# A New Nonlinear Photothermal Iterative Theory for Port-Wine Stain Detection

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Theoretical Analysis

#### 2.1. Theoretical Model

#### 2.2. Nonlinear Thermal Diffusion Equation

^{2}). The fundamental equation of the photoacoustic imaging theory is based on the thermal diffusion equation; therefore, the thermal diffusion equation and boundary conditions in the sample are [28,29]:

#### 2.3. Iterative Numerical Method for Solving the Nonlinear Heat Diffusion Equation

## 3. Numerical Results and Discussion

^{2}/s for the 2219 aluminum alloy sample.

^{2}and the number of iterations is greater than or equal to 6, the fundamental frequency amplitude converges to 7.6485 K. In addition, when the light energy is 8 mJ/cm

^{2}and the number of iterations is greater than or equal to 10, the fundamental frequency amplitude converges to 19.8287 K. Moreover, when the light energy is 10 mJ/cm

^{2}and the number of iterations is greater than or equal to 12, the fundamental frequency amplitude converges to 31.3493 K. These results imply that the proposed method is converges well.

## 4. Conclusions

- (1)
- The rates of change with frequency, thickness, and optical energy intensity are larger for higher−order harmonics than lower-order harmonics; higher−order harmonics are more sensitive to sample detection than lower-order harmonics.
- (2)
- For the same parameter values, the proposed new numerical iterative method has greater sensitivity and a wider frequency band than the perturbation method. Furthermore, the calculation time of our proposed method will not drastically increase when additional high−order harmonics are included.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) Schematic of multilayered skin; (

**b**) schematic of skin with different growth phases of a cancerous lesion; (

**c**) theoretical model for photoacoustic detection of skin tissue.

**Figure 2.**Variation in the amplitude with frequency on the posterior surface of a wine−discolored sample when the sample thickness varies: (

**a**) fundamental frequency wave; (

**b**) second harmonic (low frequency).

**Figure 3.**Variation in the amplitude of the posterior surface of the wine−discolored samples with frequency for different sample thicknesses: (

**a**) fundamental frequency wave; (

**b**) second harmonic (high frequency).

**Figure 4.**Variation in the harmonic amplitude with frequency for each order of harmonic for different sample thicknesses.

**Figure 5.**Variation in the harmonic amplitude with light energy for different sample thicknesses. (

**a**) Fundamental frequency wave; (

**b**) second harmonic; (

**c**) third harmonic; (

**d**) fourth harmonic.

**Figure 8.**The variation in the second harmonic amplitude with number of iterations for different light energies.

Layers | d (mm) | β (mm^{−1}) | σ (mm^{−1}) | g | ρ (g/cm^{−3}) | C (J/(g. K)) | K_{0} (mW/(cm. K)) |
---|---|---|---|---|---|---|---|

Stratum corneum | 0.01 | 0.00091 | 18.95 | 0.8 | 1.2 | 3.59 | 2.4 |

Living epidermis | 0.08 | 0.13 | 18.95 | 0.8 | 1.2 | 3.59 | 2.4 |

Papillary dermis | 0.10 | 0.105 | 11.65 | 0.8 | 1.09 | 3.35 | 4.2 |

Upper blood plexus | 0.08 | 0.15875 | 15.485 | 0.818 | 1.09 | 3.35 | 4.2 |

Reticular dermis | 1.50 | 0.105 | 11.65 | 0.8 | 1.09 | 3.35 | 4.2 |

Deep blood plexus | 0.07 | 0.4443 | 46.165 | 0.962 | 1.09 | 3.35 | 4.2 |

Dermis | 0.16 | 0.105 | 11.65 | 0.8 | 1.09 | 3.35 | 4.2 |

Hypodermis | 3.00 | 0.009 | 11.44 | 0.9 | 1.21 | 2.24 | 1.97 |

Muscle tissues | 3.00 | 0.029 | 7.13 | 0.9 | 1.075 | 3.5 | 4.5 |

PWS | 0.001~1.5 | 0.15875 | 46.7 | 0.99 | 1.0 | 3.6 | 5.3 |

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**MDPI and ACS Style**

Cao, N.; Liang, H.; Zhang, R.; Li, Y.; Cao, H.
A New Nonlinear Photothermal Iterative Theory for Port-Wine Stain Detection. *Int. J. Environ. Res. Public Health* **2022**, *19*, 5637.
https://doi.org/10.3390/ijerph19095637

**AMA Style**

Cao N, Liang H, Zhang R, Li Y, Cao H.
A New Nonlinear Photothermal Iterative Theory for Port-Wine Stain Detection. *International Journal of Environmental Research and Public Health*. 2022; 19(9):5637.
https://doi.org/10.3390/ijerph19095637

**Chicago/Turabian Style**

Cao, Na, Hongtao Liang, Ruoyu Zhang, Yanhua Li, and Hui Cao.
2022. "A New Nonlinear Photothermal Iterative Theory for Port-Wine Stain Detection" *International Journal of Environmental Research and Public Health* 19, no. 9: 5637.
https://doi.org/10.3390/ijerph19095637