Time constants in thermal laser medicine .2. Distributions of time constants and thermal relaxation of tissue

The thermal response of a semi-infinite medium in air, irradiated by laser light in a cylindrical geometry, cannot accurately be approximated by single radial and axial time constants for heat conduction. This report presents an analytical analysis of heat conduction where the thermal response is expressed in terms of distributions over radial and axial time constants. The source term for heat production is written as the product of a Gaussian shaped radial term and an exponentially shaped axial term. The two terms are expanded in integrals over eigenfunctions of the radial and axial parts of the Laplace heat conduction operator. The result is a double integral over the coupled distributions of the two time constants to compute the temperature rise as a function of time and of axial and radial positions. The distribution of axial time constants is a homogeneous slowly decreasing function of spatial frequency (nu) indicating that one single axial time constant cannot reasonably characterize axial heat conduction. The distribution of radial time constants is a function centred around a distinguished maximum in the spatial frequency (lambda) close to the single radial time constant value used previously. This suggests that one radial time constant to characterize radial heat conduction may be a useful concept. Special cases have been evaluated analytically, such as short and long irradiation times, axial or radial heat conduction (shallow or deep penetrating laser beams) and, especially, thermal relaxation (cooling) of the tissue, For shallow penetrating laser beams the asymptotic cooling rate is confirmed to be proportional to [(t)(0.5) - (t - t(L))(0.5)] which approaches 1/t(0.5) for t >> t(L), where t is the time and t(L), is the laser pulse duration. For deep penetrating beams this is proportional to 1/(t - t(L)). For intermediate penetration, i.e. penetration depths about equal to spot size diameters, this is proportional to l/(t - t(L))(1.5). The double integral has been evaluated numerically and the results have been compared with the various approximations available including the new results and the single time constant model. The present analysis completes our previous work, presents a closed-form formulation for the non-ablative thermal response of laser irradiated tissue and provides insight into the practical value of using time constants for representing heat conduction effects, in particular for the rate of cooling of the tissue surface.