# Introduction

Magnetic fluid hyperthermia (MFH) is a noninvasive treatment that destroys cancer cells by heating a ferrofluid-impregnated malignant tissue with an AC magnetic field while causing minimal damage to the surrounding healthy tissue. The strength of the magnetic field must be sufficient to induce hyperthermia, but it is also limited by the human ability to safely withstand it. The ferrofluid material used for hyperthermia should be one that is readily produced and is nontoxic while providing sufficient heating. We examine six materials that have been considered as candidates for MFH use. Examining the heating produced by nanoparticles of these materials, barium ferrite and cobalt ferrite are unable to produce sufficient MFH heating, that from iron-cobalt occurs at a far too rapid rate to be safe, while fcc iron-platinum, magnetite and maghemite are all capable of producing stable controlled heating. We simulate the heating of ferrofluid-loaded tumors containing nanoparticles of the latter three materials to determine their effects on tumor tissue. These materials are viable MFH candidates since they can produce significant heating at the tumor center yet maintain the surrounding healthy tissue interface at a relatively safe temperature.

Cancer is a leading cause of human deaths [56,57]. Current treatments, such as surgery and chemotherapy, can have undesirable side effects, including harm to the surrounding healthy tissue. Hyperthermia is an alternative treatment that can destroy cancerous cells by significantly elevating the temperature of tumor cells while keeping that of the surrounding healthy tissue at a reasonable level [32]. One method to induce hyperthermia is by use of ferrofluids, which are colloidal suspensions of magnetic nanoparticles in a nonpolar medium. These fluids can be magnetically targeted to cancerous tissue after intravenous application [56]. The magnetic particles extravasate into the tumor due to the high microvascular permeability and interstitial diffusion in neoplastic tissue [58]. Thereafter, the magnetic nanoparticles are heated by exposing the tumor to a high frequency alternating magnetic field, causing thermonecrosis of the embedding tissue. This process is called magnetic fluid hyperthermia (MFH) [12,40,44,56,59,60].

In order to examine the potential of hyperthermia as a viable alternative to chemotherapy and radioactive treatment, it is necessary to define what such a treatment would hope to accomplish. Temperatures in the range of 41-45 C are enough to slow or halt the growth of cancerous tissue, but such heating can also damage healthy cells [32]. Thus, an ideal hyperthermia treatment should sufficiently increase the temperature of the tumor cells while maintaining the healthy tissue temperature below 41 C. Ferrofluid-based thermotherapy can be also accomplished through thermoablation, which typically heats tissues up to 56 C to cause their necrosis, coagulation, or carbonization by exposure to a noninvasive radio frequency AC magnetic field [12]. Local heat transfer from the nanoparticles increases the tissue temperature and ruptures the cell membranes [49,61].

Iron oxide nanoparticles such as magnetite, or its oxidized form maghemite, are the most biocompatible agents for MFH [44]. These particles are typically coated with a biocompatible polymer to prevent their aggregation and biodegradation for in vivo applications. Platinum and nickel are also magnetic nanoparticles, but are toxic and vulnerable to oxidation [57]. MFH employing fine magnetic particles was first investigated by Gilchrist et al. [9]. This work was followed by several in vitro and in vivo experiments to confirm the feasibility of magnetic particle use for MFH [34,49,50,62]. Numerical investigations have also allowed researchers to understand and improve MFH therapy in soft biological tissue by using models consisting of multiple homogeneous regions [46] that contain tumor and normal tissue [45,50,51]. These models have provided approaches for the proper particle dosage and distribution in the tumor, and the optimal particle properties and magnetic field strengths that minimize the side effects of MFH on healthy tissues.

For optimal MFH treatment, ferrofluid dosage should be minimal and yet provide sufficient heating. This depends upon factors such as the magnetic anisotropy constant of the nanoparticles, and the strength and frequency of the AC field. Previous investigations have considered specific ferrofluids to determine the optimal particle type and size [4,5,12] and the thermal response of tissue [60]. However, the literature does not provide guidance about the influence of both particle type and size on MFH under typical clinical conditions. Therefore, we focus on the appropriate use of magnetic nanoparticles (MNPs) to heat soft tissue using an AC magnetic field [5] in this context.

