Diffusion Equation Approximation Research Articles

Investigation and comparison on the interaction of laser and bio-tissue based on two classical models, the beam broaden model and diffusion approximation equation model, theoretically and experimentally

Three methods are used for a numerical solution, the Monte Carlo method, diffusion approximation equation model, and beam broaden model based on Beer–Lambert’s law equation. The comparison between the first two methods is reported theoretically, and the latter is a better choice in the high-density tissue. However, the comparison between the third method and the first or the second method is rarely reported. Two classical theoretical models describing the interaction between the laser and the bio-tissue are analyzed and compared to determine which is more suitable for analyzing the interaction, the beam broaden model or diffusion approximation equation model. Intensity distribution is simulated and compared for the two models. Temperature distribution and thermal damage are investigated theoretically and experimentally for both models. The differences and the reasons are analyzed. The diffusion approximation equation model is more suitable for analyzing the mechanism between the laser and the bio-tissue based on the degree of fitting between the simulated and experimental data. Theoretical analyses for the two models are carried out in detail. The comparison between the two models is rarely reported, and it is reported in this article for the first time, theoretically and experimentally. This report provides a better choice for quickly analyzing the interaction mechanism between the laser and the bio-tissue.

Uniqueness and numerical inversion in bioluminescence tomography with time-dependent boundary measurement

In the paper, an inverse source problem in bioluminescence tomography (BLT) is investigated. BLT is a method of light imaging and offers many advantages such as sensitivity, cost-effectiveness, high signal-to-noise ratio and non-destructivity. It thus has promising prospects for many applications such as cancer diagnosis, drug discovery and development as well as gene therapies. In the literature, BLT is extensively studied based on the (stationary) diffusion approximation (DA) equation, where the distribution of peak sources is reconstructed and no solution uniqueness is guaranteed without proper a priori information. In this work, motivated by solution uniqueness, a novel dynamic coupled DA model is proposed. Theoretical analysis including the well-posedness of the forward problem and the solution uniqueness of the inverse problem are given. Based on the new model, iterative inversion algorithms under the framework of regularizing schemes are introduced and applied to reconstruct the smooth and non-smooth sources. We discretize the regularization functional with the finite element method and give the convergence rate of numerical solutions. Several numerical examples are implemented to validate the effectiveness of the new model and the proposed algorithms.

Nonlocal approximation of nonlinear diffusion equations

We show that degenerate nonlinear diffusion equations can be asymptotically obtained as a limit from a class of nonlocal partial differential equations. The nonlocal equations are obtained as gradient flows of interaction-like energies approximating the internal energy. We construct weak solutions as the limit of a (sub)sequence of weak measure solutions by using the Jordan-Kinderlehrer-Otto scheme from the context of 2-Wasserstein gradient flows. Our strategy allows to cover the porous medium equation, for the general slow diffusion case, extending previous results in the literature. As a byproduct of our analysis, we provide a qualitative particle approximation.

RADIATIVE TRANSFER: ASYMPTOTIC SOLUTION OF THE KINETIC EQUATION OF RADIATION PROPAGATION, <i>N</i>-th ORDER ASYMPTOTIC APPROXIMATION AND IMPROVED BOUNDARY CONDITIONS

In the article, new asymptotic approximation of the \(n\)-th order is obtained and proposed to be used in calculations of radiation propagation in optically thick media without scattering; the asymptotic approximation is simpler and more precise than the known diffusion approximation. It is shown, that for optically thick media the asymptotic solution of the kinetic equation of radiation propagation without scattering is asymptotic expansion of the exact integral solution of that kinetic equation. The rigorous derivation of the diffusion approximation equation is obtained. Improved boundary conditions, which are essential for practical application in calculations of radiation propagation, are derived.

Deterministic particle approximation of aggregation diffusion equations with nonlinear mobility

We consider a class of aggregation–diffusion equations on unbounded one-dimensional domains with Lipschitz nonincreasing mobility function. We show strong [Formula: see text]-convergence of a suitable deterministic particle approximation to weak solutions of a class aggregation–diffusion PDEs (coinciding with the classical ones in the no vacuum regions) for any bounded initial data of finite energy. In order to prove well-posedness and convergence of the scheme with no BV or no vacuum assumptions and overcome the issues posed in this setting by the presence of a mobility function, we improve and strengthen the techniques introduced in [S. Daneri, E. Radici and E. Runa, Deterministic particle approximation of aggregation–diffusion equations on unbounded domains, J. Differential Equations 312 (2020) 474–517].

