Bone cancers are neoplasms that can affect relatively young subjects (with age between 10 and 25 years old) with an incidence and survival rate of about 35% [1]. Bone tumours are classified depending on the tissue of origin. The preferred sites of origin are the extremities of long bones. Among the malignant neoplasms, two of the most relevant targets in orthopedic oncology are the Osteosarcomas (OS) and the Fibrosarcomas (FS). These aggressive cancers cause impairment and pain to the patients, thus ruining their quality of life [1, 2]. Furthermore, their pathologic, chaotic and fast growth damages lead to healthy bone tissue fracturing. Hence, following the diagnosis, early and effective countermeasures must be sought to prolong the patient prognosis [1]. Currently, surgical intervention is the gold-standard. In particular, osteotomies and resection procedures are carried out to try to directly eliminate cancer cells [2]. However, since the margin of the tumours is often not well defined, the recurrence rate can be around 15% [1, 2]. Moreover, as a consequence of osteoteomies, additional damages to the bone are produced, and a surgical gap is seen after the intervention. From a clinical perspective, an auto- or allogeneic graft, or a synthetic biomaterial (e.g. bone cement, bioglasses, a bioceramic or a polymer) is required to restore the functionality of the bone tissue [1, 2].

All these limitations lead to the investigation and testing of innovative and less invasive therapies [3]. As preferred for other kinds having tumours, the so-called neo-adjuvant chemotherapy and radiotherapy have been studied [2, 3]. However, the high systemic toxicity of drugs and the radio-resistance of bone cancers, especially OS, limit the effectiveness of these novel approaches [1, 2]. Therefore, in order to improve the aforementioned clinical strategies, hyperthermia treatment (HT) was proposed [3]. HT is a thermal therapy used in oncology and its goal is to increase the temperature of the given cancer tissue from 37 °C to 42 °C-45 °C for at least 60 min [3-5]. The effect of this temperature increase is the damaging of cell proteins, and the irreversible modification of the DNA repair mechanisms, while the local blood flow and sensitivity to chemotherapy and radiotherapy are also increased [3]. Hyperthermia can be induced in several ways, e.g. using ultrasounds (US) or electromagnetic (EM) energy [6]. In the case of bone tumours, the use of US is discouraged by their poor effectiveness, which is due to the less penetration depth and the significant reflection at the bone interface [6]. Regarding EM-based hyperthermia devices for orthopaedic oncology purposes, applicators working in the microwave frequency bands for medical application (i.e. 434 MHz, 915 MHz, 2.45 GHz) demonstrated poor effectiveness and drawbacks [6], even though the target tumour tissue was directly exposed to the radiation during the intervention [5-7]. However, the possibility of using radio-frequency (RF) magnetic fields (MF) to perform the hyperthermia treatments is now widely investigated [8]. Indeed, given a working frequency of hundreds of kHz, and the transparency of the body tissues, magnetic hyperthermia (MHT) ensures a higher penetration depth of the field and allows to treat deep-rooted tumours, with reduced overheating of the non-target healthy tissues [5-9]. Considering the urgent and fundamental clinical requirement of having a prosthetic implant, and given the potentialities and advantages of MHT, the possibility of implanting a magneto-responsive material, to be used both as tissue scaffold and as thermo-seed to perform the interstitial and local thermal therapy, was investigated [4]. A schematic representation of the rationale of such an approach is presented in Fig. (1). In particular, at first, the use of ferromagnetic cylindrical bars, inserted in the operated tumour margins, and heated under the action of an external MF, was tested [10]. However, poor biocompatibility may limit this form of MHT. Therefore, a plethora of magnetic biomaterials has been developed [4, 5, 9]. The synthesis of nanocomposites of magnetic nanoparticles (MNPs), polymers or ceramics (for example, using dip-coating or impregnation methods), or the manufacturing of iron-doped bioceramics, is currently under study [9]. Magnetic implants for bone tumour hyperthermia were developed using ferromagnetic or superparamagnetic nanoparticles [5]. The former kind of MNPs has an average particle radius higher than 250 nm, it presents a non-zero coercive field, and its heat losses are related to the hysteresis phenomenon [4, 10]. On the other hand, the latter kind of particles has typical dimensions below 25 nm, does not present a residual magnetization, and the heat is generated by the fluctuations of the dipole moment in response to a time-varying magnetic field [5]. The superparamagnetic MNPs are the most suitable candidates for the synthesis of magnetic thermo-seeds [4, 5]. Moreover, considering the in vivo implantation, the long-term stability and the mechanism of interaction with the tissue are pivotal aspects [4]. The magnetic bioactive ceramic, which was considered in this work, is an osteoconductive biomaterial, which means that, when implanted, it is capable of releasing ions which create a firm apatite layer at the interface with the surrounding tissues, as already experimentally demonstrated in the literature [11-13]. Furthermore, the presence of iron increases the porosity, which can act as a topological stimulus for the osteogenic cells, and can enhance the stiffness of the biomaterials, i.e. increasing their Young’s modulus. All these properties allow estimating the in-vivo life span of the implant to be longer than 12 weeks [13].

