qv/2πr –Moment µ = A∙I = πr 2 ∙qv/2πr = qvr/2 = qmvr/2m mv×r ~ angular momentum: let mvr “=“ S β = q/2m; for electron, β e = -e/2m e (negative for e) µ B = β e S ; S is in units of ħ –Key 1/m relationship between µ and m µ B (the electron Bohr Magneton) = -9.27 10 -24 J/T µ P (the proton Bohr Magneton) = 5.05 10 -27 J/T****

4. Magnitude of the Dipole Moments Key relationship: |µ| ~ 1/m Source of principle difference between –EPR experiment –NMR experiment This is why EPR can be done with cheap electromagnets and magnetic fields ~ 10mT not requiring superconducting magnets

5. Torque on the Dipole in the Magnetic Field B 0 Torque, τ, angular force τ = -∂E/∂θ = µB 0 sin θ = -| µ × B | For moment of inertia I I ∂ 2 θ/∂ 2 t = -|µ|B 0 sin θ ~ -|µ|B 0 θ for small θ This is an harmonic oscillator equation (m∂ 2 x/∂ 2 t+kx=0) Classical resonant frequency (ω 0 =√(k/m)) ω 0 =√(|µ|B 0 /I)

6. Damped Oscillating Moment: Pulsed Experiment Solution to above is ψ = A exp(iωt) + B exp(-iωt): Undamped, Infinite in time, Unphysical To I ∂ 2 θ/∂ 2 t + µB 0 θ =0 add a friction term Γ∂θ/∂t to get: I ∂ 2 θ/∂ 2 t + Γ∂θ/∂t + µB 0 θ =0 ω = iΓ/2I ± (4IµB 0 -Γ 2 ) 1/2 /2I so that  (t) = A exp(-t Γ/2I - i ((4IµB 0 -Γ 2 ) 1/2 /2I)t ) + B exp(-t Γ/2I + i ((4IµB 0 -Γ 2 ) 1/2 /2I)t ) Lifetime: T ~ 2I/ Γ Redfield Theory: Γ~µ 2 so T 1 s &T 2 s α 1/μ 2 Interaction with environment “Reality Term”

7. Damped Driven Oscillating Moment: Continuous Wave Measurement If we add a driving term to the equation so that I ∂ 2 θ/∂ 2 t + Γ∂θ/∂t + µB 0 θ =B 1 exp(iω 1 t), In the steady state θ(t) = B 1 exp(iω 1 t)/[ω 0 2 -ω 1 2 +2i ω 1 Γ/2I] ~B 1 exp(iω 1 t)/[2ω 0 (ω 0 -ω 1 )+2i ω 1 Γ/2I] a resonant like profile with ω 0 2 =µB/I, proportional to the equilibrium energy. Re [θ(t)]~B 1 / ω 1 2 [(Δω) 2 + (Γ/2I) 2 ] for ω 0 ~ω 1 Lorentzian shape, Linewidth Γ/2I

8. Plot

9. Profile of MRI Resonator Traditionally resonator thought to amplify the resonant signal The resonant frequency ω 0 and the linewidth term Γ/2I are expressed as a ratio characterizing the sharpness of the resonance line: Q= ω 0 /(Γ/2I) This is also the resonator amplification term ω 0 /Q also characterizes the width of frequencies passed by the resonator: ω 0 /Q = Δ ω

10. Q.M. Time Evolution of the Magnetization S is the spin, an operator H is the Hamiltonian or Energy Operator In Heisenberg Representation: ∂S/ ∂t = 1/iћ [H,S], Recall, H= -µ B ∙B =- β e S ∙B. [H,S i ]=HS i -S i H=-β e (S j S i -S i S j )B j = β e iћS x B ∂S/ ∂t = β e S x B; follows also classically from the torque on a magnetic moment ~ S x B as seen above. Bloch Equations follow with (let M = S, M averaging over S): ∂S/ ∂t = β e S x B -1/T 2 (S x î +S y ĵ)-1/T 1 (S z -S 0 )ķ

11. Density Matrix Define an Operator ρ associating another operator S with its average value: = Tr(ρS) ; ρ=Σ|a i ><a i |; basically is an average over quantum states or wavefunctions of the system ρ : Density Matrix: Characterizes the System ∂ρ/∂t = 1/iћ [H, ρ ], 1/iћ [H 0 +H 1, ρ ], ρ*= exp(iH 0 t/ ћ) ρ exp(-iH 0 t/ ћ)=e + ρ e - ; H 1 *=e + H 1 e - ∂ρ*/∂t = 1/iћ [H* 1 (t), ρ*(t) ], ρ*~ ρ*(0) +1/iћ ∫dt’ [H*(t’) 1, ρ*(0)]+ 0 (1/iћ) 2 ∫dt’dt”[H* 1 (t’),[H* 1 (t”), ρ*(0)] Here we break the Hamiltonian into two terms, the basic energy term, H 0 = -µ∙B 0, and a random driving term H 1 =µ∙B r (t), characterizing the friction of the damping

