**1. Introduction**

Cancer is a group of diseases in which uncontrolled growth and invade of abnormal cells into other parts of the body [1]. The World Health Organization says nearly one in six died from cancer in 2020 [2]. Thus, it is essential to detect the cancerous cells at their early stage and destroy them to minimize possible damage to the surrounding healthy tissue. So, various diagnosis techniques such as computerized tomography (C.T.) scan, magnetic resonance imaging (MRI), ultrasound, X-ray, positron emission tomography (PET) scan, etc., and treatment such as surgery, chemotherapy, laserbased photo-thermal therapy, etc., have been developed in the past.

Laser-based photo-thermal therapy has gained unprecedented growth among all available treatment techniques in the past few decades. In this therapy, the temperature of the tissue is increased above the pre-defined threshold value using the laser,

which destroys the cancerous cells [3]. In other words, the photon energy has been absorbed by the laser-irradiated biological tissue, which subsequently increases its internal energy; as a result, the temperature of the tissue increases. Amongst the various types of laser, the short-pulse laser has gained substantial importance because it can transmit high energy in a very short interval of time (order of femtosecond to picosecond) [3]. This advantage helps in increasing the temperature to the desired level in the confined region. Therefore, the short-pulse laser has been usually used to destroy cancerous cells during laser-based photo-thermal therapy. It is worth noting that tissue gets thermally damaged when its temperature crosses 43°C [4]. So, the major challenge in the medical field is to destroy the cancerous cells without damaging the surrounding healthy tissue. Thus, it is essential to understand the thermal characteristics of the laser-irradiated biological tissue to improve the efficacy of laser-based photo-thermal therapy.

The modeling of laser-tissue interaction is challenging because it is absorbing as well as scattering in nature due to its constituents such as water, hemoglobin, melanin RBCs, cell membrane, etc. [3, 5]. So, Various mathematical models such as Beer-Lambert's law, diffusion approximation theory, radiative transfer equation (RTE), etc., have been developed to model the laser-tissue interaction. Among these mathematical models, the Beer-Lambert's law ð Þ *I* ¼ *Io* exp ð Þ �*κz* is the simple and relatively straightforward mathematical model, which is solved by the researchers to determine the intensity (*I*) variation inside the laser-irradiated biological tissue [6, 7]. Here, *κ* represents the absorption coefficient. However, the limitation of this model is that it assumes that the biological tissue is purely absorbing in nature. Few researchers used the diffusion approximation theory to determine the intensity distribution by considering that the biological tissue is highly scattering in nature [8, 9]. So, assuming the biological tissue is either purely absorbing or highly scattering misleads the results. Therefore, the researchers [4] have suggested using the RTE (Eq. 1), which is the most appropriate mathematical model for modeling the laser-tissue interaction [10].

$$\frac{1}{c}\frac{\partial I}{\partial t} + \frac{dI}{ds} = -\beta I + \kappa I\_b + \frac{\sigma\_\circ}{4\pi} \int\_{4\pi} I \mathcal{Q}(\Omega, \ \Omega') d\Omega' \tag{1}$$

where, *c*,*β*,*σ<sup>s</sup>* and ∅ represents the speed of light in the medium, extinction coefficient (sum of scattering and absorption coefficient), scattering coefficient, and scattering phase function, respectively. The first and second term on the left-hand side of Eq. (1) represents the temporal and spatial variation of intensity, respectively. On the other hand, the first, second, and third terms on the right-hand side of Eq. (1) denote the attenuation of intensity due to the absorption and out-scattering, augmentation of intensity due to the emission and in-scattering, respectively.

Eq. (1) is the integrodifferential equation, and getting the analytical solution is quite challenging due to the presence of the in-scattering terms (third term of the right-hand side of Eq. (1)). So, researchers have developed various numerical methods, such as the discrete transfer method (DTM) [11], finite volume method (FVM) [12], discrete ordinate method (DOM) [13], etc., to convert this integrodifferential equation into a simpler form to get the numerical solution. The various researchers performed the comparative studies between these numerical methods and found that the DOM is computationally most efficient, uses comparatively lesser memory and the effective wave speed predicted by DOM comes closer to the actual speed of light [14–16]. So, the DOM has been employed in the present study to solve the RTE.

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

To understand the thermal behavior of the laser-irradiated biological tissue, we need to solve the bio-heat transfer equation (Eq. 2) wherein the energy generated by the laser acts as the source terms.

$$
\rho c\_v \frac{\partial T}{\partial t} = -\nabla \bullet \overrightarrow{q} + \rho c\_b \rho\_b c\_b (T\_b - T) + Q\_m - \nabla \bullet \overrightarrow{q\_r} \tag{2}
$$

where *ρ*, *cv*, *ωb*, *ρb*, and *cb* denote the density of the tissue, specific heat of the tissue, blood perfusion, density of blood, and specific heat of blood, respectively.

