## 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, `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)

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}).

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}/

`T`

_{1}) = 1 – exp (–

`T`

_{avail})/(

`NT`

_{1}).

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

`Ψ`∝ √(

`T`

_{avail}/

`T`

_{R})(1 – exp(–

`T`

_{R}/

`T`

_{1})):

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 `T`_{1} using `τ` = `T`_{R} / `T`_{1} and `T` = `T`_{avail} / `T`_{1}, which gives

`Ψ`∝ √( (

`T`)/(

`τ`) )(1 – exp(–

`τ`))

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

`τ`) [√(

`T`/

`τ`)(1 – exp(–

`τ`))] = (exp(–

`τ`))/(2)√( (

`T`)/(

`τ`

^{3}) )(1 + 2

`τ`– exp(

`τ`))

which gives

`τ`– exp(

`τ`) = 0.

The solution of this equation is

`τ`= – (1)/(2) –

`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` = `W`exp(`W`), i. e., it is the inverse function of `f`(`W`) = `W`exp(`W`)). Hence, the optimal choice for `T`_{R} is (at least theoretically):

`T`

_{R}≈ 1.256

`T`

_{1}.

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 `T`_{1} and 1.5773 `T`_{1} and values between 95 % and 100 % of the maximum SNR for `T`_{R} between about 0.7293 `T`_{1} and 2.0882 `T`_{1} (values determined by numerical evaluation and illustrated in the first figure at the top).

So, practically, choosing `T`_{R} to be `T`_{1} 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 `T`_{1}, 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 `T`_{1} 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 `T`_{1}-weighted or `T`_{2}-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.)*

pdf version available

URL: <https://dtrx.de/od/mmm/mmm_20180628.html>