12. The Damping Term (Redfield) ∂ρ*/∂t= 1/iћ [H*(t’) 1, ρ*(0)]+ 0 (1/iћ) 2 ∫dt’[H* 1 (t),[H* 1 (t-t’), ρ*(t’)] Each of the H* 1 has a term in it with H= -µ∙B r and µ ~ q/m ∂ρ*/∂t~ (-) µ 2 ρ* This is the damping term: ρ*~exp(- µ 2 t with other terms) Thus, state lifetimes, e.g. T 2, are inversely proportional to the square of the coupling constant, µ 2 The state lifetimes, e.g. T 2 are proportional to the square of the mass, m 2

13. Consequences of the Damping Term The coupling of the electron to the magnetic field is 10 3 times larger than that of a water proton so that the states relax 10 6 times faster No time for Fourier Imaging techniques For CW we must use 1.Fixed stepped gradients 1.Vary both gradient direction & magnitude (3 angles) 2. Back projection reconstruction in 4-D

14. Major consequence on image technique µ electron =658 µ proton (its not quite 1/m); Proton has anomalous magnetic moment due to “non point like” charge distribution (strong interaction effects) m proton = 1836 m electron Anomalous effect multiplies the magnetic moment of the proton by 2.79 so the moment ratio is 658

15. Relaxtion times: Water protons: Longitudinal relaxation times, T 1 also referred to as spin lattice relaxation time ~ 1 sec T 2 transverse relaxtion time or phase coherence time ~ 100s of milliseconds Electrons T 1e 100s of ns to µs T 2e 10s of ns to µs

16. Magnetization: Magnetic Resonance Experiment Above touches on the important concept of magnetization M as opposed to spin S From the above, M=Tr(ρS), Thus magnetization M is an average of the spin operator over the states of the spin system These can be –coherently prepared or, – as is most often the case, incoherent states –Or mixed coherent and incoherent

17. Magnetization As above, we define M=, the state average of spin If 1/T 1,1/T 2 =0 equations: ∂M/ ∂t = β e M x B; B=B z B x =cos(ω 0 t) B y =sin(ω 0 t) ω 0 = β e B z / ℏ i.e., the magnetization precesses in the magnetic field

18. L-C Resonator C L Zeeman (B 0 ) Magnetic Field Directions: Bz Bx By

19. Resonator Functions 1.Generate Radiofrequency/Microwave fields to stimulate resonant absorption and dispersion at resonance condition: h =  B 0 2.Sense the precessing induction from the spin sample 1.Spin sample magnetization M couples through resonator inductance 2.Creates an oscillating voltage at the resonator output

20. Two modes Continuous wave detection Pulse

21. Continuous Wave (CW) Elaborate Narrow band with high Q (ω 0 /Q = Δ ω) highly tuned resonator. For the time being ω 0 is frequency, not angular frequency Narrow window in ω is swept to produce a spectrum Because resonator is highly tuned sweep of B0, Zeeman field h ω 0 /μ=B= B 0 +B SW : Narrow window swept through resonance

22. Bridge Generally the resonator is part of a “bridge” analogous to the Wheatstone bridge to measure a resistance by balancing voltage drops across resistance and zeroing the current

23. Homodyne Bridge Resonator tuned to impedance of 50Ω Thus, when no resonance all signal from Sig Gen absorbed in resonator When resonance occurs energy is absorbed from RF circuit: RESISTANCE resonator impedance changes => imbalance of “bridge” giving a signal

24. For this “Bridge” balance with system impedance (resistance): 50 Ω Resonance involves absorption of energy from the RF circuit, appearing as a proportional resistance Bridge loses balance: resonator impedance no longer 50 Ω. Signal: 250 MHz voltage Mixers combine reference from SIG GEN Multiply Acos  t*cos  t= A[cos(  t+cos(   t]=A + high frequency which we filter Demodulating the Carrier Frequency.

