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 T_{1},
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 T_{2}^{*}
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 T_{2}^{*} but is affected by T_{2}.
In a given voxel, T_{2}^{*} will be smaller than T_{2} 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/T_{1}* *where R is the resolution of the imaging
experiment, in order to resolve the bolus displacement. For a given T_{2}^{*}
the half-height bandwidth of the broadening is 2/ T_{2}^{*}. and if the
read-out gradient is G, this corresponds to a resolution of R = 2/g
G T_{2}^{*} , which leads to the lower limit on the measurable velocity of

V_{b} = 2/g GT_{1}T_{2}^{*}.
[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 m_{1}V [2]

where m_{1 }is proportional to the first moment of the gradient (m_{1}
= 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

m_{1} = GTT_{d} [3]

and the flow induced phase shift is given by f = g GTT_{d}V. In order to measure slow flow, a sufficiently
large first moment m_{1}, 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 T_{2} before forming
an echo, the bipolar gradient pulse can have a duration of the order of T_{2}/2,
and the time between the pulses T_{d} can be as long as T_{1} for
stimulated echoes, which implies from Eq. [3] that

m_{1} = GT_{2}T_{1}/2 [4]

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

V_{p} = 2F /g GT_{2}T_{1},
[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** **T_{2}^{*}
is** **smaller** **than** **T_{2}. Therefore, phase** **methods** **are**
**inherently** **more** **sensitive** **to** **slower** **velocities than
bolus tracking when T_{2}^{*}< T_{2}.