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

: The magnetic nanoparticles (MNPs) with decreasing heating efﬁciency (characterized by speciﬁc loss power, SLP ) with temperature increase, especially around the Curie temperature ( T 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.


Introduction
After being introduced to tumors, magnetic nanoparticles (MNPs) can heat and cure tumors under an alternating magnetic field (AMF) with few side effects.The efficiency of the therapy has been verified in several clinical trials [1][2][3][4].For further improving the therapeutic effects, the MNPs with suitable Curie temperature (T 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].
In previous work, we reported ZnCoCrFeO MNPs with T C of 29-103 • C [10].The Zn-CoCrFeO 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.
In this paper, a tumor simulation model considering the temperature-dependent SLP was established to predict the heating capacity of MNPs in hyperthermia and their performances in forming a uniform and stable temperature distribution in the tumor.The method was applied to 46 • C hyperthermia of ZnCoCrFeO [14], Fe 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.

Temperature-Dependent SLP Measurements
The MNPs were manufactured by the methods used in previous studies [10,14] 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.
The SLP describes the heat, converted from the energy of AMF, per unit time and unit MNP mass, as Equation (1) [18].
where, c 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].The SLP describes the heat, converted from the energy of AMF, per unit time and unit MNP mass, as Equation (1) [18].
where, cMHG 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]; mMHG is the mass of MHG (mg); mMNP is the mass of MNPs (mg).The T-t data and SLP data of MNP61 were cited from reference [14].

Heating Performance Simulations
The model is shown in Figure 1b, i.e., a sphere tumor (15 mm radius) surrounded by healthy liver tissue (90 mm radius).The MNPs were assumed to be evenly distributed in the tumor without displacement and loss.Two Pennes bioheat transfer equations for simulating the temperature changes in tumor and healthy tissue are as Equations ( 2) and (3) [21].

𝜌
The initial condition is as Equation (4).(c) Convergence analysis of mesh number about the final temperatures of tumor center (T center ), with 1200 as the chosen mesh number.

Heating Performance Simulations
The model is shown in Figure 1b, i.e., a sphere tumor (15 mm radius) surrounded by healthy liver tissue (90 mm radius).The MNPs were assumed to be evenly distributed in the tumor without displacement and loss.Two Pennes bioheat transfer equations for simulating the temperature changes in tumor and healthy tissue are as Equations ( 2) and (3) [21].
The initial condition is as Equation (4).
At the interface of the tumor and healthy tissue, the temperature and heat flow are continuous.In addition, the inner and outer boundary conditions are as Equations ( 5) and (6).

∂T(r, t) ∂r r=0
= 0 (5) where, T is the temperature ( • C); r is the radius from the center (m); R 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].
where, the subscript P represents MNPs; φ is the volume fraction of MNPs in the tumor, which can be calculated through φ = C/ρ P ; and the C is the concentration of MNPs in the tumor (kg/m 3 or mg/mL, MNP mass/tumor volume).
The Q m and Q b rely on the body temperature change.The Q m (W/m 3 ) is as Equation ( 8) [24].
where, Q m0 is the Q m when the tissue is at 37 ) is as Equation ( 9) [25].
where, the subscript b represents blood; w b is the blood perfusion rate (1/s); T α is the temperature of arterial blood ( • C), here assumed as 37 • C. The physical parameters used for tissues and MNPs are shown in Table 1.
where, SLP(T) is the specific loss power of MNPs (W/kg), obtained in the experiments; C is the MNP concentration (kg/m 3 , MNP mass/tumor volume).The Pennes bioheat transfer equations were solved by the PDEPE function in MATLAB.The mesh numbers of time and spatial dimensions were equal.The convergence of mesh numbers was analyzed by comparing the final temperatures of the tumor center at different mesh numbers in Figure 1c.The MNP concentration in the convergence analysis was 35 mg/mL.The mesh number of 1200 was selected.

SLP-T Relationships
The temperature-time (T-t) curves of the MHG of ZnCoCrFeO MNPs are shown in Figure 2a-c.As the startup temperature increases, the temperature rising amplitude gradually decreases to zero, indicating that the MNPs have self-regulating temperatures.However, the elevated temperatures can exceed the T 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 magnetizationtemperature 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).
where, a, b and c are the fitting parameters, summarized in Table 2.

