Page 167 - Bio Engineering Approaches to Cancer Diagnosis and Treatment

7.4 Electromagnetic fields application in drug delivery 165

In above equation H, µ , and dv are external magnetic field, vacuum magnetic µ
0
permeability, and finite element, respectively. Since the magnetic nanocarrier size is 0
very small, the magnitude of the external magnetic field and the internal magnetic
field (magnetization) of the magnetic nanocarriers are approximately uniform. As a
result, the magnetic energy of a nanocarriers Eq. (7.12) is obtained from:
1 (
U =− µ ∀ MH ) (7.13)
⋅
m 2 0 MNP Um=−12µ ∀MNPM→⋅H→
0
By combining Eqs. (7.11) and (7.13), the amount of force applied to the MNP is
obtained from:
1
F = µ ∀ ∇ ( M H ) (7.14)
⋅
m 2 0 MNP F→m=12µ ∀MNP∇M→⋅H→
0
∀ MNP is the volume of the nanoparticle. To calculate the magnetic energy (Um), it ∀MNP
is necessary to determine the magnitude of the internal magnetic field (magnetiza-
tion) by applying Eq. (7.14). The internal magnetic field of an object is based on two
inductive magnetic fields under the influence of the external magnetic field and the
hysteresis. In paramagnetic materials, the induction magnetic field due to the exter-
nal magnetic field is determined from the Longevin’s law as follow [22]:
M = coth  3χ H  − 1 (7.15)
M s   M   3χ H
s
M s MMs=coth3χHMs−13χHMs
where χ and M are the magnetic susceptibility and saturation magnetization, respec-
s
tively. The saturation state is a situation in which the increase in the external field
strength has no effect on the magnetization. The internal induction field of the nano-
carriers is calculated in terms of the intensity of the external magnetic field (Eq. 7.15)
and is plotted in Fig. 7.5.
As shown in Fig. 7.5, for the nonintense magnetic field (3χ HM < 2.5), the rela- 3 χH/Ms<2.5
/
s
tion between the internal magnetic field and the strength of the external magnetic field
is almost linear and hence the right-hand side of Eq. (7.15) is approximated as [22]:
M = χ H ⇒ M = χ H (7.16)
M s M s MMs=χHMs⇒M=χH
Based on experiment, the magnitude of the induction magnetic field of nanocar-
riers is calculated from:
χ
M = H
1+ D χ (7.17) M=χ1+DmχH
m
The amount of demagnetization factor (D ) for spherical objects is one-third. Dm
m
For a strong magnetic field (H → ∞), saturation magnetization is obtained (M ), and H→∞
s
Eq. (7.15) is rewritten [22]:
M =⇒ M = M (7.18)
1
M s s MMs=1⇒M=Ms