**1. Introduction**

In spite of the certainly wide variety of magnetic nanoparticles (MNPs) with di fferent geometries, compositions and functionalizations [1–8] that have become available in recent years, and of the number of applications that have been devised for them [1,9–12], when the goal is to provide a nanocarrier suitable for targeted chemotherapy, there is still room for progress. On one hand, the methods for obtaining MNPs need to be more cost- and time-e ffective, eco-friendly, and scalable. On the other, the nanoparticles themselves should be improved in terms of maximizing the magnetic moment per particle and providing novel surface properties while exploring their potential as hyperthermia agents that would allow them to combine therapies in the near future.

In this context, cancer is one of the fields of application where magnetic nanoparticles certainly appear as most promising [12–18]. Aside of drug delivery, whereby functionalized magnetic nanoparticles are loaded with the chosen drug and some targeting molecule, such an antibody, driven to the site of action, maintained there by continuous magnetic fields and eventually set for delivery by some external action, magnetic hyperthermia appears as a realistic application of MNPs [19–27]. For that purpose, MNPs are dispersed at a suitable concentration in an aqueous solution and located in the target place for action. There, they are subjected to an alternating magnetic field (with induction of tens mT and frequency of several hundred kHz). Subsequently, the magnetic nanoparticles increase the temperature of the microenvironment in which they are immersed inducing apoptosis of the tumor cells, usually more sensitive to temperature increase compared to healthy cells [28–30].

MNPs are generally produced either by the co-precipitation of iron salts in basic aqueous media possibly stabilized by biocompatible surfactants/polymers or by the thermal decomposition of organometallic precursors in high-boiling nonpolar organic solvents at elevated temperatures (200–360 ◦C), allowing a grea<sup>t</sup> control of the size of the MNPs, their monodispersity and uniformity [31]. However, these methods have some drawbacks mainly derived from the high temperatures used, organic solvents or poor solubility of the nanoparticles in water.

Many of these drawbacks are overcome in biomimetic MNPs (BMNPs). These are produced by the mediation of magnetosome membrane-associated proteins (MAPs) from magnetotactic bacteria, and *in vitro* experiments have been demonstrated to control the size (and thus the magnetic moment per particle), shape, and surface properties of the nanoparticles [32,33]. BMNPs production can be scaled up *in vitro* in eco-friendly, cost-e ffective magnetite precipitation experiments run at room temperature and 1 atm total pressure by the simple addition of the recombinant protein. Promising BMNPs have been obtained by the mediation of MamC protein from *Magnetococcus marinus* MC-1, since these BMNPs are (i) superparamagnetic at room and body temperature while they present a saturation magnetization of 61 emu/g (at 500 Oe and 25 ◦C); (ii) are larger than most commercial MNPs and/or other biomimetic magnetites, although still single magnetic domain, showing higher blocking temperature and slower magnetization increase, and thus, larger magnetic moment per particle; (iii) contain up to 4.5 wt% of MamC, which provides functional groups allowing for functionalization; and (iv) adopt the isoelectric point of MamC (pHiep 4.4), and are strongly negatively charged at physiological pH (pH 7.4); this property allows the coupling and release of molecules to be pH-dependent. Thus, at physiological pHs, they bind to positively charged molecules (such as doxorubicin, DOXO hereafter) through electrostatic interactions, which are weaker at acidic pHs (such as those found in tumor microenvironments), allowing the release of the adsorbed molecules, and (v); they are fully cytocompatible and hemocompatible, but when they are coupled with DOXO they display dose-dependent cytotoxicity [33].

Recall that single-domain magnetic nanoparticles are characterized by a spin configuration such that in the absence of an external magnetic field, all spins are oriented parallel to each other and parallel or antiparallel to a crystallographic direction, called the easy (or anisotropy) axis [34–36], so that each particle is characterized by a large magnetic moment. Let us imagine for the moment that the particles are immobile, fixed in a non-magnetic matrix, with their easy axes oriented randomly. At very low temperature (or at very high frequencies of the external magnetic field, if this is non-stationary), the transition between the two orientations along the easy axes can only be achieved by the application of su fficiently large external magnetic field, so that the magnetization-H curve will take the form of a square hysteresis cycle, according to the Stoner-Wohlfarth model [35,37]. As temperature rises above 0 K, the transition between the two orientations can be thermally activated, and the coercivity tends to be reduced, eventually making hysteresis negligible. The same e ffect will be observed when the field frequency or the particle size is very low, since the coercive field *H*C is related to the anisotropy field *Hk* (in turn related to *K*eff, the e ffective magnetic anisotropy constant of the particles, and their saturation magnetization *M*S as *Hk* = 2 *K*eff /μ0*M*S) by [35]:

$$H\_{\mathbb{C}} = 0.48 H\_{\mathbb{K}} (1 - \kappa^{0.8}) \tag{1}$$

with

$$\kappa = \frac{k\_{\rm B}T}{K\_{\rm eff}V} \ln\left(\frac{k\_{\rm B}T}{4\mu\_{0}H\_{0}M\_{\rm S}fV\tau\_{0}}\right) \tag{2}$$

where *kBT* is the thermal energy, *V* is the particle volume (or, in the case of coated particles, the volume of the magnetizable core), μ0 is the vacuum magnetic permeability, *H*0 is the field amplitude, *f* is the field frequency, and the characteristic time τ0 depends on such quantities as temperature, saturation magnetization or anisotropy constant. It will be assumed to be a constant in the range of 10−10–10−<sup>9</sup> s. If it is admitted that it suffices with *H*C being 0.01–0.1*H*k to have reversible cycles and absence of hysteresis, then the combination of *H*0, *f*, *V*, and *M*S must be such that κ = 0.97–0.75.

