Standard Diffusion Approximation Research Articles

On the ambipolar diffusion formulation for ion-neutral drifts in the non-negligible drift velocity limit.

The ambipolar diffusion approximation is used to model partially ionized plasma dynamics in a single-fluid setting. To correctly apply the commonly used version of ambipolar diffusion, a set of criteria should be satisfied including the requirement that the difference in velocity between charges and neutral species (known as drift velocity) is much smaller than the thermal velocity, otherwise the drift velocity will drive a non-negligible level of further collisions between the two species. In this paper, we explore the consequences of relaxing this assumption. We show that a new induction equation can be formulated without this assumption. This formulation reduces to the ambipolar induction equation in the case the drift velocity is small. In the large drift velocity limit, the feedback of the drift velocity on the collision frequency results in decreased diffusion of the magnetic field compared with the standard ambipolar diffusion approximation for the same parameters. This has a natural consequence of reducing the frictional heating that can occur. Applying this to results from flux emergence simulations where the expansion of the magnetic field leads to strong adiabatic cooling of the partially ionized chromosphere resulted in a noticeable reduction in the magnitude of the predicted drift velocities. This article is part of the theme issue 'Partially ionized plasma of the solar atmosphere: recent advances and future pathways'.

Multi-frequency spatial frequency domain imaging: a depth-resolved optical scattering model to isolate scattering contrast in thin layers of skin.

Current methods for wound healing assessment rely on visual inspection, which gives qualitative information. Optical methods allow for quantitative non-invasive measurements of optical properties relevant to wound healing. Spatial frequency domain imaging (SFDI) measures the absorption and reduced scattering coefficients of tissue. Typically, SFDI assumes homogeneous tissue; however, layered structures are present in skin. We evaluate a multi-frequency approach to process SFDI data that estimates depth-specific scattering over differing penetration depths. Multi-layer phantoms were manufactured to mimic wound healing scattering contrast in depth. An SFDI device imaged these phantoms and data were processed according to our multi-frequency approach. The depth sensitive data were then compared with a two-layer scattering model based on light fluence. The measured scattering from the phantoms changed with spatial frequency as our two-layer model predicted. The performance of two models solutions for SFDI was consistently better than the standard diffusion approximation. We presented an approach to process SFDI data that returns depth-resolved scattering contrast. This method allows for the implementation of layered optical models that more accurately represent physiologic parameters in thin tissue structures as in wound healing.

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.

Information encoded in gene-frequency trajectories

In this work we present a systematic mathematical approximation scheme that exposes the way that information, about the evolutionary forces of selection and random genetic drift, is encoded within gene-frequency trajectories. We determine approximate, time-dependent, gene-frequency trajectory statistics, assuming additive selection. We use the probability of fixation to test and illustrate the approximation scheme introduced. For the case where the strength of selection and the effective population size have constant values, we show how a standard diffusion approximation result, for the probability of fixation, systematically emerges when increasing numbers of approximate trajectory statistics are taken into account. We then provide examples of how time-dependent parameters influence gene-frequency statistics.

On the numerical treatment and dependence of thermohaline mixing in red giants

In recent years much interest has been shown in the process of thermohaline mixing in red giants. In low and intermediate mass stars this mechanism first activates at the position of the bump in the luminosity function, and has been identified as a likely candidate for driving the slow mixing inferred to occur in these stars. One particularly important consequence of this process, which is driven by a molecular weight inversion, is the destruction of lithium. We show that the degree of lithium destruction, or in some cases production, is extremely sensitive to the numerical details of the stellar models. Within the standard 1D diffusion approximation to thermohaline mixing, we find that dfferent evolution codes, with their default numerical schemes, can produce lithium abundances that differ from one another by many orders of magnitude. This disagreement is worse for faster mixing. We perform experiments with four independent stellar evolution codes, and derive conditions for the spatial and temporal resolution required for a converged numerical solution. The results are extremely sensitive to the timesteps used. We find that predicted lithium abundances published in the literature until now should be treated with caution.

Population genetics inference for longitudinally-sampled mutants under strong selection.

