# Numerical Simulation of Enhancement of Superficial Tumor Laser Hyperthermia with Silicon Nanoparticles

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

^{2}at a wavelength of 808 nm was found to result in temperature rise up to 20 °С and death of the living cells [43].

^{2}[50]), especially for healthy tissues surrounding the tumor. Employing pulsed irradiation has no advantages due to the decrease of safety energy density limits with shortening pulse duration [50,51]. Although silicon possesses significant cubic nonlinear susceptibility responsible for the light self-action [52], simple estimations demonstrate that for a laser pulse energy of the order of safety threshold [51], the relative variation of absorption for bulk silicon is in the order of 10

^{−4}.

## 2. Materials and Methods

#### 2.1. Geometry and Optical Parameters of the Model

#### 2.2. Optical Characteristics of SiNPs Employed in the Simulations

_{m}(in mg/mL), given by the relation:

_{i}of particles with diameter d

_{i}can be expressed via the known size distribution function f

_{i}and mass concentration C

_{m}as follows:

_{s,i}, absorption cross-section σ

_{a,i}(Figure 3c), and anisotropy factor g

_{i}for i-th size fraction of silicon nanocrystals.

**Figure 3.**(

**a**,

**b**) Size distributions f

_{i}for SiNPs obtained by ablation of SiNW arrays in water (

**a**) and in ethanol (

**b**) [44]; (

**c**) dependencies of scattering and absorption cross-section of a spherical SiNP at wavelength of 633 nm on its diameter calculated by Mie theory.

^{NP}= 3.8823 and extinction coefficient k

^{NP}= 0.0196 for crystalline silicon [66]. Relative variations of these values with temperature increase up to 10 °С for bulk silicon are 10

^{−}

^{3}and 10

^{−5}, respectively [67,68]. Hence, the thermooptical effect on the SiNPs embedded into the tumor is expected to be negligible. Thermooptical effects in the biotissue are also extremely small. According to Ref. [69], the value of dμ

_{a}/dT for biotissues does not exceed 0.02 cm

^{−1}K

^{−1}. Considering the typical temperature increase in the treated biotissues and their absorption coefficients (see Table 1 and Table 2), the possible relative variation in absorption is estimated to not exceed 7% for tumor with embedded 1 mg/mL of eSiNPs and an order of magnitude smaller for tumor with 7 mg/mL wSiNPs. The refractive index for tumor tissue at the wavelength of 633 nm is assumed to be equal to n

^{tum}= 1.4 [57,70,71]. Optical properties of skin layers and tumor tissue are presented in Table 1. It is worth nothing that all optical parameters values were measured for a non-polarized light source [57].

_{а}, μ

_{s}, and g of the tumor with SiNPs originate from the error of size distribution measurement that were estimated as 6.0% and 10.4% for wSiNPs and eSiNPs, respectively. This yields an estimated error of 2% in scattering and absorption characteristics evaluated by Mie formulas, which caused no relevant change in the obtained results.

#### 2.3. Simulations of Light Absorption Distribution

^{7}photons uniformly distributed within the circle of radius r at the biotissue surface. All photons enter the tissue normal to its surface. The computational grid for the absorption map consisted of 200 × 200 × 500 voxels (the voxel size was $\Delta x\times \Delta y\times \Delta z$ = 0.1 mm × 0.1 mm × 0.02 mm).

_{ext}(x,y,z) within the medium, the normalized volumetric density of absorbed energy should be multiplied by the beam intensity:

#### 2.4. Calculation of Volumetric Temperature Distribution

_{p}is the biotissue heat capacity at a constant pressure; k is the thermal conductivity, Q

_{met}is the speed of metabolic heat generation the per unit volume, Q

_{ext}is the external source power density, which in our case is a result of the laser radiation absorption, and Q

_{perf}is power density of the heat loss caused by perfusion (blood transfer through capillaries and extracellular spaces in tissue [82]). Despite great progress achieved during more than 70 years since its first proposal and various improvements and refinements of the equation (see, e.g., [83,84,85]), the Pennes equation is, nevertheless, applicable for the stationary case of continuous-wave irradiation of the biotissue [83]:

