Biomedical Engineering Reference
In-Depth Information
in T
2
). In other words, the refocusing pulses cannot compensate for increased
dephasing by larger nanoparticles. The fl at region of the static dephasing regime
is due to
R
2
being limited by
R
2
*
. The decreasing
R
2
with increasing diameter in
the visit-limited regime results in the refocusing pulses being able to refocus the
dephasing caused by the nanoparticles. Also apparent in Figure 1.7 is that
R
2
in
the slow-motion regime exhibits a dependence on the inter-echo delay of the spin
echo sequence [53] .
In a homogeneous magnetic fi eld, it is possible to determine which regime
applies to a sample by comparing
R
2
to
R
2
*
. If these values are identical, then one
is in the motional averaging or static dephasing regime, but if they are different
then one is in the visit-limited regime [53, 54]. This approach has been employed
for determining the physical characteristics of MRSw biosensor systems [59 - 61] .
However, as discussed above, the
T
2
*
of bench-top relaxometers is rarely larger
than 5 ms, resulting in a lower limit for
R
2
*
of 200 s
− 1
. This means that, on bench-
top relaxometers,
R
2
will never be equivalent to
R
2
*
except at extremely high iron
concentrations. For example, a typical solution of nanoparticles such as CLIO-47
has an
R
2
of 40 m
M
− 1
s
− 1
, so for
R
2
*
to equal
R
2
the concentration of iron would
need to exceed 5 m
M
, which is 50-fold higher than typical iron conditions. The
relationship between
R
2
*
and fi eld homogeneity is important to bear in mind
when selecting instruments for characterizing MRSws. Fortunately, the echo time
dependence of
R
2
allows an easy method for determining whether one is in the
motional averaging or visit- limited regime.
The conditions used to generate the analytical models that explain the
dependence of
R
2
on particle size were similar to the conditions used for
MRSw assays. That is, the concentration of iron was held constant while
R
2
was monitored as a function of nanoparticle diameter. The analytical models
have been shown to accurately predict the dependence of
R
2
on parameters that
a biosensor designer can control, such as iron concentration, temperature,
magnetic susceptibility, particle size, and particle size [54]. Interestingly, all
of these parameters remain relatively constant for a given MRSw in comparison
to particle size, which dominates the change in
R
2
. The same group which
developed the analytical models was the fi rst to demonstrate that these models
could be used to explain the behavior of a system of clustering superparamagnetic
particles [62]. Their experimental system consisted of superparamagnetic nanopar-
ticles that clustered due to a change in the pH of the solution. After an initial
phase that was attributed to a stabilization of the dispersed particles,
R
2
was seen
to increase with agglomeration until a plateau was reached prior to a decrease in
R
2
with agglomeration. The shape of the
R
2
response as the particles agglomerated
generally matched the expected trend for the increase in average nanoparticle size,
which was similar to the shape of both dashed lines in Figure 1.7. Additionally,
Roch
et al.
demonstrated a general quantitative agreement between the measured
and expected
R
2
values. Similar exercises have since been carried out by subse-
quent authors to validate the qualitative nature of the T
2
response they were
observing, and to determine which regime their nanoparticle assays fell within
[20, 59, 61].
