Modeling and numerical simulation of light propagation through biological tissue with implanted structures

This dissertation proposes several numerical techniques for simulating laser pulse propagation through biological tissue with implants for sensing applications. The purpose of these implants is to enhance and condition the optical signals for better detection of the received signal. This work contributes to the development of methods for sensing and characterization of tissue properties and measuring concentrations of substances in blood or tissue fluid, thus making it possible to monitor these concentrations and detect anomalies. The research was carried out in three major stages. In stage 1, a technique for simulating laser pulse propagation through tissue, which addresses some of the drawbacks of existing methods, was developed. The outcome of this stage was an efficient algorithm for solving the transient photon transport equation (PTE), which governs light propagation through tissue. The proposed algorithm was first implemented for the one-dimensional case and later extended for the two- and three-dimensional cases. This algorithm was also extended to inhomogeneous media. The one-dimensional PTE is an integro-differential equation of four variables: distance, local zenith angle, local azimuthal angle and time. First, the original PTE was mapped to a moving reference frame co-moving with the incident pulse. This transformation eliminated the partial derivative term with respect to time in the original equation. The dependence on the local azimuthal angle was then removed using the discrete ordinates method, which resulted in a set of coupled three-variable integro-differential equations. A Laguerre expansion was then used to represent the time dependency of this reduced PTE. With the Laguerre expansion, any arbitrary input pulse shape can be represented using a few polynomials, and also the causality is preserved. This step resulted in a two-variable integro-differential equation for each Laguerre coefficient. The dependence on the local zenith angle was removed by the use of the discrete ordinates method, thus resulting in a set of single-variable uncoupled differential equations. The Runge-Kutta-Fehlberg (RKF) method was then used to solve for the radiance. In the proposed technique, all the sampling points in the time domain were obtained in a single execution of the algorithm, rather than having repeated executions for each time step as in time marching techniques used in most of the existing solution methods. This was made possible by expanding the time dependence using a Laguerre basis, thus making the proposed algorithm much faster when the intensity profile is required at a particular point or on a plane over a time interval. Also, since the RKF method was used to solve the final reduced equation, intensity profiles at several points and planes over the whole time spectrum were obtained in one execution of the algorithm. In addition, the causality of the system was implicitly imposed by the causal Laguerre polynomials. The use of the Runge-Kutta-Fehlberg method with respect to the spatial variable makes the extension to inhomogeneous media simple and straightforward. For the multi-dimensional cases, the Laguerre expansion was used to represent the time dependency as in the one-dimensional case. The discrete ordinates method was then used to solve for the radiance using a finite volume approach. In stage 2, a technique for mapping the photon transport equation to Maxwell's equations was developed. No work has been reported to date which addresses the problem of coupling the photon transport equation to Maxwell's equations. Since light propagation through tissue is modeled using the PTE and that through implants is modeled using Maxwell's equations, this mapping was required for simulating light propagation through tissue with implanted structures. The PTE solves for the radiance only. However, Maxwell's equations require electric and magnetic fields along with their phases. Therefore, the radiance profile obtained by solving the PTE had to be converted to an electromagnetic field, which involves constructing the phase from the radiance profile. For this purpose, the transport-of-intensity equation was solved using the full multigrid algorithm. In the final stage, the numerical simulation of laser pulse propagation through biological tissue with implanted structures for sensing applications was carried out. Even though implanted structures within biological tissue have very useful and promising applications in the field of biomedical engineering, no work on the theoretical analysis and simulation of such compound structures has been reported in the research literature. In this dissertation, two examples, a metal screen with a slit implanted in tissue and a photonic crystal structure implanted in tissue, are considered. These simulations were carried out by integrating the work carried out in stages 1 and 2. The algorithm developed in stage 1 for solving the PTE was applied to simulate pulse propagation through the tissue layers. At the tissue-implant interface, the mapping of the PTE to Maxwell's equations, developed in stage 2, was applied. Electromagnetic propagation through the implanted structure was modeled using Maxwell's equations.