The proton spin phase factor of the gradient field is known to found the Fourier encoding system matrix for MR image formation. The phase factors of short-range proton magnetic field, long-range electron magnetic field, and movement in a gradient field also connect MR physics of relaxation, magnetism and transport with tissue biology of cellularity, biomolecularity and vascularity. Therefore, spin phase factors unify explanation of image formation and tissue contrasts.
Spin phase biophysics unifies understanding of MR image formation and tissue contrasts
TARGETED AUDIENCE:
MRI physicists, engineers and clinical/biological scientists with introductory MRI physics knowledge
OBJECTIVE:
to gain a unified biophysical interpretation of MR image formation and tissue contrasts.
PURPOSE:
MRI is known for its rich tissue contrasts. The contrast mechanisms involve complex biophysical processes that connect the MRI signal in a voxel of a cubic millimeter size to millions of cells and billions of molecules in that voxel. The physical gap between the macroscopic – microscopic worlds is so large that most MR tissue contrasts are only empirically regarded as multiparametric with little mechanistic explanation. This lecture attempts to address this need to better understand tissue contrasts for improving MRI utilization. A unified approach to understanding MRI physics is described using spin phase of the fundamental Larmor precession and using basic statistical calculations.
Physics:
MRI signal is generated by the spin precession of magnetic nuclei, which are protons in most MRI, in a magnetic field. A proton’s contribution to MRI signal at time t is determined by its phase factor that depends on the magnetic field B it experiences:
e^(-iφ),φ=γBt.
The magnetic field consists of field components that are engineered using currents in coils (main field B_0, radiofrequency (RF) field b_1, and gradient field G∙r) and field components from tissue constituents including nuclei (b_n) and electrons (b_e). After RF excitation, the phase of Larmor precession is determined by
B=B_0+G∙r+b_n+b_e
The engineered field components are used for image formation: B_0 sets the stage for Larmor precession and G∙r enables spatial encoding. The tissue field components generate image contrasts.
Biology:
Because protons carry a very weak magnetic moment (μ_p=0.0015μ_B), b_n is only substantial when a proton is very close to the signal generating proton, i.e., short-range proton-proton interaction. Therefore, only when protons are held together by macromolecules and other cellular contents (cellularity) for some nanosecond duration or correlation time, their interaction is significant in affecting MRI signal. This is called relaxation, and it reflects tissue local cellularity.
On the other hand, electrons carry a large magnetic moment (μ_e=-1.0011μ_B), b_e is substantial over a large (macroscopic) spatial range, i.e., the electron-proton interaction is long-range. Because molecular electron cloud structures tend to align most electron magnetic moments in anti-parallel pairs, b_e is determined by unpaired electrons and generally by electron cloud properties (biomolecularity). This is called magnetism, and it reflects both the tissue local property (chemical shift when the observer proton is inside the molecular electron cloud) and tissue nonlocal property (magnetic susceptibility when the observer proton is outside the molecular electron cloud).
In addition to the proton-proton and electron-proton interactions discussed above, proton motion/transport including flow and diffusion (vascularity) provides another fundamental mechanism for tissue contrasts. When a gradient field is on, proton transport affects the MRI signal phase. The phase factor due to proton-proton interaction, electron-proton interaction and proton transport can be studied in the following unified manner.
Computation.
During MRI signal detection (millisecond time scale), protons have gone through millions of random bombardments of thermal motion. We can perform averaging over the observation time and over the voxel to analyze the MRI signal behavior:
⟨e^(-iφ) ⟩=e^(-i⟨φ⟩-1/2 (⟨φ^2 ⟩-⟨φ⟩^2 ) ).
Here the phase factor average is evaluated using cumulant expansion up to the second order.
Echo/Image formation:
The interleaving RF field and inhomogeneous static field causes echo formation, which can be plotted on phase graph. The gradient field enables Fourier encoding with a spatially linear phase,
φ(t,r)=γ∫_0^t▒G(t^' ) ∙rdt^'=2πk∙r.
Relaxation/cellularity:
Averaging the short-range proton-proton interactions over a macroscopic voxel space, the first order of the phase factor average is null, and the second order generates signal attenuation due to relaxation,
1/2 (⟨φ^2 ⟩-⟨φ⟩^2 )=1/2 γ^2 ⟨b_n^2 ⟩τt=t/T_2 .
Magnetism/biomolecularity:
Averaging the long-range electron-proton interactions over a macroscopic voxel space, the first order of the phase factor average is an average phase due to magnetic susceptibility and chemical shift,
⟨φ⟩=γ⟨b_e ⟩t,
and the second order is signal attenuation due to magnetic dephasing,
1/2 (⟨φ^2 ⟩-⟨φ⟩^2 )=1/2 γ^2 (⟨b_e^2 ⟩-⟨b_e ⟩^2 ) t^2=t/(T_2^' ).
Transport/vascularity:
Transport is sensitized with a gradient field (q(t)=∫▒γG dt',M=∫▒γGt dt,b=∫▒q^2 dt). The first order of the phase factor average generates a motion phase due to flow velocity u,
⟨φ⟩=Mu,
and the second order generates signal attenuation due to diffusion D,
1/2 (⟨φ^2 ⟩-⟨φ⟩^2 )=bD.
DISCUSSION:
Other contrasts may be studied similarly, including tissue biomechanical properties that can be studied using motion phase measurements. Additional molecular processes may be included in the spin phase factor analysis. For example, proton magnetic energy loss during relaxation that can be used to study longitudinal (T1) relaxation due to proton-proton interaction at the Larmor frequency; and proton magnetic energy transfer during relaxation can be used to study chemical exchange.
CONCLUSION:
The proton phase factor can provide a unified explanation of relaxation, transport, and magnetism, and MRI signal (s) of a voxel (ΔV) with spine density (m ̅) can be modeled as, s=m ̅ΔVe^(-i(γ⟨b_e ⟩t+Mu) ) e^(-(t/T_2 +t/(T_2^' )+bD) ).These tree major MRI contrast mechanisms reflect the three major tissue properties: cellularity, vascularity, and biomolecularity.
Wang Y. Principles of Magnetic Resonance Imaging: physics concepts, pulse sequences & biomedical applications. North Charleston, SC 29406: CreateSpace Publishing; 2012 (updated 2018).
Hennig J. Echoes - How to generate, recognize, use or avoid them in MR-imaging sequences Part I: fundamental and not so fundamental properties of spin echoes. Concepts in Magnetic Resonance 1991;3:125-143.
The word file with proper equations can be found here https://www.dropbox.com/s/tolkb40o80gtbmp/SpinPhysics.docx?dl=0