Multiple Coherence Pathways

Simple spin echo TETE TETE abc d spin echo 90 y 180 x

TETE TETE abc d Hahn echo 90 y 90 x TMTM TETE e f stimulated echo Hahn (90-90) and stimulated ( ) echoes Hennig Fig. 2

Repeated flip  =90 o

Repeated flip  =40 o

What is an echo? Signal peak (in time) cause by net alignment of magnetization Spin echoes: perfect alignment of isochromats –Any distribution of isochromats is refocused More generally: perfect alignment is not required to have a peak in signal –Hahn, stimulated echoes to not have isochromats aligned –Magnetization is “bunched up” on one side of xy plane –Many echoes require distribution is isochromats Unlike NMR, heavy dephasing (distribution) is the norm in MRI –MRI insufficient inhomogeneity to maintain long-term coherence –Instead, use gradients to reliably dephase (spoil) and rely on short- term coherences Can we find a representation that is better than isochromat vectors?

Shortcomings of vector representation Vector representation (e.g., Bloch): [M x M y M z ] Problems: 1.Evolution of magnetization (in absence of RF) has 2 independent components (transverse & longitudinal), but vectors have 3 2.Fundamentally treats single isochromats, where MRI essentially always encounter distributions This is why echo evolution is so complicated to depict using vectors (both temporally and spatially) Phase graph representation addresses both of these issues

Alternate representation of magnetization Problem 1: Evolution of magnetization has 2 independent components (transverse & longitudinal), but vectors have 3 Replace:[M x M y M z ] With:[F=M x +iM y M z ] In absence of RF, F and M z evolve independently relaxation, precession represented by scalar multiples no need to worry about coupling between M x, M y

Alternate representation of magnetization Problem 1: Evolution of magnetization has 2 independent components (transverse & longitudinal), but vectors have 3 Replace:[M x M y M z ] With:[F=M x +iM y M z ] Effect of RF pulse: F + = F cos 2 (  /2) + F* sin 2 (  /2) - i M z sin(  ) M z + = M z cos 2 (  /2) - M z sin 2 (  /2) - i (F-F*) sin(  ) 0o0o 180 o 90 o Single RF pulse acts like 3 separate pulses

flip angle (degrees) fraction Fractional components in arbitrary RF pulse

Configuration theory (coherence pathways) Problem 2: Vectors fundamentally represent single isochromats, where MRI essentially always encounter distributions MzMz MzMz MxMx MxMx * typos in Hennig? Hennig, Fig 4 Hennig, Eqs 8-11

Configuration theory (coherence pathways) What do F n, F n *, Z n represent? This is just a useful decomposition of the magnetization (e.g., like Fourier decomposition of an image/object) Decomposition coefficient = how much magnetization expresses this structure Hennig calls “configurations” (others call “coherences”) Each configuration is a potential echo (allow it to rephase, signal is proportional to its coefficient) No mystical properties (e.g., quantum mechanics not needed)! Hennig, Fig 4

RF pulses Echo formation time phase evolution exchange between configurations Track flow of magnetization between configurations F n + = F n-1 cos 2 (  /2) + F n * sin 2 (  /2) + Z n sin(  ) (F n *) + = F n+1 * cos 2 (  /2)+ F n-1 * sin 2 (  /2) + Z n * sin(  ) Z n + = Z n cos(  ) + (F n * - F n ) sin(  ) (see Eq 13-15)

Track flow of magnetization between configurations

Time evolution of signal dynamics

Differs from previous via starting conditions (i.e., preparatory pulses)

Time evolution of signal dynamics Differs from first via flip angle