Approximation of random diffusion equations by nonlocal diffusion equations in free boundary problems of one space dimension

We show how the Stefan type free boundary problem with random diffusion in one space dimension can be approximated by the corresponding free boundary problem with nonlocal diffusion. The approximation problem is a slightly modified version of the nonlocal diffusion problem with free boundaries considered in [J. Cao, Y. Du, F. Li and W.-T. Li, The dynamics of a Fisher–KPP nonlocal diffusion model with free boundaries, J. Functional Anal. 277 (2019) 2772–2814; C. Cortazar, F. Quiros and N. Wolanski, A nonlocal diffusion problem with a sharp free boundary, Interfaces Free Bound. 21 (2019) 441–462]. The proof relies on the introduction of several auxiliary free boundary problems and constructions of delicate upper and lower solutions for these problems. As usual, the approximation is achieved by choosing the kernel function in the nonlocal diffusion term of the form [Formula: see text] for small [Formula: see text], where [Formula: see text] has compact support. We also give an estimate of the error term of the approximation by some positive power of [Formula: see text].

The effect of modulation frequency for frequency domain diffuse optic tomography (FDDOT)

In this study, it was illustrated that extremely high frequency (EHF) Ka-band (26.5–40 GHz) modulated frequency domain diffuse optic tomography biomedical optic imaging modality is superior to the generally accepted 100 MHz modulation frequency case. The effect of modulation frequency was shown with reconstructed images for two different modulation frequency simulation cases. Ka EHF-band frequency range covers 26.5–40 GHz. In this simulation study 33 GHz modulation frequency was selected. Forward model problem photon fluencies were generated for 30 equally separated sequential phase delays. Each phase delay has different photon fluence distributions inside the imaging geometry. 3.1 cm grid sizes were set in the x, y, z cartesian grid coordinate system with 31 × 31 × 31 xyz grid elements. To test the advantage of EHF-band modulation frequecy, inverse problem solution algorithm was done and inclusion images were reconstructed for each modulation frequency simulation case. Original inclusion was embedded inside the imaging geometry at (15–16, 15–16, 15–16) x, y, z coordinate system in the three-dimensional (3D) cubic spatial form. Homogeneuous tissue background photon absorption, scattering, and anisotropy coefficients were selected as µa = 0.1 cm−1, µs = 100 cm−1, and g = 0.9. Four sources and four detectors were placed on the back-reflection geometry. Forward model problem was built between sources, voxels, and detectors. Forward model problem photon fluencies were generated based on the diffusion equation approximation of the radiative transport equation formula in the frequency domain. Original inclusion was reconstructed by simply applying mathematical pseudoinverse problem solution function. It was observed that, reconstructed inclusion image at the 33 GHz Ka-band modulation frequency simulation case is superior to the reconstructed inclusion image at the 100 MHz modulation frequency simualtion case. Since there are too many frequency selection opportunities, the best Ka-band frequency was selected to demonstrate here. Thus, 33 GHz and 100 MHz modulation frequencies were tested against each other. These two different electronic modulation frequencies were tested and compared with each other.

A robust numerical approximation of advection diffusion equations with nonsingular kernel derivative

In this article we aim to approximate linear time fractional advection diffusion equations (TFADE) with Atangana-Baleanu- Caputo(ABC) derivative using local meshless method and Laplace transformation(LT). The method comprises of three steps. In the first step the the time variable is eliminated using LT. In the second step the reduced problem is solved using local meshless method. In the third step the solution of TFADE with ABC derivative is retrieved from local meshless methods solution by representing it as Bromwich integral. We then approximate the integral using some suitable quadrature rule. The stability and convergence of the method are discussed. The local meshless method is utilized to overcome the ill-conditioning issue of the interpolation matrices in global meshless methods and to over come the shape parameters sensitivity. Also in comparison with time stepping methods the LT is employed and contour integration technique is utilized to deal with the ABC derivative, which circumvent the calculation of costly convolution integrals in the approximation of ABC derivative, and also avoids the effect of time step on the stability and accuracy. Some test problems are considered in one and two dimensions to validate the proposed numerical method. The two dimensional problem is solved in regular and irregular domains. The computational experiments confirms that this method is computationally efficient and highly accurate for such type of problems.

