Simplified Spherical Harmonics Equations Research Articles

Multigroup form of simplified spherical harmonics (SPN) equations for neutron transport

SPN equations offer an attractive and superior alternative to standard diffusion approximation of neutron transport equation. From the standpoint of numerical solvers, SPN equations significantly reduce the computational cost as compared to solving full PN equations. This makes it more appealing for analysis of advanced reactor designs. Recently, we have reported the mathematically consistent derivation of second order differential form of SPN equations utilizing the trial function proposed by Pomraning. We also improved the conventional SPN model by deriving appropriate interface and boundary conditions. Further, we formally demonstrated that the new SPN model is identical to the zeroth layer of newly proposed GSPN model. However, the mathematical framework established so far was restricted to one speed neutrons. In the current work, we extend our SPN model to multigroup transport formalism. We also derive the relevant interface and boundary conditions.

On Simplified Spherical Harmonics (SPN) and Generalized SPN (GSPN) formulations

Simplified PN (SPN) equations are derived based on variational method using a modified version of ansatz originally proposed by Pomraning. New boundary and interface conditions are derived for the SPN equations using the corresponding angular flux expression. The equivalence of equations, thus derived, along with its interface and boundary conditions with those of a specific case of Generalized SPN (GSPN), put forward by Chao, is brought out.

Time-dependent simplified spherical harmonics formulations for a nuclear reactor system

The steady-state simplified spherical harmonics equations (SPN equations) are a higher order approximation to the neutron transport equations than the neutron diffusion equation that also have reasonable computational demands. This work extends these results for the analysis of transients by comparing of two formulations of time-dependent SPN equations considering different treatments for the time derivatives of the field moments. The first is the full system of equations and the second is a diffusive approximation of these equations that neglects the time derivatives of the odd moments. The spatial discretization of these methodologies is made by using a high order finite element method. For the time discretization, a semi-implicit Euler method is used. Numerical results show that the diffusive formulation for the time-dependent simplified spherical harmonics equations does not present a relevant loss of accuracy while being more computationally efficient than the full system.

A block Arnoldi method for the SPN equations

ABSTRACTThe simplified spherical harmonics equations are a useful approximation to the stationary neutron transport equation. The eigenvalue problem associated with them is a challenging problem from the computational point of view. In this work, we take advantage of the block structure of the involved matrices to propose the block inverse-free preconditioned Arnoldi method as an efficient method to solve this eigenvalue problem. For the spatial discretization, a continuous Galerkin finite element method implemented with a matrix-free technique is used to keep reasonable memory demands. A multilevel initialization using linear shape functions in the finite element method is proposed to improve the method convergence. This initialization only takes a small percentage of the total computational time. The proposed eigenvalue solver is compared to the standard power iteration method, the Krylov-Schur method and the generalized Davidson method. The numerical results show that it reduces the computational time to solve the eigenvalue problem.

Solving analytically the simplified spherical harmonics equations in cylindrical turbid media.

A methodology is presented for analytically solving simplified spherical harmonics equations (SPN) in a finite homogeneous absorbing and scattering cylindrical medium. The SPN equations are a reliable approximation to the radiative transfer equation for describing light propagation inside turbid media. The equations consist of a set of coupled partial differential equations (PDEs). The analytical solution developed here is for a steady-state isotropic point source located at an arbitrary point inside a cylindrical turbid medium. Partial-reflection boundary conditions are considered, as they realistically model the refractive index mismatch between a turbid medium and its surroundings (air), as occurs in practice in biomedical optics. The eigen method is used to decouple the set of SPN PDEs. The methodology is applied to the SP3, which has proved to be sufficiently accurate in practice, but it is readily generalizable to higher orders. The solution is compared with the analytical solution of the diffusion equation as well as to gold standard Monte Carlo simulations for validation, against which it shows good agreement. This work is important, as it provides an additional tool for validating numerical solutions of SPN equations for curved geometries, namely, cylindrical shapes, which are often used in practice.

