**5. Results and discussion**

The results obtained using the solutions presented in Sections 3 and 4 have been discussed in this section. First, the spatial variation of the two components of the total intensity (collimate and diffused) was calculated at the location of the point of laser irradiation (Location 1 having coordinates of *x* = 0.98 mm and *y* = 2 mm) and the center of the domain (Location 2 having coordinates of *x* = 0.98 mm and *y* = 0.98 mm) to understand the laser-tissue interaction. After that, the temperature rise due to single-pulse and train of pulses was calculated at the same two locations. Then, the comparative study between the Fourier and non-Fourier (C-V and DPL) heat conduction model were carried out. Furthermore, numerical results were compared with the analytical results obtained using the FIT.

When light propagates through the biological tissue, the intensity has two components: collimated and diffused. The collimated component of the intensity represents the variation of the ballistic photons, which don't undergo any multiple scattering events, and its spatial variation at different time instants is shown in **Figure 2a**. The figure shows that the collimated intensity is maximum at *y* = 2 mm because it is the point of laser irradiation (Location 1). As expected, the collimated intensity decreases exponentially following the Beer-Lambert law. The collimated intensity is only available when time is less than or equal to the pulse width; otherwise, the value of the collimated intensity is zero.

Variation of the diffuse component of light intensity with respect to the depth (*y*) of the tissue phantom at various time instants is shown in **Figure 2b**. At the initial time instant (*t* = 4.6667 ps), the magnitude of diffuse light is significantly much smaller than the collimated intensity. Furthermore, in contrast to the profile seen for the collimated component wherein the maxima were observed at the top surface of the tissue phantom (the point of laser irradiation), the maxima of the diffuse

### **Figure 2.**

*Spatial variation of collimated (a) and diffuse (b) components of light along a section passing through* x *= 0.98 mm [3].*

### *Modeling of Laser-Irradiated Biological Tissue DOI: http://dx.doi.org/10.5772/intechopen.106794*

component are observed at points below the top surface inside the body of the tissue phantom. This trend can be attributed to the scattering properties of the tissue domain, which affect the propagation of the diffuse component of the light intensity.

**Figure 3a** shows the temporal variation of the temperature at the same two locations considered in the biological tissue subjected to single pulse laser irradiation. As expected, the change in temperature at Location 1 (the point of laser irradiation) is more significant in comparison to Location 2. It is also to be seen from the figure that the rate of temperature rise is quite rapid during the first 5 ps, which is almost equal to the width of the laser pulse employed for irradiation since the laser power is available for this complete duration. Furthermore, the temperature rise achieves a constant value after nearly 20 ps. Because of this observation, the local temperature profile at *t* = 180 ps was used as the initial temperature field for solving the bio-heat transfer equation (Eq. 10) for determining the temperature distribution within biological tissue subjected to a train of laser pulses.

The Pennes bio-heat transfer equation was solved to determine the temperature distribution inside the biological tissue subjected to the train of pulses (repetition rate: 1 kHz). **Figure 3b** shows the temporal variation of temperature at the same two locations. The figure shows that the temperatures at these locations increase during the first one second, and thereafter a decay in the temperature values can be seen. It is expected since the total duration of laser irradiation is only one second. The temperature rise is comparatively much higher at Location 1, where the laser pulse strikes the sample, than in the rest of the region within the biological tissue. The temperature drop after one second is also steeper at the top wall of the tissue phantom as it is directly exposed to the ambient conditions.