_{met}= 420 W/cm

^{3}is the volumetric rate of metabolic heat generation [86], and the last term of Equation (11) describes the perfusion heat loss power density, with ρ

_{bl}= 1060 kg/m

^{3}, C

_{bl}= 3770 J/(kg∙К) [87,88], T

_{bl}= 37.2 °С [89,90], ω

_{bl}(Table 3) being the blood density, heat capacity, temperature, and perfusion coefficient (transfer of blood through capillaries and extracellular spaces [82]), respectively. The values of thermal conductivity coefficient k for different biotissues are presented in Table 3. Generally, the thermal parameters of nanoparticles can differ from those for bulk material due to the variation of the phonon spectrum and well-developed surface [91]. However, previously, we have shown that Raman spectra of the employed SiNPs coincide with the spectrum for crystalline silicon [34], while in the current study the volume fraction of the SiNPs does not exceed 0.003 and SiNP heating is expected to occur at several degrees. Based on this, we consider no variations of thermal parameters both for the SiNPs and for the tumor with embedded SiNPs.

^{®}software package using the finite element method [93,94,95]. The computational grid was tetrahedral with the minimum and maximum element sizes of 0.15 mm and 1 mm, respectively. Tetrahedral mesh allows accounting for the complicated shape of the ellipsoidal tumor boundaries when solving the equation. It also significantly reduces the calculation time by varying the grid voxel size depending on the typical scale of the simulated structure (larger voxels for subcutaneous fat and smaller voxels for skin and tumor) compared to the grid with rectangular voxels.

^{®}environment, we used linear interpolation of the discrete function Q

_{ext}(x,y,z) on the tetrahedral grid nodes. Constant temperature of 37.0 °С was set at the rear boundary of the medium located within the biological tissue (z = 10 mm), while at the skin–air interface convection was taken into account [96]:

^{2}∙K) and air temperature T

_{air}= 25.0 °С. Temperature distribution at the lateral boundaries T

_{sides}(z) was evaluated from the solution of the 1D stationary bioheat transfer equation in the infinitely wide medium containing no tumor and in the absence of heat sources in biotissue:

## 3. Results and Discussion

#### 3.1. The Effect of eSiNPs and wSiNPs on Radiation Absorption Efficiency

_{eff}(see Table 2). Moreover, the dependencies q(x = 0, y = 0, z) for eSiNPs with the mass concentration C

_{m}= 5 mg/mL and for wSiNPs with the mass concentration C

_{m}= 1 mg/mL are characterized by quite similar behavior, while the values of µ

_{eff}are very close for these two cases.

_{fit}(z) ∝ exp(−µ

_{eff}z), which describes attenuation of light in diffusive regime [75], provides a good fit to the simulated dependencies q(x = 0, y = 0, z) within the tumor region (Figure 4b). The exponential fit is shown for one curve in Figure 4b using the corresponding value of µ

_{eff}taken from Table 2. Note that the slope of the function q(x = 0, y = 0, z) increases with the concentration of SiNPs as well as its amplitude defined by the absorption coefficient µ

_{a}. Therefore, the effect of the concentration increase on the absorption efficiency is not obvious at first sight: the higher the concentration, the more pronounced are both the absorption and the signal drop. Assuming both the amplitude and the longitudinal scale of the function q(x = 0, y = 0, z), we propose the parameter A for the estimation of the light absorption efficiency within the whole tumor node with the embedded SiNPs that is defined as the product of the absorption coefficient in the tumor over the thickness of the “skin layer” of radiation penetration into the tumor:

_{m}, the absorption efficiency almost does not depend on C

_{m}for wSiNPs, while for eSiNPs the increase in concentration provides a noticeable rise of the parameter A. However, at very large eSiNPs concentrations (C

_{m}> 10 mg/mL) the growth of A(C

_{m}) slows down and obviously tends to saturate. In general, the increase in absorption efficiency obtained by embedding wSiNPs does not exceed 25% at their highest concentration compared to that for the tumor solely, and it is about 40% and 60% for eSiNPs at concentrations of 3 mg/mL and 7 mg/mL, respectively.

