Supporting analyses of experimental results were provided using the Buffington-Thomas-Edwards (BTE) thermal model.
30 31 Absorption coefficients for cell monolayers without and with artificial pigmentation were calculated
32 from measured bulk absorbances (each of seven wells read in triplicate) using a microtiter plate reader (Genios model; Tecan USA., Research Triangle Park, NC). Absorbance filters were either 460 ± 5 or 810 ± 5 nm. An average absorbance for plate and medium alone was subtracted from average absorbances of plate, medium, and cells (with or without melanosomes). The subtractive analysis minimized the incorporation of surface reflection and absorption from media and well-plate into our measured absorption coefficients. A spectrometer (Ocean Optics) was used to measure absorption of exposure medium (1-cm cuvette). Thickness of the bottom of well-plates (growth surface) was found to be 1.57 ± 0.02 mm (cross-sectional slicing). The absorption coefficient for the well-plate was computed using bulk absorbance readings from the microtiter plate reader. However, this value was probably an overestimate, as a subtractive method was not used.
The BTE thermal model numerically computed an approximate solution to the bioheat equation, expressed in cylindrical coordinates by
equation 1 . The solution to this initial value problem was performed using a finite-difference method for time-dependent partial differential equations.
33 The cell culture was modeled as a three-layer structure of essentially infinite radial extent. Layer axial dimensions representative of the measured values for cell culture media, cell monolayer, and well-plate used were 100 μm, 7 ± 1.5 μm, and 1.6 mm, respectively. Each layer was assumed to be homogeneous in optical and thermal properties, each of which is listed in
Table 2 . No losses by scattering were considered.
\[{\rho}c\ \frac{{\delta}{\nu}}{{\delta}t}{=}\ \frac{{\kappa}}{r}\ \frac{{\delta}{\nu}}{{\delta}r}{+}\ \frac{{\delta}}{{\delta}r}\left({\kappa}\ \frac{{\delta}{\nu}}{{\delta}r}\right)\ {+}\ \frac{{\delta}}{{\delta}z}\left({\kappa}\ \frac{{\delta}{\nu}}{{\delta}z}\right)\ {+}A\]
In
equation 1 , ν = ν
(z,
r,
t) represents the temperature rise (in Kelvin) in the cell culture and well-plate as a function of time and position,
A =
A(z,
r,
t), is referred to as the source term and represents energy from the laser absorbed per unit time and volume (in J · cm
−3 · s
−1), κ = κ
(z) is the thermal conductivity (in J · cm
−1 · s
−1 · K
−1),
c =
c(
z) is the specific heat of the layer (in J · g
−1 · K
−1), ρ= ρ
(z) is the layer density (in g · cm
−3), with
z specifying the axial coordinate in the tissue. For a cell culture sample, we assumed no perfusion and that the culture was in thermal equilibrium with the surrounding atmosphere.
For our model, the time-dependent solution to
equation 1was determined for a source term that provides a time-dependent description of the linear absorption of optical energy as a function of depth in the tissue, complete with spectral and radial dependence of energy being absorbed. Surface boundary conditions were addressed by an equation that is critical to the correct prediction of surface temperatures within the skin,
34 and includes a Lewis approximation.
35 The experiment modeled consisted of a layer of cell culture media covering a strongly absorbing monolayer of cells. This complete boundary condition ensured that the evaporative, radiative, and convective energy losses were correctly incorporated into the analysis.
Damage to the tissue was evaluated through the Arrhenius damage integral
34 given by
equation 2 :
\[{\Omega}(z,r){=}C\ {{\int}_{t_{1}}^{t_{2}}}\mathrm{exp}\left(\ \frac{{-}E}{RT}\right)\ dt\]
In
equation 2 ,
R is the universal gas constant,
T represents the absolute temperature of a given coordinate at a given time,
T(
z,
r,
t), and
t 1 and
t 2 represent the initial and final simulation times used in the solution of
equation 1 . The variable
C is a normalizing rate constant (in seconds) and
E the activation energy for a reactive process (in J · mole
−1). Values for
C and
E are reported in the literature
34 and are based on differing assumptions about tissue type and geometry. Without empirically determined values for C and E specific to our cell culture system, we chose to use 1 × 10
44 · s
−1 and 2.93 × 10
5 J · mole
−1, respectively.
5 34 Values of this damage integral approaching Ω = 1, indicate irreversible thermal damage to the cells at a given location for our study. Overall, our choice in selected constants and threshold parameters were consistent with established models in the literature that have been experimentally validated.
36 37