MRI: Boltzmann, Magnetism, and the Magnetic Moment Ensemble

Boltzmann Magnetization, Magnetism, and the Magnetic Moment Ensemble

Boltzmann magnetism, also known as equilibrium nuclear magnetism, describes the nuclear magnetism of a spin sample when it attains thermal equilibrium within a magnetic field. This equilibrium is dynamic, with constant energy exchange between the spins and their environment, yet the overall magnetization remains stable.

Every proton has the same magnetic moment, denoted as mu (\mu), calculable via:

\mu = \gamma \hbar \frac{1}{2}

Where:

• \gamma is the gyromagnetic ratio, unique to each nucleus and indicative of its magnetic sensitivity.

• \hbar is the reduced Planck's constant, a fundamental constant in quantum mechanics.

In the absence of a magnetic field, proton spins are randomly oriented. Introducing a magnetic field causes the spins to align in one of two states:

• Spin up: Aligned with the field (low energy state), also known as the parallel state.

• Spin down: Opposed to the field (high energy state), also known as the anti-parallel state.

The spin-up state is energetically favorable, but the energy difference between the states is small relative to the sample's thermal energy. Consequently, spins distribute nearly equally between these states, with a slight excess in the spin-up state dictated by Boltzmann statistics.

Polarization

Polarization is the fraction of spins in excess in the aligned, low-energy state. Higher polarization yields a stronger NMR signal.

It is expressed as:

Polarization = \frac{\gamma \hbar B_0}{2kT}

Where:

• \gamma is the gyromagnetic ratio.

• \hbar is the reduced Planck's constant.

• B0 is the external magnetic field strength. Increasing B0 improves polarization and signal strength.

• k is Boltzmann's constant.

• T is the sample temperature in Kelvin. Lower temperatures increase polarization.

Calculating Equilibrium Magnetization

The equilibrium magnetization is found by summing the magnetic moments of the unpaired spins.

Boltzmann magnetism is:

M_0 = N \mu \cdot Polarization

Where:

• N is the number of spins in the sample. Higher spin density increases magnetization.

• \mu is the magnetic moment of a single spin.

Substituting values, the total magnetization is:

M0 = N \gamma^2 \hbar^2 B0 / 4kT

Simplifying with \chi_0, the static nuclear susceptibility:

M0 = \chi0 B_0

Where:

\chi_0 = N \gamma^2 \hbar^2 / 4kT

\chi0 reflects the induced magnetic moment in a given field and is inversely proportional to temperature, aligning with Curie's Law. Higher \chi0 indicates greater responsiveness to the magnetic field.

Verifying units confirms dimensional consistency with the magnetic moment's SI unit.

NMR Experiment

An NMR experiment perturbs the sample's equilibrium and monitors the return to Boltzmann equilibrium. This return follows:

• Transverse magnetization decays to 0 at the T2 rate due to spin-spin interactions and field inhomogeneities.

• Longitudinal magnetization returns to Boltzmann equilibrium at the T1 rate, influenced by spin-lattice interactions.

The signal is proportional to the number of spins, the magnetic moment, the contributing spin fraction, and the precession frequency \omega = \gamma B0. Doubling B0 quadruples the signal because signal ∝ B_0^2.

Bloch Equation

The Bloch equation details magnetization dynamics. In a magnetic field B, a magnetization vector M precesses around B in a conical path. The cone angle is twice the angle between B and M.

Clarification on Proton Representation

Protons exist in spin-up or spin-down Zeeman states. Longitudinal magnetization quantifies the proton distribution in each state. The precession signal is energy released during state transitions, akin to electronic transitions but with energy separation proportional to B_0.

Simplest NMR experiment

The simplest NMR experiment involves placing a sample in B0 and applying B1 perpendicular to B*0, causing magnetization precession. A 90-degree rotation maximizes the detectable signal. The resulting free induction decay (FID) is detected as magnetization returns to equilibrium—a 'ping' or pulse/acquire method.

This necessitates noting the Larmor oscillator phase to set the excitation axis and signal phase. NMR/MRI pulse sequences exploit this.

However, in practice, the NMR signal decays faster than predicted due to B_0 field inhomogeneity. Even minor field variations significantly impact experiments, even with water samples (T2 ≈ 1 second). Field imperfections dramatically affect the detected signal, reducing resolution and distorting spectral lines.

