174
11 Simulation of the Thermal Processes
At the interface between the ith and (i + 1)th layers (z = h i (x, y)), the following
continuity conditions of the heat flow and temperature are met:
λ i
∂ T i
∂z
− λ (i+1)
∂ T (i+1)
∂z
| z=h i (x,y) = 0
(11.5)
T i − T (i+1)
| z=h i (x,y) = 0,
(11.6)
where h i (x, y) is determined from (4.1).
When solving set of (11.1)–(11.6), we obtain
(a) the temperature distribution over layers along the propagation direction of the
laser beam; and
(b) the dependence of the temperature on the optical properties of the biological
tissue, which can be used to study the effect of the temperature field on the
electrophysical parameters of the biological tissue for the case in vivo.
For further investigation and analysis of the dependences obtained, we will use
numerical methods.
11.3 Numerical Calculations Using a Model Medium
and Conclusions
To numerically solve the set of (11.1)–(11.6), we construct an implicit iteration
scheme on a spacetime mesh, the boundary conditions for temperature being replaced
by their finiteanalogs [9]. We consider the model medium that is shown in Fig. 6.1 and
has the following parameters [10]: the characteristic layer thicknesses are d 2 = 65 ·
10
−6 , n
◦
2 = 1.50, n
◦
3 = 1.40, n
◦
4 = 1.35, n
◦
5 = 1.40 n
◦
1 = 1,χ 1 = 0, χ 2 = χ 3 = χ 4 =
χ 5 = 10
−5 , the wavelength is λ = 0.63 µm (center of the line of a He−Ne laser).
The arbitrarily specified constants are a 1 = −0.0024, b 1 = 0.020, a 2 = 0.021, b 2 =
0.030, a 3 = 0.041, b 3 = 0.051, c 1 = c 2 = c 3 = 10
−2 . The values of parameters a 1 ,
b 1 , a 2 , b 2 , a 3 , b 3 , c 1 , c 2 and c 3 are chosen for the interface of each layer so that the
surface shape are as close as possible to the interface shape of the corresponding
layer in the structure of human skin, the thermal conductivity (W/(m K)), the specific
heat J/(kg), and the density ×10
−3 (kg/m
3
) are 0.498, 3.2 and 1 for the first layer,
0.266, 3.7 and 1.6 for the second layer, 0.530, 3.6 and 1 for the third layer, 0.266, 3.7
and 1.6 for the fourth layer, and the heat-transfer coefficient is 0.009 W cm
−2 K
−1 .
The calculations were performed for two-layer particles simulating red corpuscles.
Each layer was taken to have ten particles, the speed of blood flow in the dermis is
15 mL/(min 100 g), pulse duration is 20c, the radiation power density is 1W/m
2 .
Figure 11.1a and b shows the time-dependent temperature distribution in the
direction of the incident radiation (z axis) for a multilayer light-absorbing and scattering medium that simulates human skin and its components at various refractive
indices. The upper layer of the simulated biological tissue (epidermis) is seen to be
11 Simulation of the Thermal Processes
At the interface between the ith and (i + 1)th layers (z = h i (x, y)), the following
continuity conditions of the heat flow and temperature are met:
λ i
∂ T i
∂z
− λ (i+1)
∂ T (i+1)
∂z
| z=h i (x,y) = 0
(11.5)
T i − T (i+1)
| z=h i (x,y) = 0,
(11.6)
where h i (x, y) is determined from (4.1).
When solving set of (11.1)–(11.6), we obtain
(a) the temperature distribution over layers along the propagation direction of the
laser beam; and
(b) the dependence of the temperature on the optical properties of the biological
tissue, which can be used to study the effect of the temperature field on the
electrophysical parameters of the biological tissue for the case in vivo.
For further investigation and analysis of the dependences obtained, we will use
numerical methods.
11.3 Numerical Calculations Using a Model Medium
and Conclusions
To numerically solve the set of (11.1)–(11.6), we construct an implicit iteration
scheme on a spacetime mesh, the boundary conditions for temperature being replaced
by their finiteanalogs [9]. We consider the model medium that is shown in Fig. 6.1 and
has the following parameters [10]: the characteristic layer thicknesses are d 2 = 65 ·
10
−6 , n
◦
2 = 1.50, n
◦
3 = 1.40, n
◦
4 = 1.35, n
◦
5 = 1.40 n
◦
1 = 1,χ 1 = 0, χ 2 = χ 3 = χ 4 =
χ 5 = 10
−5 , the wavelength is λ = 0.63 µm (center of the line of a He−Ne laser).
The arbitrarily specified constants are a 1 = −0.0024, b 1 = 0.020, a 2 = 0.021, b 2 =
0.030, a 3 = 0.041, b 3 = 0.051, c 1 = c 2 = c 3 = 10
−2 . The values of parameters a 1 ,
b 1 , a 2 , b 2 , a 3 , b 3 , c 1 , c 2 and c 3 are chosen for the interface of each layer so that the
surface shape are as close as possible to the interface shape of the corresponding
layer in the structure of human skin, the thermal conductivity (W/(m K)), the specific
heat J/(kg), and the density ×10
−3 (kg/m
3
) are 0.498, 3.2 and 1 for the first layer,
0.266, 3.7 and 1.6 for the second layer, 0.530, 3.6 and 1 for the third layer, 0.266, 3.7
and 1.6 for the fourth layer, and the heat-transfer coefficient is 0.009 W cm
−2 K
−1 .
The calculations were performed for two-layer particles simulating red corpuscles.
Each layer was taken to have ten particles, the speed of blood flow in the dermis is
15 mL/(min 100 g), pulse duration is 20c, the radiation power density is 1W/m
2 .
Figure 11.1a and b shows the time-dependent temperature distribution in the
direction of the incident radiation (z axis) for a multilayer light-absorbing and scattering medium that simulates human skin and its components at various refractive
indices. The upper layer of the simulated biological tissue (epidermis) is seen to be