Despite the research interest in the morphological, chemical and biochemical characterization of such a functional nanocomposite, the theoretical and engineering aspects, related to its possible (and optimal) use as an interstitial hyperthermia agent, have not yet been fully investigated. Recently, the heat dissipation mechanism of magnetic implants, produced using superparamagnetic MNPs, was investigated at working frequencies from 100 kHz to 700 kHz [5, 11]. The non-linear nature of the HT was first elucidated [14-16]. The authors highlighted, through the multi-physics simulations of a spherical magnetic implant, exposed to a RF magnetic field, that as the temperature of the system at the nanoparticles location increases, their magnetic susceptibility decreases, due to the enhancement of the thermal energy barrier increases. Therefore, the tendency of the thermo-seed to dissipate heat decreases as the system temperature increases, reaching a quasi-steady state regime. Even though peculiar anatomical cases were numerically investigated by Lodi et al. [17], the implant geometry has been considered in a simplified way. It should be pointed out that for clinical and practical cases, the synthetic bone grafts can have complex forms and shapes, but the minimal and most fit shape for an orthopaedic application is that of a cylinder [4, 10]. Therefore, in this paper, the non-linear and multiphysics model [14, 17] is modified to account for the heat transfer aspects and solved, using a commercial Finite Element Method (FEM) software for the case of a cylindrical magnetic implant of hydroxyapatite, in order to define a set of geometric and operative parameters for the hyperthermia treatment, using these novel thermo-seeds.

Fig. (1). Schematic representation of the hyperthermia treatment of bone tumours using the implanted magnetic scaffolds as thermo-seeds after resection of the tumour.

The commercial FEM software Comsol Multiphysics v5.5 (Comsol Inc., Burlington, MA, USA) was used to simulate the hyperthermia treatment of bone tumours using magnetic scaffolds as thermo-seeds. The coupled electromagnetic and thermal problems were solved, as summarized in Fig. (4).

Assuming a homogeneous initial temperature distribution, the magnetic properties of the implant and tissues were evaluated by using Eq.s (12) and (13). Then Maxwell’s equations were solved in the frequency domain using the AC/DC module [17, 34]. The power which is stored and dissipated by the magnetic implant was derived by using Eq.s (9), whilst the dielectric losses were computed using Eq. (8). The overall electromagnetic power Q_EM was obtained according to Eq. (3). After the evaluation of the thermal properties at the current temperature, the new temperature distribution at the following time step was evaluated through Eq. (2) by means of the Bio-Heat Transfer module. The temperature distribution was used to calculate the new value of the EM properties and then the resolution steps were repeated until the final time, t _f, was reached.

By using the resolution scheme in Fig. (4), the hyperthermia treatment of the residual bone tumour cells was simulated for 100 min for a cylindrical prosthesis with variable dimensions. The volume fraction of MNPs in the biomaterial was kept at a constant value (see Tab. 3.2) and the electromagnetic properties were assumed to follow the proposed model. Considering that the biological target is a 0.5 mm thick annular region of residual tumour cells, the height (h _sc) and the radius (r _sc) of the implant varied from 2.5 mm to 10 mm, with a 2.5 mm interval, therefore, sixteen combinations of the possible implant size were tested.

Fig. (4). Scheme of the resolution steps for the multi-physic problem.

