# Gold Nanoparticles as a Photothermal Agent in Cancer Therapy: The Thermal Ablation Characteristic Length

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## Abstract

**:**

## 1. Introduction

## 2. Photothermal Model

## 3. Numerical Results and Discussion

#### 3.1. Influence of the Laser Power and of the Volume of Nanoparticles

#### 3.2. The Thermal Ablation Characteristic Length

#### 3.3. Sensitivity Analysis

#### 3.4. Aggregates of Nanoparticles

- $d\le {d}_{elec}$: electromagnetic and thermal coupling,
- ${d}_{elec}\le d\le {d}_{therm}$: thermal coupling,
- $d\ge {d}_{therm}$: no coupling.

#### 3.5. Discussion of Material Parameters and Configuration

## 4. Conclusions

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Slice of the geometry (

**a**) and of the associated mesh (

**b**) of a spherical nanoparticle embedded in a spherical cell.

**Figure 2.**Temperature in the $z=0$ plane, in the whole computational domain (

**a**) and zoom around the gold nanoparticle (

**b**). The volumes of the nanoparticle and the cell are: $0.52\times {10}^{-3}\phantom{\rule{3.33333pt}{0ex}}\mathsf{\mu}$m${}^{3}$ and $4.2\times {10}^{3}\phantom{\rule{3.33333pt}{0ex}}\mathsf{\mu}$m${}^{3}$, respectively. The cell’s membrane has spatial extension from −10 $\mathsf{\mu}$m to 10 $\mathsf{\mu}$m, and the diameter of the nanoparticle is 100 nm.

**Figure 3.**Map of the temperature (in ${}^{\circ}$C) as a function of both the distance to the center of nanoparticle in the cell (i.e., distance $\le 10\phantom{\rule{3.33333pt}{0ex}}\mathsf{\mu}$m) and the incident laser power (${P}_{w}\in [0.5;2.0]$ W). The diameter of the gold nanoparticle is ${D}_{np}=100$ nm (${V}_{np}=0.52\times {10}^{-3}\phantom{\rule{3.33333pt}{0ex}}\mathsf{\mu}$m${}^{3}$).

**Figure 4.**Temperature (in ${}^{\circ}$C) as a function of the distance from the center of the nanoparticle in the cell (i.e., distance $\le 10\phantom{\rule{3.33333pt}{0ex}}\mathsf{\mu}$m) and as a function of the volume of nanoparticle (${V}_{np}\in [0.1;1.5]\times {10}^{-3}\phantom{\rule{3.33333pt}{0ex}}\mathsf{\mu}$m${}^{3}$). The diameter of the nanoparticle varies from 50 nm–150 nm, and the laser power is ${P}_{w}=1.0$ W.

**Figure 5.**Evolution of the temperature in the cell for two volumes of nanoparticles: ${V}_{p1}=0.91\times {10}^{-3}\phantom{\rule{3.33333pt}{0ex}}\mathsf{\mu}$m${}^{3}$, ${V}_{p2}=0.52\times {10}^{-3}\phantom{\rule{3.33333pt}{0ex}}\mathsf{\mu}$m${}^{3}$; and four laser powers: ${P}_{w11}=0.9$ W, ${P}_{w12}=0.6$ W, ${P}_{w21}=1.9$ W, ${P}_{w22}=1.3$ W. The ablation threshold is shown.

**Figure 6.**Maps of the temperature in the cell as a function of the nanoparticle volume (

**a**) and of the laser power (

**b**). The laser power and the volume of the nanoparticle are adapted respectively in order to produce a selected maximum temperature in the particle ${T}_{np,max}=47{\phantom{\rule{3.33333pt}{0ex}}}^{\circ}\mathrm{C}$.

**Figure 7.**Map of the temperature in the cell for a system of three embedded nanoparticles of diameter ${D}_{np,i}=100$ nm. (

**a**) ${d}_{1,2}={d}_{1,3}=10$ nm: strong coupling. (

**b**) ${d}_{1,2}={d}_{1,3}=400$ nm: weak coupling.

**Figure 8.**Temperature as a function of the distance to the center of the nanoparticle of diameter ${D}_{np,1}=100$ nm, for $N=1$ and a combination of $N=3$ identical nanoparticles. The inter-distances are respectively ${d}_{1,2}={d}_{1,3}=10$ nm, ${d}_{1,2}={d}_{1,3}=400$ nm, ${d}_{1,2}={d}_{1,3}=700$ nm. The temperature is plotted along the x-axis (

**a**) and along the y-axis (

**b**).

**Table 1.**Values of the parameters for the photothermal model for the two media (cell, gold nanoparticle) with ${j}^{2}=-1$.

$\mathit{\rho}$ (kg·m${}^{-3}$) | ${\mathit{C}}_{\mathit{p}}$ (m${}^{2}\xb7$s${}^{-2}\xb7$K${}^{-1}$) | $\mathit{\kappa}$ (kg·m·s${}^{-3}\xb7$K${}^{-1}$) | ${\mathit{\u03f5}}_{\mathit{r}}$ (at 830 nm) | |
---|---|---|---|---|

cell | 1090 | 2185 | 1.20 | 2.04 |

Au | 19,300 | 129 | 310 | −26.61 + j1.67 |

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**MDPI and ACS Style**

Grosges, T.; Barchiesi, D. Gold Nanoparticles as a Photothermal Agent in Cancer Therapy: The Thermal Ablation Characteristic Length. *Molecules* **2018**, *23*, 1316.
https://doi.org/10.3390/molecules23061316

**AMA Style**

Grosges T, Barchiesi D. Gold Nanoparticles as a Photothermal Agent in Cancer Therapy: The Thermal Ablation Characteristic Length. *Molecules*. 2018; 23(6):1316.
https://doi.org/10.3390/molecules23061316

**Chicago/Turabian Style**

Grosges, Thomas, and Dominique Barchiesi. 2018. "Gold Nanoparticles as a Photothermal Agent in Cancer Therapy: The Thermal Ablation Characteristic Length" *Molecules* 23, no. 6: 1316.
https://doi.org/10.3390/molecules23061316