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### Introduction to Magnetic Resonance Imaging

Howard Halpern

Basic Interaction

- Magnetic Moment m (let bold indicate vectors)
- Magnetic Field B
- Energy of interaction:
- E = -µ∙B = -µB cos(θ)

Magnetic Moment

- μ ~ Classical Orbital Dipole Moment
- Charge q in orbit with diameter r, area A = πr2
- Charge moves with velocity v; Current is qv/2πr
- Moment µ = A∙I = πr2∙qv/2πr = qvr/2 = qmvr/2m
- mv×r ~ angular momentum: let mvr “=“ S
- β = q/2m; for electron, βe = -e/2me (negative for e)
- µB= βeS ; S is in units of ħ
- Key 1/m relationship between µ and m
- µB (the electron Bohr Magneton) = -9.27 10-24J/T
- µP(the proton Bohr Magneton) = 5.05 10-27 J/T****

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

Torque on the Dipole in the Magnetic Field B0

- Torque, τ, angular force
- τ = -∂E/∂θ = µB0 sin θ = -| µ × B |
- For moment of inertia I
- I ∂2θ/∂2t = -|µ|B0 sin θ ~ -|µ|B0θ for small θ
- This is an harmonic oscillator equation (m∂2x/∂2t+kx=0)
- Classical resonant frequency (ω0=√(k/m)) ω0=√(|µ|B0/I)

Damped Oscillating Moment: Pulsed Experiment

- Solution to above is ψ = A exp(iωt) + B exp(-iωt): Undamped, Infinite in time, Unphysical
- To I ∂2θ/∂2t + µB0θ =0 add a friction term Γ∂θ/∂t to get:

I ∂2θ/∂2t + Γ∂θ/∂t + µB0θ =0

ω = iΓ/2I ± (4IµB0-Γ2)1/2/2I so that

(t) = A exp(-t Γ/2I - i ((4IµB0-Γ2)1/2/2I)t ) +

B exp(-t Γ/2I + i ((4IµB0-Γ2)1/2/2I)t )

Lifetime: T ~ 2I/ Γ

Redfield Theory: Γ~µ2 so T1s &T2sα 1/μ2

Interaction with environment

“Reality Term”

Damped Driven Oscillating Moment: Continuous Wave Measurement

- If we add a driving term to the equation so that I ∂2θ/∂2t + Γ∂θ/∂t + µB0θ =B1exp(iω1t),
- In the steady state θ(t) = B1exp(iω1t)/[ω02-ω12+2i ω1 Γ/2I]

~B1exp(iω1t)/[2ω0(ω0-ω1)+2i ω1 Γ/2I]

a resonant like profile with

ω02=µB/I, proportional to the equilibrium energy.

Re [θ(t)]~B1/ ω12 [(Δω)2+ (Γ/2I)2] for ω0~ω1

Lorentzian shape, LinewidthΓ/2I

Profile of MRI Resonator

- Traditionally resonator thought to amplify the resonant signal
- The resonant frequency ω0and the linewidthterm Γ/2I are expressed as a ratio characterizing the sharpness of the resonance line: Q=ω0/(Γ/2I)
- This is also the resonator amplification term
- ω0/Q also characterizes the width of frequencies passed by the resonator: ω0/Q = Δω

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= -µB∙B=- βeS ∙B.

[H,Si]=HSi-SiH=-βe(SjSi-SiSj)Bj= βeiћS x B

∂S/ ∂t = βeS 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 = βeS x B -1/T2(Sxî +Syĵ)-1/T1(Sz-S0)ķ

Density Matrix

- Define an Operator ρassociating another operator S with its average value:
- <S> = Tr(ρS) ; ρ=Σ|ai><ai|; 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 [H0+H1, ρ],
- ρ*= exp(iH0t/ћ) ρexp(-iH0t/ћ)=e+ ρe-; H1*=e+H1e-
- ∂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, H0 = -µ∙B0,

and a random driving term H1=µ∙Br(t), characterizing the friction of the damping

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∙Br and µ ~ q/m
- ∂ρ*/∂t~ (-) µ2 ρ* This is the damping term:
- ρ*~exp(- µ2t with other terms)
- Thus, state lifetimes, e.g. T2, are inversely proportional to the square of the coupling constant, µ2
- The state lifetimes, e.g. T2 are proportional to the square of the mass, m2