For the geometry shown in Fig. (2), a magnetic field with a strength of 12 kAm ^-1, working at the frequency of 300 kHz, as used earlier [27], was assumed to be applied to heat the system. The volumetric heat dissipation, due to the exposure to the RF EM field, can be distinguished in two contributions, according to Eq. (3). It is interesting to quantify which mechanism dominates, i.e. whether the dielectric losses or the power which is dissipated by the MNPs which are embedded in the implant, prevails in the heating during the HT. As a result, Q_m >> Q_e and Q_EM ~ Q_m were obtained. This physical condition is due to two main reasons, namely, as usual, the induced electric field is one order of magnitude lower than the amplitude of the applied magnetic field [7], and to the huge amount of heat which is dissipated by the MNPs contained in the implant. It was demonstrated that this kind of material can produce a power per volume unit of about 10 ⁶ Wm ^-3 [17], which is comparable to the calculated average value of ~6·10 ⁶ Wm ^-3. Knowing the total electromagnetic power, the final outcome of the HT strongly depends on the blood tissue perfusion, as previously discussed in Sect. 2. For the magnetic implant, to produce any thermal effect, it is necessary that |Q_EM| > |ρ_bC_p,bω_bΔT|, taking into account that in the Pennes’ equation, the perfusion acts as a sink term (hence ΔT < 0). As can be noticed from Tab. 3.3, at the starting temperature, the power per unit volume, which is subtracted by the vascular beds of the tissues, ranges from 664 Wm ^-3, for the healthy bone, to a maximum of 1·10 ⁶ Wm ^-3, for the tumoural region, respectively. It is worth noting that, at the initial time and temperature, the EM power, which is stored and dissipated in the target area, is higher than the energy per time and volume unit, which is removed by the system volume. At this point, a successful treatment is expected. However, the perfusion contribution in Eq. (2) is not constant. We accounted for this rather complex physiopathological condition in our simulations with Eq.s (14) and (15). Therefore, according to Eq. (15), when the temperature in the tumour volume reaches 42 °C, the perfusion doubles its value (up to 1.2 s ^-1), further limiting the thermal therapy. In order to assess the effectiveness of the considered magnetic hydroxyapatite thermo-seed, the spatio-temporal evolution of the system in Fig. (3) must be investigated.

In Fig. (5) the 2D temperature profile at t=80 min is shown. The highest temperature values were found along the direction of the applied magnetic field. The power, dissipated by the MNPs embedded in the biomaterials, quickly diffuses to the surrounding fracture gap. This is due to the relatively high thermal conductivity of the hydroxyapatite material, with respect to the biological materials, as reported in Table. 3. However, the radial heating pattern presents a sudden decrease of about 3 mm away from the implant surface, tending to become homogeneous for distances higher than 20 mm. This trend resembles, from a qualitative point of view, the steady-state temperature profile, which is prescribed by Eq. (4) [10].

Fig. (5). Temperature distribution at t=80 min. The implant has dimensions h sc=7.5 mm, r sc=10 mm.

From Fig. (6) it can be noticed that the size of the implant strongly affects the average temperature in the tumour region. The quasi-steady state temperature is achieved after 13 min for each of the considered cases. For h _sc=2.5 mm, the temperature is about 37.1 °C for any radius value of the implant. When the height is doubled, the temperature of the residual cells increases only up to 37.5 °C, with a very slight dependence on the radius (i.e. a variation of about 0.5 °C). When the magnetic cylinder has a height of 7.5 mm, the temperature increases to 39.24 °C ± 0.3 °C. Finally, if h _sc=10 mm, the temperature in the target region is 44.71 °C, i.e. over the therapeutic threshold value of 42 °C for neoplastic cells. Such temperature level, which is maintained for 90 minutes, according to the literature, is sufficient to initiate the cascade of biological events, which could enhance the effectiveness of chemotherapy and radiotherapy [35, 36]. The average heating rates are similar to those found in previous studies [11, 14-17], but, from a quantitative point of view, are consistent with the experimental results found in other studies [26, 27]. Indeed, the magnetic hydroxyapatite implant, with properties reported

in Tab. 3.2, which was considered a case-study in this work, could reach 40 °C in 1 min of heating when exposed to the same magnetic field in a saline solution [28]. Regarding the relatively poor influence of the implant radius on the final temperature in the tumour, it must be pointed out that the external magnetic field is assumed to be applied along the y-axis of the system shown in Fig. (2), therefore, a preferred direction for the heat diffusion exists. These findings are corroborated by the temperature distribution shown in Fig. (5). Then, from the findings of Fig. (6), it is possible to infer that the design and the choice of the exposure parameters are crucial for the quality of the treatment. Therefore, the induction heating system should be designed, considering that the implant geometry strongly affects the conveyed energy and the heat transfer to the tumour region.

