# A Computational Study on Magnetic Nanoparticles Hyperthermia of Ellipsoidal Tumors

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

^{8}A∙m

^{−1}∙s

^{−1}for safety purposes [29,30]. For intravenous injection, the magnetic field may also facilitate the nanoparticles magnetic guidance towards the desired location [31,32]. Due to the AMF, the injected nanoparticles generate heat via mainly two mechanisms, usually referred to as Néelian and Brownian relaxations. Based on this, Rosenzweig [33] developed a theoretical heat generation model that takes into account the magnetic properties of the nanoparticles as well as properties associated with the applied magnetic field. The efficiency of MNPs to convert electromagnetic energy into heat may be quantified experimentally, usually by specific loss power (SLP), also referred to as specific absorption rate (SAR) [15,34]. The most common types of MNPs used to date are superparamagnetic iron oxide nanoparticles (SPIONs) such as magnetite Fe

_{3}O

_{4}and maghemite γ-Fe

_{2}O

_{3}[35,36,37], due to their biocompatibility [38] upon injection in human tissues. Several other nanoparticle types have also been reported, such as metallic alloys [39,40] and carbon nanotubes [41]. The distribution of the nanoparticles reaching the tumor may generally vary from approximately uniform distributions throughout the whole tumor to fully localized ones inside specific tumor regions as observed from real tumor sections [42,43]. The distributions depend on several factors such as injection parameters, the tumor’s physiological and mechanical properties, and the physicochemical properties of the nanoparticles [44,45,46].

## 2. Materials and Methods

#### 2.1. Geometrical Description

- (i)
- oblate spheroids with semi-axis a > b
- (ii)
- prolate spheroids with semi-axis a < b

_{h}is taken approximately eight times larger than the semi-major axis of the tumor with the highest aspect ratio. Due to the rotational symmetry of the geometries, the present thermal problem can be solved as an axisymmetric problem instead of a 3D one, which substantially decreases the computational cost of the numerical simulations [99].

#### 2.2. Bio-Heat Transfer Analysis

_{n}and ρ

_{b}denote the densities of the tissues and the blood respectively, c

_{n}and c

_{b}are the corresponding heat capacities, T(x,y,t) is the local tissue temperature, k

_{n}is the tissue thermal conductivity, w

_{b}is the blood perfusion rate, and T

_{b}= 37 °C is the blood temperature.

_{met,n}is the internal heat generation rate per unit volume associated with the metabolic heat production. Finally, Q

_{s}is the power dissipation density by the MNPs. It is assumed no leakage of MNPs to the surrounding healthy tissue. Therefore, Q

_{s}is only applied to the cancerous region filled with the MNPs, similarly to several earlier investigations, e.g., [15,67,69,92]. Following the assumptions by [42,44,92,95,101], a uniform distribution of nanoparticles is assumed. The temperature of the surface of the healthy tissue region (see Figure 2), which is far away the tumor surface as explained in the previous section, is set to T

_{s}= T

_{b}[69,75]. For the initial temperature condition we set T(x,y,t = 0) = T

_{b}. A symmetry (zero heat flux) boundary condition is applied on the x- and y-axes (line OA and line OB respectively in Figure 2). Also, heat flux and temperature continuity are considered on the tumor-healthy tissue interface as follows [12,93]:

**n**is the direction vector perpendicular to the tumor surface. To carry out numerical simulations for the different AR cases, typical tissue thermophysical values (for example prostate tissue) are chosen from earlier works [36,63,67,71,94] and shown in Table 2. Note that we have assumed that the perfusion rate is independent from region and temperature [31,58]. We have also assumed that the thermophysical properties of the tumor and the healthy tissue are the same, based on earlier works [36,67,71,94] For a complete mathematical formulation of heat transfer in a solid body and the implementation of “internal heat generation” terms within a numerical setting, the interested reader can be referred to [102].