25. More Field modulation Very low frequency “baseline” drift: 1/f noise Solution: Zeeman field modulation Add to B 0 a B mod =cos  mod t term where  mod is ~audio frequency. Detect with a Lock-in amplifier whose output is the input amplitude at a multiple (harmonic) of  mod First harmonic ~ first derivative like spectrum

26. First hamonic spectrum

27. Pulse Offensive lineman’s approach to magnetic resonance Depositing a broad-band pulse into the spin system and then detect its precession and its decay times CW: driving a harmonic oscillator with frequency  and measuring amplitude response Pulse: striking mass on spring and measuring oscillation response including many modes each with –A frequency –An amplitude –Fourier transform give spectral frequency response

28. Broadband Pulse Real Uncertainty principle in signal theory: ΔνΔT=1 Exercise: For a signal with temporal distribution F(t)=exp(-t^2) what is the product of the FWHH of the temporal distribution and that of its Fourier transform? What is that of F(t)=exp(-(at)^2)?

29. MRI broadband ΔT=1μs Δν=1 MHz Water proton gyromagnetic ratio: 42 MH/T => 1μs pulse give 0.024 T excitation (.24 KGauss)

30. EPR Broadband 1 ns pulse => 1 GHz excitation Electron gyromagnetic ratio: 28 GHz/T 1 ns pulse gives 0.035 T wide excitation NMR lines typically much narrower than EPR EPR is a hard way to live

31. Pulse bridge Simpler particularly for MRI Requires same carrier demodulation Requires lower Q for broad pass band Excites all spins in the passband window so, in principle, more efficient acqusition

32. Magnetic field gradients encode position or location Postulated by Lauterbur in landmark 1973 Nature paper Overcomes diffraction limit on 40 MHz RF –λ = 7.5 m Gradient: G i = dB/dx i where x 1 =x; x 2 =y; x 3 =z B is likewise a vector but generally take to be B z.

33. Standard MRI z slice selection z dimension distinguished by pulsing a large gradient during the acqusition G z Δz = ∂ B z / ∂ zΔz = ΔB = h Δ ν/ β e Δ ν = G z Δz/h β e If Δ ν > ω 0 /Q the frequency pass band of the resonator, then the sensitive slice thickness is defined by Δz= h β e ω 0 /QG z

34. X and Y location encoding For standard water proton MRI, within the z selected plane Apply Gradient Gx-Gy plane Frequency and phase encoding

35. Phase encoding Generally one of the Gx or Gy direction generating gradients selected. Say Gy The gradient is pulsed for a fixed time Δt before signal from the precessing magnetization is measured. Magnetization develops a phase proportional to the distance along the y coordinate Δφ=2πyG y /β p where β p is the proton gyromagnetic ratio.

36. Frequency Encoding The orthogonal direction say x has the gradient imposed in that direction during the acquisition of the magnetization precession signal This shifts the frequency of the precession proportional to the x coordinate magnitude Δω=xG x

37. Signal Amplitude vs location The complex Fourier transform of the voltage induced in the resonator by the precession of the magnetization gives the signal amplitude as a function of frequency and phase The signal amplitude as a function of these parameters is the amplitude of the magnetization as a function of location in the sample

38. EPR: No time for Phase Encoding Generally use fixed stepped gradients Both for CW and Pulse Tomographic or FBP based reconstruction

39. Imaging: Basic Strategy Constraint: Electrons relax 10 6 times faster than water protons Image Acquisition: Projection Reconstruction: Backprojection

40. Key for EPR Images Spectroscopic imaging Obtain a spectrum from each voxel EPRI usually uses an injected reporter molecule Spectral information from the reporter molecule from each voxel quantitatively reports condition of the fluids of the distribution volume of the reporter

41. Projection Acquisition in EPR Spectral Spatial ObjectSupport ~

42. =α=α Imposition of a Gradient on a spectral spatial object (a) here 3 locations each with spectra acts to shear the spectral spatial object (b) giving the resolved spectrum shown. This is equivalent to observing (a) at an angle  (c). This is a Spectral- Spatial Projection.

43. Projection Description With s(B sw, Ĝ) defined as the spectrum we get with gradients imposed

44. More Projection The integration of f sw is carried out over the hyperplane in 4-space by Defining with The hyperplane becomes with

45. And a Little More So we can write B = B sw + c tanα Ĝ∙x

46. So finally it’s a Projection with

47. Projection acquisition and image reconstruction Image reconstruction: backprojections in spectral-spatial space Resolution: δx=δB/G max THUS THE RESOLUTION OF THE IMAGE PROPORTIONAL TO δB each projection filtered and subsampled; Interpolation of Projections: number of projections x 4 with sinc(?) interpolation, Enabling for fitting Spectral Spatial Object Acquisition: Magnetic field sweep w stepped gradients (G) Projections: Angle  : tan(  =G*  L/  H

48. The Ultimate Object: Spectral spatial imaging

49. Preview: Response of Symmetric Trityl (deuterated) to Oxygen So measuring the spectrum measures the oxygen. Imaging the spectrum: Oxygen Image