The traditional model in SE imaging to which the main question refers considers four combinations of TR and TE values:
The overall signal intensity (S) of a SE sequence can be shown to be approximately
where [H] is the spin (proton) density and K is a scaling factor. By considering the individual terms it is readily apparent that T1 effects are connected to TR and T2 effects are connected to TE, and that [H] effects are always present.
When TE is made short compared to T2, the ratio TE/T2 → 0, so the T2-weighting term e−TE/T2 → e−0 → 1. In other words, T2 effects largely disappear. Conversely, when TE is made long compared to T2, the importance of the exponential weighting term increases.
Another way to understand the effect of TE on T2-weighting is to consider the signals generated by two tissues with different T2 values. When TE is short, the echo occurs when there has been little time for T2-decay to have taken place and hence the tissues are not differentiated. If TE is long, the relative differences in signal decay between the two tissues become more noticeable, and hence more "T2-weighting."
Similar arguments can be made for the interplay between TR and T1. When TR is long compared to T1, the T1-weighting term e−TR/T1 → 0, so T1 effects disappear. At long TR's tissues with different T1 values have all had time to recover from the 90° excitation pulse, so their signals are not dramatically different. Conversely, short TR's accentuate "T1-weighting".
Finally, when TR is long and TE is short, both T1 and T2 effects are minimized. The only remaining factor is the spin-density [H], which becomes the dominant weighting for that combination of parameters.
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Long T1 materials are dark on T1-weighted images, but long T2 materials are bright on T2-weighted images. And vice versa. Why don't these behave the same way?