^{−8}. To speed up the simulations we ran them in parallel mode on a system of distributed processors by geometric domain and associated field decomposition, as described in [105].

#### 2.3. Heat Generation by the Magnetic Nanoparticles

^{3}) from a monodispersed magnetic fluid filled with superparamagnetic particles subjected to an alternating magnetic field [15] is given by:

_{0}= 4$\mathsf{\pi}$ × 10

^{−7}H∙m

^{−1}is the vacuum magnetic permeability, f is the frequency of the magnetic field with magnetic field intensity H

_{0}(A∙m

^{−1}), χ

_{0}corresponds to the magnetic susceptibility of the magnetic fluid defined by

_{i}the initial susceptibility given by

_{d}the domain magnetization of a suspended particle, V

_{m}= 4/3R

^{3}is the magnetic volume for a nanoparticle of radius R, k

_{B}= 1.381 × 10

^{−23}J∙K

^{−1}is the Boltzmann constant and T is the absolute temperature. In Equation (10), φ is the solid volume fraction of the nanoparticles. The power dissipation is accomplished via two physical processes, the Néel and Brownian relaxation. Because these processes take place in parallel, the effective relaxation time in Equation (5) is given by

_{B}the Brownian characteristic relaxation time

_{H}is taken as the hydrodynamic volume of the nanoparticle related to V

_{m}as V

_{H}= (1 + δ/R)

^{3}V

_{m}where δ is the thickness of a sorbed surfactant layer (δ = 2 nm according to Rosensweig [33]). In Equation (13) τ

_{N}is the Néel characteristic relaxation time given by [15]:

_{0}≈ 10

^{−9}s is an attempt time [15,49] and $\mathcal{K}$ is the anisotropy constant (J/m).

#### 2.4. Tissue Thermal Damage

^{−1}), E

_{a}the activation energy (J∙mol

^{−1}) and R the gas constant. The temperature T(x,y,t) in Equation (15) is in Kelvin. Ω = 1 means that the damage process is 63.2% complete [21,54] and the tissue may be assumed to be irreversibly damaged [54,106]. The values of the frequency factor and activation energy depend upon the cell line. For the computational results of the present investigation, the constituent cells of the tissue are assumed to be the AT1 subline of Dunning R3327 rat prostate cells with the corresponding values obtained from earlier works [76,92], namely: A = 2.99 × 10

^{37}s

^{−1}and E

_{a}= 244.8 kJ∙mol

^{−1}.

#### 2.5. Mesh and Timestep Sensitivity Analysis

## 3. Computational Results and Discussion

_{3}O

_{4}) nanoparticles are selected as heat mediators assuming typical magnetic properties, as shown in Table 4. The magnetic field properties are also presented in Table 4. Note that for the selected H

_{0}and f values we find H

_{0}f = 1.496 × 10

^{9}A∙m

^{−1}∙s

^{−1}, which falls between the limits of Atkinson-Brezovich (4.85 × 10

^{8}A∙m

^{−1}∙s

^{−1}) and Dutz-Hergt (5 × 10

^{9}A∙m

^{−1}∙s

^{−1}) criterions [29,30]. Also, the nanoparticles volume fraction we used is low. Therefore, the effective tumor parameters of MNPs-saturated tissue are nearly identical to tumor parameters without nanoparticles that are used in the model. By substituting these parameters in Equation (8) we find Q

_{s}= 1.91 × 10

^{5}W/m

^{3}, which is within the range of earlier publications [63,65,68].

