# Prediction of Gold Nanoparticle and Microwave-Induced Hyperthermia Effects on Tumor Control via a Simulation Approach

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

^{+}CD8

^{+}T lymphocytes by DC-mediated antigen presentation. In addition, ICD is associated with increased infiltration of lymph nodes by B cells, which produce tumor-specific antibodies [19]. Hyperthermia also inhibits DNA repair, including proper processing and amendment of double-strand breaks (DSBs), making it a potent radio- and chemosensitizer, for various types of cancer, including tumors of head and neck, bladder, breast, and cervix [18].

_{2}outer layer.

## 2. Materials and Methods

#### 2.1. Effect of the Size, Shape, and Structure of the Nanoparticles on Their Absorption Efficiency

_{2}layer), as shown in Figure 4.

#### 2.2. Simulations for Nanoparticle and Microwave-Induced Hyperthermia and Hyperthermic Cell Death

#### 2.3. Nanoparticle- and Microwave-Induced Hyperthermia

^{2}. The beam is considered as a continuous wave (CW), and the incoming wave is plane. Water was set as the surrounding environment for both types of nanoparticles. The optical properties of gold were retrieved from Rakic [32], and the ones of silica from the built-in library of Comsol.

_{0}is the free space wavenumber. Finally, in Equation (12), n is the vector perpendicular to the surface of the nanoparticle, and ${\mathit{E}}_{b}=\mathit{E}{e}^{-i{k}_{0}x}$ is the background electric field.

_{i}is the variable of concentration of the nanoparticles, D

_{i}is the diffusion coefficient of each material, and R

_{i}is a generation term that was set at 30 nanoparticles/m

^{3}. The concentration of NPs injected into the tumor is 40 μg/mL [33]. The solution is injected into the center of the tumor. The last equations, Equations (14) and (15), represent the diffusion of the heat produced by the nanoparticles, inside the tumor and the surrounding region, where $\mathit{q}=-k\nabla T$ (k = the thermal conductivity), u is the normal vector, ρ and ρ

_{b}are the densities of each tissue and of the blood, respectively, C

_{p}, and C

_{p,b}are the specific heat capacities of tissues and blood, T and T

_{b}are the temperatures of tissue and arterial blood, ω

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

_{met}is the metabolic heat source. The values of the various parameters are presented in Table 1.

#### 2.4. Estimation of Hyperthermic Cell Death

_{s}, and D

_{τ}is a threshold value of maximum cell death for cultures that have suffered minimal thermal damage. The parameters $\overline{{k}_{f}}$, ${k}_{b}$, ${T}_{k}$, ${k}_{s}$, and ${D}_{\tau}$ were estimated using the function “fmincon,” which finds the minimum of constrained nonlinear multivariable functions, provided by MATLAB; experimental data for melanoma were obtained from Blanco-Andujar et al. [39] and Feng et al. [40], and for prostate cancer were derived from Huang et al. [41]. The optimized parameters are presented in Table 4. In slow cell death, the cells can be considered either as “dead” or “not dead.” As a result, in slow cell death, there is not a vulnerable phase, and ${k}_{f}={k}_{b}=0$.

#### 2.5. Tumor Growth Model

## 3. Results

^{3}. In the diffusion study, the NPs diffuse radially, following a Gaussian form, outwards both to the tumorous (r ≤ 3 mm) and the surrounding healthy tissue (r > 3 mm), forming a concentration gradient (Figure 12b). The distribution of the NPs in the cancerous region has taken place in such a way that the induced thermal effect would not damage the healthy tissue. Concluding the first part of our study, we have produced thermal profiles of the tumorous and the surrounding healthy tissue at the 10th minute of the heating procedure (Figure 12c). The intratumoral temperature surpasses the threshold of cell damage, while the temperature of the surrounding healthy tissue, between 3 and 6 mm, is raised to a degree that can cause damage.