Analytical solution of the simplified spherical harmonics equations in spherical turbid media

We present for the first time an analytical solution for the simplified spherical harmonics equations (so-called SPN equations) in the case of a steady-state isotropic point source inside a spherical homogeneous absorbing and scattering medium. The SPN equations provide a reliable approximation to the radiative transfer equation for describing light transport inside turbid media. The SPN equations consist of a set of coupled partial differential equations and the eigen method is used to obtain a set of decoupled equations, each resembling the heat equation in the Laplace domain. The equations are solved for the realistic partial reflection boundary conditions accounting for the difference in refractive indices between the turbid medium and its environment (air) as occurs in practical cases of interest in biomedical optics. Specifically, we provide the complete solution methodology for the SP3, which is readily applicable to higher orders as well, and also give results for the SP5. This computationally easy to obtain solution is investigated for different optical properties of the turbid medium. For validation, the solution is also compared to the analytical solution of the diffusion equation and to gold standard Monte Carlo simulation results. The SP3 and SP5 analytical solutions prove to be in good agreement with the Monte Carlo results. This work provides an additional tool for validating numerical solutions of the SPN equations for curved geometries.

Analytical Green's function for the fluorescence simplified spherical harmonics equations in turbid medium.

It is more complicated to write the analytical expression for the fluorescence simplified spherical harmonics ( SPN) equations in a turbid medium, since both the processes of the excitation and emission light and the composite moments of the fluence rate are described by coupled equations. Based on an eigen-decomposition strategy and the well-developed analytical methods of diffusion approximation (DA), we derive the analytical solutions to the fluorescence SPN equations for regular geometries using the Green's function approach. By means of comparisons with the results of fluorescence DA and Monte Carlo simulations, we have shown the effectiveness of our proposed method and the expected advantages of the SPN equations in the case of small source-detector separation and high absorption.

On radiative transfer using synthetic kernel and simplified spherical harmonics methods in linearly anisotropically scattering media

The Synthetic Kernel (SKN) method is employed to a 3D absorbing, emitting and linearly anisotropically scattering inhomogeneous medium. Standard SKN approximation is applied only to the diffusive components of the radiative transfer equations. An alternative SKN (SKN⁎) method is also derived in full 3-D generality by extending the approximation to the direct wall contributions. Complete sets of boundary conditions for both SKN approaches are rigorously obtained. The simplified spherical harmonics (P2N−1 or SP2N−1) and simplified double spherical harmonics (DPN−1 or SDPN−1) equations for linearly anisotropically scattering homogeneous medium are also derived. Resulting full P2N−1 and DPN−1 (or SP2N−1 and SDPN−1) equations are cast as diagonalized second order coupled diffusion-like equations. By this analysis, it is shown that the SKN method is a high-order approximation, and simply by the selection of full or half range Gauss–Legendre quadratures, SKN⁎ equations become identical to P2N−1 or DPN−1 (or SP2N−1 or SDPN−1) equations. Numerical verification of all methods presented is carried out using a 1D participating isotropic slab medium. The SKN method proves to be more accurate than SKN⁎ approximation, but it is analytically more involved. It is shown that the SKN⁎ with proposed BCs converges with increasing order of approximation, and the BCs are applicable to SPN or SDPN methods.

Adaptive hp finite element method for fluorescence molecular tomography with simplified spherical harmonics approximation

Recently, the simplified spherical harmonics equations (SPN) model has attracted much attention in modeling the light propagation in small tissue geometries at visible and near-infrared wavelengths. In this paper, we report an efficient numerical method for fluorescence molecular tomography (FMT) that combines the advantage of SPN model and adaptive hp finite element method (hp-FEM). For purposes of comparison, hp-FEM and h-FEM are, respectively applied to the reconstruction process with diffusion approximation and SPN model. Simulation experiments on a 3D digital mouse atlas and physical experiments on a phantom are designed to evaluate the reconstruction methods in terms of the location and the reconstructed fluorescent yield. The experimental results demonstrate that hp-FEM with SPN model, yield more accurate results than h-FEM with diffusion approximation model does. The phantom experiments show the potential and feasibility of the proposed approach in FMT applications.

