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 V_b 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

V_b = 2/g GT₁T₂^*. [1]

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 [2]

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 T_d. Then

m₁ = GTT_d [3]

and the flow induced phase shift is given by f = g GTT_dV. 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 T_d. 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 T_d can be as long as T₁ for stimulated echoes, which implies from Eq. [3] that

m₁ = GT₂T₁/2 [4]

Thus, from Eqs. [2] and [4], the minimum velocity measurable by the phase method is

V_p = 2F /g GT₂T₁, [5]

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 V_p is smaller than V_b is because T₂^* is smaller than T₂. Therefore, phase methods are inherently more sensitive to slower velocities than bolus tracking when T₂^*< T₂.