Biomedical Engineering Reference
In-Depth Information
where τ
0
is the attempt time (generally taken to be 10
−9
seconds),
V
m
is the volume of magnetic material,
k
B
is Boltzmann's con-
stant, and
T
is the absolute temperature. The approximate criti-
cal diameters for superparamagnetism for several common
magnetic nanoparticle materials are included in Figure 17.2.
Although remnant magnetization and hysteresis behavior are
eliminated in superparamagnetic particles, significant losses
can still occur through moment relaxation mechanisms.
The following description and equations are based on reviews
provided by Rosensweig (Rosensweig 2002) and Hergt et al.
(Hergt et al. 2002). Relaxation of superparamagnetic particles
is described by two mechanisms: Brownian and Néelian relax-
ation, illustrated in Figure 17.2. Brownian relaxation involves
the physical rotation as the overall particle moment aligns with
the external field, creating frictional losses. Néelian relaxation
involves the rotation and losses of the individual moments
within the particle. Characteristic time constants can be used to
describe each process:
where χ′ and χ″ are the in-phase and out-of-phase components of
the ferrofluid magnetic susceptibility, respectively. Substitution
and integration of Equation 17.7 yields the cyclic change in
internal energy, which can then be multiplied by the frequency
to give the volumetric power generation:
2
P
= ′′χµπ
(
f H
).
(17.10)
0
0
Thus the rate of heating is dependent on the out-of-phase
component of susceptibility (lagging the applied field) and the
incident power density, which is the term in parentheses. This
expression is equivalent to SAR in watts per cubic meter of fluid
(or tissue). This can be easily converted into more standard units
of cubic centimeters or grams tissue. In addition, absorption for
magnetic nanoparticles is often expressed in terms of watts per
mass iron, which can also be obtained through simple conver-
sions. This value is often termed SAR
Fe
or specific loss power
(SLP). Both SAR and SLP will be used throughout the remainder
of the chapter, and it is important to keep the distinction straight.
The ferrofluid susceptibility is dependent on both nanopar-
ticle and field properties, so it is helpful to express this term
through more fundamental parameters. Frequency dependence
can be given by:
τ=
η
V
kT
3
H
Brownian:
(17. 3)
B
B
Γ
π
e
K V
kT
uM
B
2
12
πτ
πτ
χ
f
Néelian:
τ
=
τ
,
Γ
=
(17. 4)
N
0
χ
′′
=
2
Γ
(17.11)
0
+
(
f
)
2
where η is the viscosity of the suspending medium,
V
H
is the
hydrodynamic volume of the particle (including coatings), and Γ
is used to represent the ratio of anisotropic to thermal energies.
These two processes occur simultaneously, governing behavior
much like two resistors in parallel. The shorter time constant
will thus have a tendency to dominate, and the effective relax-
ation time (τ) can be found by:
where χ
0
is the equilibrium susceptibility, which can be con-
servatively estimated by the chord susceptibility, following the
Langevin equation (Rosensweig 2002):
3
1
µ
MHV
kT
0
s
am
χχ
ξ
=
coth()
ξ
−
,
ξ
=
(17.12)
0
i
ξ
B
where χ
i
is the initial susceptibility and ξ is the Langevin param-
eter. The initial susceptibility is determined by differentiating
the Langevin relationship:
τ=
ττ
τ+τ
BN
BN
.
(17. 5)
Magnetic work is traditionally expressed as the product of the
field strength and the change in magnetic induction. The work
performed by the external field is going to result in a change
in internal energy (
U
). Taking the fundamental relationship
between induction (
B
), magnetization (
M
), and applied field:
2
χ=
µφ
MV
kT
0
s
m
.
(17.13)
i
3
B
Equations 17.10 through 17.13 then provide the capability to
predict SAR based on nanoparticle and field parameters, which
is often expressed in the simplified form, which follows. In this
form, SAR depends on nanoparticle concentration through the
volume fraction in Equation 17.12. However, if it is converted
to watts per gram magnetic material, the SLP will be constant
for a given frequency and field strength (i.e., no concentration
dependence).
BHM
a
)
(17. 6)
=
µ
0
(
+
and integrating by parts, the cyclic increase in internal energy
can be found by:
0
∫
U
=−µ
M dH
.
(17.7)
2
1(2)
πτ
+πτ
f
W
m
For a sinusoidal alternating magnetic field, the time-dependent
field and magnetization can be expressed in terms of the field
strength and frequency:
2
SAR
=µ πχ
f H
~
.
(17.14)
0
0
0
2
3
f
Relaxation behavior depends strongly on nanoparticle size, and
heating demonstrates a peak efficiency at a specific radius, depend-
ing on the magnetic material. Bulk values for saturation magne-
tization (
M
s
) and anisotropy (
K
u
) for some relevant materials are
Ht
()
=
H
cos(
2π
ft
)
(17. 8)
a
Mt
()
=
H
(
χ
′
cos(
2
π
ft
)
+ ′′
χ
sin(
2
π
ft
)
)
(17.9)
a
