Proton-Density Weighting
INCORRECT. A short TR maxmises signal differences due to T1 differences between tissues. This is used in T1-weighting.
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INCORRECT. A short TR maxmises signal differences due to T1 differences between tissues. This is used in T1-weighting. A long TE maxmises signal differences due to T2 differences between tissues. This is used in T2-weighting. So, a short TR with a long TE would produce a confused mix of contrast properties in an image, and the signal-to-noise ratio would be reduced. This type of contrast is not required in MRI.
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CORRECT. A long TR minimises signal differences due to T1 differences between tissues, and provides allows more magnetisation to recover in the z-direction after each signal measurement, ready for the next signal measurement. A short TE minimises signal differences due to T2 differences between tissues, and provides a higher signal level since less decay of the precessing net magnetisation in the xy plane has occurred. The resultant contrast properties of an image are called proton-density (or spin-density) weighting.
Note that we only ever speak of weighted images: There is always some T2 and T1 weighting present in any conventional MRI image. Proton-density weighted images are not often required in MRI; the proton density does not differ very much from one tissue to the next, and the contrast between tissues is low.
Further reading on this topic:
Books: MRI The Basics p88, MRI From Picture to Proton p34
Online: Medcyclopaedia