van Gemert, 1996  Title: Time constants in thermal laser medicine:
II. Distributions of time constants and thermal relaxation of
tissue.
Authors: van Gemert MJ, Lucassen GW, Welch
AJ
Journal: Phys Med Biol 1996 Aug;41(8):1381-99
PMID: 8858726, UI: 97011734
Affiliated institution: Laser Center,
Academic Medical Center, Amsterdam, The Netherlands.
Cited in: Dierickx
The thermal response of a semi-infinite medium in
air, irradiated by laser light in a cylindrical geometry, cannot
accurately be approximately by single radial and axial time constants
for heat conduction. This report presents an analytical analysis
of hear 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 (v) 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-tL)0.5] which approaches
1/t0.5 for t >> tL, where t is the time and tL is the laser
pulse duration. For deep penetrating beams this is proportional
to 1/(t-tL). For intermediate penetration, i.e. penetration depths
about equal to spot size diameters, this is proportional to 1/(t-tL)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.
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