#### 3.2. The Effect of eSiNPs and wSiNPs on Tumor Heating

_{ext}, the volumetric heating rate from inner heat sources, in the bioheat transfer equation. For therapeutic applications, it is important to initiate tissue thermal destruction as a result of irradiation only in the tumor, not in surrounding normal tissues. In this context, the laser power P was varied to obtain temperatures above the threshold of T

_{42}= 42 °C in the entire tumor volume while trying to minimize the temperature increase in surrounding healthy tissue above T

_{42}. Such a situation is referred to as optimal heating. The analysis of the choice of P ensuring tumor hyperthermia with minimal overheated normal tissue volume was performed for two beam sizes: one corresponding to the transversal tumor size and one exceeding it for two times aiming to ensure sufficient heating of the tumor boundary region.

#### 3.2.1. Beam Size Equals the Tumor Transversal Size

^{2}[50]) in biotissue with and without SiNPs. This combination of r and I parameters ensures heating of the entire tumor volume to temperatures above T

_{42}in the presence of eSiNPs at concentrations higher than 5 mg/mL, although it is worth mentioning that, besides quite a high value of intensity itself, it requires longer than 400 s to reach the stationary heating mode considered in simulations [5,39].

_{42}at the beam axis through the whole tumor length (Figure 6a). However, the side regions of the tumor are left underheated, which can be seen from the transversal temperature profiles at depth z = 1 mm (Figure 6b, black solid line). The profiles have the bell shape owing to high tumor thermal conductivity, which smooths the temperature distribution despite the size of the beam exactly fitting the tumor width, and the intensity distribution is uniform.

_{42}, thus providing sufficient heating of the entire tumor for all considered concentrations (Figure 6b). The application of wSiNPs results in less efficient additional heating of the tumor: though the whole volume is heated over T

_{42}, the maximal temperature is reached only within a few hundred microns at the tumor top, and the back half of the tumor is sufficiently less heated compared to the front part. Note that the maximal temperature value obtained with the largest concentration of wSiNPs (7 mg/mL) is almost the same as for the eSiNPs at a concentration of 5 mg/mL. These results can be explained by the increase of absorption efficiency of eSiNPs with their concentration while the increase of C

_{m}has almost no effect on the absorption efficiency in the presence of wSiNPs (Figure 4c).

#### 3.2.2. Beam Size Exceeds the Tumor Transversal Size

^{2}seems to be extremely high from the point of view of the safety requirements which demands intensities of about 200 mW/cm

^{2}or less [50]. Obviously, the increase in beam radius allows to deliver the same energy to the tissue with lower intensity. Figure 7 depicts the temperature profiles for beam with radius of 5 mm and laser power of 165 mW. The corresponding intensity is 210 mW/cm

^{2}, which is significantly lower than the maximum permissible radiation intensity in the order of 500 mW/cm

^{2}for only heating effects to occur.

_{42}in the entire tumor volume given that without SiNPs this threshold temperature is not reached. However, it can be expected that due to the complex structure of biotissue the temperatures in real tissue may vary, and the obtained increase is not so significant as compared to the value of 4.5 °C obtained in the case of a narrow beam. A comparison with Figure 6 shows that employing the beam with the diameter exceeding the tumor width allows to arrange more uniform transversal temperature distribution in the tumor region.

^{2}) [50] can be violated in the tumor region provided that the surrounding healthy tissues are not over-exposed.

#### 3.2.3. Comparison of Flat Beam and Gaussian Beam Irradiation

_{0}= 110 mW and r = 2.5 mm (precisely of the tumor diameter) (cf. Figure 6 and Figure 10a,b) and precisely up to 42 °C at P

_{0}= 165 mW and r = 4 mm (cf. Figure 7 and Figure 10c,d, note that the beam radius for uniform circular beam is larger than that for the Gaussian beam, which indicates even less tumor heating efficiency in the latter case). The reasons for this effect are obvious: (i) more than 37% of the laser radiation is absorbed outside the tumor if the beam radius exceeds the tumor radius, and (ii) due to a strongly non-uniform heat rate the heat flux is more prominent than in the case of a uniform intensity distribution. Thus, the simulation of the hyperthermia by Gaussian beam evidences that the circular flat laser beam has certain advantages over the Gaussian one.

