What’s special about 1.256 431 208 626… in MRI?
Consider the following simple question in magnetic resonance imaging (MRI) or spectroscopy (MRS): Given a fixed total measurement time, Ttotal, (e. g., a typical breath-hold duration of 16 s or a maximum accepted sequence duration of 10 min) and the possibility to fit into this duration an acquisition with several repeated read-outs (that are to be averaged to increase the data quality), what is the optimum balance between the repetition time (TR or TR) and the number of averaged signals (also called simply the number of averages, N, or the number of excitations, NEX or Nex)? (For simplicity, let’s consider only pulse sequences with 90 ° excitations (e. g., spin-echo sequences), in which all available magnetization is flipped to the transverse plain in each repetition.)
The solution (for 90° excitations)
In general, a single MRI acquisition (without averaging) requires M repeated excitations separated by TR (e. g., M could be the number of phase-encoding steps for conventional spin-echo acquisitions or M = 1 for single-shot EPI or one-dimensional MR spectroscopy). Hence, the acquisition time without data averaging is Tacq = MTR, and, neglecting for now that the number of averages must be integer, this number of averages is N = Ttotal / Tacq = Ttotal / (MTR). In the following, the actual relevant time parameter is, thus, Tavail = Ttotal / M, the time available for each required excitation, and the number of averages is N = Tavail / TR. We can also express the repetition time by these parameters as:
On the one hand, averaging data increases the obtainable signal-to-noise ratio (SNR or Ψ) proportional to the square root of the number of averages:
On the other hand, increasing the number of averages (at fixed total measurement duration!) results in a shortened time for longitudinal relaxation: TR = Tavail / N and, thus, in less SNR:
The resulting total SNR is proportional to the product of both factors, i. e.,
The SNR (as a function of TR at constant total measurement time) has, thus, a maximum in an intermediate range; for very short TR (and, consequently, many averages), the longitudinal magnetization cannot relax sufficiently, which reduces the available signal; for very long TR, the relaxation is approximately complete anyway, but the number of averages decreases.
To calculate the optimum TR (or N), TR and Tavail are best expressed in terms of T1 using τ = TR / T1 and T = Tavail / T1, which gives
The maximum of this expression with respect to τ is obtained by setting its derivative to zero, i. e.
The solution of this equation is
where W–1 is the lower branch of the Lambert W-function (or product-log or Omega function, which gives the solution for W in z = Wexp(W), i. e., it is the inverse function of f(W) = Wexp(W)). Hence, the optimal choice for TR is (at least theoretically):
Fortunately, the function to be maximized has a rather broad maximum, and one obtains values between 99 % and 100 % of the maximum SNR for TR between about 0.9937 T1 and 1.5773 T1 and values between 95 % and 100 % of the maximum SNR for TR between about 0.7293 T1 and 2.0882 T1 (values determined by numerical evaluation and illustrated in the first figure at the top).
So, practically, choosing TR to be T1 gives still a nearly optimal SNR; if a larger number of signal acquisitions is preferred, e. g., for statistical evaluation, then TR may even be shortened to 0.75 T1, and if – on the other hand – the base SNR of every single acquisition becomes very low (but is to be reconstructed before signal averaging) then TR may be chosen to be 2 T1 to increase the quality of each single (not averaged) data set.
The presented result is not new; in fact, it has been derived before (at least numerically) in several publications. An early example is, e. g., a publication by R. R. Ernst and R. E. Morgan (1973).
The presented analysis is valid only for simple 90° excitations; FLASH sequences or steady-state pulse sequences which refocus the transverse magnetization as well will exhibit a different behavior.
Of course, optimizing TR as described above is not an option if T1-weighted or T2-weighted images are to be acquired. In these cases, the optimal value of TR depends on the desired contrast and not on the total SNR.
(This is a shortened version of a slightly longer pdf document that contains some more details and discusses also a few special cases.)