SLOW FLOW NMR LIMITS: Bolus tracking versus phase methods.
Two methods for flow measurement which have been widely used are the time-of-flight and the phase methods. The time-of-flight method is based on detecting spin displacement during the time between initial excitation and signal measurement. This displacement is limited by the velocity of the flowing material and the lifetime of the tag. In addition, information outside of the bolus is not accessible without further experiments. The second class of methods detect the alteration of the phase of transverse spin magnetization induced by spin motion along a magnetic field gradient. We have always taken it for granted that phase methods could measure lower velocities than the bolus tracking methods but had not looked into this question. Here we discuss the circumstances when phase methods can measure slower velocities than the bolus-tracking methods. The discussion is in terms of stimulated echo experiments where the lifetime of the tag is T1, but the conclusion is equally applicable to spin echo and gradient- echo experiments.
The difference arises because the position of the bolus is defined in the frequency encoding direction and its accuracy depends on the effective T2* caused by static field homogeneity but not the applied gradient whereas the velocity phase-encoding is analogous to the spatial phase-encoding in spin warp imaging experiments, and is not affected by T2* but is affected by T2. In a given voxel, T2* will be smaller than T2 because of inhomogeneous magnetic field within a voxel caused by susceptibility changes in the sample and imperfect shimming.
First, we consider the bolus-tracking method in which the velocity Vb has to be greater than R/T1 where R is the resolution of the imaging experiment, in order to resolve the bolus displacement. For a given T2* the half-height bandwidth of the broadening is 2/ T2*. and if the read-out gradient is G, this corresponds to a resolution of R = 2/g G T2* , which leads to the lower limit on the measurable velocity of
Vb = 2/g GT1T2*. 
Here we are assuming that, because of broadening, the bolus has to move a distance R before it can be accurately distinguished from its original position.
The phase methods depend on the phase shift of the transverse magnetization of spins moving with a constant velocity and a component V along the velocity encoding gradient G being given by
f = g m1V 
where m1 is proportional to the first moment of the gradient (m1
= ò tG(t) dt with
ò G(t) dt = 0). A simple velocity encoding method consists of a bipolar gradient pulse, with gradient amplitude G, duration T, and the time between the pulses Td. Then
m1 = GTTd 
and the flow induced phase shift is given by f = g GTTdV. In order to measure slow flow, a sufficiently large first moment m1, has to be used to give a measurable phase shift. For bipolar gradient pulses this can be accomplished by increasing either the area of the gradient pulses GT or the separation of the pulses Td. Now for the phase method, with the transverse magnetization evolving for a time T2 before forming an echo, the bipolar gradient pulse can have a duration of the order of T2/2, and the time between the pulses Td can be as long as T1 for stimulated echoes, which implies from Eq.  that
m1 = GT2T1/2 
Thus, from Eqs.  and , the minimum velocity measurable by the phase method is
Vp = 2F /g GT2T1, 
where F is some minimum phase shift distinguishable from zero.
Because we are not directly considering the effects of the signal/noise ratio, the exact value of F is arbitrary in this discussion, just as the value of the minimum spatial resolution R as defined by the half-height was arbitrary. The main reason why Vp is smaller than Vb is because T2* is smaller than T2. Therefore, phase methods are inherently more sensitive to slower velocities than bolus tracking when T2*< T2.