# Simulating Evaluation Method on Heating Performances of Magnetic Nanoparticles with Temperature-Dependent Heating Efficiencies in Tumor Hyperthermia

^{1}

^{2}

^{*}

## Abstract

**:**

_{C}), are expected to realize the self-regulated temperature hyperthermia of the tumor. However, the actual decrease of the SLP is gradual, resulting in the deviation of self-regulated temperatures from the measured T

_{C}. So far, no method is available for evaluating the heating performances of those MNPs. Here, by simulating the temperature-dependent SLP, the heating performances of MNPs are evaluated from three clinically concerning aspects: the capacity for effective heating, the temperature uniformity in the tumor, and the temperature stability under environmental changes such as MNP loss or tumor progression. The developed methods were applied to ZnCoCrFeO, Fe

_{3}O

_{4}, and γ-Fe

_{2}O

_{3}MNPs. It was found that the uniform temperature distribution relies on lowering the heating power in the inner regions of the tumor, and the stable control of temperature depends on the dynamic adaptation of the heating power to the tumor temperature change. The proposed method may be used to predict the heating ability of MNPs and help the selection of MNPs for hyperthermia.

## 1. Introduction

_{C}) attracted attention for better temperature control [5]. The T

_{C}is the critical temperature at which the MNPs transform from ferromagnetic to paramagnetic and lose heating abilities under AMF. By selecting the MNPs with proper T

_{C}, the temperature in the tumor can be self-regulated at preferred temperature ranges of different therapies, removing the burdens of thermometers and cooling systems [5]. In addition, these MNPs can produce uniform temperature distribution all over the tumor and lead to consistent therapeutic outcomes [6,7]. By now, MNPs with T

_{C}of 37–93 °C have been manufactured. Their ability to self-regulate temperatures has been verified both in vitro [8,9,10] and in vivo [11,12,13,14].

_{C}of 29–103 °C [10]. The ZnCoCrFeO MNPs with T

_{C}of 61 °C could self-regulate temperatures at 43–44 °C in vivo [14]. The MnZnFeO ferrite MNPs with T

_{C}of 89.6 °C raised the temperature of tumors in nude mice to 42.8 °C [11]. However, the ZnCoCrFeO MNPs with T

_{C}of 37.5 °C elevated the water temperature to 43.8 °C [10]. Similarly, the FeCuZnMgO ferromagnetic MNPs with T

_{C}of 43 °C achieved 45 °C in vivo [13]. The ZnFeO MNPs with T

_{C}of 93 °C obtained 41.5 °C in cellular magnetic heating experiments [9]. Clearly, the deviations exist between T

_{C}and self-regulated temperature, suggesting the insufficiency of using T

_{C}to predict the heating performances of MNPs. Theoretically, the ideal MNPs should have high heating efficiency (represented by specific loss power, SLP) below T

_{C}, and the SLP instantly drops to zero at T

_{C}. Thus, the temperature can be controlled at T

_{C}. However, the actual decline of SLP around T

_{C}is usually gradual, leaving residual SLP and extra heating abilities over T

_{C}[14]. It seems necessary to consider the temperature-dependent SLP for predicting the heating abilities, rather than just relying on the T

_{C}and a single SLP value. However, no such evaluation standard can be found yet.

_{3}O

_{4}[15,16] and γ-Fe

_{2}O

_{3}MNPs [17]. It was found that the MNPs with a steeper decrease of heat source power densities within 37–46 °C can reduce the heating power in the inner tumor regions and automatically adjust the heating power to the tumor temperature changes, possessing better uniformities and stabilities.

