The filter for spin echo T1 is $S=1-e^{\frac{\mathrm{-TR}}{\mathrm{T1}}}$. A pulse sequence has a constant TR and tissues have different T1s. For a given pulse sequence, when plotting the signal due to T1 from different tissues the TR remains constant and T1 is the dependent variable. The graph shows signal resulting from "T1 weighting," in a spin echo sequence with a given TR. Change the TR of the pulse sequence with the slider or text box and the graph moves from side to side providing different contrasts for different ranges of T1.

Contrast between tissues with different T1 relaxation times is the difference in the signals between these tissues To distinguish grey matter from white matter the curve should be steep in the range of T1s between grey and white matter.

Likewise, a sequence's sensitivity to T1 changes due to pathology is based on the slope of the curve at that T1. If the curve is steep, small changes in T1 will result in large changes in relative signal.

Select different tissue types to see how their relative signals change with TR.

The curves can be plotted on a linear axis (T1) or a logarithmic axis (lnT1). On the linear axis, changes along the x axis represent a change in T1. On the logarithmic axis, a change along the x axis represents the fractional change in T1, or $\frac{\Delta \mathrm{T1}}{\mathrm{T1}}$. This is because for $\frac{{\mathrm{T1}}_a}{{\mathrm{T1}}_b}\approx 1$, $\ln \left({\mathrm{T1}}_a\right)-\ln \left({\mathrm{T1}}_b\right)=\frac{{\mathrm{T1}}_a-{\mathrm{T1}}_b}{{\mathrm{T1}}_a}$.

The T1 contrast for absolute changes in T1 is the slope of the graph with the linear x-axis. The T1 contrast for fractional changes in T1 is the slope of graph with the logarithmic axis. There is debate as to whether contrast is best defined based on absolute or fractional changes, so both are provided.

What is apparent is that the contrast is different across different ranges of T1. The TR needs to be chosen so there is contrast in the range of T1s of clinical interest.

What TR would be good for distinguishing grey and white matter? Liver from fat? Spleen from liver? Visually, TRs from 500 to 1200 result in good contrast over the range of typical soft tissues (fat, muscle, grey matter, white matter).

Can you maximize contrast? Do you need to?

You can determine the TR which will result in the maximum T1 contrast between two tissues if you know the T1 values of the tissues. The difference in signal is $e^{\frac{\mathrm{-TR}}{a}}-e^{\frac{\mathrm{-TR}}{b}}$ where a and b are the T1 values of the tissues in question. You can determine the maximum difference by setting the 1st derivative to zero and solving for TR, which yields $\mathrm{TR}=\frac{\mathrm{ab\; ln}\left(\raisebox{1ex}{$a$}\!\left/ \!\raisebox{-1ex}{$b$}\right.\right)}{a-b}$.

For distinguishing grey and white matter, the optimal TR is 878 ms. How about for distinguishing muscle and fat? It is 320 ms.

This is complicated, and it is normally not useful to maximize the contrast between normal structures unless you are after an antaomic image. You cant maximize contrast between all tissues with a single TR. What is useful is being sensitive to changes from normal. When looking for pathology, the filter must have a steep slope at the T1 of the tissue of interest.

You can determine the TR which will result in the maximal contrast, or steepest slope, for a given T1 by setting the second derivative of $S=1-e^{\frac{\mathrm{-TR}}{\mathrm{T1}}}$ to zero and solving for TR. This is the TR which will produce the most contrast near a specific T1 value.

For the graph plotted on the linear T1 axis, the curve is steepest at T1 = TR/2. For the graph potted on the lnT1 axis, the curve is steepest at T1 = TR.

For maximal sensitivity to changes in white matter T1 (based on abolute changes in T1) choose a TR that is twice the T1 of white matter. This will make the curve steepest at the T1 of white matter.

For maximal sensitivity to changes in white matter based on fractional changes in T1 choose a TR that is equal to the T1 of white matter. This will make the curve steepest at the T1 of white matter and will most accentuate their differences.

Its frustrating that based on the x-axis you prefer, the optimal TR changes by a factor of 2!

Note that the optimal TR for distinguishing grey and white matter (TR of 878 ms) is NOT the optimal TR for detecting changes in the T1 of white matter (TR of 1360 ms). This means a T1 spin echo sequence optimized for segmenting grey matter structures is not the most sensitive sequence for detecting, for example, increased T1 in the white matter secondary to demyelination or edema. Likewise, the optimal TR for distinguishing muscle and fat (TR of 320 ms) is not the most sensitive TR for detecting changes in the T1 of muscle (TR of 840 ms).

This is OK. You can't optimize a sequence for everything. A TR of, for example, 700 ms does a good job of distinguishing normal tissues AND detecting changes from normal over a wide range of tissue T1 values.