Takeaways - RF spoiling: why not just use random phases?

Answer to the student¶

	Quadratic $\phi_n = \phi_0\,n(n+1)/2$	Random $\phi_n$
Destroys coherent transverse magnetization?	Mostly (depends on $\phi_0$ )	Yes, on average
Reaches a steady state?	Yes. Exactly the same signal every TR	No. Fluctuates every TR
Bias against Ernst	Small, deterministic, $T_2$ -dependent, correctable	None on average
TR-to-TR variation	Zero	Large (grows with $T_2$ /TR and flip angle)
Image artifacts	None in steady state (contrast bias only)	Ghosting / noise-like streaks along phase-encode
Helped by more dummy scans?	Yes, until steady state	No
Reproducible between scans?	Yes	Only if the seed is fixed

The quadratic formula isn’t just one way to scramble phases. It is built so that each TR is a shifted copy of the previous one across the voxel’s dephasing distribution. That symmetry is what produces a steady state. A random sequence scrambles phases just as well, but it has no such symmetry, so the signal never settles, and in imaging, an unsettled signal means corrupted k-space.

Discussion questions for class¶

On the φ₀ plot, why is $\phi_0 = 0$ identical to gradient-only spoiling? What about a linear phase schedule, $\phi_n = n\phi_0$ ? (Hint: a constant $\Delta\phi$ is just a relabelling of the isochromats.)
The random scheme is unbiased on average. Suggest an acquisition where that would actually be an advantage. (Hint: think about what happens if you average many repetitions, or acquire a non-imaging, single-voxel signal over many TRs.)
Why does the random scheme’s TR-to-TR variation grow with $T_2$ and flip angle?
The quadratic scheme’s residual bias depends on $T_2$ . How would that affect variable-flip-angle $T_1$ mapping, and how could you correct for it?
In the image, why are the artifacts spread along the phase-encode direction only, and not the readout direction?

Reproducing and extending¶

The simulation code is in spoilsim/ (≈150 lines of NumPy) and is tested in tests/. To rebuild the book:

pip install -r requirements.txt
pytest -q
myst build --html --execute     # static site in _build/html
myst start --execute            # live preview

Things to try: change TR or the tissue parameters in spoilsim/phantom.py, add a centric phase-encode ordering in spoilsim/imaging.py (the transient then lands in the centre of k-space), or compare $\phi_0 = 50^\circ$ with $117^\circ$ .

References¶

Zur Y, Wood ML, Neuringer LJ. Spoiling of transverse magnetization in steady-state sequences. Magn Reson Med. 1991;21(2):251–263.
Preibisch C, Deichmann R. Influence of RF spoiling on the stability and accuracy of T1 mapping based on spoiled FLASH with varying flip angles. Magn Reson Med. 2009;61(1):125–135.