Analytical solutions to the simplified spherical harmonics equations using eigen decompositions

We develop a modified method to simplify the analytical solutions to the simplified spherical harmonics equations (SP(N)). The scheme decouples the SP(N) partial differential equations into independent equations using eigen decompositions and calculates the Green's function of the photon migrations based on the eigenvectors and eigenvalues. In contrast to the established solutions that are based on the original coupled equations, the proposed derivation is theoretically concise and universally extendable to other regular geometries. We validate the proposed method in comparison with Monte-Carlo simulations for an infinite scattering medium and a circular geometry as an example of the boundary value problems.

Diffuse optical tomographic imaging of biological media by time-dependent parabolic SPN equations: a two-dimensional study

We investigate the problem of retrieving the optical properties (absorption and scattering) of biological tissue from a set of optical measurements. A diffuse optical tomography (DOT) algorithm that incorporates constrained optimization methods is implemented. To improve image quality, the DOT algorithm exploits full time-domain data. The time-dependent parabolic simplified spherical harmonics equations (TD-pSPN) are used as the forward model. Time-dependent adjoint variables are resorted to in the calculation of the gradient of the objective function. Several numerical experiments for small geometric media with embedded inclusions that mimic small animal imaging are performed. In the experiments, optical coefficient values are varied in the range of realistic values for the near-infrared spectrum, including high absorption values. Single and multiparameter reconstructions are performed with the diffusion equation and higher orders of the TD-pSPN equations. The results suggest the DOT algorithm based on the TD-pSPN model outperforms the DE, and accurately reconstructs optical parameter distributions of biological media both spatially and quantitatively.

Comparison between radiative transfer theory and the simplified spherical harmonics approximation for a semi-infinite geometry

In this study, the third-order simplified spherical harmonics equations (SP3), an approximation of the radiative transfer equation, are solved for a semi-infinite geometry considering the exact simplified spherical harmonics boundary conditions. The obtained Green's function is compared to radiative transfer calculations and the diffusion theory. In general, it is shown that the SP3 equations provide better results than the diffusion approximation in media with high absorption coefficient values but no improvement is found for small distances to the source.

On simplified spherical harmonics equations for the radiative transfer equation

The radiative transfer equation (RTE) arises in a wide range of applications in sciences and engineering. Due to the high dimension and complicated form of the RTE, it is challenging to solve the equation directly. In the literature, several approximation methods were developed for the RTE. One approximation method, the simplified spherical harmonics (SP N ) method, provides an efficient way to generate good approximate solutions of the RTE with high absorption and small geometries. The main purpose of the paper is to study well-posedness of the simplified spherical harmonics system. The weak formulation used in the proof of the solution existence and uniqueness provides a starting point for developing the Galerkin finite element method to solve the simplified spherical harmonics system.

Light propagation from fluorescent probes in biological tissues by coupled time-dependent parabolic simplified spherical harmonics equations: erratum

An exponent was missing in the denominator of the definition of the Henyey-Greenstein phase function. Besides, the normalization constant of the phase function was presented for the three dimensional case only. We give the correct expression in this erratum for the two- and three-dimensional cases. In addition, we correct for a forgotten scalar product operator in Eq. (15). Those corrections do not alter the results or the conclusions of the paper.

Light propagation from fluorescent probes in biological tissues by coupled time-dependent parabolic simplified spherical harmonics equations

We introduce a system of coupled time-dependent parabolic simplified spherical harmonic equations to model the propagation of both excitation and fluorescence light in biological tissues. We resort to a finite element approach to obtain the time-dependent profile of the excitation and the fluorescence light fields in the medium. We present results for cases involving two geometries in three-dimensions: a homogeneous cylinder with an embedded fluorescent inclusion and a realistically-shaped rodent with an embedded inclusion alike an organ filled with a fluorescent probe. For the cylindrical geometry, we show the differences in the time-dependent fluorescence response for a point-like, a spherical, and a spherically Gaussian distributed fluorescent inclusion. From our results, we conclude that the model is able to describe the time-dependent excitation and fluorescent light transfer in small geometries with high absorption coefficients and in nondiffusive domains, as may be found in small animal diffuse optical tomography (DOT) and fluorescence DOT imaging.