When exposed to such an AC field, the MNPs dissipate magnetic energy into heat through both Brownian and N ́eel relaxations. Besides the strength and frequency of the alternating magnetic field [63], the magnetic properties of an MNP also play an important role in heat generation and dissipation [4, 5]. We account for the particle size distribution, saturation magnetization, and the material anisotropy constant. Since iron nanoparticles have both a large saturation magnetization [64] and a high Curie temperature (of 1043 K) [64], we consider iron and iron compound nanoparticles as primary choices for MFH [4,5]. Changing the nanoparticle size can significantly alter the ability of an MNP to generate heat [5,65], making the determination of an optimum nanoparticle size necessary for a specified set of conditions.

Here, we present a thermodynamic analysis of ferrofluid magnetic heating and com- pare the performance of six different types of ferrofluids, namely those containing magnetite ($$Fe_2O_3$$), maghemite ($$\gamma-Fe_2O_3$$), iron-platinum ($$FePt$$), iron-cobalt ($$FeCo$$), barium-ferrite ($$BaFe_2O_4$$), and cobalt-ferrite ($$CoFe_2O_4$$). We examine their performance for different magnetic field strengths, frequencies and particle radii. Thereafter, we investigate the heating of a tumor and the surrounding healthy tissue with suitable MFH ferrofluid candidates.

# How Magnetic Heating Works

## The Physics

For a constant density system and an adiabatic process [44],

\begin{align} \large dU = \delta Q + \delta W = \delta W = \overrightarrow {H}\cdot d\overrightarrow {B} \end{align}

where $$dU$$ denotes the internal energy change, $$\delta Q$$ the heat input, $$\delta W$$ the work done on the system, $$\overrightarrow {H}$$ the magnetic field intensity, and $$\overrightarrow {B}$$ the magnetic field. To get the total change in internal energy, we do the following:

\begin{align} \large \Delta U = -\mu _{0}\oint MdH \end{align}

Here, $$\mu_0$$ denotes the permeability of free space and M the magnetization of the material. Since an oscillating magnetic field is required to produce the Brownian and Ńeel relaxations during MFH, we assume that

\begin{align} \large H(t) = H_0 cos(\omega t) \end{align}

Using the Langevin equation the magnetization of the nanoparticles

\begin{align} \large M_0(t) = M_{sat}\phi(cot(L(t)) - 1/L(t)) \end{align}

where $${M_{sat}}$$ denotes saturation magnetization of the material. The Langevin parameter $$L(t)$$ is defined as

\begin{align} \large L(t) = (4\pi R^3/3)\mu_0 M_{sat}H(t)/(kT) \end{align}

where R is the nanoparticle radius, k the Boltzmann constant and T the absolute temparature (K). The ferrofluid susceptibility is given by

\begin{align} \large \chi_0 = M_0(t)/H(t) \end{align}

and

\begin{align} \large \chi^{'} = \chi_0 /(1 + (\omega \tau)^2) \end{align}

and

\begin{align} \large \chi^{''} = (\omega \tau \chi_0)/(1 + (\omega \tau)^2) \end{align}

where $$\chi_{'}$$ and $$\chi_{''}$$ denote the real and imaginary components of the complex ferrofluid susceptibility $$\chi = \chi^{'} - i\chi^{''}$$. Here, $$\tau$$ refers to the relaxation time of the ferrofluid that is dependent on its material properties, and $$\omega$$ the angular frequency of the AC magnetic field. This leads to the particle magnetization of

\begin{align} \large M(t) = H_0(\chi^{'}cos(\omega t) + \chi^{''}sin(\omega t)) \end{align}

Substituting the above into our equation for internal energy, we get

\begin{align} \large \Delta U=\omega \mu _{0}H_{0}^{2}X''\int _{0}^{2\pi / \omega }\sin ^{2}\left( \omega t\right) dt \end{align}

The power dissipation, P, due to magnetic heating is the product, $$f\Delta U$$ becomes

\begin{align} \large P = f\Delta U = \mu_0 \pi \chi^{''}f H_0^2 \end{align}<