## 4. Comparison with Experiments

^{3}, normal lymph node volume 105.6 ± 43.37 mm

^{3}). Twenty days after tumor transplantation, magnetite nanoparticles, of average core magnetite size D = 10 nm, were injected from the rabbit tongue. The average nanoparticle uptake from the cancerous lymph was approximately 4 mg ± 1 mg. For the hyperthermia treatment, a transistor inverter was used with frequency 118 kHz and magnetic field strength of 384 Oe or 30.6 kA/m. From a histological section of the swollen lymph, we approximated the tumor shape with a prolate spheroid that we fitted on top of the tumor. Two tumor-shaped approximations are considered as shown in Figure 11a,b. In Case A we find AR ≈ 1.8, and for case B, AR ≈ 2.2. Inserting the tumor volume value in Equation (4) we calculate a ≈ 5.1 mm and from Equation (2) we find b ≈ 9.18 mm for Case A. In Case B we find a ≈ 4.78 mm and b ≈ 10.44 mm. From the values reported by Hamaguchi et al. [86] and using Rosensweig’s theory (Equations (8)–(14)) we find the heat dissipated by the nanoparticles equal to 2.1 × 10

^{5}W/m

^{3}. For the blood perfusion we use 1.3 × 10

^{−3}s

^{−1}within the range of earlier works [63,92,93,94]. The treatment temperature simulation results, for Case A and Case B, are shown in Figure 11c,d, respectively. For the 4 mg dosage, the predictions are in qualitative agreement with the temperature measurements by Hamaguchi et al. [86]. Some small differences are observed between the numerical result of Case A and Case B, with Case A being slightly closer to the measurements. It should be pointed it out that Hamaguchi et al. [86] report that the 4 mg nanoparticle uptake from the cancerous lymph has approximately ±1 mg uncertainty in the measurement. Interestingly, if we use a 5 mg dosage for Case A and Case B our results are in better agreement with the experimental temperature measurements by Hamaguchi et al. [86].

^{3}and was heated for 600 s. In their work, iron oxide nanoparticles (IONP) of 10–20 nm in diameter were. The IONPs were exposed to magnetic field strengths between 20 and 50 kA/m (rms) at 162 kHz. Pearce et al. [92] report that the transient temperature was recorded at a location called “center” and another location separated by 3 mm, called “tip”. They also mention that the center probe location was placed as close as possible to the approximate center of the tumor. A redrawn histologic section of the tumor in Pearce et al. [92] is shown in Figure 12. As in the previous experimental comparison, we approximated the tumor shape with a prolate spheroid that we fitted on top of the tumor. Two tumor shape approximations were considered, as shown in Figure 12a,b. For Case A we found AR ≈ 1.29 and for case B, AR ≈ 1.6. We then found a ≈ 3.9 mm and b ≈ 5.1 mm for Case A and for Case B we find a ≈ 3.6 mm and b ≈ 5.8 mm. The experimental temperature measurements close to the tumor center (probe location center) and about 3 mm from the tumor center (probe location tip), are shown in Figure 12c,f. According to Pearce et al. [92], the value of heat generated by the nanoparticles was “…adjusted somewhat until the experiment maximum transient temperature (or steady state) temperature record from the embedded probes was closely approximated by the numerical model result.”. They also report that the same approach was followed for the blood perfusion: “…adjusted to improve match to the measurements…”. The numerical results given by [92] are shown in Figure 12 with broken lines. The adjusted by Pearce et al. [92] value for the generated heat by the nanoparticles was 1.1 × 10

^{6}W/m

^{3}. For the adjusted perfusion, according to Pearce et al. [92], the initial tumor perfusion, 3 × 10

^{−3}s

^{−1}was increased to as much as 7 × 10

^{−3}s

^{−1}, as required to match experimental results. If we follow the Pearce et al. [92] approach of adjusting the heat generated and the perfusion rate we find good agreement with the measurements for the probe location center, as shown in Figure 12c (Case A), using the values of 1.75 × 10