Consequences of the Damping Term

- The coupling of the electron to the magnetic field is 103times larger than that of a water proton so that the states relax 106 times faster
- No time for Fourier Imaging techniques
- For CW we must use
- Fixed stepped gradients
- Vary both gradient direction & magnitude (3 angles)

2. Back projection reconstruction in 4-D

Major consequence on image technique

- µelectron=658 µproton (its not quite 1/m);
- Proton has anomalous magnetic moment due to “non point like” charge distribution (strong interaction effects)
- mproton= 1836 melectron
- Anomalous effect multiplies the magnetic moment of the proton by 2.79 so the moment ratio is 658

Relaxtion times:

- Water protons: Longitudinal relaxation times, T1 also referred to as spin lattice relaxation time ~ 1 sec

T2 transverse relaxtion time or phase coherence time ~ 100s of milliseconds

- Electrons

T1e 100s of ns to µs

T2e 10s of ns to µs

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

Magnetization

- As above, we define M=<S>, the state average of spin
- If 1/T1,1/T2=0 equations:∂M/ ∂t = βeMx B; B=Bz
- Bx=cos(ω0t)
- By=sin(ω0t)
- ω0= βeBz/ℏ
- i.e., the magnetization precesses in the magnetic field

Resonator Functions

- Generate Radiofrequency/Microwave fields to stimulate resonant absorption and dispersion at resonance condition: hn=mB0
- Sense the precessing induction from the spin sample
- Spin sample magnetization M couples through resonator inductance
- Creates an oscillating voltage at the resonator output

Two modes

- Continuous wave detection
- Pulse

Continuous Wave (CW)

- Elaborate
- Narrow band with high Q (ω0/Q = Δω) highly tuned resonator. For the time being ω0is 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 ω0/μ=B= B0 +BSW : Narrow window swept through resonance

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

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

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 Acoswt*coswt= A[cos(w+w)t+cos(w-w)t]=A + high frequency which we filter
- Demodulating the Carrier Frequency.

More

- Field modulation
- Very low frequency “baseline” drift: 1/f noise
- Solution: Zeeman field modulation
- Add to B0 a Bmod=coswmodt term where wmod is ~audio frequency.
- Detect with a Lock-in amplifier whose output is the input amplitude at a multiple (harmonic) of wmod
- First harmonic ~ first derivative like spectrum

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 w 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

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)?

MRI broadband

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

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

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

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: Gi= dB/dxi where x1=x; x2=y; x3=z
- B is likewise a vector but generally take to be Bz.

Standard MRI z slice selection

- z dimension distinguished by pulsing a large gradient during the acqusition
- GzΔz = ∂Bz/∂zΔz = ΔB = hΔν/βe
- Δν = GzΔz/hβe
- If Δν > ω0/Q the frequency pass band of the resonator, then the sensitive slice thickness is defined by Δz=hβeω0/QGz

X and Y location encoding

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

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=2πyGy/βpwhere βp is the proton gyromagnetic ratio.

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=xGx

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

EPR: No time for Phase Encoding

- Generally use fixed stepped gradients
- Both for CW and Pulse
- Tomographic or FBP based reconstruction

Constraint: Electrons relax 106 times faster than water protons

Image Acquisition: Projection

Reconstruction: Backprojection

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

Projection Acquisition in EPR

- Spectral Spatial Object Support ~

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 a (c). This is a Spectral-Spatial Projection.

=α

Projection Description

- With s(Bsw, Ĝ) defined as the spectrum we get with gradients imposed

More Projection

- The integration of fsw is carried out over the hyperplane in 4-space by

Defining

with

The hyperplane becomes

with

Projection acquisition and image reconstruction

Image reconstruction: backprojections in spectral-spatial space

Resolution: δx=δB/Gmax

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 a: tan(a)=G*DL/DH

Spectral spatial imaging

Preview: Response of Symmetric Trityl

(deuterated) to Oxygen

- So measuring the spectrum
- measures the oxygen. Imaging the spectrum: Oxygen Image

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