Biomedical Engineering Reference
In-Depth Information
magnetic fi eld generated by a superparamagnetic particle on the resonant fre-
quency of hydrogen nuclei in adjacent water molecules. When
Δ
ω
τ
CP
>
1, then the
particles are termed “ strongly ” magnetized, but when
1 the particles are
termed “weakly” magnetized. Since, for a typical relaxometer,
Δ
ω
τ
CP
<
τ
CP
is no shorter
than tens of microseconds,
must be less than 10
5
for the particles to be within
the weakly magnetized regime. Therefore, most superparamagnetic nanoparticles
used for magnetic relaxation assays are in the strongly magnetized theoretical
region because
Δ
ω
Δ
ω
(
∼
1
×
1 0
7
) is large compared to the inverse of achievable echo
times (1/
CP
= 10
3
). This means that the inter-echo delay is always longer than the
amount of dephasing that occurs at the surface of a particle. Particles with weaker
magnetizations (
τ
10
3
) induce less dephasing and are, within the theoretical
regime, referred to as “ weakly ” magnetized.
Another characteristic of superparamagnetic nanoparticle solutions that is used
to differentiate physical behavior is the diffusion time, or travel time, of water (
Δ
ω
∼
τ
D
)
relative to the inter-echo time of the pulse sequence,
τ
CP
. Nanoparticle solutions
are in the long echo limit when the
τ
D
is signifi cantly less than
τ
CP
.
τ
D
can be
determined by the relationship:
R
D
2
τ
D
=
(1.3)
where
D
is the time taken for a water molecule to diffuse the distance of a nanopar-
ticle radius,
R
, and
D
is the diffusion constant of water (10
− 9
m
2
s
- 1
). Here,
τ
D
can
be thought of as the time taken for a water molecule to pass a hemisphere of a
nanoparticle, or a “fl yby” time. When
τ
CP
, then the nanopar-
ticle system is within the short echo limit. Typical CPMG sequences have echo
times on the order of hundreds of microseconds to several milliseconds, and
therefore the short echo limit cannot be approached unless the nanoparticle diam-
eter approaches 1000 nm. The most common MRSw biosensors are within the
“long echo limit” because the length of the inter-echo delays (
τ
D
is much larger than
τ
0.25 ms) is
longer than the time taken for a water molecule to diffuse pass the hemisphere
of a nanoparticle (0.2 - 100
τ
CP
>
s).
As the particle size of a solution of superparamagnetic particles at fi xed iron
concentration is increased, there is an initial increase in
R
2
, followed by a plateau
and a later decrease (Figure 1.7). The regime on the left-hand side of the curve
has been termed the motional averaging regime, the regime in the middle the
static dephasing regime, and the regime on the right the visit-limited, or slow-
motion regime [57]. The boundaries between the motional averaging and visit-
limited regimes can be determined by generating plots such as that shown in
Figure 1.7, or they can be determined by the relationship between
μ
Δ
ω
and
τ
D
. If
Δ
1
then the system is in the visit-limited regime. As the diameter of the particles
increase in the motional averaging regime, the refocusing echos in the CPMG
pulse sequence (used to measure T
2
) cannot effi ciently refocus the magnetization
that has been dephased by the nanoparticles - hence the increase in
R
2
(or decrease
ω
τ
D
<
1, then the system is in the motional averaging regime, but if
Δ
ω
τ
D
>