Longitudinal allele frequency data are becoming increasingly prevalent. Such samples permit statistical inference of the population genetics parameters that influence the fate of mutant variants. To infer these parameters by maximum likelihood, the mutant frequency is often assumed to evolve according to the Wright–Fisher model. For computational reasons, this discrete model is commonly approximated by a diffusion process that requires the assumption that the forces of natural selection and mutation are weak. This assumption is not always appropriate. For example, mutations that impart drug resistance in pathogens may evolve under strong selective pressure. Here, we present an alternative approximation to the mutant-frequency distribution that does not make any assumptions about the magnitude of selection or mutation and is much more computationally efficient than the standard diffusion approximation. Simulation studies are used to compare the performance of our method to that of the Wright–Fisher and Gaussian diffusion approximations. For large populations, our method is found to provide a much better approximation to the mutant-frequency distribution when selection is strong, while all three methods perform comparably when selection is weak. Importantly, maximum-likelihood estimates of the selection coefficient are severely attenuated when selection is strong under the two diffusion models, but not when our method is used. This is further demonstrated with an application to mutant-frequency data from an experimental study of bacteriophage evolution. We therefore recommend our method for estimating the selection coefficient when the effective population size is too large to utilize the discrete Wright–Fisher model.

Convolution model of the diffuse reflectance for layered tissues

We present an explicit model for the diffuse reflectance due to a collimated beam of light incident normally on layered tissues. This model is derived using the corrected diffusion approximation applied to a layered medium, and it takes the form of a convolution with an explicit kernel and the incident beam profile. This model corrects the standard diffusion approximation over all source-detector separation distances provided the beam is sufficiently wide compared to the scattering mean free path. We validate this model through comparison with Monte Carlo simulations. Then we use this model to estimate the optical properties of an epithelial layer from Monte Carlo simulation data. Using measurements at small source-detector separations and this model, we are able to estimate the absorption coefficient, scattering coefficient, and anisotropy factor of epithelial tissues efficiently with reasonable accuracy.

Modeling boundary measurements of scattered light using the corrected diffusion approximation

We study the modeling and simulation of steady-state measurements of light scattered by a turbid medium taken at the boundary. In particular, we implement the recently introduced corrected diffusion approximation in two spatial dimensions to model these boundary measurements. This implementation uses expansions in plane wave solutions to compute boundary conditions and the additive boundary layer correction, and a finite element method to solve the diffusion equation. We show that this corrected diffusion approximation models boundary measurements substantially better than the standard diffusion approximation in comparison to numerical solutions of the radiative transport equation.

Correcting the diffusion approximation at the boundary

The diffusion approximation to the radiative transport equation applies for light that has propagated deeply into an optically thick medium, such as biological tissue. It does not accurately model light near boundaries where measurements of scattered light are often taken. Here, we compute a correction to the diffusion approximation at the boundary. This correction requires only small modifications to the standard diffusion approximation used in biomedical optics. In particular, one needs only to compute the coefficients in the boundary condition for the diffusion approximation and an additive correction. We give explicit procedures for these computations. Using numerical results for the steady-state plane-parallel slab problem, we show that this corrected diffusion approximation is a much better approximation than the standard diffusion approximation for modeling the reflectance and transmittance.

Firing-Rate Response of a Neuron Receiving Excitatory and Inhibitory Synaptic Shot Noise

The synaptic coupling between neurons in neocortical networks is sufficiently strong so that relatively few synchronous synaptic pulses are required to bring a neuron from rest to the spiking threshold. However, such finite-amplitude effects of fluctuating synaptic drive are missed in the standard diffusion approximation. Here exact solutions for the firing-rate response to modulated presynaptic rates are derived for a neuron receiving additive excitatory and inhibitory synaptic shot noise with exponential amplitude distributions. The shot-noise description of the neuronal response to synaptic dynamics is shown to be richer and qualitatively distinct from that predicted by the diffusion approximation. It is also demonstrated how the framework developed here can be generalized to multiplicative shot noise so as to better capture effects of the inhibitory reversal potential.

Study of diffuse reflectance based on a hybrid diffusion approximation and simulating experiment

A modified diffusion approximation is investigated. In this theoretical model, diffusion coefficient D of the standard diffusion approximation is substituted by diffusion coefficient Dasym of P3 approximation. Analytic solution RHybrid(ρ) of this hybrid diffusion for reflectance in single source approximation and steady-state case with extrapolated boundary is obtained; influences of the transport albedo a′ on the standard diffusion approximation RDA(ρ) and the hybrid diffusion approximation RHybrid(ρ) are compared; recovering of optical parameters from diffuse reflectance RHybrid(ρ) is demonstrated, and the errors and application ranges are studied. The results show that this hybrid diffusion approximation RHybrid(ρ) can be used in strong absorption media and at short source-detector separation.

Radiative transport in the delta-P1 approximation for semi-infinite turbid media.