## 2. Materials and Methods

#### 2.1. Temperature-Dependent SLP Measurements

_{0.54}Co

_{0.46}Cr

_{0.7}Fe

_{1.3}O

_{4}, Zn

_{0.54}Co

_{0.46}Cr

_{0.65}Fe

_{1.35}O

_{4}and Zn

_{0.54}Co

_{0.46}Cr

_{0.6}Fe

_{1.4}O

_{4}MNPs with T

_{C}of 37.5 °C, 56.0 °C and 61.0 °C are signed as MNP

_{37}, MNP

_{56}, and MNP

_{61}. The SLP data were calculated, based on the temperature-time (T-t) data from the calorimetry experiments. The experimental setup (Figure 1a) is as follows. The MNPs were fixed by hydrogel in a tube as magnetic hydrogel (MHG) for uniform and stable dispersion [14]. The tube was placed in a Dewar vacuum flask for reducing heat dissipation. Then, they were surrounded by circulating water with controllable temperature. The fiber optic thermometer (Fotemp-Trafo FTT-0100, Optocon, Dresden, Germany) monitored the central temperature of 2 mL MHG of 80 mg/mL (MNPs mass/hydrogel volume) in the tube. Before calorimetry measurements, the inner startup temperatures were monitored by the fiber optic thermometer and controlled by adjusting the circulating water temperature. When the startup temperatures were stable, the AMF generator (GUF-30T, Shenqiu Yongda High Frequency Equipment Co., Ltd., Zhoukou, China) heated the MNPs at 400 ± 5 Oe and 100 ± 5 kHz for 3 min to obtain the T-t curves. The startup temperatures were set from 20 °C to 75 °C with a 5 °C interval. The measurement was repeated four times for each kind of MNPs at each startup temperature. The mean values of T-t curves were used for the calculations of SLP data.

_{MHG}is the specific heat capacity of MHG (2.30 ± 0.18 J/(g·°C)) [14]; T is the temperature (°C); t is the time (s); dT/dt is the initial slope of the T-t curve (the linear fragment at 10–20 s of temperature rising), provided the temperature distribution within the sample is homogeneous and the initial thermal losses are negligible [19,20]; m

_{MHG}is the mass of MHG (mg); m

_{MNP}is the mass of MNPs (mg). The T-t data and SLP data of MNP

_{61}were cited from reference [14].

#### 2.2. Heating Performance Simulations

_{T}is the radius of the tumor (15 mm); R

_{L}is the radius of healthy liver tissue (90 mm); ρ is the density (kg/m

^{3}); c is the specific heat capacity (J/(kg·°C)); k is the thermal conductivity (W/(m·°C)); Q

_{m}is the power density of metabolic heat generation (W/m

^{3}); Q

_{b}is the power density of heat dissipation by blood perfusion effect (W/m

^{3}); PDP is the power density of MNPs [7], i.e., the power dissipation of MNPs per unit of volume (W/m

^{3}); the subscript T represents the tumor; the subscript L represents the healthy liver tissue; the subscript TP represents the tumor region with evenly distributed MNPs. The ρ, c, and k of the tumor region change with the MNP concentration as Equation (7) [22,23].

_{P}; and the C is the concentration of MNPs in the tumor (kg/m

^{3}or mg/mL, MNP mass/tumor volume).

_{m}and Q

_{b}rely on the body temperature change. The Q

_{m}(W/m

^{3}) is as Equation (8) [24].

_{m0}is the Q

_{m}when the tissue is at 37 °C (W/m

^{3}).

_{b}(W/m

^{3}) is as Equation (9) [25].

_{b}is the blood perfusion rate (1/s); T

_{α}is the temperature of arterial blood (°C), here assumed as 37 °C.

^{3}, MNP mass/tumor volume).

## 3. Results and Discussion

#### 3.1. SLP-T Relationships

_{C}, indicating that the MNPs have residual heating abilities over T

_{C}. The initial slopes of the T-t curves are extracted for SLP calculations (Equation (1)). In addition, the SLPs at different temperatures are shown in Figure 2d. With the increase of temperature, the SLP decreases gradually around the T

_{C}, ranging for dozens of degrees. Above the T

_{C}, considerable residual SLP exists, possibly resulting from the underestimated T

_{C}by measurement methods. The T

_{C}is normally determined by locating the maximum first derivatives of the thermogravimetric mass-temperature curves [10,14] or the minimum first derivatives of the magnetization-temperature curves [8]. So, the obtained T

_{C}value represents the majority of the MNPs. While the MNPs, in fact, always have a wide range of T

_{C}(some over 100 °C [10,14]). The measurement methods are practical for determining the representative T

_{C}but may result in inaccurate prediction of heating performances. For accurate predictions, the SLPs at different temperatures should be considered. The SLP-T data were fitted with Gaussian curves to provide the heat source in simulations with continuous coupling relationships. The fitting equation is as Equation (11).

#### 3.2. Heating Performances

#### 3.2.1. Temperature Rises and Distributions

_{center}) and the edge (T

_{edge}) would be used to represent the final temperature range within the tumor in the simulations hereafter.

