This paper extends our work on applying the Finite Element Method (FEM) to the propagation of light in tissue. We address herein the topics of boundary conditions and source specification for this method. We demonstrate that a variety of boundary conditions stipulated on the Radiative Transfer Equation can be implemented in a FEM approach, as well as the specification of a light source by a Neumann condition rather than an isotropic point source. We compare results for a number of different combinations of boundary and source conditions under FEM, as well as the corresponding cases in a Monte Carlo model.

The use of optical radiation in medical physics is important in several fields for both treatment and diagnosis. In all cases an analytic and computable model of the propagation of radiation in tissue is essential for a meaningful interpretation of the procedures. A finite element method (FEM) for deriving photon density inside an object, and photon flux at its boundary, assuming that the photon transport model is the diffusion approximation to the radiative transfer equation, is introduced herein. Results from the model for a particular case are given: the calculation of the boundary flux as a function of time resulting from a delta-function input to a two-dimensional circle (equivalent to a line source in an infinite cylinder) with homogeneous scattering and absorption properties. This models the temporal point spread function of interest in near infrared spectroscopy and imaging. The convergence of the FEM results are demonstrated, as the resolution of the mesh is increased, to the analytical expression for the Green's function for this system. The diffusion approximation is very commonly adopted as appropriate for cases which are scattering dominated, i.e., where mu s >> mu a, and results from other workers have compared it to alternative models. In this article a high degree of agreement with a Monte Carlo method is demonstrated. The principle advantage of the FE method is its speed. It is in all ways as flexible as Monte Carlo methods and in addition can produce photon density everywhere, as well as flux on the boundary. One disadvantage is that there is no means of deriving individual photon histories.

In order to quantify near-infrared spectroscopic (NIRS) data on an inhomogeneous medium, knowledge of the contribution of the various parts of the medium to the total NIRS signal is required. This is particularly true in the monitoring of cerebral oxygenation by NIRS, where the contribution of the overlying tissues must be known. The concept of the time point spread function (TPSF), which is used extensively in NIRS to determine the effective optical pathlength, is expanded to the more general inhomogeneous case. This is achieved through the introduction of the partial differential pathlength, which is the effective optical pathlength in the inhomogeneous medium, and an analytical proof of the applicability of the modified Beer-Lambert law in an inhomogeneous medium is shown. To demonstrate the use of partial differential pathlength, a Monte Carlo simulation of a two-concentric-sphere medium representing a simplified structure of the head is presented, and the possible contribution of the overlying medium to the total NIRS signal is discussed.

The optical properties of samples of bone from pig skull have been measured over the wavelength range 650-950 nm. The scattering phase function was measured on thin samples of the bone using a goniometer, and a value for the mean cosine g, of the scattering angle, was calculated. The scattering and absorption coefficients, mu s and mu a were then determined from measurements of diffuse reflectance and transmittance made with a pair of integrating spheres, by a step-wise search through a table of diffuse reflectance and transmittance versus mu a and mu s generated by a Monte Carlo model incorporating the measured scattering phase function. Values for g measured on six samples varied from 0.925 +/- 0.014 at 650 nm to 0.945 +/- 0.013 at 950 nm. Corresponding values for mu a and mu s measured on 18 samples were mu a = 0.04 +/- 0.002 mm-1, mu s = 35 +/- 0.7 mm-1 at 650 nm to mu a = 0.05 +/- 0.002 mm-1, mu s = 24 +/- 0.6 mm-1 at 950 nm.

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In this paper we show how to derive the mean and variance of the transilluminated signal obtained in models of light propagation in tissue, based both on a stochastic Monte Carlo method and on a deterministic diffusion approximation. The theoretical treatment of the Monte Carlo model applies only to integrated intensity measurements, whereas the diffusion approximation gives an estimator for the time-dependent case as well. We present results that show the accurate prediction of Monte Carlo statistics, and propose that the diffusion approximation is therefore a suitable mechanism for incorporating noise into modelling procedures.

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We present a Finite Element (FE) model for calculation of photon propagation in highly scattering tissues. The model can either be used for time domain measurements, where the temporal distribution of transmitted light after an ultra short input pulse is measured, or for frequency domain measurements, where the input is a frequency modulated light source, and the phase shift and modulation depth between the input and output signal are measured.The FE model is used on inhomogeneous objects to investigate the effect of scattering and absorbing inhomogeneities on boundary measurements in both the time and frequency domain. The time and frequency versions are validated by comparing the results with data from analytical calculafions and from a Monte-Carlo model. Experimental mean time of flight measurements made on a tissue equivalent phantom are compared with the model data. The experimental setup consists of a cylindrical homogeneous tissue phantom containing scattering and absorbing rods. The input pulses generated by a picosecond laser are inserted into the sample at 16 equally spaced surface positions consecutively. The response signals are measured at 16 intermediate locations using a streak camera.By comparing data from a two-dimensional and a three-dimensional FE model we derive a conversion factor for integrated intensity and mean time of flight that will permit the reconstruction of 3D data from a cylinder using a 2D FE model. The reconstruction is demonstrated on data generated with the FE forward model. INTRODUCFIONNear infrared spectroscopy offers the possibility to monitor blood oxygenation and metabolic rates noninvasively in vivo by measuring absorption changes'. This is done by delivering laser light to one point on the surface of a patient, and detecting the transmitted signal collected from another surface point at a distance of several centimetres. In order to quantitate optical pathlength in the tissues, ultrashort pulses of light are employed and the mean time of flight of the exiting light measured. The shape of this impulse response function (the time point spread function, TPSF) depends on the tissue absorption and scauenng properties. Absorption causes attenuation and shortening of the TPSF, while scattering leads to broadening and delay of the TPSF, as the photon pathlengths increase. Biological tissues are generally scatter-dominated, and the mean photon pathlength is several times the geometric distance between the light input and output position. 0-8194-11 15-9/93/$6.00 SPIE Vol. 1888 / 179 Downloaded From: http://proceedings.spiedigitallibrary.org/ on 05/14/2015 Terms of Use: http://spiedl.org/terms