Excitation-resolved fluorescence tomography with simplified spherical harmonics equations

Fluorescence tomography (FT) reconstructs the three-dimensional (3D) fluorescent reporter probe distribution inside biological tissue. These probes target molecules of biological function, e.g. cell surface receptors or enzymes, and emit fluorescence light upon illumination with an external light source. The fluorescence light is detected on the tissue surface and a source reconstruction algorithm based on the simplified spherical harmonics (SPN) equations calculates the unknown 3D probe distribution inside tissue. While current FT approaches require multiple external sources at a defined wavelength range, the proposed FT method uses only a white light source with tunable wavelength selection for fluorescence stimulation and further exploits the spectral dependence of tissue absorption for the purpose of 3D tomographic reconstruction. We will show the feasibility of the proposed hyperspectral excitation-resolved fluorescence tomography method with experimental data. In addition, we will demonstrate the performance and limitations of such a method under ideal and controlled conditions by means of a digital mouse model and synthetic measurement data. Moreover, we will address issues regarding the required amount of wavelength intervals for fluorescent source reconstruction. We will explore the impact of assumed spatially uniform and nonuniform optical parameter maps on the accuracy of the fluorescence source reconstruction. Last, we propose a spectral re-scaling method for overcoming the observed limitations in reconstructing accurate source distributions in optically non-uniform tissue when assuming only uniform optical property maps for the source reconstruction process.

Analytical solutions of the simplified spherical harmonics equations

We derived analytical solutions of the simplified spherical harmonics equations, an approximation of the radiative transfer equation, for infinitely extended scattering media. The derived equations are simple (sum of exponential functions) and quickly evaluated. We compared the solutions with Monte Carlo simulations in the steady-state and time domains and found much better agreement compared to solutions of the diffusion equation, especially for large absorption coefficients, short time values, and small distances from the source.

Light transport in biological tissue using three-dimensional frequency-domain simplified spherical harmonics equations

The accuracy of the commonly used diffusion approximation as used in diffuse optical tomography is known to be limited in cases involving strong absorption and in these situations a higher ordered approximation is necessary. In this study, a light transport model has been developed based upon the three-dimensional frequency-domain simplified spherical harmonics (SPN) approximation for orders up to N = 7. The SPN data are tested against a semi-infinite multi-layered Monte Carlo model. It has been shown that the SPN approximation for higher orders (N >1) provides an increase in accuracy over the diffusion equation specifically near sources and at boundaries of regions with increased optical absorption. It is demonstrated that the error of fluence calculated near the sources between the diffusion approximation and the SPN model (N = 7) can be as large as 60%, therefore limiting the use of the diffusion approximation for small animal imaging and in situations where optical changes near sources are critical for tomographic reconstructions.

Light transport in biological tissue based on the simplified spherical harmonics equations

In this work, we demonstrate the validity of the simplified spherical harmonics equations to approximate the more complicated equation of radiative transfer for modeling light propagation in biological tissue. We derive the simplified spherical harmonics equations up to order N = 7 for anisotropic scattering and partially reflective boundary conditions. We compare numerical results with diffusion and discrete ordinates transport solutions. We find that the simplified spherical harmonics methods significantly improve the diffusion solution in transport-like domains with high absorption and small geometries, and are computationally less expensive than the discrete ordinates transport method. For example, the simplified P 3 method is approximately two orders of magnitude faster than the discrete ordinates transport method, but only 2.5 times computationally more demanding than the diffusion method. We conclude that the simplified spherical harmonics methods can accurately model light propagation in small tissue geometries at visible and near-infrared wavelengths, yielding transport-like solutions with only a fraction of the computational cost of the transport calculation.

Simplified Spherical Harmonics in the Variational Nodal Method

The multigroup simplified spherical harmonics equations with anisotropic scattering are derived from a variational principle that preserves nodal balance. The resulting equations are discretized using a Ritz procedure with spatial trial functions that are complete polynomials within the nodes and on the interfaces. The resulting equations are cast in a response matrix form and incorporated as an option of the variational nodal spherical harmonics code VARIANT. Fixed source and multigroup eigenvalue calculations are performed on benchmark problems. The accuracy and computational efficiency of spherical harmonic and simplified spherical harmonic approximations are compared, and the compensating effects of spatial and angular truncation errors are examined. The results indicate that in most situations, simplified and standard spherical harmonics results of the same order are in close agreement, while the use of simplified spherical harmonics substantially reduces computing costs.