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 Magnetic hyperthermia is the name given to an experimental cancer treatment. It is based on the fact that magnetic nanoparticles, when subjected to an alternating magnetic field, produce heat. As a consequence, if magnetic nanoparticles are put inside a tumor and the whole patient is placed in an alternating magnetic field of well-chosen amplitude and frequency, the tumor temperature would raise. This could kill the tumor cells by necrosis if the temperature is above 45°C, or could improve the efficiency of chimiotherapy if the temprature is raised around 42°C. This treatment is tested on humans only in Germany, but research is done in several laboratories around the world to test and develop this technique.

Generalities and definition
A general feature of many magnetic materials is to display a magnetic hysteresis when it is measured at a positive magnetic field, then negative, then positive again. The area of this hysteresis loop is dissipated in the environement under the form of thermal energy. This is the energy used in magnetic hyperthermia. The power dissipated by a magnetic material subjected to an alternative magnetic field is often called "Specific Aborption Rate" (SAR) in the community of magnetic hyperthermia; it is expressed in W/g of nanoparticles. The SAR of a given material is then simply given by: $$SAR=Af$$, where A is the area of the hysteresis loop and f the frequency of the magnetic field. A is expressed in J/g and is also called "specific losses" of the material. It should be noted that this expression is always true, whatever can be the complexity of determining A. Indeed, as will be more detailed below, A depends on a very complex manier on all the properties of the magnetic material. In the case of magnetic nanoparticles, it depends on their magnetocristalline anisotropy K, their volume V, the temperature T, the frequency of the magnetic field f, its amplitude $$H_{max}$$ and the volumic concentration of the nanoparticles which induces magnetic interactions between the nanoparticles.

Influence of nanoparticle size on their domain structure
The size of nanoparticles have a great influence on their magnetic domains. Small sizes nanoparticles are composed of a single domain. Larger ones are composed of several domains minimizing the magnetostatic energy. At intermediate sizes, they display a beautiful magnetic structure called vortex. A rough approximation to determine the size above which a magnetic nanoparticles is not single-domain any more is when its size is above the typical domain wall dimension in the magnetic material, which ranges from a few to a few tens of nanometers. The nature of the domain structure have a profond influence of the hysteresis of the magnetic nanoparticles and, as a consequence of their hyperthermia properties.

Basic mechanisms involved in the magnetization of magnetic single-domain nanoparticles
The goal of this part is to present the basic mechanisms which must be taken into account to describe the reversal of single domain-nanoparticles.

Reversal by Brownian motion
In hyperthermia application, the nanoparticles are in a fluid, the blood. During in vitro hyperthermia measurements they are generally dispersed in a liquid and form a ferrofluid. They move randomly in the fluid, a phenomenom called Brownian motion. When a magnetic field is applied to them, magnetic nanoparticles rotate and progressively align with the magnetic field, similarly to a compass. The time taken for a magnetic nanoparticle to align is given by the Brown relaxtion time : $$ \tau_B = \frac{3 \eta V}{k_BT}$$, where $$\eta$$ is the solvant viscosity. The delay between the magnetic field rotation and the magnetization rotation leads to an hysteresis.

Reversal by thermal activation
The magnetization of a nanoparticle can spontanenously change of orientation under the influence of thermal energy, a phenomenon called superparamagnetism. The typical time between two orientation changes is given by the Néel relaxation time $$ \tau_N = \tau_0 e^{\frac{K V}{k_B T}}$$, where $$\tau_0$$ is an attemp frequency with a value around 10^{-9}-10^{-10}.

Reversal by the suppresion of the anisotropy barrier by a magnetic field
The magnetization of the nanoparticle is also reversed when an applied magnetic field is large enough to suppress the energy barrier between the two equilibrium positions, a phenomenom which is known as the Stoner-Wohlfarth model of magnetization reversal.

The linear response theory
The linear response theory is only valid when the response of the magnetic material is linear with the applied magnetic field and can be thus written under the form $$M=\chi H$$, where $$\chi$$ is the complex susceptibility of the material. It is thus valid when the applied magnetic field is much smaller than the magnetic field needed to saturate the magnetization of the nanoparticle or than the one needed to suppress the energy barrier between the two equilibrium positions in the Stoner-Wohlfarth model. Is is able to take into account both the reversal by thermal activation and the reversal by Brownian motion. The linear response theory uses an average relaxation time $$\tau$$, given by $$\frac{1}{\tau} = \frac{1}{\tau_B} -\frac{1}{\tau_N} $$. The out of phase component of complex susceptibility is then given by $$\chi '' = \frac{\mu_0 M_S^2 V}{3 k_B T} \frac{2 \pi f \tau_N}{1+(2 \pi f \tau_N)^2}$$. The hysteresis loop is then an ellipse with an area given by $$A = \pi \mu_0 H^2 \chi ''$$.