The efficacy of laser-based photo-thermal therapy for selective destruction of cancerous cells depends on accurately predicting the temperature distribution inside the laser-irradiated biological tissue. Because of this, various heat conduction models, e.g., Fourier, C-V, DPL, etc., have been developed by researchers in the past. The need for the non-Fourier heat conduction model is realized because the photons travel with a finite speed through the biological tissues, contrary to the infinite speed of thermal wave propagation assumed in the Fourier model. So, we compared the temperature

### **Figure 3.**

*(a) Temporal profile of temperature rise in tissue subjected to single pulse laser irradiation at* x *= 0.98 mm; (b) Temporal profile of temperature subjected to a train of pulse at* x *= 0.98 mm [3].*

distribution obtained using the Fourier and non-Fourier heat conduction models to study the thermal response of laser-irradiated biological tissue phantom. **Figure 4** shows the two-dimensional temperature distribution inside the laser-irradiated biological tissue at different time instants for Fourier and non-Fourier (C-V and DPL) models. While a nearly uniform diffusion of heat is to be seen in the contour plots corresponding to the Fourier model, a distinct wave nature in the thermal profiles is associated with the predictions of non-Fourier (C-V and DPL) heat conduction models. The oscillations in temperature values penetrate within the body of the tissue phantom as time progresses. The C-V heat conduction model predicts the maximum temperature values at any given time instant, followed by the DPL-based model. The C-V model also results in maximum amplitude of oscillations seen in the thermal wavefront compared to the DPL model.

The numerical results obtained using the FVM were compared with analytical results obtained using the FIT. **Figure 5** shows the analytical and numerical results for different heat conduction models (Fourier and non-Fourier). The temporal profiles of temperature without markers represent the results based on the FIT-based analytical solution, and those with markers represent the numerical results. The figure shows that the results obtained using numerical simulations predict relatively lower values of temperatures at Locations 1 and 2 in comparison with those obtained based on the

**Figure 4.** *Two-dimensional temperature distribution at different time instants (i) Fourier (ii) C-V (iii) DPL model [21].*

*Modeling of Laser-Irradiated Biological Tissue DOI: http://dx.doi.org/10.5772/intechopen.106794*

#### **Figure 5.**

*Comparison of temporal profiles of temperature predicted using Fourier and non-Fourier heat conduction models at Location 1 (a) and 2 (b) (Without marker: Analytical results; With marker: numerical results) [22].*

analytical approach. It is also to be seen from the figure that the Fourier heat conduction model predicts relatively lower temperature values than those calculated using non-Fourier conduction models, which is to be attributed to the infinite speed of propagation of thermal waves considered in the conventional Fourier model.

Furthermore, the C-V heat conduction model predicts the highest temperature values (shown in the inset of **Figure 5a** and **b** for better clarity), while the DPL model predicts the temperatures that lie in between Fourier and C-V heat conduction models. The observed trend may be explained based on the C-V heat conduction model only considers the effect of the thermal relaxation time associated with heat flux (*τq*). In contrast, the dual-phase lag model considers the coupled effects of nonzero values of relaxation times related to the temperature gradient (*τT*) and heat flux (*τq*). In physical terms, the phase lag associated with the temperature gradients, i.e., *τ<sup>T</sup>* tends to suppress the amplitude of the thermal wavefront. Because the value of *τ<sup>T</sup>* is zero in the C-V heat conduction model, the resultant temperature profiles are relatively free of such dampening effects. Hence, the absolute values of temperatures predicted based on this form of non-Fourier heat conduction model are expected to be higher than predicted using the DPL model, wherein these two relaxation times are considered non-zero. The profiles shown in **Figure 5** support this observation wherein the magnitude of temperatures as predicted using the DPL model lies between those obtained using the C-V heat conduction model (maximum values) and the Fourierbased heat conduction model (minimum values).

**Figure 5b** shows the temporal variation of temperature at Location 2. The profiles corresponding to the Fourier heat conduction model show a sudden drop in temperature values immediately after the laser power is switched off (*t* > 1.0 s). On the other hand, because of the phase lag terms associated with temperature gradients and/or heat flux, the temperature profiles predicted using the non-Fourier heat conduction models (hyperbolic and DPL) show nearly constant values of temperature for a relatively long period of time (*t* ≤ 11 s) before the drop in temperature values starts. It indicates the prolonged (sustained) effects of thermal energy deposited at a given spatial location within the body of the tissue phantom, according to non-Fourier heat conduction models.