^{6}W/m

^{3}and 2.5 × 10

^{−3}s

^{−1}. It should be pointed out that at t = 0 we have used the experimentally measured temperature (32 °C), while in the numerical model in [92] a higher temperature of approximately 36 °C was assumed by Pearce et al. [92], without providing an explanation for this choice. This perhaps explains the differences between our adjusted values with the ones by Pearce et al. [92]. Good agreement with the measured temperature and our model is also observed for the tip location, seen in Figure 12e, while in the prediction by Pearce et al. [92], the computational model gives higher temperatures than the experiment at this location. For the tumor geometry of Case B, we use the adjusted heat generated and blood perfusion values from Case A and compare our predictions with the experiments in Figure 12d (center location) and Figure 12f (tip location). Of course, due to the larger AR of the tumor than in Case A, the maximum temperatures are somewhat lower but reasonably close to the measurements. Unfortunately, due to the large range of two simultaneous parameters, namely, the nanoparticle diameter (10 to 20 nm) and the applied magnetic field (20 to 50 kA/m) reported in Pearce et al. [92], we could not apply Rosensweig’s theory as we did for Hamaguchi et al. [86]. Subsequently, we compared the cumulative equivalent minutes at 43 °C (CEM43) of our model with the CEM43 measurements and model predictions reported by Pearce et al. [92]. According to Pearce et al. [92], the CEM43 in discrete interval form is written as

_{CEM}is the time scaling ratio, 43 °C is the reference temperature and Δt

_{i}(min) is spent at temperature T

_{i}(°C). In their work R

_{CEM}= 0.45 was chosen. Using Equation (16) for our model predictions in Figure 12 we obtain CEM43 values close to the calculated by Pearce et al. [92], as shown in Table 5.

^{3}. The tumor volume in Ling et al. [91] was calculated from Equation (4). Then 0.1 mL of a mixture that was injected in the tumor tissue containing 10 wt% Fe

_{3}O

_{4}powder, 36 wt% polymethylmethacrylate (PMMA) powder and 54 wt% MMA liquid. A magnetic field of frequency 626 kHz and strength 28.6 kA/m was applied for 180 s. Ling et al. [91] report that the ferrous powder diameter was in the range of D = 20–50 nm. An ultrasound image section of the tumor from Ling et al. [91] is shown at the top of Figure 13. The red dashed line corresponds to the tumor boundary as marked by Ling et al. [91]. For the comparison, we recognize two cases. Given the tumor volume mentioned above for Case A (Figure 13a) we find AR ≈ 2.5 with a ≈ 3.5 mm and b ≈ 8.8 mm, while for Case B, AR ≈ 2.82 with a ≈ 3.37 mm and b ≈ 9.5 mm. Subsequently, we applied Rosensweig’s theory to estimate the amount of heat dissipated by the nanoparticles. We initially calculated the solid volume fraction of the nanoparticles in the tumor volume. The density of the injected mixture can be calculated from the following equation

_{Fe3O4}= 5180 kg/m

^{3}[33], ρ

_{MMA}= 940 kg/m

^{3}[111], ρ

_{PMMA}= 1180 kg/m

^{3}[112] and the corresponding weight fractions are mentioned above. Substituting the values in the above equation we find ρ

_{mixture}= 1450 kg/m

^{3}. The 0.1 mL mixture injected in the 453 mm

^{3}tumor volume has a mass of 0.1 mL × 1450 mg/mL = 145 mg. Since 10 wt% of the mixture is the ferrous mass, the nanoparticles weight in the tumor is 14.5 mg. This corresponds a solid volume fraction in the tumor of φ ≈ 6.16 × 10

^{−3}. Also, Ling et al. [91] measured the magnetization saturation of the nanoparticles M

_{s}= 1560 A/m (or mass magnetization saturation σ

_{s}= 0.3 emu∙g

^{−1}). According to Rosensweig [33], the domain magnetization (M

_{d}in Equations (10) and (11)) is given by M

_{d}= Μ

_{s}/φ, where substitution in the present case gives M

_{d}≈ 253 kA/m. If we select the value of 35 nm for the ferrous powder average size in the range 20–50 nm reported by Ling et al. [91], and use the above-mentioned values, we find (from Rosensweig’s theory) 1.7 × 10

