First of all what does the term "time of flight" refer to? Time of flight is a general term used to indicate that what we're interested in is the time it takes for a particle to travel through a medium. It could be electrons through a semiconductor, ions through a vacuum (time of flight spectroscopy)... but in the case of vascular imaging, we're interested in particles with spin moving through the plane (or slab) of the MRI image.
The issue with any angiogram is contrast... that is, we want to make the vessels stand out or become more conspicuous relative to their surroundings. In conventional angiography, this can be accomplished with DSA technique. With CTA, the density of the iodinated contrast is much denser than surrounding tissue, resulting in improved contrast...
...but with MRA, the situation is sort of backwards. Improved contrast of the vessels is achieved primarily by making everything else dark. For instance, in the cross-section of the calf below, the TR is set short enough to keep the different spins in the tissues from recovering much longitudinal magnetization at all prior to the next excitation pulse, causing a weak signal from the "saturated" protons.
So... although the first image acquired in a particular slice may have the normal T1WI appearance you might be used to... if you hit that slice with an RF pulse again , and then again, and then again... the poor spins get so bombarded with RF pulses that they reach a steady state where they are not recovering much transverse magnetization at all, and a ghosty image slice remains.
Normally in a GRE imaging, a "spoiler" mechanism is used to prior to the start of the next TR to prevent buildup of transverse magnetization... but in this situation we want to saturate out the tissue to make the vessels bright, and we have to turn that normal spoiler off.
An important point to note is that since our goal is a GRE sequence with very short TRs to saturate out everything but the vessels, our MRA sequence will be inherently T1-weighted (short TR GRE). This has the potential to lead to some confounders with any materials that are inherently T1 bright (like subacute hemorrhage).
So where does the blood in the vessel fit into all of this? Well... as the proton spins are being bombarded in the slice of interest, the vessel is constantly bringing in "new" protons with inflowing blood. These, not having had a chance to be barraged with the high-repetition RF pulses, don't have a build-up of transverse magnetization, have a higher longitudinal magnetization, and let off a bright signal... and BAM! you've got contrast.
...but there is a limit to how low you can go on the TR. Take it down too far, and you may give the perpendicularly inflowing blood too much time in the plane to get hit with the RF pulses, decreasing contrast. You could fix this with thinner sections (but this means more time in the scanner, and more potential for motion).
Another important parameter is the flip angle. The higher the flip angle, the more the longitudinal magnetization is tipped into the transverse plane, and the greater the saturation.
So what if we only want to see arterial flow?, or venous flow?, and not have both superimposed on each other in a moosh? This is what the saturation band in MRA is for. According to the reasoning above, any new spins moving into the plane perpendicularly would produce bright signal... arterial or venous. Fortunately, the arteries and veins usually move in opposite directions, so in order to get rid of either one of them for a better look, all you have to do is apply the same RF bombardment principles at the level just above or below the one you're imaging (the saturation band), and all the inflowing spins will already have their magnetization maxed and won't contribute to flow-related enhancement.
1. "MRI Principles" Mitchell DG and Cohen MS. 2nd ed (2004) pp 318-324
2. Tatli S, Lipton MJ, Davison BD, et al. "MR Imaging of Aortic and Peripheral Vascular Disease" Radiographics 2003; 23:S59-78