We have developed an analytic solution for spatially resolved diffuse reflectance within the deltaP1 approximation to the radiative transport equation for a semi-infinite homogeneous turbid medium. We evaluate the performance of this solution by comparing its predictions with those provided by Monte Carlo simulations and the standard diffusion approximation. We demonstrate that the delta-P1 approximation provides accurate estimates for spatially resolved diffuse reflectance in both low and high scattering media. We also develop a multi-stage nonlinear optimization algorithm in which the radiative transport estimates provided by the delta-P1 approximation are used to recover the optical absorption (microa), reduced scattering (micros'), and single-scattering asymmetry coefficients (g1) of liquid and solid phantoms from experimental measurements of spatially resolved diffuse reflectance. Specifically, the delta-P1 approximation can be used to recover microa, micros', and g1 with errors within +/- 22%, +/- 18%, and +/- 17%, respectively, for both intralipid-based and siloxane-based tissue phantoms. These phantoms span the optical property range 4 < (micros' /microa) < 117. Using these same measurements, application of the standard diffusion approximation resulted in the recovery of microa and micros' with errors o f +/- 29% and +/- 25%, respectively. Collectively, these results demonstrate that the delta-P1 approximation provides accurate radiative transport estimates that can be used to determine accurately the optical properties of biological tissues, particularly in spectral regions where tissue may display moderate/low ratios of reduced scattering to absorption (micros'/microa).

Laser‐induced thermal response and characterization of nanoparticles for cancer treatment using magnetic resonance thermal imaging

Spherical nanoparticles with a gold outer shell and silica core can be tuned to absorb near-infrared light of a specific wavelength. These nanoparticles have the potential to enhance the treatment efficacy of laser-induced thermal therapy (LITT). In order to enhance both the potential efficacy and safety of such procedures, accurate methods of treatment planning are needed to predict the temperature distribution associated with treatment application. In this work, the standard diffusion approximation was used to model the laser fluence in phantoms containing different concentrations of nanoparticles, and the temperature distribution within the phantom was simulated in three-dimensions using the finite element technique. Magnetic resonance temperature imaging was used to visualize the spatiotemporal distribution of the temperature in the phantoms. In most cases, excellent correlation is demonstrated between the simulations and the experiment (<3.0% mean error observed). This has significant implications for the treatment planning of LITT treatments using gold-silica nanoshells.

Migration and Proliferation Dichotomy in Tumor-Cell Invasion

We propose a two-component reaction-transport model for the migration-proliferation dichotomy in the spreading of tumor cells. By using a continuous time random walk (CTRW), we formulate a system of the balance equations for the cancer cells of two phenotypes with random switching between cell proliferation and migration. The transport process is formulated in terms of the CTRW with an arbitrary waiting-time distribution law. Proliferation is modeled by a standard logistic growth. We apply hyperbolic scaling and Hamilton-Jacobi formalism to determine the overall rate of tumor cell invasion. In particular, we take into account both normal diffusion and anomalous transport (subdiffusion) in order to show that the standard diffusion approximation for migration leads to overestimation of the overall cancer spreading rate.

Investigation of light propagation models to determine the optical properties of tissue from interstitial frequency domain fluence measurements

Four models, standard diffusion approximation (SDA), single Monte Carlo (SMC), delta-P1, and isotropic similarity (ISM), are developed and evaluated as forward calculation tools in the estimation of tissue optical properties. The inverse calculation uses the ratio of the fluences and phase difference at two locations close to an intensity modulated isotropic source to recover the reduced scattering coefficient mus' and the absorption coefficient mua. Diffusion theory allows recovery of optical properties (OPs) within 5% for media with mus'mua>10. The performance of the delta-P1 model is similar to SDA, with limited enhanced accuracy. The collimation approximation may limit the use of the delta-P1 model for spherical geometry, and/or the fluence may not be accurately calculated by this model. The SMC model is the best, recovering OPs within 10% regardless of the albedo. However, the necessary restriction of the searched OPs space is inconvenient. The performance of ISM is similar to that of diffusion theory for media with mus'mua>10, and better for 1<mus'mua<10, i.e., determines absorption within 5% and reduced scattering within 20%. In practice, satisfactory estimates (within 5 to 10%) can be achieved using SDA to recover mus' and ISM to recover mua for media with mus'mua>5.

Determination of the optical properties of tissue-simulating phantoms from interstitial frequency domain measurements of relative fluence and phase difference

We estimated the absorption and reduced scattering coefficients of tissue-simulating phantoms from interstitial measurements of the phase difference and relative amplitude signals at two distances from a sinusoidally modulated isotropic source. It was found that absorption and reduced scattering coefficients can be recovered within 10% and slightly over 10% respectively, using either the data collected by two detectors 3mm apart or by two detectors 5mm apart with light collected by one detector attenuated by a neutral density filter. This accuracy was achieved over a wide range of optical properties, mu(a)=0.008 to 0.17mm(-1) and mu(s)'=0.3 to 1.8mm(-1). Additional factors affecting accuracy including source anisotropy, uncertainty in fiber placement, phase amplitude crosstalk, and the forward light propagation model (the combined isotropic similarity model and standard diffusion approximation versus the modified spherical harmonics method) were studied by Monte Carlo simulations (first two factors) and experiments (last two factors).