Temperature Rises and Distributions
The temperature changes in time and spatial dimensions of the three ZnCoCrFeO MNPs are shown in Figure 3a-c.In addition, the corresponding temperature rises of tumor centers and the final temperature distributions are summarized in Figure 3d.The temperatures rise from 37 °C (body temperature).So, the SLP over 37 °C (Figure 2d) contributes to the heating processes and should be focused on.The temperature increase in the tumor stops after about 10 min, which can be seen from the central temperatures in Figure 3d (lower abscissa).Furthermore, the final temperature distributions, in Figure 3d  The temperature changes in time and spatial dimensions of the three ZnCoCrFeO MNPs are shown in Figure 3a-c.In addition, the corresponding temperature rises of tumor centers and the final temperature distributions are summarized in Figure 3d.The temperatures rise from 37 • C (body temperature).So, the SLP over 37 • C (Figure 2d) contributes to the heating processes and should be focused on.The temperature increase in the tumor stops after about 10 min, which can be seen from the central temperatures in Figure 3d (lower abscissa).Furthermore, the final temperature distributions, in Figure 3d (upper abscissa), would last for another 50 min in the one-hour clinical hyperthermia treatment, mainly determining the curing effects [1][2][3][4].Thus, in the following heating performance evaluations, the final temperature distributions would be focused on.In the tumor, the maximum and minimum temperatures appear at the center and the edge, marked in Figure 3d.So, the final temperatures at the center (T center ) and the edge (T edge ) would be used to represent the final temperature range within the tumor in the simulations hereafter.In the following simulations of heating performances, three kinds of MNPs in references with temperature-dependent SLP (TC unknown) are also included.They are signed after the authors' names as MNPNemala (Fe3O4) [15], MNPRegmi (Fe3O4) [16], and MNPBeković (γ-Fe2O3) [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 TC (MNPnone) with the SLP of 6 W/g at any temperature, and the ideal MNPs (TC 46 °C, MNPideal) with the SLP of 6 W/g under TC and 0 W/g above TC.The chosen 6 W/g is within the 5-7 W/g of MNP56 and MNP37.For convergence consideration, the SLP decrease of MNPideal 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.In the following simulations of heating performances, three kinds of MNPs in references with temperature-dependent SLP (T 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.
Tedge of MNPideal, 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 selfregulating abilities.At extortionate concentrations, the Tcenter of ZnCoCrFeO MNPs with lower TC, 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 Tcenter of 46 °C, are chosen in Figure 5a,c and will be used in the following analysis.Since the MNPideal has a large range of concentrations for Tcenter of 46 °C, a high concentration of 100 mg/mL is selected, which can increase the Tedge over 42 °C.At the proper concentrations, the PDP-T curves are calculated by Equation (10) and shown in Figure 5b,d,f.

Capacities
For heating capability evaluation, the proper MNP concentration for heating the tumor center to 46 • C is searched for each kind of MNP.The T 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.
After confirming their abilities to heat the tumor to 46 • C, whether the corresponding AMF conditions and concentrations are safe still needs to be investigated.High AMF intensity or frequency may cause pain [1].Normally, the safe criteria for AMF are "intensity (H) × frequency (f ) < 5 × 10 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.After confirming their abilities to heat the tumor to 46 °C, whether the corresponding AMF conditions and concentrations are safe still needs to be investigated.High AMF intensity or frequency may cause pain [1].Normally, the safe criteria for AMF are "intensity (H) × frequency (f) < 5 × 10 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 In clinical hyperthermia applications, the AMF conditions and concentrations can be adjusted according to patient tolerance.If the patient is more sensitive to the pain, the AMF intensity needs to be reduced [1].Consequently, the MNP concentration requires elevation for ensuring effective heating.The balance of the AMF and MNP concentration values refers to Equation (10).In Equation (10), the PDP(T) equals SLP(T) × C, where the SLP is proportional to the square of AMF intensity (H 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.