_{2}nanosheets with a mean hydrodynamic diameter of 200 nm were experimentally shown to be able to heat water to 30 °C within 600 s. In [103], palladium hydride nanoparticles suspended in water gained additional heating at 10 °C during 180 s at NIR laser power of 154 mW. However, they require a sophisticated fabrication technique, while the pulsed laser ablation of silicon nanostructures in liquids seems to be a promising, scalable technology. As for inert material studies, gold nanoparticles irradiated with NIR light permit selective tumor hyperthermia at 42−43 °C without side effects to surrounding healthy tissues due to the spatial and temporal control of heat generation (selective accumulation and irradiation of heat nanosources) [104]. Nevertheless, silicon demonstrates more prominent biodegradation compared to gold, since silicon presents in living organisms as a natural trace compound.

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**Schematic of multilayer human skin model with an embedded tumor node employed in Monte Carlo simulations.

**Figure 4.**(

**a**) Distributions of normalized volumetric density of absorbed energy q(x = 0, y = 0, z) along the laser beam axis z for different types of SiNPs and their different concentrations for the beam of radius r = 5 mm. Dashed vertical lines depict the tumor edges. (

**b**) Distribution of q(x = 0, y = 0, z) in the tumor with the exponential fit to the curve for wSiNP at mass concentration of 3 mg/mL. The fitting is shown with a dashed black line. Plots designations are the same for (

**a**) and (

**b**) fragments. (

**c**) Dependence of the absorption efficiency parameter A on the concentration for wSiNPs and eSiNPs in the tumor.

**Figure 5.**Cross-section x, y = 0 of normalized volumetric density of absorbed energy q(x,y,z) for beam radius of 5 mm: (

**a**) in absence of SiNPs in nBCC; (

**b**,

**c**) BCC with eSiNPs (

**b**) and wSiNPs (

**c**) at concentration of 3 mg/mL; (

**d**,

**e**) BCC with eSiNPs (

**d**) and wSiNPs (

**e**) at concentration of 7 mg/mL.

**Figure 6.**Temperature distribution in biotissue with SiNPs embedded in the tumor at the indicated mass concentrations upon irradiation by laser beam with radius of 2.5 mm and power of 110 mW: (

**a**) in-depth temperature dependence at the beam axis T(x = 0, y = 0, z); (

**b**) transversal temperature dependencies at the depth z = 1 mm corresponding to the tumor center. Red horizontal line indicates the hyperthermia threshold temperature; grey vertical lines show the tumor edges.

**Figure 7.**Temperature distribution in biotissue with SiNPs embedded in the tumor at the indicated mass concentrations upon irradiation by laser beam with radius of 5 mm and power of 165 mW: (

**a**) in-depth temperature dependence at the beam axis T(x = 0, y = 0, z); (

**b**) transversal temperature dependencies at the depth z = 1 mm corresponding to the tumor center. Red horizontal line indicates the hyperthermia threshold temperature; grey vertical lines show the tumor edges.

**Figure 8.**The dependencies of temperature at (x = 0, y = 0, z = 1 mm), at the center of tumor (filled markers) and at (x = 0, y = 3 mm, z = 1 mm), in healthy skin neighboring the tumor at the same depth (hollow markers) on the laser power: (

**a**) beam radius 2.5 mm; (

**b**) beam radius 5 mm. Red horizontal line depicts the hyperthermia threshold temperature.

**Figure 9.**Cross-section x,y = 0 of normalized volumetric density of absorbed energy q(x,y,z) for Gaussian beam radius of 4 mm: in absence of SiNPs in nBCC (

**a**); BCC with eSiNPs (

**b**) and wSiNPs (

**c**) at SINP mass concentration of 5 mg/mL.

**Figure 10.**Temperature distribution in biotissue with SiNPs embedded in the tumor at the mass concentrations of 5 mg/mL upon irradiation by Gaussian laser beam with radius of 2.5 mm and power of 110 mW (

**a**,

**b**), radius of 4 mm and power of 165 mW: (

**a**,

**c**) axial temperature profile at the beam axis T(x = 0, y = 0, z); (

**b**,

**d**) transversal temperature profile at the depth z = 1 mm corresponding to the tumor center. Red horizontal line indicates the hyperthermia threshold temperature of 42 °C; dashed vertical lines show the tumor edges.