_{C}unknown) are also included. They are signed after the authors’ names as MNP

_{Nemala}(Fe

_{3}O

_{4}) [15], MNP

_{Regmi}(Fe

_{3}O

_{4}) [16], and MNP

_{Beković}(γ-Fe

_{2}O

_{3}) [17]. The SLP data, extracted from references [15,16,17], are shown in Figure 4a. They are fitted by Equation (11) for the curves in Figure 4a. Furthermore, the fitting parameters are shown in Table 3. In addition, two assumed MNPs are also simulated, in order to exhibit the improvements and limitations in heating performances of the actual MNPs. They are the MNPs without T

_{C}(MNP

_{none}) with the SLP of 6 W/g at any temperature, and the ideal MNPs (T

_{C}46 °C, MNP

_{ideal}) with the SLP of 6 W/g under T

_{C}and 0 W/g above T

_{C}. The chosen 6 W/g is within the 5–7 W/g of MNP

_{56}and MNP

_{37}. For convergence consideration, the SLP decrease of MNP

_{ideal}was set as a linear drop from 6 W/g at 45.99 °C to 0 W/g at 46.00 °C. Their SLP-T curves are shown in Figure 4b.

#### 3.2.2. Capacities

_{center}s and T

_{edge}s at different MNP concentrations (Cs) are shown in Figure 5a,c,e marked as envelope curves. The MNP

_{none}and MNP

_{ideal}in Figure 5e have the same temperature distributions below 17.39 mg/mL. Over this concentration, the temperatures of the MNP

_{none}increase linearly, while the T

_{center}of MNP

_{ideal}self-regulates at 46 °C. The exceeded concentration only elevates the T

_{edge}of MNP

_{ideal}, indicating the whole tumor approaches 46 °C and the temperature distribution may be more uniform. The temperature-concentration curves of the actual MNPs, in Figure 5a,c, show upper convex shapes, indicating they have part of the self-regulating abilities. At extortionate concentrations, the T

_{center}of ZnCoCrFeO MNPs with lower T

_{C}, in Figure 5a, is closer to 46 °C, suggesting better self-regulating abilities. For the sake of comparing the heating performances of the different MNPs, the proper concentrations, providing the T

_{center}of 46 °C, are chosen in Figure 5a,c and will be used in the following analysis. Since the MNP

_{ideal}has a large range of concentrations for T

_{center}of 46 °C, a high concentration of 100 mg/mL is selected, which can increase the T

_{edge}over 42 °C. At the proper concentrations, the PDP-T curves are calculated by Equation (10) and shown in Figure 5b,d,f.

^{9}A/(m·s)” [28]. High MNP concentrations may lead to cytotoxicity [14]. The maximum ever applied MNP concentration, recorded in clinical trials, is 280 mg/mL [29]. With regards to the ZnCoCrFeO MNPs, the AMF conditions of 100 kHz and 400 Oe (32 kA/m) in calorimetry experiments and the chosen MNP concentrations of 30–95 mg/mL in Figure 5a may be acceptable. For the MNPs in Figure 5c, the chosen AMF conditions of MNP

_{Nemala}(375 kHz, 11.28 kA/m) [15], MNP

_{Regmi}(395 kHz, 5.6 kA/m) [16], and MNP

_{Beković}(100 kHz, 15 kA/m) [17], as well as the MNP concentrations of 3–27 mg/mL in Figure 5c may also be acceptable. So, these MNPs may be capable of hyperthermia. Noticeably, the safe AMF criteria depend on the body diameter exposed to the AMF, and the acceptable MNP concentration relates to the MNP composition, the biocompatibility of the surface coating, and so on. So, the capability of the specific MNPs will need detailed evaluations in clinical applications. Overall, by simulating the temperature-dependent heating efficiencies, the required concentration of MNPs can be determined, and the safety can be discussed.

^{2}), and correlates to the AMF frequency (f), according to Rosensweig’s model [30]. Thus, multiple combinations of H, f and C values can lead to the same PDP-T curves in Figure 5b,d, which result in effective heating. In addition, the combination with lower overall side effects can be selected among them. For instance, if the MNPs have excellent biocompatibility, raising the concentration to four times and reducing the H to half will lower the pain risk of high AMF but still maintain the T

_{center}at 46 °C.

#### 3.2.3. Uniformities

_{none}(Figure 6e,f), indicating more uniform temperature distributions. In addition, the MNP

_{37}has the best uniformities among the investigated actual MNPs. However, they cannot provide a temperature distribution like the MNP

_{ideal}, which forms 46 °C in over half of the tumor in Figure 6f.

_{center}) and the tumor edge (T

_{edge}), as well as the exceeded temperatures in 20% volume (T

_{20}), 50% volume (T

_{50}), 90% volume (T

_{90}) of the tumor are concerned [29]. To quantitatively evaluate the uniformities, the dimensionless parameter U

_{i}is defined, shown in Equations (12) and (13).