^{3}. The temperature for the hyperthermic cell death of the prostate tumor is set at 50 °C and 48 °C for the melanoma. The use of the three-state model helps us observe the fast cell death during the first 30 min of heating for the melanoma and 15 min (Figure 14a) prostate cancer (Figure 14c). The reduced viability due to the slow apoptotic cell death for the next 48 h after treatment is shown in Figure 14b–d. From the cell death and the tumor growth model, we obtained the evolution of tumor size during time (Figure 15). The results depicted in Figure 15 indicate the reduction of tumor size when the patient receives therapy every 4 days for seven sessions in the Melanoma case, and on the 0th, 2nd, and 6th days in the Prostate case. In the first two days of each 4-day interval, the cells were damaged and underwent apoptotic cell death. In the next two days, the cancer cells recovered and started to proliferate again. This resulted in a slight increase in the tumor volume after the hyperthermic treatment. Therefore, tumor cells appear to develop resistance to thermotherapy, as in the case of chemoradiotherapy [50,51], leading to tumor regrowth. In a clinical setting, this can contribute to cancer’s recurrence and relapse [50].

## 4. Discussion

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Absorption spectra of gold nanoparticles of different diameters (10–1000 nm). The cross section increases, but the peak lies in the region of 500 nm for all curves.

**Figure 2.**The absorption cross section of gold nanoparticles as a function of their diameter. The red line is a fitting curve based on the function y = αx

^{p}, where α = 2.9 × 10

^{−4}and p = 1.46. The incident wavelength is assumed to be 532 nm, which is a common laser wavelength corresponding to the second harmonic of Nd:YAG lasers and is also in the region of maximum absorption of gold nanoparticles.

**Figure 3.**Absorption efficiency spectrum of a gold nanoshell as a function of the thickness of the gold layer. The spectrum is red-shifted as the nanoshell thickness decreases. The particle diameter is assumed to be 30 nm, as in the case of the hyperthermia simulations.

**Figure 4.**Absorption efficiency spectrum of a gold nanoparticle covered with a TiO

_{2}layer as a function of the thickness of the titanium layer. The spectrum is red-shifted as the thickness of the dielectric layer increases. The particle diameter is assumed to be 30 nm.

**Figure 5.**Absorption cross section of a gold nanoparticle in the form of a prolonged ellipsoid, for different values of the ratio of the principal semi axes $\left(1\le r\le 5\right)$. The cross section is increased and red-shifted as the ratio of the principal axes increases.

**Figure 6.**Workflow for the estimation of tumor shrinkage using (

**a**) gold nanoparticle-induced hyperthermia and (

**b**) using microwave-induced hyperthermia.

**Figure 7.**Corresponding geometries and meshes for the simulation of plasmon resonance. (

**a**,

**c**), AuNP with a 40 nm diameter surrounded by 225 nm of water; (

**b**,

**d**) AuSiO

_{2}NP with a 30 nm diameter (20 nm SiO

_{2}, 5 nm Au layer) surrounded by 225 nm of water.

**Figure 10.**Simulation of microwave-induced temperature effects on prostate tissue. (

**a**) Temperature distribution at the 9th min of treatment where the antenna is tuned at 2.4 GHz with power of 10 W. (

**b**) Temperature distribution at the 30th min of heating where the antenna was tuned at 433 MHz with power of 30 W. The unit of the contours is in Celsius degrees, and the unit of x- and y-axes in meters.

**Figure 12.**Resonant wavelength of two different size nanoparticles. (

**a**) The blue line corresponds to the 40 nm AuNP and the orange line to the 30 nm Au-SiNP. (

**b**) Distribution of NPs inside the tumorous tissue 1 h after the injection in the center of the tumor. (

**c**) Temperature rise during the first 10 min of NP laser-induced heating.

**Figure 13.**Simulation of microwave-induced temperature effects for melanoma tissue. (

**a**) Temperature distribution at the 9th minute of treatment where the antenna is tuned at 2.4 GHz with power of 10 W. (

**b**) Temperature distribution at the 30th minute of heating and the antenna is tuned at 433 MHz with 100 W power. The unit of the contours in Celsius degrees, and the unit of x- and y-axes in meters.

**Figure 14.**Simulation of tumor response to hyperthermia treatment. (

**a**,

**c**) Depiction of the fast cell death that occurs during the treatment for melanoma and the prostate cancer cells. (

**b**,

**d**) Depiction of post-treatment slow cell death for melanoma and the prostate cancer cell, respectively. The experimental data have been taken from Blanco-Andujar et al. [39] and Feng et al. [40] for melanoma, and Huang et al. [41] for prostate cancer.

**Figure 15.**Estimated results for tumor shrinkage upon hyperthermia complete treatment (3–7 sessions) for two cancer types. The therapeutic sessions for Melanoma were repeated every 4 days. In the Prostate tumor, the time interval between the first two sessions was 2 days, while the interval between the 2nd and 3rd session was 4 days. For Melanoma, the expected values considering HSP90 inhibition were included.