An artificial neural network model for accurate and efficient optical property mapping from spatial-frequency domain images

Conventional optical property mapping by spatial-frequency domain imaging (SFDI) is prone to relatively low efficiency due to the iterative nature of nonlinear curve-fitting based on light transfer model, such as diffusion approximation equation. This study aims at expediating the prediction of optical property mapping with high accuracy by developing an artificial neural network (ANN) model coupled with SFDI. A dataset was first generated using Monte Carlo simulations based on Graphic Processing Unit. The ANN training was then conducted and the model was optimized for multiple factors (learning rate, batch size and number of neurons). Experiments with simulation samples and a solid phantom were carried out to verify the performance of the ANN model for predicting optical property mapping. Results showed that the normalized mean absolute errors of absorption coefficient (μa) and reduced scattering coefficient (μ's) were 0.18 % and 0.027 %, while the root mean square errors were as low as 0.01 and 0.14. Optical properties of the solid phantom demonstrated that the proposed ANN model retained the accuracy, and was about three orders of magnitude faster than the inverse curve-fitting model. Finally, the ANN model was tested on optical property measurement of ‘Golden Delicious’ apples with three different bruising levels (non-, light and severe bruising). Results indicated that the ANN model could measure apple μa and μ's accurately and efficiently, and the measured μ's mapping was capable of detecting early bruising in apples. Contrast of bruise feature in μ's value between bruised and non-bruised tissues was significantly improved, enhancing apple bruising detection.

Photon phase shift imaging research on frequency domain diffuse optic tomography

Diffuse optic imaging is an important biomedical optic research tool. Diffuse optic tomography (DOT) modality needs progressive philosophical approaches for scientific contribution. Technological developments and philosophical approaches should both go forward. Phase-shift based frequency domain diffuse optical tomography (FDDOT) method was well established in the literature. The instruments were tested for brain neurofunctional imaging. A mixture of AC laser intensity and phase data were used at these works. According to those works, deep tissue resolution was improved by only using phase data. Because phase data is only related to the photon mean free path in imaging tissue media. Besides this advantage, laser intensity data is also affected by noisy background light and electrical artifacts. Another most important advantage of only using phase data can be explained as time-resolved temporal change which can be directly related to the phase shift of modulated frequency source. In this work, the FDDOT imaging method which uses phase shift data were tested for simulation. Laser source-driven forward model problem weight matrix simulation data was given to the simple pseudo-inverse-based inverse problem solution algorithm for one inclusion example. The inclusion image was reconstructed and demonstrated successfully. Forward model problem weight functions inside the tissue simulation media were calculated and used based on the phase shifts at the same core modulation frequency. 100 MHz modulation frequency was selected due to its FDDOT standard. 13 sources and 13 detectors were placed on the back-reflected imaging surface. 40 x, y, z cartesian coordinate grid elements were used in the image reconstruction algorithm. Photon absorption coefficient: μa = 0.1 cm−1, and scattering coefficient: μs = 100 cm−1 values were set for simulation background. One inclusion object was embedded inside the background imaging tissue simulation environment. x, y, z cartesian coordinate grid sizes were selected for 1 μm for each direction. Photon phase shift fluencies were added to the forward model problem. The forward model problem was built according to the frequency domain photon migration diffusion equation (DE) approximation. The pseudoinverse mathematical inverse problem solution function was applied to test the results. The embedded inclusion object was reconstructed successfully with the high-resolution image quality. The philosophical approach has future promising DOT imaging capability. The phase shift version of the FDDOT modality has an important advantage for future purposes.

Epidemic modeling: Diffusion approximation vs. stochastic differential equations allowing reflection

A birth–death process is considered as an epidemic model with recovery and transmittance from outside. The fraction of infected individuals is for huge population sizes approximated by a solution of an ordinary differential equation taking values in [Formula: see text]. For intermediate size or semilarge populations, the fraction of infected individuals is approximated by a diffusion formulated as a stochastic differential equation. That diffusion approximation however needs to be killed at the boundary [Formula: see text]. An alternative stochastic differential equation model is investigated which instead allows a more natural reflection at the boundary.