An approximate approach to the general solution of the problem, not using models based on the Stoner-Wohlfarth one is the so-called linear-response theory, where, keeping the assumption of immobile particles, it is found that the hysteresis cycle has its origin in the Néel-Brown relaxation, characterized by a relaxation time τN, which is a measure of the time taken by the system to return to equilibrium after application of a step magnetic field, or half the time needed for spontaneous inversion of magnetic moment orientation [38–40]. It is given by:

$$
\tau\_{\rm N} = \tau\_0 \exp\left(\frac{K\_{\rm eff} V}{k\_{\rm B} T}\right) \tag{3}
$$

This brings about a delay between magnetization and field, or an imaginary component of the magnetic susceptibility, and manifests again in a finite area hysteresis cycle, as long as the frequency of the field remains in the vicinity of 1/<sup>τ</sup>N. This approach is only strictly valid for low applied magnetic field strength or highly anisotropic particles.

For hyperthermia applications, the particles are typically dispersed in an aqueous solution, and hence they can rotate under field inversions so that viscous friction is an additional source of phase delay between magnetization and external field, and hence, an additional relaxation contribution to hysteresis. It is called Brownian relaxation, and it is characterized by a time τB:

$$
\pi\_{\rm B} = \frac{3\eta V\_{\rm H}}{k\_{\rm B}T} \tag{4}
$$

where η is the viscosity of the medium, and *V*H its hydrodynamic volume (including, if any, that of the coating layer) [19,41,42]. Taking the value of *K*eff equal to 25 kJ/m<sup>3</sup> for magnetite, the Brownian relaxation times for biomimetic (~35 nm in diameter, reported below) and purely inorganic particles (~28 nm) are, respectively, 14.4 μs, and 7.4 μs, and orders of magnitude higher in the case of magnetization reversal. This means that for the particle sizes and frequencies involved, the magnetic moment is frozen in the particle and the only heating source is friction, according to the linear response model. Considerations on the validity of the relaxation approach just described can be found in References [43–46]. In the context of the linear response theory, if the alternating magnetic field has a frequency in the vicinity of the reciprocal of the mentioned times (*f* in the order of 70 kHz and 135 kHz, respectively), the imaginary component, χ", of the complex magnetic susceptibility is maximum, and this is important, as the dissipated power d*W*/d*t* is proportional to this quantity [28]:

$$\frac{\text{d}W}{\text{d}t} = \mu\_0 \pi \chi'' H\_0^2 \tag{5}$$

Another important aspect, not always considered, regards the stability of the particles in the suspension, compromised not only by the colloidal interactions but also by the magnetic dipolar ones. Hence the need for properly controlling the stability, as the hyperthermia response degrades for aggregated systems [22,47] or when the monodomain range is surpassed [36,48].

Experimentally, the quantity of interest is the so-called Specific Absorption Rate (SAR), or heat released per unit mass of magnetic material (*m*) and per second:

$$SAR = \left(\frac{CV\_s}{m}\right)\frac{\text{d}T}{\text{d}t} \tag{6}$$

*C* being the volume heat capacity of the suspension, *Vs* its volume and d*T*/d*t* the rate of temperature increase [28,39]. Typical *SAR* values are in the order of tens to hundreds of W/g. In order to perform a comparison between different materials without the interference of the details of the specific device used (very often lab-made), a quantity is defined as a measure of the magnetothermal performance of a given suspension, namely, the Intrinsic Loss Power (ILP), given as:

$$ILP = \frac{SAR}{fH\_0^2} \tag{7}$$

with typical values in the order of 10−<sup>9</sup> Hm2kg−1.

In this work, we explore the possibility of maximizing the hyperthermia effect (the rate of temperature rise, in fact) by combining the two types of magnetic particles (BMNPs and MNPs), differing in size and other properties. The goal of the present study is to provide a composition that could be used in the future as a platform for combining drug delivery and hyperthermia. The former particles, BMNPs, mediated by MamC have been chosen because of their demonstrated ability to function as nano-transporters of drugs, being the nanoassembly stable at physiological pH while it destabilizes releasing the drug under acidic pH values, which naturally occur in the tumor environment. The second ones, MNPs, have been chosen because of their potential as hyperthermia agents [33]. By mixing the two systems, we expect to deliver a system that, in the future may prove useful to combine both treatments, i.e targeted chemotherapy plus targeted hyperthermia by using the same platform. To the best of our knowledge, the hyperthermia of mixed systems has never been investigated, but advantages are foreseen regarding the possibility of optimizing the response.
