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Hyperthermia and ablation 279

relaxation—Néel relaxation (rotation of the magnetic moments) and Brown(ian) relaxa-
tion (rotation of the particles themselves within the fluid)—which occur simultaneously in
ferrofluids. In superparamagnetic MNPs only Néel relaxation occurs.
The time constants of these relaxation processes are (Urdaneta, 2015)

4πηr 3
τ Brownian 5 h ; τ Neel 5 τ 0 e KV=κT ; τ 21 5 τ 21 1 τ 21 ; ð8:16Þ
Brownian
Neel
kT
where r h is the hydrodynamic radius of a particle (its coating is larger than the radius
of the MNP), η is the dynamic viscosity of the carrying fluid, T is the temperature,
K is an anisotropy constant, V is the magnetic volume, k B is Boltzmann constant, and
29
τ 0 5 10 s. For hyperthermia at near 42 C, the temperature dependence of the relax-

ation times may be neglected.
Recall (Chapter 1: Physical, Mathematical, and Numerical Modeling) that the
temporary magnetization, in complex representation (harmonic working conditions) is
M 5 χH, where H is the magnetic field strength, and the frequency-dependent mag-
netic susceptibility is (Rosensweig, 2002) χ fðÞ 5 χ 0 , with the equilibrium
1 2 jωτ Browninan
magnetic susceptibility χ 0 , and the angular velocity ω 5 2πf. Using this constitutive
model for the combined Néel and Brown relaxations yields the real and imaginary
parts of the magnetic susceptibility
χ
0 ωr
χ 5 Re χ 5 2 ; χ '' 5 Im χ 5 χ 0 2 : ð8:17Þ
0
1 1 ωrðÞ 1 1 ωrðÞ
The equilibrium susceptibility is given by

2 2
3 1 μ φM V M μ M H 0 V M
d
d
χ 5 χ i ξ coth ξ 2 ξ ; χ 5 0 3k B T ; ξ 5 0 k B T ; ð8:18Þ
i
0
where H 0 is the incident magnetic field, φ is the volume fraction of MNPs, and M d is
the domain magnetization.
The elementary change in the internal energy produced by the magnetic work inter-
action in magnetic media without permanent magnetization is (Chapter 1: Physical,
Mathematical, and Numerical Modeling) dU 5 H dB 5 μ 0 H d(H 1 M(H)) that, for
the MNPs of concern, integrated over a cycle yields (Rosensweig, 2002)

I
ΔU 52 μ 0 MUdH; ð8:19Þ

jωt
which, in the time-domain, MtðÞ 5 Re χH 0 e 5 H 0 χ cos ωtðÞ 1 χ '' sin ωtðÞð 0 Þ,yields
2π=ω
ð
2
2
ΔUj cycle 5 μ H χv sin ωtðÞdt; ð8:20Þ
0
0
0