Backward differentiation formula finite difference schemes for diffusion equations with an obstacle term

Abstract Finite difference schemes, using backward differentiation formula (BDF), are studied for the approximation of one-dimensional diffusion equations with an obstacle term of the form $$\begin{equation*}\min(v_t - a(t,x) v_{xx} + b(t,x) v_x + r(t,x) v, v- \varphi(t,x))= f(t,x).\end{equation*}$$For the scheme building on the second-order BDF formula, we discuss unconditional stability, prove an $L^2$-error estimate and show numerically second-order convergence, in both space and time, unconditionally on the ratio of the mesh steps. In the analysis an equivalence of the obstacle equation with a Hamilton–Jacobi–Bellman equation is mentioned, and a Crank–Nicolson scheme is tested in this context. Two academic problems for parabolic equations with an obstacle term with explicit solutions and the American option problem in mathematical finance are used for numerical tests.

GADMA: Genetic algorithm for inferring demographic history of multiple populations from allele frequency spectrum data.

BackgroundThe demographic history of any population is imprinted in the genomes of the individuals that make up the population. One of the most popular and convenient representations of genetic information is the allele frequency spectrum (AFS), the distribution of allele frequencies in populations. The joint AFS is commonly used to reconstruct the demographic history of multiple populations, and several methods based on diffusion approximation (e.g., ∂a∂i) and ordinary differential equations (e.g., moments) have been developed and applied for demographic inference. These methods provide an opportunity to simulate AFS under a variety of researcher-specified demographic models and to estimate the best model and associated parameters using likelihood-based local optimizations. However, there are no known algorithms to perform global searches of demographic models with a given AFS.ResultsHere, we introduce a new method that implements a global search using a genetic algorithm for the automatic and unsupervised inference of demographic history from joint AFS data. Our method is implemented in the software GADMA (Genetic Algorithm for Demographic Model Analysis, https://github.com/ctlab/GADMA).ConclusionsWe demonstrate the performance of GADMA by applying it to sequence data from humans and non-model organisms and show that it is able to automatically infer a demographic model close to or even better than the one that was previously obtained manually. Moreover, GADMA is able to infer multiple demographic models at different local optima close to the global one, providing a larger set of possible scenarios to further explore demographic history.

Conditioning of implicit Runge–Kutta integration for finite element approximation of linear diffusion equations on anisotropic meshes

The conditioning of implicit Runge–Kutta (RK) integration for linear finite element approximation of diffusion equations on general anisotropic meshes is investigated. Bounds are established for the condition number of the resulting linear system with and without diagonal preconditioning for the implicit Euler (the simplest implicit RK method) and general implicit RK methods. Two solution strategies are considered for the linear system resulting from general implicit RK integration: the simultaneous solution where the system is solved as a whole and a successive solution which follows the commonly used implementation of implicit RK methods to first transform the system into a number of smaller systems using the Jordan normal form of the RK matrix and then solve them successively.For the simultaneous solution in case of a positive semidefinite symmetric part of the RK coefficient matrix and for the successive solution it is shown that the conditioning of an implicit RK method behaves like that of the implicit Euler method. If the smallest eigenvalue of the symmetric part of the RK coefficient matrix is negative and the simultaneous solution strategy is used, an upper bound on the time step is given so that the system matrix is positive definite.The obtained bounds for the condition number have explicit geometric interpretations and take the interplay between the diffusion matrix and the mesh geometry into full consideration. They show that there are three mesh-dependent factors that can affect the conditioning: the number of elements, the mesh nonuniformity measured in the Euclidean metric, and the mesh nonuniformity with respect to the inverse of the diffusion matrix. They also reveal that the preconditioning using the diagonal of the system matrix, the mass matrix, or the lumped mass matrix can effectively eliminate the effects of the mesh nonuniformity measured in the Euclidean metric. Illustrative numerical examples are given.

Efficient prediction of cell size in solid polymeric foams by numerically solving the diffusion approximation of light scattering equation

This paper investigates alternative approaches to accurately calculate the cell size of solid polymeric foams by means of light scattering (LS) experimental data. To this end, static optical transmission measurements have been carried out in foams with different densities, cell sizes and based on different polymers. Using these data cell size has been calculated by solving numerically the diffusion approximation equation of LS that takes into account both scattering and absorption mechanisms. From the analysis performed we have proved that light absorption becomes significant in solid polymeric foams.

Modeling of diffuse reflectance of light in heterogeneous biological tissue to analysis of the effects of multiple scattering on reflectance pulse oximetry.

