Exact solutions to the cancer laser ablation modeling

TL;DR

The paper tackles the problem of modeling light diffusion, heating, and tissue damage during focal laser ablation in heterogeneous breast and prostate tissues. It achieves this by deriving exact analytical solutions to a coupled PDE system that blends the diffusion-approximation of the radiative transfer equation, the Pennes bioheat equation, and Arrhenius-type tissue damage, using Duhamel’s principle and Fourier–Bessel expansions. Key contributions include closed-form expressions for the fluence rate and temperature, a rigorous assessment of the diffusion approximation's applicability, and insights into source localization and optimal exposure timing to protect healthy tissue. The results provide a cost-free analytical framework to guide FLA planning and may reduce reliance on purely numerical simulations in treatment design.

Abstract

The present paper deals with the study of the fluence rate over both healthy and tumor tissues in the presence of focal laser ablation (FLA). We propose new analytical solutions for the coupled partial differential equations (PDE) system, which includes the transport equation modeling the light penetration into biological tissue, the bioheat equation modeling the heat transfer and its respective damage. The present building could be the first step to the knowledge of the mathematical framework for biothermophysical problems, as well as the main key to simplify the numerical calculation due to its no cost. We derive exact solutions and simulate results from them. We discuss the potential physical contributions and present respective conclusions about (1) the validness of the diffusion approximation of the radiative transfer equation; (2) the local behavior of the source of scattered photons; (3) the unsteady-state of the fluence rate; and (4) the boundedness of the critical time of the thermal damage to the cancerous tissue. We also discuss some controversial and diverging hypotheses.

Figures (4)

• Figure 1: Schematic representations of the laser-tissue system: the laser emission tip $\Gamma_\mathrm{tip} =\{(x,y)\in \mathbb R^2 :\,\,x^2+y^2<r_{\mathrm{f}}^2 \} \times \{0\}$, the light-emitting cylinder $]0;r_\mathrm{f}[\times ]0;L[$, the tumor $\Omega_\mathrm{i} =]0; r_\mathrm{i}[\times ] -\ell ;\ell[$, and the surrounding healthy tissue $\Omega_\mathrm{o}= \{(x,y)\in\mathbb R^2:\,\ x^2+y^2<r_\mathrm{i} ^2 \}\times ]\ell;L[\cup \{(x,y)\in\mathbb R^2:\,\, r_\mathrm{i}^2<x^2+y^2<r_\mathrm{o} ^2 \}\times ]0;L[$. Left: Sagittal view. Right: Cylindrical coordinates $(r,z)$.
• Figure 2: Graphical representations of the source $S$ for the wattage set at 5W (in blue): with wavelengths of 810nm (solid line) and 980nm (dashed line) and at 1.3W (in red): with wavelengths of 980nm (dashed line) and 1064nm (solid line). (a) Plot with linear axes of the breast source. (b) Plot using a logarithmic scale for the $y$-axis of the breast source. (c) Linear plot of the prostate source. (d) Semilog plot of the prostate source. For details see Table \ref{['tabmax']}.
• Figure 3: (a) Breast radial graphical representations of the fluence rate $\phi$ for the wavelength of 810nm and wattage set at 5W (blue solid line) and for 980nm and wattage set at 5W (blue dashed line) and at 1.3W (red dashed line). (b) Plot for the tumor-adipose breast tissue for the wavelength of 980nm and wattage set at 1.3W. (c) Prostate radial graphical representations of the fluence rate $\phi$ as in (a). (d) Plot for the tumor-healthy prostate tissue for the wavelength of 810nm and wattage set at 5W.
• Figure 4: (a) Breast radial graphical representations of the fluence rate $\phi$ for the wavelength of 810nm and wattage set at 5W (blue solid line) and for 980nm and wattage set at 5W (blue dashed line) and at 1.3W (red dashed line). (b) Plot for the tumor-adipose breast tissue for the wavelength of 980nm and wattage set at 1.3W. (c) Prostate radial graphical representations of the fluence rate $\phi$ as in (a). (d) Plot for the tumor-healthy prostate tissue for the wavelength of 980nm and wattage set at 1.3W.