The term on the left-hand side of Eq. (2) represents the time rate of change of energy inside the biological tissue. On the other hand, the first, second, third, and fourth terms on the right-hand side of Eq. (2) represent the diffusion term due to the net heat transfer across the boundaries, blood perfusion term due to the energy exchange between the tissue and surrounding capillary blood vessels, metabolic heat generation term, and the divergence of the radiative heat flux which represents net radiative heat loss from the medium.

When Fourier's law of heat conduction (Eq. 3) is substituted into Eq. (2), the resulting equation is known as the Fourier model-based bio-heat transfer equation, also known as Pennes bio-heat transfer equation.

$$
\overrightarrow{q} = -k\nabla T\tag{3}
$$

The limitation of the Pennes bio-heat transfer equation is that it assumes that arterial blood temperature is almost uniform throughout the tissue domain while the venous blood temperature equals that of the local tissue temperature [17]. However, the assumption of thermal equilibrium between the blood vessel and the surrounding tissue is not valid when the diameter of the blood vessel is more than 500 μm (i.e., large blood vessel). In that case, the energy equation has been separately solved for the tissue and blood vessel domain to get the accurate temperature distribution. The interested reader can find the numerical solution of the energy equation to determine the temperature distribution inside the laser-irradiated biological tissue embedded with the large blood vessel elsewhere [18].

Few researchers experimentally demonstrated that Fourier's law of heat conduction under predicts the temperature value that does not accurately match the experimental data because of the inherent assumption of the infinite speed of thermal wave propagation through the biological tissue [4, 19, 20]. Furthermore, the Fourier model gives inaccurate results in an area where used the short-pulse laser, the temperature is in the range of cryogenic, studying the thermal response of non-homogeneous structures, e.g., biological tissue [21]. To overcome the limitations of Fourier heat conduction models, researchers have modified Eq. (3) by considering the relaxation time associated with the heat flux (*τq*) and temperature gradient (*τT*), and it is known as the dual-phase lag (DPL) model (Eq. 4).

$$
\overrightarrow{q} + \tau\_q \frac{\partial \overrightarrow{q}}{\partial t} = -k \left\{ \nabla T + \tau\_T \frac{\partial (\nabla T)}{\partial t} \right\} \tag{4}
$$

When *τ<sup>T</sup>* is equal to zero, the DPL model becomes the Cattaneo-Vernotte (C-V) equation. However, the limitation of this model is that it doesn't consider the effects of microstructural interactions in a non-homogeneous medium, e.g., biological samples [21]. The DPL model becomes the conventional Fourier's law of heat conduction when both relaxation times are equal to zero. So, the DPL model is the generalized form of the non-Fourier heat conduction model. Thus, the numerical and analytical solutions of the DPL model-based bio-heat transfer equation have been obtained using the FVM and finite integral technique (FIT), respectively, which are discussed in Sections 3 and 4 respectively.

As mentioned earlier, the last term on the right-hand side of Eq. (2) represents the divergence of radiative heat flux (Eq. (5)), which is obtained by solving the Eq. (1).

$$\nabla \bullet \overrightarrow{q\_r} = \kappa (4\pi I\_b - G) = \kappa \left(4\sigma T^4 - G\right) \tag{5}$$

where, *Ib* and *<sup>G</sup>* <sup>¼</sup> <sup>P</sup>*<sup>M</sup> <sup>m</sup>*¼1*ωmI <sup>m</sup>*represent the black body intensity and the incident intensity. Here *ωm*denotes the angular weight in the particular direction *m,* and *M* is the total number of divisions of the 4*π* solid angle.

It is to be noted here that the time scale used for solving the RTE is the order of picosecond because the light propagates through the biological tissue generally has the same order of time scale. However, at such time scales, the contribution of the diffusion term, blood perfusion term, and metabolic heat generation term (the first three terms on the right-hand side of Eq. (2)) to the increment of the temperature of the tissue are insignificant as compared to the divergence of the radiative heat flux. However, the time scale of the order of the millisecond is generally used for solving the bio-heat transfer equation to capture the effect of these terms. Therefore, it is a multi-time scale problem because the two different time scales have been used to solve the RTE and bio-heat transfer equation. The solution procedure for solving this multi-time scale problem has been discussed in Section 2.
