M3 - Monologues, MRI, Miscellanea

What’s special about 1.256 431 208 626… in MRI?

posted on 2018-06-28 (by Olaf Dietrich)

Consider the following simple question in magnetic resonance imaging (MRI) or spectroscopy (MRS): Given a fixed total measurement time, T_total, (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 T_R) and the number of averaged signals (also called simply the number of averages, N, or the number of excitations, NEX or N_ex)? (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)

Dependence of the signal-to-noise ratio $Ψ$ on the repetition time $T_{ ext{R}}$ at fixed total measurement time with a maximum at $T_{ ext{R}} / T_1 = 1.256 431 208 626…$ — Dependence of the signal-to-noise ratio `Ψ` on the repetition time `T`_R at fixed total measurement time with a maximum at `T`_R / `T`₁ = 1.256 431 208 626…

In general, a single MRI acquisition (without averaging) requires M repeated excitations separated by T_R (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 T_acq = MT_R, and, neglecting for now that the number of averages must be integer, this number of averages is N = T_total / T_acq = T_total / (MT_R). In the following, the actual relevant time parameter is, thus, T_avail = T_total / M, the time available for each required excitation, and the number of averages is N = T_avail / T_R. We can also express the repetition time by these parameters as:

T_R = T_acq / M = T_total / (M N) = T_avail / N.

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:

Ψ ∝  N  =  T_avail / T_R .

Dependence of the SNR $Ψ$ on the number $N$ of averages (left) and the $T_{ ext{R}}$ (right, at fixed total measurement time) due to averaging neglecting the influence of $T_{ ext{R}}$ on $T_1$ relaxation — Dependence of the SNR `Ψ` on the number `N` of averages (left) and the `T`_R (right, at fixed total measurement time) due to averaging neglecting the influence of `T`_R on `T`₁ relaxation

On the other hand, increasing the number of averages (at fixed total measurement duration!) results in a shortened time for longitudinal relaxation: T_R = T_avail / N and, thus, in less SNR:

Ψ ∝ 1 – exp(–T_R / T₁) = 1 – exp –T_availNT₁.

Dependence of the SNR $Ψ$ on the number $N$ of averages (left, at fixed total measurement time) and the $T_{ ext{R}}$ (right) due to $T_1$ relaxation neglecting the influence of signal averaging — Dependence of the SNR `Ψ` on the number `N` of averages (left, at fixed total measurement time) and the `T`_R (right) due to `T`₁ relaxation neglecting the influence of signal averaging

The resulting total SNR is proportional to the product of both factors, i. e.,

Ψ ∝  T_avail / T_R (1 – exp(–T_R / T₁)):

Resulting dependence of the SNR $Ψ$ on $T_{ ext{R}}$ at constant total measurement time (and thus, implicitly, on the number of averages) considering both averaging and relaxation effects — Resulting dependence of the SNR `Ψ` on `T`_R at constant total measurement time (and thus, implicitly, on the number of averages) considering both averaging and relaxation effects

The SNR (as a function of T_R at constant total measurement time) has, thus, a maximum in an intermediate range; for very short T_R (and, consequently, many averages), the longitudinal magnetization cannot relax sufficiently, which reduces the available signal; for very long T_R, the relaxation is approximately complete anyway, but the number of averages decreases.

To calculate the optimum T_R (or N), T_R and T_avail are best expressed in terms of T₁ using τ = T_R / T₁ and T = T_avail / T₁, which gives

Ψ ∝  Tτ (1 – exp(–τ))

The maximum of this expression with respect to τ is obtained by setting its derivative to zero, i. e.

0 = ∂∂τ [ T / τ (1 – exp(–τ))] = exp(–τ)2 Tτ³ (1 + 2τ – exp(τ))

which gives

1 + 2τ – exp(τ) = 0.

The solution of this equation is

τ = – 12 – W_–1(–1 / (2  e )) = 1.256 431 208 626…

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 T_R is (at least theoretically):

T_R ≈ 1.256 T₁.

Fortunately, the function to be maximized has a rather broad maximum, and one obtains values between 99 % and 100 % of the maximum SNR for T_R between about 0.9937 T₁ and 1.5773 T₁ and values between 95 % and 100 % of the maximum SNR for T_R 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 T_R to be T₁ gives still a nearly optimal SNR; if a larger number of signal acquisitions is preferred, e. g., for statistical evaluation, then T_R 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 T_R 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 T_R 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 T_R 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.)

Categorized: MRI, NMR, SNR, physics
pdf version available
URL: <https://dtrx.de/od/mmm/mmm_20180628.html>

M3 – Monologues, MRI, Miscellanea

What’s special about 1.256 431 208 626… in MRI?

The solution (for 90° excitations)

M³ – Monologues, MRI, Miscellanea

What’s special about 1.256 431 208 626… in MRI?