_{none}(Figure 6f). The temperature distribution of the MNP

_{none}is taken as the baseline, corresponding to U

_{i}= 0. Higher U

_{i}means the temperature is closer to T

_{center}. U

_{i}= 1 means extremely uniform. The uniformity indexes of corresponding MNPs are shown in Figure 7a. The results suggest that the ZnCoCrFeO MNPs with lower T

_{C}have higher uniformity at each position. The MNP

_{ideal}and MNP

_{none}present the upper and lower limits of uniformities. Among the actual MNPs, the MNP

_{37}has the highest U

_{90}, indicating the best therapeutic outcome [29,31].

_{none}with constant SLP as an example, when a small amount of MNPs is placed in a tumor as a unit and starts to generate heat, part of the generated heat will elevate the temperature of the local tissue, and the rest will be transferred to the ambient tissues. If three equidistant units are arranged in a line around the tumor center, the central part will receive most of the energy for temperature rise, thus higher temperature at the center. So, when the sphere tumor is uniformly filled with such units of MNP

_{none}, the temperature decreases from the tumor center to the edge, as seen in Figure 6e,f. Once the SLP of MNPs decreases with the increased temperatures, the powers of the inner hot units are restricted automatically, reducing the central and overall temperatures. To maintain the same T

_{center}, a higher concentration is needed. In addition, compensatory elevations in outer units’ powers present as higher PDPs in outer regions. The phenomenon can be seen through the PDP-T curves of the actual MNPs and MNP

_{ideal}in Figure 6 (right ordinate). Thus, the MNPs with steeper PDP-T curves form more uneven PDP distributions with higher values around the tumor center, helping to equalize the absorbed energies in different regions and leading to better temperature uniformity.

_{C}. Bagaria, et al. simulated different secondary polynomial MNPs’ radial distributions [32]. Liangruksa, et al. compared the homogenous, exponential, and Gaussian MNP distributions [33]. Zhang, et al. divided the tumor into multi regions with different MNP concentrations [34]. They all proved that, by distributing more MNPs around the tumor center patterns, the PDP at the center is higher, leading to more uniform temperature distributions. So, the temperature uniformities of the MNPs with low T

_{C}can be further improved by optimizing the distribution of the MNPs in the tumor.

#### 3.2.4. Stabilities

_{T}) affects the heat dissipation efficiency from tumor to peripheral tissue. Larger R

_{T}provides a smaller surface area over volume ratio, leading to more heat retention and higher temperatures in the tumor. Besides, the blood perfusion rate in tumors (w

_{bT}), which relates to heat dissipation along blood vessels, varies with the tumor stages and also impacts the temperature distribution significantly [36]. So, the varying PDP, R

_{T}and w

_{bT}mainly contribute to the instability of hyperthermia.

_{center}of 46 °C in Figure 5 are taken as the base. The PDP, R

_{T}in Equation (2) and w

_{bT}in Equation (9) are separately changed in the simulations and the resultant changes in T

_{center}are compared as stability evaluations, as shown in Figure 7b. For the influences of heat source change, 20% increase and decrease of PDP are simulated. For the impact of tumor progression or recovery, 20% increase and decrease of R

_{T}are simulated. For the effect of tumor blood system recovery, 10 times of w

_{bT}is simulated in reference to the w

_{bT}range of liver (0.000833–0.02289 s

^{−1}) [26]. In the results, less change in T

_{center}(closer to 46 °C) represents better stability. The increases of PDP and R

_{T}cause overheating. The decreases of PDP and R

_{T}, as well as the elevation of w

_{bT}, lead to insufficient heating. The changes of T

_{center}are less for the ZnCoCrFeO MNPs with lower T

_{C}, indicating higher stabilities. The actual MNPs are better than the MNP

_{none}. The MNP

_{ideal}can self-regulate the T

_{center}at 46 °C for all situations.