For all of the tested implant sizes, the temperature in the non-target tissues, the average temperature in the surgical gap and in the healthy bone is of about 39.82 °C and 37.25 °C, respectively. Despite the moderately high power, which is generated by the induced currents due to the induced electric field (see Fig. 7), the action of the magnetic implant is limited and local. The most relevant aspect is that the bone tissue is not harmed by the thermal therapy, thus ensuring possible healing of the injured body region.

However, the findings shown in Fig. (5) are not sufficient to complete the quantitative description of the quality of the hyperthermia treatment of bone tumours performed using magnetic thermo-seeds. By using the derived temperature data, the equivalent thermal dose at 43 °C was calculated using Eq. (17). The curves for the delivered therapeutic heat are shown in Fig. (8). For magnetic implants having a height of h _sc < 7.5 mm, for any given radius value, r _sc, the CEM43 is equal to zero, according to Eq. 17.

Fig. (6). Average temperature in the tumour region over time for all the height and radius values of the simulated implants.

Fig. (7). Volumetric heat dissipated by the dielectric and magnetic heating mechanism during the 100 min hyperthermia treatment under the action of a 12 kAm -1 magnetic field, working at the frequency of 300 kHz.

Fig. (8). Cumulative equivalent thermal dose minutes at 43 °C for the cylindrical magnetic implants.

When the prosthesis height increases to 2.5 mm, coherent with the findings reported earlier (5), the CEM43 is 54.25. This value is one order of magnitude lower than the thermal dose which would be required to have the minimum biological effect [37, 38]. The highest CEM43 values, equal to 3257, for tumour tissues, were obtained when h _sc= 10 mm. The range of this clinically relevant figure of merit is consistent with the previous findings reported [17, 38, 39] (i.e. from several hundred to several thousand for similar exposure times). Thanks to the proposed non-linear temperature modeling of the hyperthermia treatment, in order to treat a 0.5 mm thick region of residual bone tumour cells, it was found that the bigger the implant, the more effective the treatment.

This work addressed the problem of establishing a quantitative basis to simulate the hyperthermia treatment of bone tumours using magnetic thermo-seeds, taking into account the non-linear and multiphysics nature of the heat delivery and transfer during this oncologic thermal treatment. The proposed model overcomes the limitations of the available theoretical frameworks [10, 20] by providing a modified numerical resolution scheme for the coupled electromagnetic-heat transfer phenomena when both the EM and thermal properties are dependent on the system temperature. In particular, different from the model shown in a previous study [11, 14-17],the description of the magnetic properties of the implant, which determine the heat losses during the exposure to the RF field, is simplified to lower the computational burden of the numerical experiments. Furthermore, this work investigated the relevancy of the implant geometry to the effectiveness of the hyperthermia treatment.

Among the clinical advantages of the interstitial hyperthermia treatment, there is the possibility to prevent the limb amputation, thanks to the local control of the tumour recurrence, due to the improvement of the radio- and chemotherapy effectiveness, and to the possibility of operating a guided tissue regeneration. However, despite these benefits, the translation of the thermo-seeds to the clinical practice raises some issues. At first, the studied interstitial hypertermia treatment with magnetic scaffolds requires an invasive and complicated thermometry, as prescribed by the guidelines [4, 23]. Considering that potential infections and the temperature sampling would be limiting factors, it is crucial to develop non-invasive alternative methodologies for the treatment monitoring, such as magnetic resonance imaging or thermoacoustic probing with ultrasounds [37, 38]. Moreover, the long-term biocompatibility, and the possible inflammatory effects, are fundamental issues to be investigated for increasing the quality of the treatment from the biological and clinical point of views [4, 5, 34, 40-42].

Future works could investigate the crucial role of blood tissue perfusion in conditioning the outcome of the hyperthermia treatment [43-45]. The proposed non-linear model could also be used to develop a control strategy to optimize the treatment [46].