Analytical thermal-optic model for laser heating of biological tissue using the hyperbolic heat transfer equation

Abstract

In this paper, we solve in an analytical way the thermal-optic coupled problem associated with a 1D model of non-perfused homogeneous biological tissue irradiated by a laser beam. We consider a laser pulse duration of 200 micros and study the temperatures of areas very close to the point of laser beam application. We consider that these values of the temporal and spatial variables mean that the problem has to be solved by means of the hyperbolic heat conduction model instead of the classic or parabolic model. We therefore obtain the solution using both models and apply the temperature profiles obtained to a specific biological tissue for comparison. Finally, we theoretically study the effect of the thermal relaxation time on the temperature profiles in the tissue for both heating and cooling phases (i.e. during and after laser application).

Modeling the heating of biological tissue based on the hyperbolic heat transfer equation

“…Using in Eq. for the speed of these steps, which exactly agrees with the known finite speed of the thermal wave in the cornea [10]. This fact demonstrates the wave character of the HHTE model.…”

“…Note that Eq. (10) shows that q(x, t) depends on the full history of the temperature gradient within the time interval [0, t], something which is entirely absent in the much simpler expression for the Fourier flux in Eq. ( 2).…”

“…shows the numerical estimates for temperatures at various tissue penetration depths obtained as a function of time[10]. This figure represents the hyperbolic (solid line) and parabolic (dashed line) temperature evolution at the four locations x = 0.01, 0.1, 0.5 and 1 mm for the thermokeratoplasty technique.…”

Analytical Solution for Three-Dimensional Hyperbolic Heat Conduction Equation with Time-Dependent and Distributed Heat Source

“…Ahmadikia et al [10] solved one dimensional thermal wave and panes bioheat transfer models using Laplace transform under laser irradiation. Trujillo et al [11] applied Laplace scheme to obtain a general solution of hyperbolic temperature profile in a one-dimensional model of nonperfused homogeneous biological tissue under laser heating. Zobier and Chaudhry [12] derive an analytical model for computation of temperature and heat flux distribution in a semi-infinite solid under spatially decaying, time-dependent laser source, using Laplace transform.…”

“…The method of solution is Eigen function expansion procedure. For demonstration, following the works [11], the solution, is used for simulation of a continues laser interaction in a cubical tumor tissue. The innovations of the solution are:  Using any numerical procedure such as inverse procedure in Laplace method for separable simple function.…”