T1, T2 & T2*: Relaxation Mechanisms

From a tipped vector back to equilibrium

After a radiofrequency pulse tips the net magnetization away from the static field, the spin system is no longer at thermal equilibrium. Relaxation is the set of processes that restore it: the longitudinal component grows back toward its equilibrium value $M_0$, while the transverse component dephases and decays. These two processes are governed by different physics and proceed on different timescales, and the time constants that describe them, T1 and T2, are the dials that produce nearly every flavor of clinical MR contrast. Tissues differ far more in T1 and T2 than in proton density: gray and white matter differ by only a few percent in spin density but by roughly 30 percent in T1 at 1.5 T. By choosing when we sample (echo time, TE) and how often we excite (repetition time, TR), we convert those relaxation differences into image contrast.

The Bloch equations with relaxation

Felix Bloch added phenomenological relaxation terms to the equation of motion for the magnetization vector $\mathbf{M}$. The longitudinal component $M_z$ relaxes toward $M_0$ with time constant T1, and each transverse component decays toward zero with time constant T2. Ignoring the precession terms, the relaxation behavior is captured by two separable first-order differential equations whose solutions are the workhorses of MR signal modeling.

The molecular origin: BPP theory

Why does free water have a T1 of several seconds while the same protons bound near a macromolecule relax in tens of milliseconds? The Bloembergen-Purcell-Pound (BPP) theory answers this. Relaxation is driven by fluctuating local magnetic fields, arising chiefly from dipolar coupling to neighboring protons as molecules tumble. The key quantity is the rotational correlation time $\tau_c$, the characteristic time over which a molecule reorients and the local field loses memory of its previous value. Longitudinal relaxation needs spectral power near the Larmor frequency $\omega_0$ (and at $2\omega_0$), because a spin flip changes the Zeeman energy and that energy must be carried by a fluctuation at the transition frequency. The available power at a given frequency is the spectral density $J(\omega)$, the Fourier transform of the field autocorrelation function.

The rate $1/T_1$ is proportional to $J(\omega_0)$, which Eq. 2.3 maximizes when $\omega_0 \tau_c \approx 1$; T1 therefore passes through a minimum when the tumbling rate matches the Larmor frequency. Fast-tumbling free water ($\tau_c \approx 10^{-12}$ s) and nearly rigid solids (large $\tau_c$) both relax inefficiently and have long T1, while mid-range molecules and water transiently bound to macromolecules sit near the minimum and relax quickly. T2 instead depends on $J(\omega_0)$ and the zero-frequency term $J(0)=2\tau_c$, which reflects slow or static components that dephase spins without an energy-conserving flip. For slowly tumbling environments $J(0)$ keeps growing while $J(\omega_0)$ has collapsed, so solids and large immobile macromolecules have very short T2 yet long T1.

Field dependence and representative values

The BPP framework predicts the robust rule that T1 lengthens with increasing $B_0$ while T2 is relatively field-independent over the clinical range. For most tissues at 3 T, T1 is roughly 20 to 40 percent longer than at 1.5 T; T2 changes only modestly, typically shortening by a few to about fifteen percent. The table gives representative values for sequence planning and for sanity-checking measured maps. Published numbers scatter with temperature, method, and microstructure, so treat them as representative; the ordering, however, is dependable: $T_1$(CSF) is much greater than $T_1$(gray) greater than $T_1$(white) greater than $T_1$(fat).

| Tissue | T1 at 1.5 T (ms) | T1 at 3 T (ms) | T2 (ms, approx.) | | --- | --- | --- | --- | | White matter | 560 to 650 | 830 to 1100 | 70 to 90 | | Gray matter | 920 to 1100 | 1330 to 1820 | 80 to 100 | | CSF | 3000 to 4000 | 3700 to 4500 | 1500 to 2200 | | Fat | 260 to 380 | 370 to 420 | 60 to 130 | | Skeletal muscle | 870 to 1000 | 1130 to 1420 | 30 to 50 | | Liver | 560 to 590 | 800 to 810 | 40 to 50 |

T2 versus T2*: irreversible and reversible dephasing

Equation 2.2 describes true T2: irreversible loss of coherence from spin-spin interactions, the random time-varying dipolar fields that scramble phase unpredictably. In a real magnet the measured free-induction-decay envelope falls off faster, because protons in different locations also see static field offsets from main-field imperfections, susceptibility differences at tissue interfaces, and chemical shift. These add reversible dephasing characterized by $T_2'$, and combine with true T2 to give the observed $T_2^{*}$ of Eq. 2.3. Because the offsets contributing to $T_2'$ are static, a 180 degree refocusing pulse reverses their accumulated phase and recovers that signal as a spin echo.

Use the RelaxationExplorer below to make these curves concrete. Set a 90 degree excitation and watch $M_z$ recover along the T1 exponential while $M_{xy}$ decays along the T2 exponential. Compare a long-T1 tissue such as CSF with a short-T1 tissue such as fat, then toggle the field-inhomogeneity contribution to see the steeper T2* envelope drop beneath the T2 curve.

Turning relaxation into contrast

Sequence timing converts relaxation differences into image intensity. With a spin-echo readout the available signal combines proton density $\rho$, T1 weighting through TR, and T2 weighting through TE, previewing the full signal equation.

• Short TR (comparable to tissue T1) maximizes the spread in $(1 - e^{-\mathrm{TR}/T_1})$, so short-T1 tissue recovers more between excitations and appears brighter: T1 weighting. Pair with short TE.
• Long TE (comparable to tissue T2) maximizes the spread in $e^{-\mathrm{TE}/T_2}$, so long-T2 tissue retains more signal and appears brighter: T2 weighting. Pair with long TR.
• Long TR with short TE suppresses both relaxation weightings, leaving signal proportional to $\rho$: a proton-density image.

Clinical consequences: contrast agents, iron, and hemorrhage

Gadolinium-based contrast agents are paramagnetic chelates that create large fluctuating local fields, accelerating relaxation of nearby water protons. They shorten both T1 and T2, but because tissue T1 is much longer than T2, a small added relaxation rate produces a large fractional change in T1 and only a modest one in T2. The net effect on a T1-weighted image is bright enhancement wherever the agent accumulates, such as in blood-brain-barrier breakdown. The relaxivity $r_1$ in Eq. 2.5 is the per-millimolar efficiency of the agent; clinical chelates have $r_1$ near 4 to 5 per mM per s at 1.5 T, so enhancement scales with concentration and relaxivity, not dose alone.

To summarize the chain of reasoning: molecular tumbling sets $\tau_c$; $\tau_c$ and field strength set T1 and T2 through the spectral density; static inhomogeneity adds $T_2'$ to give the measured T2*; and the sequence designer chooses TR and TE to weight images by these constants. Contrast agents and endogenous iron shift the relaxation rates directly, turning the same physics into diagnostic enhancement and susceptibility contrast. With these relationships in hand you can predict, before scanning, how any tissue will appear under a given sequence.