M³ – Monologues, MRI, Miscellanea

Is T₂ necessarily shorter than T₁?

If you are familiar with MRI (or NMR in general), then probably also with the relaxation time constants T₁ and T₂. These tissue-specific (or substance-specific) constants describe how fast the nuclear magnetization returns to its equilibrium value, M₀, after excitation by a pulsed radio-frequency (rf) field. Shortly summarized, T₁ describes the exponential recovery of the longitudinal magnetization M_L (i. e., of the magnetization parallel to the external static magnetic field B₀), and T₂ describes the exponential decay of the transverse magnetization M_T (that is precessing in the plane orthogonal to B₀).

Typically (as shown in the first figure), T₂ values of tissue are considerably lower than T₁ values, i. e., the transverse magnetization decays quicker than the longitudinal relaxation needs for recovery. For most tissues in vivo, T₁ varies between about 300 ms and 3 s, while T₂ varies between about 10 ms and 200 ms. Longer T₂ relaxation times (up to about 3 s as well) are found for liquids.

So one may ask if there are good physical reasons for T₂ values being shorter than or – at most – equal to T₁. As a physicist, I’d start with checking some extreme cases, e. g., assuming that T₂ is much longer than T₁, i. e. T₂ ≫ T₁. Then the longitudinal magnetization can fully recover while at the same time some transverse magnetization would be preserved. As a result, the magnitude of the total magnetization √( M_L² + M_T² ) would become greater than the equilibrium value M₀ – which is physically impossible.

To analyze these properties of T₁ and T₂ in more detail, longitudinal and transverse relaxation can also be plotted together in a diagram showing the transverse magnetization on the horizontal axis and the longitudinal magnetization on the vertical axis. These diagrams show the evolution of the magnetization (for experts: in a rotating frame of reference) after a 90° rf pulse (all trajectories start at the lower right corner of the diagram). The left-hand side of the following figure shows three cases T₂ = T₁ / 2 (blue), T₂ = T₁ (green), and T₂ = 2 T₁ [sic!] (cyan); and all three curves show a “benign” behavior in that they lie in the shaded area below the black circle segment M_L² + M_T² = M₀². This means that the magnitude of the total magnetization vector is always smaller than M₀.

However, the right-hand side of this figure shows what’s happening if T₂ becomes greater than 2 T₁ – in this case, T₂ = 3 T₁ (red curve): Now the magnitude of the total magnetization vector increases above the physical limit of M₀, i. e., the red line crosses the black border of physically benign behavior!

In fact, it can be shown (see appendix if interested) that the maximum T₂ value, for which the red curve stays always below the black line, is exactly T₂ = 2 T₁. And as so often, almost everything that is physically possible is also realized in nature (although the case T₁ < T₂ < 2 T₁ is really extremely rare), as described by Malcolm H. Levitt in his highly recommendable NMR text book “Spin dynamics” (2nd ed., section 11.9.2, note 13):

The case where T₂ > T₁ is encountered when the spin relaxation is caused by fluctuating microscopic fields that are predominantly transverse rather than longitudinal. One mechanism which gives rise to fields of this form involves the antisymmetric component of the chemical shift tensor (not to be confused with the CSA). […] Molecular systems in which this mechanims is dominant are exceedingly rare (see F. A. L. Anet, D. J. O’Leary, C. G. Wade and R. D. Johnson, Chem. Phys. Lett., 171, 401 (1990)).

So, the answer to the title question is: No, T₂ can in fact be greater than T₁ in very special circumstances, but it can never be greater than 2 T₁.

Appendix

The maximum T₂ value, for which the red curve stays always below the black line, is exactly T₂ = 2 T₁. This can be seen by analyzing the inequality

First, we divide by M₀² and set T₂ = αT₁ as well as β = exp(–t / T₁), yielding

which is (after subtraction of 1 and division by β)

β is by definition (for positive t) between 0 and 1, so the right-hand side of the last inequality is a linear function descending from 2 to 1 (i. e. always  ≤ 2). Its left-hand side has very different shapes depending on α: it is increasing from 0 to 1 for 0 < α ≤ 2 (since then the exponent 2 / α – 1 ≥ 0); but it is going to infinity for β → 0 if α > 2 (since then the exponent 2 / α – 1 < 0). So, the last inequality will not hold in the latter case for sufficiently small values of β, which means that non-physical behavior occurs if α > 2 or, using the definition from above, if T₂ > 2 T₁.