contrast

mri · part three · why anything looks like anything

Parts one and two build a map of where the protons are. That map is nearly useless — brain is about 80% water everywhere, so a pure proton-density picture of a head is a faintly textured blob. Every clinical MR image is instead a picture of how fast the magnetisation recovers and decays, and which of those you see is not a property of the scanner. It is a choice, made by somebody, in the two or three numbers of a schedule.

1 · A tissue is three numbers

Strip away the anatomy and a tissue, to an MRI, is T₁ (how quickly magnetisation recovers along the field after being knocked over), T₂ (how quickly it loses coherence in the transverse plane) and proton density. Here are the ones this page uses, measured:

Stanisz et al. 2005, Table 1 — 3 T, 37 °C

	tissue	T₁ / ms	T₂ / ms

Three things to know before trusting a digit of that. These are in vitro measurements on excised tissue — bovine white and grey matter, mouse liver and muscle, human blood — not in-vivo human numbers. The same table's own literature column disagrees with several: it gives grey-matter T₁ as 1470 ± 50 ms against the 1820 ± 114 ms measured here, a 24% disagreement in the single most-used parameter in the field, and neither is wrong — T₁ depends on preparation, temperature, method and magnetization transfer. And proton density is set to 1.0 for every tissue here, because that table does not measure it, so all the contrast below comes from T₁ and T₂ alone. Real proton-density differences in brain are small but not zero.

2 · The same tissues, any brightness you like

Three schedules, three equations. A spin echo waits TR between excitations and reads at TE:

S = PD · (1 − 2e^{−(TR−TE/2)/T₁} + e^−TR/T₁) · e^−TE/T₂

Short TR punishes long-T₁ tissue, because it has not finished recovering when the next pulse arrives — that is T₁ weighting. Long TE punishes short-T₂ tissue, because it has already faded by the time you look — that is T₂ weighting. Make TR long and TE short and you punish neither, which is the proton-density image nobody wants. Drag the schedule and watch six tissues rearrange themselves.

drive the schedule · the image is reconstructed through part two's k-space path

spin echo

signal per tissue

TR500 ms TE10 ms TI316 ms flip angle30°

…

3 · Contrast is not signal

A sequence can be bright and useless. What an image is for is the difference between two tissues, and that difference has its own landscape over the schedule — with a maximum in a different place from the signal maximum, and, more interestingly, a curve where it goes to zero. Along that curve two tissues produce exactly the same brightness and the image cannot tell them apart, however good the scanner is.

|white matter − grey matter| over the schedule · click to jump there

contrast here—

invisible at this TE when TR =—

best TR for this TE—

The dark valley running across the map is the invisibility curve: every echo time up to about 110 ms has a repetition time at which white and grey matter cancel exactly. It runs between the two places clinical imaging actually stands — T₁-weighted at the bottom left, T₂-weighted at the top right — which is another way of saying those two regions are on opposite sides of a sign change, and the "opposite" contrast of a T₁ and a T₂ image is not a coincidence of physiology but a geometric fact about this surface.

4 · Two closed forms worth knowing

The Ernst angle is the flip angle that maximises gradient-echo signal for a given TR/T₁: differentiate the signal with respect to α and you get cos α = e^−TR/T₁. It is why fast sequences use small flip angles — at TR = 5 ms a tissue with T₁ = 1 s wants about 6°, not 90°. And it maximises signal for one tissue; the angle that maximises the difference between two is somewhere else entirely.

The null time deletes a tissue. Invert the magnetisation, wait for the one you want gone to pass through zero, and excite at that instant: TI = T₁ · ln(2 / (1 + e^−TR/T₁)), which becomes the familiar T₁ · ln 2 when TR is long. This is how STIR removes fat and FLAIR removes cerebrospinal fluid — and why the same TI does not work on a scanner running a different TR.

left: gradient-echo signal against flip angle · right: recovery after inversion

Ernst angle at TR = —

null at TI = —

TR50 ms null whichliver

…

Where this stops. Three simplifications worth naming. Real sequences read many echoes per excitation (fast spin echo), and the later ones are more T₂-weighted than the equation here admits. Gradient-echo signal assumes perfect spoiling; imperfect spoiling produces steady-state free precession, which is a different equation and a different-looking image. And the biggest omission: contrast agents, which shorten T₁ in proportion to their concentration and are in a large fraction of clinical scans. All three are additions to this page rather than corrections to it. The thing every patient notices and nobody explains — why it screams — is part four.

Where this comes from

Stanisz GJ, Odrobina EE, Pun J, Escaravage M, Graham SJ, Bronskill MJ & Henkelman RM (2005). T₁, T₂ relaxation and magnetization transfer in tissue at 3T. Magn Reson Med 54:507–512. Every tissue number on this page is Table 1, "This study", 3 T column — read from the paper, uncertainties included. In vitro, 37 °C.
Ernst RR & Anderson WA (1966). Application of Fourier transform spectroscopy to magnetic resonance. Rev Sci Instrum 37:93–102. The Ernst angle, and the pulse-and-transform method the whole instrument runs on. Nobel 1991.
Bloch F (1946). Nuclear induction. Phys Rev 70:460–474. The equations every closed form on this page is checked against, by simulation, in the engine's tests.
Damadian R (1971). Tumor detection by nuclear magnetic resonance. Science 171:1151–1153. The claim that relaxation times differ between tissues — the one that made a medical instrument conceivable at all. This page is that claim, operationalised.
Zavala Bojorquez J, Bricq S, Acquitter C, Brunotte F, Walker PM & Lalande A (2017). What are normal relaxation times of tissues at 3 T? Magn Reson Imaging 35:69–80. A review of how much published relaxation times disagree, and why. Reports fat T₁ at 3 T as 366 ± 78 ms by one method and 450 ± 26 ms by another in the same breast — which is the honest state of the art, and the reason the caveat above is not false modesty.
Hennig J, Nauerth A & Friedburg H (1986). RARE imaging: a fast imaging method for clinical MR. Magn Reson Med 3:823–833. Many echoes per excitation — the sequence family whose contrast this page's single-echo equation does not quite describe.

The three sequence equations are validated against a full Bloch simulation built from part one's integrator — repeated pulses run to steady state — rather than against a textbook; the Ernst angle and the null time are checked against brute-force optima of their own equations; and the tissue table is asserted digit-for-digit against the paper. cd sci/engine-rs && cargo test --release — 39 known-answer tests.

engine: rust → wasm · no dependencies · 39 tests · part one · part two · sci