The Stoner-Wohlfarth model and the maximum area
The Stoner-Wohlfarth model allows one to calculate the hysteresis loop of magnetic nanoparticles at T=0 with the assumption that the nanoparticles are fixed in the magnetic field (the Brownian motion is neglected) and magnetically independant. Its main interest is to predict the maximum area for independant nanoparticles with given properties. Indeed, the addition of thermal energy or Brownian motion only leads to a decrease of the hysteresis loop area (see below).

The Stoner-Wohlfarth model predicts that the coercive field HCof an assembly of nanoparticles with randomly oriented axes is given by $$H_C(0) = \frac{K}{M_S}$$. The area of the hysteresis is approximatively $$A = 2\mu_0 M_S H_C(0)$$ .

Extension of the Stoner-Wohlfarth model to include temperature and time
Extensions of the Stoner-Wohlfarth model have been done to include the influence of the temperature and frequency on the hysteresis loop. These extensions are only valid is the effect of the temperature is small, i.e. if $$f >> \Tau_N$$. In this case, one can consider that the effect of the temperature is simply to reduce the coercive field of the nanoparticles. Numerical simulations have shown that, in this case, the expression of the coercive field for randomly oriented nanoparticles is $$H_C = 2 H_C(0) (0.479-0.81(\frac{k_B T}{2 K V} ln(\frac{1}{f \tau_0}))^{\frac{3}{4}})$$

Basic mechanisms involved in the magnetization of magnetic multi-domain nanoparticles
In multi-domain nanoparticles the basic ingredients to describe the magnetization reversal are the nucleation of new domains and the propagation of domain walls. Both mechanisms are strongly influenced by the structural defects at the surface or inside the nanoparticles and make difficult any quantitative prediction of the hysteresis loops shape and area from intrinsic parameters of the magnetic nanoparticles.

Models to be used for multi-domain nanoparticles
At low magnetic field, the hysteresis loop is expected to be a Rayleigh loop. In this case, the hysteresis area is $$A = \frac{4}{3} \eta H^3$$, where $$\eta$$ is the Rayleigh constant.

Producing a high frequency magnetic field
Two basic means to produce the high frequency field necessary to study hyperthermia can be used : the coil and the electromagnet. For the "coil" way, a very simple method to get the high frequency magnetic field is to use an induction furnace, which precisely used a high-frequency magnetic fieldon to heat. It is however conceived to work at a single frequency and requires a water cooling system. It is also possible to build electromagents able to work at various frequencies and without cooling system at the conditions to use resonant circuits and Litz wires.

Measuring the temperature and artefacts
A platinum or semi-conductor resistance thermometer in a high-frequency magnetic field is self-heated and leads to erroneous temperature measurements. Temperature measurements in hyperthermia can be made using alcool thermometer or optic fiber thermometers.

A colloidal solution heated by an external magnetic field will be subject to convection phenomena so the temperature inside the calorimeter is not homogeneous. Shaking of the colloidal solutions at the end of a measurement or average on several temperature probes can ensure a more accurate temperature measurement.

Iron oxide nanoparticles
The most widely used magnetic nanoparticles for hyperthermia consists in iron oxyde nanoparticles. Similar nanoparticles are used as contrast agents in magnetic resonance imaging. They are in this context called "Superparamagnetic Iron Oxide Nanoparticles", or SPION. The main interest of these nanoparticles are their biocompatibility and their stability with respect to oxidation. The nanoparticles displaying the largest hysteresis area so far are the SPIONs synthesized by magnetotactic bacterias, with A = 2.3 mJ/g although chemically synthesized nanoparticles reach values up to A = 1.5 mJ/g

Metallic nanoparticles
The higher magnetization of metallic nanoparticles of Co, Fe or FeCo compared to iron oxide increases the maximum theoretical SAR values which could be reached using them in hyperthermia applications. A = 1.5 mJ/g has been reported for FeCo nanoparticles and A = XXXX mJ/g for Co nanoparticles

The main issue with respect to metallic nanoparticles concerns their protection against oxidation.

Experiments on humans
The only hyperthermia setup suitable to treat human has been developped at the Charité Medical School, Clinic of Radiation Therapy in Berlin