Frequency domain photon migration in theδ−P1approximation: Analysis of ballistic, transport, and diffuse regimes

The standard diffusion approximation (SDA) to the Boltzmann transport equation (BTE) is commonly used to describe radiative transport for biomedical applications of frequency-domain diffuse optical imaging and spectroscopy. Unfortunately, the SDA is unable to provide accurate radiative transport predictions on spatial scales comparable to the transport mean free path and for media in which optical scattering is not dominant over absorption. Here, we develop and demonstrate the use of the delta- P1 approximation to provide improved radiative transport estimates in the frequency domain via the addition of a Dirac delta function to both radiance and phase function approximations. Specifically, we consider photon density wave propagation resulting from the illumination of an infinite turbid medium with an embedded, intensity-modulated, spherical light source. We examine the accuracy of the standard diffusion and delta- P1 approximations relative to Monte Carlo simulations that provide exact solutions to the BTE. This comparison establishes the superior accuracy of the delta- P1 approximation relative to the SDA that is most notable at distances less than 3 transport mean free paths from the source. In addition, we demonstrate that the differences in photon density wave propagation in a highly forward scattering medium (g1=0.95) vs an isotropically scattering medium (g1=0) provides a basis to define three spatial regimes where the light field is dominated by (a) unscattered/ballistic light, (b) minimally scattered light, and (c) diffusely scattered light. We examine the impact of optical properties, source modulation frequency, and numerical aperture of detection on the spatial extent and location of these regimes.

Sci‐PM Fri ‐ 02: Determination of optical properties of tissue in situ using four different frequency domain models

The following problem was investigated: implant an intensity modulated light source and two detectors in tissue. From a measurement of the ratio of fluences and relative phase at the two detectors, estimate the absorption (μa) and reduced scattering (μs′) coefficients of the tissue. An initial guess of these quantities is iteratively improved by comparison of forward calculations with measurements. The model used for forward calculation must be fast, yet accurate. Four models were compared on the basis of errors in recovered optical properties: standard diffusion approximation (SDA), single Monte Carlo (SMC), delta‐P1, and isotropic similarity model (ISM). For μs′/μa&gt;10, SDA recovered both coefficients within 5–10%, but for lower μs′/μa the errors were as large as 25%. Although the delta‐P1 model has been found to perform better under some conditions, we found no significant advantage in this application. The ISM was equivalent to SDA for μs′/μa&gt;10 and gave errors in μa less than 7% and μs′ less than 17% for low albedo. The SMC model was the best; both coefficients were recovered to within 10% regardless of the albedo. In order to apply the SMC model, a restricted optical coefficient space must be used in the inverse problem solution.

Characterization of the reduced scattering coefficient for optically thin samples: theory and experiments

Diffusion theory is currently used to characterize highly scattering media from time and/or spatially resolved measurements. However, many of the limits of validity of the diffusion approximation in its standard form have been set through assumptions taken during its derivation, and it is generally believed that the diffusion equation holds only for length scales much larger than the transport mean free path (ltr). We here present several modifications to the standard diffusion approximation to provide a simple expression for characterizing optically thin scattering slabs. The improvements concern mainly the correct modelling of the temporal and spatial distribution of the source term and the use of a frequency dependent diffusion coefficient and boundary conditions. This novel expression is put to the test against Monte Carlo simulations and experimental data taken from measurements with phantoms of known optical properties. The reduced scattering coefficient can be retrieved accurately for slab widths of the order of the mean free path.

Sampling tissue volumes using frequency-domain photon migration.

We demonstrate the use of Monte Carlo simulations to generate photon scattering density functions (PSDFs) that represent the tissue volume sampled by steady-state and frequency-domain photon migration. We use these results to illustrate how scaling laws can be developed to determine the mean sampling depth of the multiply scattered photons detected by photon migration methods that remain valid outside the bounds of the standard diffusion approximation, i.e., at small source-detector separations and in media where the optical absorption is significant relative to scattering. Using both the PSDF computation and the newly formulated scaling laws, we focus on a comprehensive description of the effects of source modulation frequency, optical absorption, and source-detector separation on the depth of the sampled tissue volume as well as the sensitivity of frequency-domain photon migration measurements to the presence of a localized absorption heterogeneity.