^{6}W/m

^{3}of dissipated heat. The comparison of the present model tumor surface temperature, using this value, with the measurements by Ling et al. [91] is shown in Figure 13c for Case A and Figure 13d for Case B. The blood perfusion value used for the Hamaguchi et al. [86] comparison is also used here. Note that in [91] the temperature is recorded using thermal imaging. The exact location of the tumor surface at which temperature was measured is not reported. The thermal images presented in Ling et al. [91] suggest that it is measured at a central surface location. For the comparison we have selected in our model a temperature at the tumor surface along the semi-minor axis (see Figure 13a or Figure 13b). Both Case A and B model predictions are close to the low measured temperature limit, with Case A being marginally better. If we select an average value of 30 nm for the powder size, which is also within the 20–50 nm range reported by Ling et al. [91], our model predictions are in good agreement with the measured temperatures as shown in Figure 13c,d.

## 5. Concluding Remarks

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**(

**a**) Virtual representation of tumors by ellipsoid geometries. (

**b**) Notation of the major and minor axis length of the spheroids. All shapes shown have the same volume and are fully symmetric around the y-axis.

**Figure 2.**Schematic representation of the axisymmetric model, where y-axis is the revolution axis and x-axis is a symmetry axis (figure not to scale). The ellipsoidal tumor is assumed to be surrounded by a significantly larger spherical healthy tissue (R

_{h}>> a or b). T

_{s}corresponds to the temperature of the outer surface of the healthy tissue.

**Figure 3.**Two representative computational meshes used in the study focused at the tumor region and the close area around it. Magnified views close to the tumor/healthy tissue boundary are also shown. Both meshes correspond to tumors with aspect ratio AR = 2.

**Figure 4.**Comparison of the present computational results for various dimensionless treatment times with the closed-form solution by Liangruksa et al. [67]. T* is the dimensionless temperature, r* the dimensionless distance from the tumor center and t* is the dimensionless time, as defined in Liangruksa et al. [67].

**Figure 5.**Treatment temperature field after 22 min of heating for oblate spheroidal tumor shapes with different aspect ratios. (

**a**) AR = 1, (

**b**) AR = 2, (

**c**) AR = 4 and (

**d**) AR = 8.

**Figure 6.**Treatment temperature field after 22 min of heating for prolate spheroidal tumor shapes with different aspect ratios. (

**a**) AR = 2, (

**b**) AR = 4 and (

**c**) AR = 8.

**Figure 8.**Treatment temperature on the tumor surface, at two tumor surface probe locations P1 and P2 shown in the scheme (

**a**) for: (

**b**) AR = 1, (

**c**) AR = 2, (

**d**) AR = 4 and (

**e**) AR = 8. All ellipsoidal tumor shapes have the same volume.

**Figure 10.**Thermal damage in three tumor shapes and for a healthy tissue region close to the tumor after 22 min of treatment: (

**a**) AR = 1, (

**b**) prolate with AR = 2 and (

**c**) oblate with AR = 2.

**Figure 11.**Two cases approximating the tumor shape from a histological cross-section by Hamaguchi et al. [86], with a prolate spheroid. Note that the tumor histological cross-section has been redrawn from the original: (

**a**) prolate spheroid shape, case A with AR ≈ 1.8, on top of the redrawn tumor and (

**b**) prolate spheroid shape, case B with AR ≈ 2.2, on top of the redrawn tumor. Plots (

**c**,

**d**) show parametric comparison of the numerically determined temperature at the tumor center with the measured temperature by [86]. Temperature data points and bars are mean values and standard deviation respectively of 5 independent experiments.

