Enhancement of light propagation depth in skin: cross-validation of mathematical modeling methods

Abstract

Various techniques to enhance light propagation in skin have been studied in low-level laser therapy. In this study, three mathematical modeling methods for five selected techniques were implemented so that we could understand the mechanisms that enhance light propagation in skin. The five techniques included the increasing of the power and diameter of a laser beam, the application of a hyperosmotic chemical agent (HCA), and the whole and partial compression of the skin surface. The photon density profile of the five techniques was solved with three mathematical modeling methods: the finite element method (FEM), the Monte Carlo method (MCM), and the analytic solution method (ASM). We cross-validated the three mathematical modeling results by comparing photon density profiles and analyzing modeling error. The mathematical modeling results verified that the penetration depth of light can be enhanced if incident beam power and diameter, amount of HCA, or whole and partial skin compression is increased. In this study, light with wavelengths of 377 nm, 577 nm, and 633 nm was used.

Appendix 1

Nomenclature

Φ:

photon density profile (W/cm²)

q:

light source function (W/cm³)

$\Re$ :

Robin function, which is the photon density profile with respect to the Dirac delta source (W/cm³)

μ _a :

absorption coefficient (cm⁻¹)

μ _s :

scattering coefficient (cm⁻¹)

g:

anisotropy factor

$d,\mu _{s} \prime ,\mu _{a}$ :

reduced scattering coefficient (cm⁻¹) ${\left( {\mu _{s} \prime {\text{ = }}{\left( {1 - g} \right)}\mu _{s} } \right)}$

κ :

diffusion coefficient (cm) ${\left( {{\kappa = 1} \mathord{\left/ {\vphantom {{\kappa = 1} {{\left( {3{\left( {\mu _{a} + \mu _{s} \prime } \right)}} \right)}}}} \right. \kern-\nulldelimiterspace} {{\left( {3{\left( {\mu _{a} + \mu _{s} \prime } \right)}} \right)}}} \right)}$

n:

refractive index

a:

reflection coefficient on the surface

c:

speed of light in a vacuum (cm/s) (c = 2.99 × 10¹⁰ cm/s)

ω :

angular modulation frequency of the source intensity(1/s)

Ω:

region of interest (skin structure)

∂Ω:

boundary of Ω (∂Ω = ∂^topΩ + ∂^bottomΩ + ∂^leftΩ + ∂^rightΩ)

∂^topΩ:

top boundary of Ω

∂^bottomΩ:

bottom boundary of Ω

∂^leftΩ:

left boundary of Ω

∂^rightΩ:

right boundary of Ω

N _b :

number of finite element basis