Analytic evaluation of diffuse fluence error in multi-layer scattering media with discontinuous refractive index

A simple analytic method of estimating the error involved in using an approximate boundary condition for diffuse radiation in two adjoining scattering media with differing refractive index is presented. The method is based on asymptotic planar fluences and enables the relative error to be readily evaluated without recourse to Monte Carlo simulation. Three examples of its application are considered: (1) evaluating the error in calculating the diffuse fluences at a boundary between two media with differing refractive index and dissimilar scattering properties (2) the dependence of the relative error in a multilayer medium with discontinuous refractive index on the ratio of the reduced scattering coefficient to the absorption coefficient ms'/ma (3) the parametric dependence of the error in the radiant flux Js at the surface of a three-layer medium. The error is significant for strongly forward biased scattering media with non-negligible absorption and is cumulative in multi-layered media with refractive index increments between layers.Comment: 21 pages 7 Figures Text further revise

Two-step verification method for Monte Carlo codes in biomedical optics applications

Significance: Code verification is an unavoidable step prior to using a Monte Carlo (MC) code. Indeed, in biomedical optics, a widespread verification procedure for MC codes is still missing. Analytical benchmarks that can be easily used for the verification of different MC routines offer an important resource.Aim: We aim to provide a two-step verification procedure for MC codes enabling the two main tasks of an MC simulator: (1) the generation of photons' trajectories and (2) the intersections of trajectories with boundaries separating the regions with different optical properties. The proposed method is purely based on elementary analytical benchmarks, therefore, the correctness of an MC code can be assessed with a one-sample t-test.Approach: The two-step verification is based on the following two analytical benchmarks: (1) the exact analytical formulas for the statistical moments of the spatial coordinates where the scattering events occur in an infinite medium and (2) the exact invariant solutions of the radiative transfer equation for radiance, fluence rate, and mean path length in media subjected to a Lambertian illumination.Results: We carried out a wide set of comparisons between MC results and the two analytical benchmarks for a wide range of optical properties (from non-scattering to highly scattering media, with different types of scattering functions) in an infinite non-absorbing medium (step 1) and in a non-absorbing slab (step 2). The deviations between MC results and exact analytical values are usually within two standard errors (i.e., t-tests not rejected at a 5% level of significance). The comparisons show that the accuracy of the verification increases with the number of simulated trajectories so that, in principle, an arbitrary accuracy can be obtained.Conclusions: Given the simplicity of the verification method proposed, we envision that it can be widely used in the field of biomedical optics.</p

Perturbative forward solver software for small localized fluorophores in tissue

In this paper a forward solver software for the time domain and the CW domain based on the Born approximation for simulating the effect of small localized fluorophores embedded in a non-fluorescent biological tissue is proposed. The fluorescence emission is treated with a mathematical model that describes the migration of photons from the source to the fluorophore and of emitted fluorescent photons from the fluorophore to the detector for all those geometries for which Green’s functions are available. Subroutines written in FORTRAN that can be used for calculating the fluorescent signal for the infinite medium and for the slab are provided with a linked file. With these subroutines, quantities such as reflectance, transmittance, and fluence rate can be calculated

Fractional Schrödinger Equation in the Presence of the Linear Potential

In this paper, we consider the time-dependent Schrödinger equation: i ∂ ψ ( x , t ) ∂ t = 1 2 ( − Δ ) α 2 ψ ( x , t ) + V ( x ) ψ ( x , t ) , x ∈ R , t &gt; 0 with the Riesz space-fractional derivative of order 0 &lt; α ≤ 2 in the presence of the linear potential V ( x ) = β x . The wave function to the one-dimensional Schrödinger equation in momentum space is given in closed form allowing the determination of other measurable quantities such as the mean square displacement. Analytical solutions are derived for the relevant case of α = 1 , which are useable for studying the propagation of wave packets that undergo spreading and splitting. We furthermore address the two-dimensional space-fractional Schrödinger equation under consideration of the potential V ( ρ ) = F · ρ including the free particle case. The derived equations are illustrated in different ways and verified by comparisons with a recently proposed numerical approach

Analytische Lösungen der Strahlungstransfergleichung und ihrer Approximationen

This thesis presents the derivation of analytical expressions of the radiative transfer equation and their approximations in the steady-state and time domains. The diffusion equation, which is the most often used approximation of the radiative transfer equation, is solved analytically for different homogeneous and layered geometries. By applying the integral transform formalism the respective form of the diffusion equation is reduced to an ordinary differential equation. At this stage the solution of the boundary value problem is obtained via standard techniques. The derived solutions were validated against other independent analytical methods found in literature and the Monte Carlo method. Analytical solutions of the simplified spherical harmonics equations are derived for homogeneous media. The Fourier transform is used to convert these coupled diffusion-like equations to a system of ordinary differential equations. The obtained solutions for infinite and semi-infinite media were validated against the finite difference method. The infinite space fluence within the transport theory is derived for an anisotropically scattering medium and different source distributions. Additionally, for the case of isotropic scattering a simplified version is given. Based on results of the infinite space fluence and by applying the eigenvalue method the radiance caused by an isotropic point source in an infinitely extended medium is obtained. The analytical solutions were compared to the Monte Carlo method. Within the stochastic nature of the Monte Carlo simulations an exact agreement was found in the steady-state and time domains