Magnetic Field Inhomogeneity

Consider two proton spins in slightly different fields: 1 Tesla and 1.000001 Tesla. Given a gyromagnetic ratio of 267.5 x 10^6 rad/T/s (42.6 MHz/T), this leads to a frequency difference (Larmor equation). The difference in frequency is 267.5 radians (or 42.6 revolutions) which means the spin in the higher field will complete 42.6 more precession cycles than the first spin every second. The spins will therefore be \pi radians or 180 degrees out of phase in 11.7 milliseconds!

Field inhomogeneity causes signal decay via a time constant T2*, diminishing tissue contrast.

Correcting the problem using pulse sequence

The sample is initially in equilibrium with a magnetic field in the positive z-direction. The B1 excitation field is on the negative y-axis, causing precession about the B1 field. At a 90-degree flip angle, B*1 is turned off. The total magnetization is now in the positive x-direction of the transverse plane, delivering a detectable signal. The dephasing and T2 decay dynamics are then observed.

The precession frequency, \omega*k, for the kth spin is the Larmor frequency plus \Delta \omega (\gamma \Delta B). \Delta B can also vary with time due to Brownian diffusion. Working in the rotating frame (spinning at \omega0), the \omega0 term is dropped. Spins only precess if their \\Delta \omega is non-zero.

Spins dephase, and total magnetization decreases following e^{-\frac{t}{T_2}}.

Mathematical Expression for Total Magnetization Vector M

The magnetization M is the vector sum of individual magnetic moments \muk. The x-component of \mu is \cos(\omegak t), and the y-component is \sin(\omegak t). Using Euler's relation, the vector can be expressed as a complex number e^{-i \omegak t}.

Assuming constant \mu, the distribution of spins with a given frequency yields a frequency distribution (f(\omega)). The transverse magnetization in time is:

M(t) = \mu \int_{-\infty}^{\infty} f(\omega) e^{-i\omega t} d\omega

This is the Fourier Transform of the frequency distribution, f(\omega). The signal decay rate reflects the frequency distribution; slow dephasing corresponds to slow decay and a narrow frequency distribution, while rapid dephasing indicates quicker decay.

The omega distribution reflects the distribution of regional magnetic field B, given that all protons have the same \gamma. Each spin experiences \Delta B due to the tissue environment and field imperfections. The Larmor equation dictates that field inhomogeneity broadens the frequency distribution, shortening the decay time.

Sources of Decay

Decay arises from:

• Intrinsic molecular dynamics of the system or tissue.

• Imperfect Field.

The signal is modulated by e^{-Rt}, where R is the decay rate and is characterized by time constant T2. It is also modulated by e^{-Rb t}, where Rb is the decay rate due to field inhomogeneity. The decay rate R* accounts for all sources of spin dephasing:

R^* = \frac{1}{T2} + Rb = \frac{1}{T2} + \gamma \Delta B = \frac{1}{T2^*}

T_2^* is the observed signal decay due to all dephasing mechanisms.

Field homogeneity improves with smaller imaging volumes. However, even with a perfectly manufactured magnet, placing a sample inside distorts the field and increases \Delta B.

Ping NMR Experiment

The ping NMR experiment excites nuclear magnetization into the transverse plane, where its decay reflects intrinsic tissue properties. However, field imperfections accelerate this decay, reducing tissue contrast.

To correct this, a 180-degree pulse is delivered shortly after the FID decays, generating a spin echo. The B_0 field is oriented out of the screen, with spin colors indicating field strengths. The 180 pulse rotates spins 180 degrees, causing rephasing.

After the 180 pulse, spins precess at a rate dictated by field strength. The 180 pulse doesn't change spatial positions or precession frequency.

The detected FID follows T2* decay. If the 180 is delivered at time tau, the spins will have dephased by a certain amount. It is expected that following the 180, the spins will take exactly that amount of time to rephase. The echo peaks decay because, unlike dephasing due to field inhomogeneity, intrinsic T2 decay is not reversible.

If the 180 is delivered at time tau, the echo will appear at time 2 tau and its peak will reach the T2 decay curve. Thus, the peak amplitude can be used to calculate T2 for T2 contrast.

The echoes will appear regardless of the excitation axis. Delivery of B*1 pulses at different orientations in the rotating frame will cause echoes to have a different phase as the FID.

Spin with his \Delta B is zero will align with his buddies at each echo.

The Math Behind a 180-Degree Pulse

Following the 90-degree pulse, spins acquire phase at time tau. After a 180-degree pulse, the imaginary component becomes negative, but the real component remains unchanged. This switches the accumulated phase from positive to negative. The total accumulated phase will equal 0 when time t