In the field of photon migration in turbid media, different Monte Carlo methods are usually employed to solve the radiative transfer equation. We consider four different Monte Carlo methods, widely used in the field of tissue optics, that are based on four different ways to build photons’ trajectories. We provide both theoretical arguments and numerical results showing the statistical equivalence of the four methods. In the numerical results we compare the temporal point spread functions calculated by the four methods for a wide range of the optical properties in the slab and semi-infinite medium geometry. The convergence of the methods is also briefly discussed.

Using diffusion theory, we show that a dual-slope method is more effective than single-slope methods or single-distance methods at enhancing sensitivity to deeper tissue. The dual-slope method requires a minimum of two sources and two detectors arranged in specially configured arrays. In particular, we present diffusion theory results for a symmetrical linear array of two sources (separated by 55 mm) that sandwich two detectors (separated by 15 mm), for which dual slopes achieve maximal sensitivity at a depth of about 5 mm for direct current (DC) intensity (as measured in continuous-wave spectroscopy) and 11 mm for phase (as measured in frequency-domain spectroscopy) under typical values of the tissue optical properties (absorption coefficient: $\sim\!0.01\,\,{\rm mm}^{ - 1}$, reduced scattering coefficient: $\sim\!1\,\,{\rm mm}^{ - 1}$). This result is a major advance over single-distance or single-slope data, which feature maximal sensitivity to shallow tissue ($ \lt\!{{2}}\,\,{\rm{mm}}$ for the intensity, $ \lt\!{{5}}\,\,{\rm{mm}}$ for the phase).

We present a two-step Monte Carlo (MC) method that is used to solve the radiative transfer equation in heterogeneous turbid media. The method exploits the one-to-one correspondence between the seed value of a random number generator and the sequence of random numbers. In the first step, a full MC simulation is run for the initial distribution of the optical properties and the “good” seeds (the ones leading to detected photons) are stored in an array. In the second step, we run a new MC simulation with only the good seeds stored in the first step, i.e., we propagate only detected photons. The effect of a change in the optical properties is calculated in a short time by using two scaling relationships. By this method we can increase the speed of a simulation up to a factor of 1300 in typical situations found in near-IR tissue spectroscopy and diffuse optical tomography, with a minimal requirement for hard disk space. Potential applications of this method for imaging of turbid media and the inverse problem are discussed.

This work presents the design and validation of an instrument for dual-slope broadband diffuse reflectance spectroscopy. This instrument affords calibration-free, continuous-wave measurements of broadband absorbance of optically diffusive media, which may be translated into absolute absorption spectra by adding frequency-domain measurements of scattering at two wavelengths. An experiment on a strongly scattering liquid phantom (milk, water, dyes) confirms the instrument’s ability to correctly identify spectral features and measure absolute absorption. This is done by sequentially adding three dyes, each featuring a distinct spectral absorption, to the milk/water phantom. After each dye addition, the absorption spectrum is measured, and it is found to reproduce the spectral features of the added dye. Additionally, the absorption spectrum is compared to the absorption values measured with a commercial frequency-domain instrument at two wavelengths. The measured absorption of the milk/water phantom quantitatively agrees with the known water absorption spectrum ($R^2=0.98$), and the measured absorption of the milk/water/dyes phantom quantitatively agrees with the absorption measured with the frequency-domain instrument in six of eight cases. Additionally, the measured absorption spectrum correctly recovers the concentration of one dye, black India ink, for which we could accurately determine the extinction spectrum (i.e., the specific absorption per unit concentration). The instrumental methods presented in this work can find applications in quantitative spectroscopy of optically diffusive media, and particularly in near-infrared spectroscopy of biological tissue.

We approach the perturbative solution to the diffusion equation for the case of absorbing inclusions embedded in a heterogeneous scattering medium by using general properties of the radiative transfer equation and the solution of the Fredholm equation of the second kind given by the Neumann series. The terms of the Neumann series are used to obtain the expression of the moments of the generalized temporal point-spread function derived in transport theory. The moments are calculated independently by using Monte Carlo simulations for validation of the theory. While the mixed moments are correctly derived from the theory by using the solution of the diffusion equation in the geometry of interest, in order to obtain the self moments we should reframe the problem in transport theory and use a suitable solution of the radiative transfer equation for the calculation of the multiple integrals of the corresponding Neumann series. Since the rigorous theory leads to impractical formulas, in order to simplify and speed up the calculation of the self moments, we propose a heuristic method based on the calculation of only a single integral and some scaling parameters. We also propose simple quadrature rules for the calculation of the mixed moments for speeding up the computation of perturbations due to multiple defects. The theory can be developed in the continuous-wave domain, the time domain, and the frequency domain. In a companion paper [J. Opt. Soc. Am. A 23, 2119 (2006) ] we discuss the conditions of applicability of the theory in practical cases found in diffuse optical imaging of biological tissues.