**Table 1.**Selected parameters for blood, dermis, epidermis, fat, tumor, and muscles tissues, as used in Equations (2)–(4).

Specific Heat Capacity Cp (J/kg/°C) | Density ρ (kg/m³) | Thermal Conductivity k (W/m∙°C) | Blood Perfusion Rate w (m³/m³∙s) | Metabolic Heat Source qm (W/kg) | Diffusivity m ^{2}/s | |
---|---|---|---|---|---|---|

Blood | 3617 [34] | 1050 [34] | 0.52 [34] | - | 1090 [35] | - |

Dermis | 3300 [36] | 1200 [36] | 0.45 [36] | 1.25 × 10^{−3} [36] | 1200 [37] | - |

Epidermis | 3590 [36] | 1200 [36] | 0.23 [36] | 0 [36] | 1200 [37] | 6.2 × 10^{−11} [38] |

Fat | 2348 [34] | 911 [34] | 0.21 [34] | 1.25 × 10^{−3} [36] | 464 [37] | - |

Tumor/Muscle | 3421 [34] | 1090 [34] | 0.49 [34] | 1.65 × 10^{−3} [34] | 991 [34] |

AuNP 20 nm (Surrounded by 225 nm of Water) (Figure 7a,c) | AuSiO_{2}NP 30 nm(Surrounded by 225 nm of Water) (Figure 7b,d) | |
---|---|---|

Max element size | 39.2 nm | 38.4 nm |

Min element size | 4.9 nm | 4.8 nm |

Max element growth rate | 1.45 | 1.45 |

Curvature factor | 0.5 | 0.5 |

Resolution of narrow regions | 0.6 | 0.6 |

Melanoma (Figure 9) | Prostate (Figure 10) | |
---|---|---|

Max element size | 5.36 mm | 5.36 mm |

Min element size | 24 μm | 24.5 μm |

Max element growth rate | 1.3 | 1.3 |

Curvature factor | 0.3 | 0.3 |

Resolution of narrow regions | 1 | 1 |

Melanoma | Prostate Cancer | |
---|---|---|

Temperature (degrees Celsius) | 48 | 50 |

$\overline{{k}_{f}}$ (min^{−1}) | 0.25481 | 0.18946 |

${k}_{b}$ (min^{−1}) | 0.66477 | 0.23063 |

${T}_{k}$ (degrees Celsius) | 40.1513 | 39.678 |

$\overline{{k}_{s}}$ (hours^{−1}) | 0.59547 | $0.316\times {10}^{-3}$ (no data) |

${D}_{\tau}$ | $0.45003\times {10}^{-3}$ | 0.208 (no data) |

Melanoma | Melanoma (HSP90 Inhibited) | Prostate Cancer | |
---|---|---|---|

α_{0} (s^{−1}) | 0.328 ± 0.003 | 0.237 ± 0.005 | 0.243 ± 0.016 |

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**MDPI and ACS Style**

Dimitriou, N.M.; Pavlopoulou, A.; Tremi, I.; Kouloulias, V.; Tsigaridas, G.; Georgakilas, A.G.
Prediction of Gold Nanoparticle and Microwave-Induced Hyperthermia Effects on Tumor Control via a Simulation Approach. *Nanomaterials* **2019**, *9*, 167.
https://doi.org/10.3390/nano9020167

**AMA Style**

Dimitriou NM, Pavlopoulou A, Tremi I, Kouloulias V, Tsigaridas G, Georgakilas AG.
Prediction of Gold Nanoparticle and Microwave-Induced Hyperthermia Effects on Tumor Control via a Simulation Approach. *Nanomaterials*. 2019; 9(2):167.
https://doi.org/10.3390/nano9020167

**Chicago/Turabian Style**

Dimitriou, Nikolaos M., Athanasia Pavlopoulou, Ioanna Tremi, Vassilis Kouloulias, Georgios Tsigaridas, and Alexandros G. Georgakilas.
2019. "Prediction of Gold Nanoparticle and Microwave-Induced Hyperthermia Effects on Tumor Control via a Simulation Approach" *Nanomaterials* 9, no. 2: 167.
https://doi.org/10.3390/nano9020167