_{T}and w

_{bT}alterations are included. Taking R

_{T}fluctuations of MNP

_{56}as examples, the PDP-T curve and original temperature range (T

_{center}and T

_{edge}) are drawn in Figure 7c. When the temperature decreases, the temperature range in Figure 7c shifts lower (blue dash lines), while the PDP-T curve remains unchanged (black solid line). At this time, the PDPs of all units in the tumor increase with the shifted temperature range, causing an elevation of the overall PDP in the tumor (blue circles), thus alleviating the effect of the environmental changes and exhibiting self-regulating behaviors. (b) For the fluctuations in the heat source, the PDP alteration is included. Taking the MNP

_{56}as an example, the PDP-T curve and original temperature range are drawn in Figure 7d. When the PDPs of all units decrease (blue solid line), the temperature ranges also decrease (blue dash lines), elevating the units’ PDPs back a little bit (blue circles). Thus, for both situations, although the mechanisms are different, a steeper PDP-T curve always leads to better stabilities. In addition, the MNP

_{37}has the steepest PDP-T curve and the best stabilities among the actual MNPs.

_{C}helps to gather the MNPs with the same SLP decrease temperature and may offer the MNPs steeper PDP-T curves, thus improving the stabilities. The evaluation methods, proposed in this paper, are applicable to other kinds of tumors with different shapes, as well as hyperthermia with different temperature ranges, such as 46–60 °C.

## 4. Conclusions

_{3}O

_{4}and γ-Fe

_{2}O

_{3}MNPs, it can be seen that, with proper concentrations, these MNPs are capable of 46 °C hyperthermia for the proposed tumor model. Moreover, the MNPs with steeper PDP-T curves within 37–46 °C have better uniformities and stabilities, relying on restricting the PDP around the tumor center and automatically regulating the PDP under temperature change.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

AMF | alternating magnetic field |

T_{C} | Curie temperature (°C) |

MHG | magnetic hydrogel |

MNP | magnetic nanoparticles |

MNP_{37}, MNP_{56}, MNP_{61} | ZnCoCrFeO MNPs with T_{C} of 37.5 °C, 56.0 °C and 61.0 °C |

MNP_{Nemala} | MNPs by Nemala et al. |

MNP_{Regmi} | MNPs by Regmi et al. |

MNP_{Beković} | MNPs by Beković et al. |

MNP_{ideal} | assumed ideal MNPs with T_{C} of 46 °C (heating efficiency is constant below T_{C} and zero above T_{C}) |

MNP_{none} | assumed MNPs without T_{C} (heating efficiency does not change with temperature) |

_{b} (subscript) | of blood |

_{L} (subscript) | of healthy liver tissue |

_{MHG} (subscript) | of MHG |

_{MNP} (subscript) | of MNPs |

_{T} (subscript) | of tumor |

_{TP} (subscript) | of tumor and MNPs |

T (°C) | temperature |

t (s) | time |

T_{center} (°C) | temperature at the center of the tumor |

T_{20}, T_{50}, T_{90} (°C) | temperatures exceeded in 20%, 50% or 90% volume of the tumor |

T_{edge} (°C) | temperature at the edge between tumor and healthy tissue |

T_{α} (°C) | the temperature of arterial blood (37 °C) |

U_{20}, U_{50}, U_{90}, U_{edge} (dimensionless) | uniformities within 20%, 50%, 90% or 100% volume of the tumor |

SLP (W/g) | specific loss power |

PDP (W/m^{3}) | power dissipation of MNPs per unit of tumor volume |

c (J/(kg·°C)) | specific heat capacity |

C (kg/m^{3} or mg/mL) | MNP concentration in tumor region (MNP mass/tumor volume) |

R_{T}, R_{L} (mm) | radiuses of tumor region and overall healthy liver region |

r (mm) | radius from the tumor center |

ρ (kg/m^{3}) | density |

k (W/(m·°C)) | thermal conductivity |

Q_{m} (W/m^{3}) | power density of metabolic heat generation |

Q_{m0} (W/m^{3}) | power density of metabolic heat generation at body temperature (37 °C) |

Q_{b} (W/m^{3}) | power density of heat dissipation by blood perfusion effect |

w_{b} (1/s) | blood perfusion rate |

ϕ | volume fraction of MNPs in the tumor |

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**Figure 1.**(

**a**) Experimental setup of calorimetry experiments; (

**b**) Tissue model of the thermal simulations, R

_{T}is the radius of the sphere tumor, R

_{L}is the radius of the healthy liver tissue; (

**c**) Convergence analysis of mesh number about the final temperatures of tumor center (T

_{center}), with 1200 as the chosen mesh number.

**Figure 2.**The T-t data of the MNP

_{37}(

**a**), MNP

_{56}(

**b**), and MNP

_{61}from reference [14] (

**c**) at different initial temperatures (20–75 °C, 5 °C intervals); n = 4 for each line, data: mean value and standard uncertainty. (

**d**) Experimental SLP data and fitted SLP-T curves of MNP

_{37}, MNP

_{56}and MNP

_{61}; the SLP data of MNP

_{61}were from reference [14]; n = 4 for each point, data: mean value and standard uncertainty.

