Radiative Transport Equation Case Study

The surgical technique of laser induced photo thermal therapy, used for selective destruction of cancerous cells inside the body of a tissue phantom, considers employing a laser source to target radiation on a tumor region, for the purposes of inducing a local hyperthermia to destroy the cancerous cells. It has been observed that upon raising the cell temperature beyond 43°C [1] for a specified amount of time, necrosis of the biological cells can be achieved. Further, coupled with the temperature rise, successful destruction of the cancerous cells also requires a specified exposer tissue [2]. Robinson et al. [3] observed that a local temperature rise beyond 53°C for an exposure time of up to 1 s would ensure a complete destruction of the cancerous
The turbid nature of the tissue phantoms coupled with the fact that the biological tissues are porous in nature consisting of numerous blood vessels of various diameters, make the job of determination of the laser intensity distributions extremely difficult. The Radiative Transport Equation (RTE) developed by Chandrasekhar [4] has been employed in the present work to solve for the light transfer inside the tissue phantoms. The RTE model, developed by Chandrasekhar [4] has been developed by considering a balance of photons inside a control volume along all possible directions of travel, by including the effects of various optical phenomenon such as emission, absorption, in-scattering and out-scattering. The balance of photons from all the possible events results in the generation of an integro-differential equation. The analytical solutions are extremely difficult to obtain, and are possibly only for situations which involve simple geometry and boundary conditions. Thus, RTE has to be solved by employing suitable numerical methods. Various researchers have developed numerical methods for solving the RTE based on techniques of Finite Volume Method [5], Discrete Ordinates Method [6], Discrete Transfer Method [7] and Monte Carlo based statistical technique [8].
The energy equations which predict the temperature fields inside tissue phantoms are commonly referred to as the bio-heat transfer equations, as they involve various terms associated with the biological heat transfer processes occurring inside a tissue specimen. The light intensity distributions obtained from solving the RTE, upon coupling with an appropriate from of the bio-heat transfer would result in the determination of the resultant temperature inside the tissue phantoms under the influence of irradiation. Thus it is critical to select a bio-heat transfer equation which would represent the physical nature of the tissue along with all the heat transfer interactions inside the biological tissue phantoms. Many forms of the bio-heat transfer equations were proposed by numerous researchers to account for the different anatomical & bio-physical heat transfer aspects inside the tissue. Each of these developed models is associated with its inherent advantages and drawbacks. The earliest form of the bio-heat transfer equation was developed by Penne [12], who considered the tissue to be a homogenous solid body with blood perfusing inside the body of the tissue. It was assumed that the temperature of the blood perfusing was equal to the venous blood