Fingertip-type pulse oximeters are popular, but their inconvenience for long-term monitoring in daily life means that other types of wearable pulse oximeters, such as reflectance pulse oximeters, need to be developed. For the purpose of developing reflection pulse oximetry, we have analyzed the light propagation in tissue to calculate and estimate the measured intensities of reflected light using the analytical and numerical solutions of the diffusion approximation equation. The reflectance of light from the biological tissue is investigated from theoretical and experimental perspectives, for light in the visible and near-infrared wavelengths. To establish the model, the calculated curves were compared with the analytical solution (AS) of the diffusion approximation equation in biological tissue. The results validated that the diffusion approximation equation could resolve the heterogeneous advanced tissue and the finite element method (FEM) could offer the simulation with higher efficiency and accuracy. Our aim has been to demonstrate the power of the FEM and AS in modeling of the steady-state diffusion approximation in a heterogeneous medium. Also, experimental data and the Monte Carlo model as a gold standard were used to verify the effectiveness of these methods.

Simulation of a fast diffuse optical tomography system based on radiative transfer equation

Studies show that near-infrared (NIR) light (light with wavelength between 700nm and 1300nm) undergoes two interactions, absorption and scattering, when it penetrates a tissue. Since scattering is the predominant interaction, the calculation of light distribution in the tissue and the image reconstruction of absorption and scattering coefficients are very complicated. Some analytical and numerical methods, such as radiative transport equation and Monte Carlo method, have been used for the simulation of light penetration in tissue. Recently, some investigators in the world have tried to develop a diffuse optical tomography system. In these systems, NIR light penetrates the tissue and passes through the tissue. Then, light exiting the tissue is measured by NIR detectors placed around the tissue. These data are collected from all the detectors and transferred to the computational parts (including hardware and software), which make a cross-sectional image of the tissue after performing some computational processes. In this paper, the results of the simulation of an optical diffuse tomography system are presented. This simulation involves two stages: a) Simulation of the forward problem (or light penetration in the tissue), which is performed by solving the diffusion approximation equation in the stationary state using FEM. b) Simulation of the inverse problem (or image reconstruction), which is performed by the optimization algorithm called Broyden quasi-Newton. This method of image reconstruction is faster compared to the other Newton-based optimization algorithms, such as the Levenberg-Marquardt one.

Block steepest descent method based on nonoverlapping domain decomposition, I

AbstractIn the present paper we propose a new iterative method for the numerical solution of system of linear algebraic equations with symmetric positive definite and positive semidefinite matrices arising from the approximation of diffusion equations by the mixed hybrid finite element method on triangular and tetrahedral meshes. The method is based on a combination of the block steepest descent idea and nonoverlapping domain decomposition algorithm. We give the formulation of the method and a simple convergence estimate in the case of regular shaped quasiuniform meshes.

An Optimal Solver for Linear Systems Arising from Stochastic FEM Approximation of Diffusion Equations with Random Coefficients

This paper discusses the design and implementation of efficient solution algorithms for symmetric linear systems associated with stochastic Galerkin approximation of elliptic PDE problems with correlated random data. The novel feature of our preconditioned MINRES solver is the incorporation of error control in the natural “energy” norm in combination with a reliable and efficient a posteriori estimator for the PDE approximation error. This leads to a robust and optimally efficient stopping criterion: the iteration is terminated as soon as the algebraic error is insignificant compared to the approximation error. The MATLAB codes used in the numerical studies are available online.

$hp$-Adaptation Driven by Polynomial-Degree-Robust A Posteriori Error Estimates for Elliptic Problems

We devise and study experimentally adaptive strategies driven by a posteriori error estimates to select automatically both the space mesh and the polynomial degree in the numerical approximation of diffusion equations in two space dimensions. The adaptation is based on equilibrated flux estimates. These estimates are presented here for inhomogeneous Dirichlet and Neumann boundary conditions, for spatially-varying polynomial degree, and for mixed rectangular-triangular grids possibly containing hanging nodes. They deliver a global error upper bound with constant one and, up to data oscillation, error lower bounds on element patches with a generic constant only dependent on the mesh regularity and with a computable bound. We numerically asses the estimates and several hp-adaptive strategies using the interior penalty discontinuous Galerkin method. Asymptotic exactness is observed for all the symmetric, nonsymmetric (odd degrees), and incomplete variants on non-nested unstructured triangular grids for a smooth solution and uniform refinement. Exponential convergence rates are reported on nonmatching triangular grids for the incomplete version on several benchmarks with a singular solution and adaptive refinement.