**Figure 3.**The time and spatial temperature changes in the tissue model of MNP

_{37}(

**a**), MNP

_{56}(

**b**) and MNP

_{61}(

**c**), at 35 mg/mL MNP concentration; (

**d**) The corresponding temperature rises of tumor centers (lower abscissa) and the final temperature distributions (upper abscissa).

**Figure 5.**The temperature range (T

_{center}-T

_{edge}) at different MNP concentrations of MNP

_{37}, MNP

_{56}and MNP

_{61}(

**a**); of MNP

_{Nemala}, MNP

_{Regmi}and MNP

_{Beković}(

**c**); of MNP

_{none}and MNP

_{ideal}(

**e**). The PDP-T curves at proper concentrations for T

_{center}46 °C of MNP

_{37}, MNP

_{56}and MNP

_{61}(

**b**); of MNP

_{Nemala}, MNP

_{Regmi}and MNP

_{Beković}(

**d**); of MNP

_{none}and MNP

_{ideal}(

**f**).

**Figure 6.**Temperature distributions (left ordinate) and PDP distributions (right ordinate) in the tumor of the ZnCoCrFeO MNPs by radius (

**a**), by volume (

**b**); of the MNPs from references by radius (

**c**), by volume (

**d**); and of the MNP

_{ideal}and MNP

_{none}by radius (

**e**), by volume (

**f**).

**Figure 7.**(

**a**) Temperature uniformities (U) of the ZnCoCrFeO MNPs, the MNPs from references, MNP

_{ideal}and MNP

_{none}. (

**b**) Stabilities (T

_{center}changes) under PDP, R

_{T}and w

_{bT}fluctuations of the ZnCoCrFeO MNPs, the MNPs from references, MNP

_{ideal}and MNP

_{none}; (

**c**) Changes of temperature ranges of the MNP

_{56}under R

_{T}fluctuations; (

**d**) Changes of temperature ranges and PDP-T curves of the MNP

_{56}under PDP fluctuations.

ρ | c | k | w_{b} | Q_{m0} | |
---|---|---|---|---|---|

kg/m^{3} | J/(kg·°C) | W/(m·°C) | 1/s | W/m^{3} | |

Tumor [7] | 1060 | 3540 | 0.52 | 0.000833 | 5790 |

Blood [26] | 1050 | 3617 | N/A | N/A | N/A |

Liver [26] | 1079 | 3540 | 0.52 | 0.0155 | 10,682 |

MNPs [7] | 5180 | 670 | 40 | N/A | N/A |

a | b | c | |
---|---|---|---|

MNP_{37} | 7.369 | 20.210 | 17.330 |

MNP_{56} | 5.196 | 28.500 | 24.070 |

MNP_{61} | 3.892 | 30.200 | 35.770 |

a | b | c | |
---|---|---|---|

MNP_{Nemala} | 46.350 | 0 | 67.240 |

MNP_{Regmi} | 8.257 | 12.480 | 41.810 |

MNP_{Beković} | 5.158 | 11.240 | 64.400 |

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## Share and Cite

**MDPI and ACS Style**

Ding, S.-W.; Wu, C.-W.; Yu, X.-G.; Dai, C.; Zhang, W.; Gong, J.-P.
Simulating Evaluation Method on Heating Performances of Magnetic Nanoparticles with Temperature-Dependent Heating Efficiencies in Tumor Hyperthermia. *Magnetochemistry* **2022**, *8*, 63.
https://doi.org/10.3390/magnetochemistry8060063

**AMA Style**

Ding S-W, Wu C-W, Yu X-G, Dai C, Zhang W, Gong J-P.
Simulating Evaluation Method on Heating Performances of Magnetic Nanoparticles with Temperature-Dependent Heating Efficiencies in Tumor Hyperthermia. *Magnetochemistry*. 2022; 8(6):63.
https://doi.org/10.3390/magnetochemistry8060063

**Chicago/Turabian Style**

Ding, Shuai-Wen, Cheng-Wei Wu, Xiao-Gang Yu, Chao Dai, Wei Zhang, and Jian-Po Gong.
2022. "Simulating Evaluation Method on Heating Performances of Magnetic Nanoparticles with Temperature-Dependent Heating Efficiencies in Tumor Hyperthermia" *Magnetochemistry* 8, no. 6: 63.
https://doi.org/10.3390/magnetochemistry8060063