Biotissue | Thickness, mm | Absorption Coefficient μ_{а}, cm^{−1} | Scattering Coefficient μ_{s}, cm^{−1} | Scattering Anisotropy Factor g |
---|---|---|---|---|

nBCC | 2 | 1 | 100 | 0.8 |

Epidermis | 0.05 | 2.5 | 240 | 0.8 |

Dermis | 1.20 | 1.5 | 150 | 0.8 |

Subcutaneous fat | 8.75 | 1.5 | 130 | 0.8 |

**Table 2.**Optical properties of nBCC in the presence of SiNPs with mass concentrations С

_{m}= 1–7 mg/mL at wavelength of 633 nm.

Biotissues | SiNP Mass Concentration C_{m}, mg/mL | Absorption Coefficient μ_{а}, cm^{−1} | Scattering Coefficient μ_{s}, cm^{−1} | Scattering Anisotropy Factor g | Effective Attenuation Coefficient μ_{eff}, cm^{−1} |
---|---|---|---|---|---|

nBCC with wSiNPs | 1 | 3.02 | 140.74 | 0.68 | 20.87 |

3 | 7.06 | 222.21 | 0.58 | 46.11 | |

5 | 11.10 | 303.69 | 0.53 | 71.57 | |

7 | 15.15 | 385.13 | 0.51 | 96.26 | |

nBCC with eSiNPs | 1 | 1.64 | 102.92 | 0.78 | 10.93 |

3 | 2.91 | 108.75 | 0.74 | 16.50 | |

5 | 4.18 | 114.58 | 0.70 | 21.99 | |

7 | 5.46 | 120.41 | 0.68 | 26.84 |

Epidermis | Dermis | Subcutaneous Fat | nBCC | |
---|---|---|---|---|

Thermal conductivity, k, W/(m∙K) | 0.24 | 0.45 | 0.185 | 0.561 |

Blood perfusion ω_{bl}, 1/s | 0 | 0.006 | 0.0008 | 0.012 |

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Sokolovskaya, O.I.; Sergeeva, E.A.; Golovan, L.A.; Kashkarov, P.K.; Khilov, A.V.; Kurakina, D.A.; Orlinskaya, N.Y.; Zabotnov, S.V.; Kirillin, M.Y.
Numerical Simulation of Enhancement of Superficial Tumor Laser Hyperthermia with Silicon Nanoparticles. *Photonics* **2021**, *8*, 580.
https://doi.org/10.3390/photonics8120580

**AMA Style**

Sokolovskaya OI, Sergeeva EA, Golovan LA, Kashkarov PK, Khilov AV, Kurakina DA, Orlinskaya NY, Zabotnov SV, Kirillin MY.
Numerical Simulation of Enhancement of Superficial Tumor Laser Hyperthermia with Silicon Nanoparticles. *Photonics*. 2021; 8(12):580.
https://doi.org/10.3390/photonics8120580

**Chicago/Turabian Style**

Sokolovskaya, Olga I., Ekaterina A. Sergeeva, Leonid A. Golovan, Pavel K. Kashkarov, Aleksandr V. Khilov, Daria A. Kurakina, Natalia Y. Orlinskaya, Stanislav V. Zabotnov, and Mikhail Y. Kirillin.
2021. "Numerical Simulation of Enhancement of Superficial Tumor Laser Hyperthermia with Silicon Nanoparticles" *Photonics* 8, no. 12: 580.
https://doi.org/10.3390/photonics8120580