Uniformities
The temperature distributions in the tumor model are shown by the radius in Figure 6a,c,e (left ordinate), and by the volume in Figure 6b,d,f (left ordinate).Along the radial direction, the temperature decreases from the center to edge.The actual MNPs (Figure 6a-d) have higher temperatures all over the tumor than the MNP 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.
In clinical trials, the final temperatures at the tumor center (T 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).
where, the subscript i can be 20, 50, 90 and edge, representing 20% volume, 50% volume, 90% volume of the tumor, and the tumor edge; the superscript 0 represents the values obtained from the MNP 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].
The improvements in uniformity result from the equalized absorbed energies (for temperature rise) within the tumor.Before interpreting the contribution of the decreased PDPs with increased temperatures in this process, the reason for the temperature distribution of "inner higher, outer lower" needs to be explained first.Taking the MNP 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 con-centration 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.
The uniformity improvement can also be accomplished by distributing more MNPs around the tumor center, regardless of the T 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.In clinical trials, the final temperatures at the tumor center (Tcenter) and the tumor edge (Tedge), as well as the exceeded temperatures in 20% volume (T20), 50% volume (T50), 90% volume (T90) of the tumor are concerned [29].To quantitatively evaluate the uniformities, the dimensionless parameter Ui is defined, shown in Equations ( 12) and ( 13).taken as the baseline, corresponding to Ui = 0. Higher Ui means the temperature is closer to Tcenter.Ui = 1 means extremely uniform.The uniformity indexes of corresponding MNPs are shown in Figure 7a.The results suggest that the ZnCoCrFeO MNPs with lower TC have higher uniformity at each position.The MNPideal and MNPnone present the upper and lower limits of uniformities.Among the actual MNPs, the MNP37 has the highest U90, indicating the best therapeutic outcome [29,31].The improvements in uniformity result from the equalized absorbed energies (for temperature rise) within the tumor.Before interpreting the contribution of the decreased PDPs with increased temperatures in this process, the reason for the temperature distribution of "inner higher, outer lower" needs to be explained first.Taking the MNPnone 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 MNPnone, 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 Tcenter, 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

Stabilities
In hyperthermia of MNPs, the reasons for temperature instabilities can be summed up as the changes of heat sources (heat generation) and the tumor properties (heat dissipation).(a) For heat sources, the AMF and the concentration influence heat generation.In clinical trials, the applied AMF intensity and frequency may fluctuate, influencing the SLP of MNPs [1,29].Furthermore, the concentrations of MNPs may change in the subsequent therapeutic procedure.For instance, in a tumor mouse model, only 3/4 of the applied MNPs can be detected after 24 h [35].Thus, the PDP of the heat source may be affected by the SLP and concentration (Equation ( 10)), altering the temperature distribution in the tumor.(b) For tumor properties, the tumor itself may progress or recover, affecting the heating process.The alteration of tumor size (R 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.
For evaluating the stabilities of MNPs, the simulations with the T 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

Figure 1 .
Figure 1.(a) Experimental setup of calorimetry experiments; (b) Tissue model of the thermal simulations, RT is the radius of the sphere tumor, RL is the radius of the healthy liver tissue; (c) Convergence analysis of mesh number about the final temperatures of tumor center (Tcenter), with 1200 as the chosen mesh number.

Figure 1 .
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 .
Figure 2. The T-t data of the MNP37 (a), MNP56 (b), and MNP61 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 MNP37, MNP56 and MNP61; the SLP data of MNP61 were from reference [14]; n = 4 for each point, data: mean value and standard uncertainty.

Figure 2 .
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.

Magnetochemistry 2022, 8 , 16 Figure 3 .
Figure 3.The time and spatial temperature changes in the tissue model of MNP37 (a), MNP56 (b) and MNP61 (c), at 35 mg/mL MNP concentration; (d) The corresponding temperature rises of tumor centers (lower abscissa) and the final temperature distributions (upper abscissa).

Table 3 .Figure 3 .
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 4 .
Figure 4. (a) The SLP data and fitted SLP-T curves of MNPNemala, MNPRegmi, and MNPBeković; the SLP data are obtained from the references [15-17]; (b) The SLP-T curves of MNPnone and MNPideal.

Figure 4 .
Figure 4. (a) The SLP data and fitted SLP-T curves of MNP Nemala , MNP Regmi , and MNP Beković ; the SLP data are obtained from the references [15-17]; (b) The SLP-T curves of MNP none and MNP ideal .

Magnetochemistry 2022, 8 , 16 Figure 6 .
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 MNPideal and MNPnone by radius (e), by volume (f).

Figure 6 .
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 .
Figure 7. (a) Temperature uniformities (U) of the ZnCoCrFeO MNPs, the MNPs from references, MNPideal and MNPnone.(b) Stabilities (Tcenter changes) under PDP, RT and wbT fluctuations of the ZnCoCrFeO MNPs, the MNPs from references, MNPideal and MNPnone; (c) Changes of temperature ranges of the MNP56 under RT fluctuations; (d) Changes of temperature ranges and PDP-T curves of the MNP56 under PDP fluctuations.

Figure 7 .
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.
. The Zn 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

Table 2 .
Gaussian fitting parameters of SLP-T curves for ZnCoCrFeO MNPs.

Table 3 .
Gaussian fitting parameters of SLP-T curves for the MNPs from references.