In tissue optics, the optical characteristics and parameters of tissues can be obtained by utilizing the diffused light from the tissue surface, which can also predict the characteristic parameters of tissue structure. This can help understand the tissue pathology degree and changes in physiological properties. Various models have been studied in light transmission modeling to describe the light transmission behavior in tissues. Common theoretical models include the Boltzmann equation, diffusion equation, and P3 equation, where the diffusion equation is a first-order approximation of the Boltzmann equation, and the P3 equation is a third-order approximation. We aim to investigate the light transmission behavior in multi-layered media and provide the P3 steady-state equation for light transmission in multi-layered biological tissues with semi-infinite thicknesses. Meanwhile, this equation is extended to the frequency domain equation and transformed into the time domain equation by Fourier transforms at different frequencies, which serves as a third-order approximation of the radiation transfer theory. Our objective is to evaluate the P3 equation accuracy compared to the first-order diffusion equation in complex multi-layered biological tissues, and thus validate the accuracy by comparisons with different parameter settings.

Based on the study of the diffusion equation describing the light transmission in multi-layered media with semi-infinite thicknesses, we combine it with the P3 equation for single-layered media to successfully build the steady-state model of the P3 equation applicable to light transmission in multi-layered biological tissues, and elucidate the boundary conditions. By adopting Fourier transforms and extrapolated boundary conditions, we obtain the functional solution of the P3 steady-state model for light transmission in multi-layered biological tissues. To validate this model, we employ Monte Carlo simulation as a standard non-experimental verification method. Additionally, we calculate and compare the steady-state and time-domain solutions of the P3 equation for light transmission in media with semi-infinite thicknesses, as well as the steady-state and time-domain solutions of the diffusion equation, with different optical parameters taken into account. We calculate the relative errors among the P3 equation, the Monte Carlo simulation, and the diffusion equation, focusing on scenarios with low and high absorption coefficients under the steady state. Meanwhile, we analyze the results at different detection distances and compare them with reference data obtained from the Monte Carlo simulation. Additionally, in the time domain, we compare the P3 equation results with the diffusion equation and Monte Carlo simulation for different parameters and distances, particularly near the peak values.

By employing the Fourier transform method, we successfully establish the P3 equation for light transmission in multi-layered media with semi-infinite thicknesses, and conduct Monte Carlo simulations to validate our model. Meanwhile, we calculate the spatially resolved reflectance and time-resolved reflectance for the P3 equation and consider cases with five and six layers of media with semi-infinite thicknesses to verify the accuracy of the steady-state P3 equation. The results demonstrate that the results of the P3 equation are consistent with those of the Monte Carlo simulations and the diffusion equation. To further validate the accuracy of the steady-state P3 equation, we compute the relative errors among the P3 equation, the Monte Carlo simulations, and the diffusion equation. Firstly, we calculate the relative errors for low absorption coefficients, revealing that there is a discrepancy between the P3 equation and the diffusion equation at close distances, while consistent results are yielded at far distances. The relative errors between the diffusion equation and the steady-state P3 equation are nearly zero in the far-field. Next, we compute the relative errors for high absorption coefficients, showing that the steady-state P3 equation is more accurate than the diffusion equation across the entire measurement range. Additionally, we investigate light transmission in two-layered media with semi-infinite thicknesses consisting of fat and muscle, confirming that the P3 equation can be applied to practical measurements in biological tissues. To verify the accuracy of the time-domain P3 equation for multi-layered media, we compare results with five- and six-layered media with semi-infinite thicknesses against Monte Carlo simulations and the diffusion equation. In regions far from the peak, the results of the P3 equation match exactly with those of the diffusion equation. In the vicinity of the peak, the results of the P3 equation for multi-layered media closely approximate the results of Monte Carlo simulations. We also emphasize that near the peak, larger absorption coefficients lead to greater errors, but the P3 equation exhibits smaller errors than the diffusion equation.

In conclusion, as a third-order approximation, the P3 equation demonstrates higher accuracy in describing light transmission in multi-layered media than the first-order diffusion equation. Our results support the importance of adopting more accurate equations such as the P3 equation to gain a better understanding of light behavior in complex multi-layered tissues. The proposed P3 equation accurately describes light transmission in biological tissues, particularly in cases with higher absorption coefficients near the peak region. Our study provides valuable insights for light transmission in multi-layered media and suggests that the P3 equation outperforms the diffusion equation in specific conditions. Further research can explore the applications of the P3 equation in various biological and clinical settings to enhance our understanding of the interaction between light and tissues and optimize relevant medical procedures.