**Figure 12.**Two cases approximating the tumor shape from a histological cross-section by Pearce et al. [92] with a prolate spheroid. Note that the tumor histological cross-section has been redrawn from the original: (

**a**) prolate spheroid shape, case A with AR ≈ 1.29, on top of the redrawn tumor and (

**b**) prolate spheroid shape, case B with AR ≈ 1.57, on top of the redrawn tumor. Comparison of the present numerical model with the 3D numerical model and experiments by Pearce et al. [92] at the tumor center (probe center) for (

**c**) Case A and (

**d**) Case B and at the probe tip (approximately 3 mm from tumor center) for (

**e**) Case A and (

**f**) Case B.

**Figure 13.**Two cases approximating the tumor shape from a histological cross-section by Ling et al. [91], with a prolate spheroid. Note that the tumor histological cross-section has been redrawn from the original: (

**a**) prolate spheroid shape, case A with AR ≈ 2.5, on top of the redrawn tumor and (

**b**) prolate spheroid shape, case B with AR ≈ 2.82, on top of the redrawn tumor. Comparison of the present model assuming two nanoparticle size values, with experimental temperature measurements at the tumor surface for (

**c**) Case A and (

**d**) Case B.

Prolate Tumors | ||
---|---|---|

Aspect ratio (AR) | a (mm) | b (mm) |

2 | 7.93 | 15.87 |

4 | 6.29 | 25.19 |

8 | 5.0 | 40.0 |

Oblate Tumors | ||

Aspect ratio (AR) | a (mm) | b (mm) |

1 | 10.0 | 10.0 |

2 | 12.5 | 6.29 |

4 | 15.87 | 3.96 |

8 | 20.0 | 2.50 |

Tissue | ρ (kg/m^{3}) | c (J/kg∙K) | k (W/m∙K) | w_{b} (s^{−1}) | Q_{met} (W/m^{3}) |
---|---|---|---|---|---|

Tumor | 1045 | 3760 | 0.5 | 1.3 × 10^{−3} | 540 |

Healthy tissue | 1045 | 3760 | 0.5 | 1.3 × 10^{−3} | 540 |

Blood | 1060 | 3770 | – | – | – |

Mesh Number | Number of Cells | Temperature Location 2 mm above Tumor Center (°C) |
---|---|---|

1 | 9500 | 41.581 |

2 | 15,740 | 41.852 |

3 | 32,781 | 41.911 |

4 | 57,468 | 41.915 |

Parameter | Value |
---|---|

M_{d} (kA∙m^{−1}) | 446 |

$\mathcal{K}$ (kJ∙m^{−3}) | 41 |

ρ_{nano} (kg∙m^{−3}) | 5180 |

R (nm) | 9.5 |

η (Pa∙s) | 6.53 × 10^{−4} |

φ | 4.8 × 10^{−4} |

f (kHz) | 220 |

H_{0} (A∙m^{−1}) | 6800 |

**Table 5.**Comparison of the CEM43 values of our model, for Case A, with the model and the experimental measurements by Pearce et al. [92].

Probe Location | CEM43 by Present Model | CEM43 by Pearce et al. [92] | |
---|---|---|---|

Model | As Determined from Measured Temperatures | ||

Center | 199,017 | 255,712 | 227,311 |

Tip | 11,814 | 79,966 | 20,494 |

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Polychronopoulos, N.D.; Gkountas, A.A.; Sarris, I.E.; Spyrou, L.A.
A Computational Study on Magnetic Nanoparticles Hyperthermia of Ellipsoidal Tumors. *Appl. Sci.* **2021**, *11*, 9526.
https://doi.org/10.3390/app11209526

**AMA Style**

Polychronopoulos ND, Gkountas AA, Sarris IE, Spyrou LA.
A Computational Study on Magnetic Nanoparticles Hyperthermia of Ellipsoidal Tumors. *Applied Sciences*. 2021; 11(20):9526.
https://doi.org/10.3390/app11209526

**Chicago/Turabian Style**

Polychronopoulos, Nickolas D., Apostolos A. Gkountas, Ioannis E. Sarris, and Leonidas A. Spyrou.
2021. "A Computational Study on Magnetic Nanoparticles Hyperthermia of Ellipsoidal Tumors" *Applied Sciences* 11, no. 20: 9526.
https://doi.org/10.3390/app11209526