Reducing shape errors in the discrete dipole approximation using effective media.

The discrete dipole approximation (DDA) simulates optical properties of particles with any given shape based on the volume discretization. These calculations cost a large amount of time and memory to achieve high accuracy, especially for particles with large sizes and complex geometric structures, such as mixed black-carbon aerosol particles. We systematically study the smoothing of the DDA discretization using the effective medium approximation (EMA) for boundary dipoles. This approach is tested for optical simulations of spheres and coated black-carbon (BC) aggregates, using the Lorenz-Mie and multiple-sphere T-Matrix as references. For spheres, EMA significantly improves the DDA accuracy of integral scattering quantities (up to 60 times), when the dipole size is only several times smaller than the sphere diameter. In these cases, the application of the EMA is often comparable to halving the dipole size in the original DDA, thus reducing the simulation time by about an order of magnitude for the same accuracy. For a coated BC model based on transmission electron microscope observations, the EMA (specifically, the Maxwell Garnett variant) significantly improves the accuracy when the dipole size is larger than » of the monomer diameter. For instance, the relative error of extinction efficiency is reduced from 4.7% to 0.3% when the dipole size equals that of the spherical monomer. Moreover, the EMA-DDA achieves the accuracy of 1% for extinction, absorption, and scattering efficiencies using three times larger dipoles than that with the original DDA, corresponding to about 30 times faster simulations.

Reproducing the morphology-dependent resonances of spheres with the discrete dipole approximation.

During the development and application of a scattering algorithm, its accuracy is normally validated by comparing with results of spherical particles given by the exact Mie theory. Being the simplest shape, sphere supports morphology-dependent resonances (MDRs), which cause sharp variations of the scattering properties in narrow size ranges. We show that MDRs may mislead the validation of any volume- or surface-discretization methods, including the discrete dipole approximation (DDA) and, thus, should be explicitly avoided. However, the brute-force DDA simulations can actually capture the narrow peaks in the extinction efficiency over the size parameter, but only if a dipole size parameter is smaller than twice the MDR width. That is much more computationally intensive than typical DDA simulations. We find that a single Lorentzian MDR peak may be split into two due to the symmetry breaking by the DDA discretization. Furthermore, instead of time-consuming high-resolution DDA simulations for reproducing MDR, we developed and validated a significantly more computationally efficient method. It is based, first, on fitting simulated data with one or two Lorentzian peaks combined with a cubic baseline. Second, we use Richardson extrapolation of peak parameters to zero dipole size, exploiting the smooth convergence of these parameters towards the reference Mie values. When applied to two MDRs with relative widths 2 × 10-3 and 9 × 10-4, the developed workflow, powered by intensive simulations, reproduces the peak positions with unprecedented accuracy - errors less than 0.07% and 0.4% of their widths, respectively. This extends the way for studying the evolution of the MDR under non-axisymmetric deformations of a sphere or a spheroid.

Additivity of integral optical cross sections for a fixed tenuous multi-particle group.

We use the volume integral equation formulation of frequency-domain electromagnetic scattering to settle the issue of additivity of the extinction, scattering, and absorption cross sections of a fixed tenuous group of particles. We show that all the integral optical cross sections of the group can be obtained by summing up the corresponding individual-particle cross sections, provided that the single-scattering approximation applies.

Method for the simulation of blood platelet shape and its evolution during activation.

We present a simple physically based quantitative model of blood platelet shape and its evolution during agonist-induced activation. The model is based on the consideration of two major cytoskeletal elements: the marginal band of microtubules and the submembrane cortex. Mathematically, we consider the problem of minimization of surface area constrained to confine the marginal band and a certain cellular volume. For resting platelets, the marginal band appears as a peripheral ring, allowing for the analytical solution of the minimization problem. Upon activation, the marginal band coils out of plane and forms 3D convoluted structure. We show that its shape is well approximated by an overcurved circle, a mathematical concept of closed curve with constant excessive curvature. Possible mechanisms leading to such marginal band coiling are discussed, resulting in simple parametric expression for the marginal band shape during platelet activation. The excessive curvature of marginal band is a convenient state variable which tracks the progress of activation. The cell surface is determined using numerical optimization. The shapes are strictly mathematically defined by only three parameters and show good agreement with literature data. They can be utilized in simulation of platelets interaction with different physical fields, e.g. for the description of hydrodynamic and mechanical properties of platelets, leading to better understanding of platelets margination and adhesion and thrombus formation in blood flow. It would also facilitate precise characterization of platelets in clinical diagnosis, where a novel optical model is needed for the correct solution of inverse light-scattering problem.

Calibration-free quantitative immunoassay by flow cytometry: Theoretical consideration and practical implementation for IgG antibody binding to CD14 receptors on human leukocytes.

We propose a calibration-free method to determine the number of receptors per cell, as well as the direct and the reverse reaction rate constants for a single receptor. The method is based on the analysis of the temporal evolution of the cells mean fluorescent intensity measured by a flow cytometer during the ligand-receptor (antigen-antibody) binding under the conditions of their comparable concentrations. We developed the kinetic approach accounting both for the delay between the dilution and the measurement and for the practical duration of the measurement itself. The method was applied to determine thenumber of CD14 receptors on human blood mononuclear (granulocytes, monocytes, lymphocytes) cells of several donors. We also obtained the direct ( k+= (5.6 ± 0.2) × 107 M-1 min-1 ) and reverse ( k-= (1.3 ± 0.2) × 10-2 min-1 ) rate constants of ligand-receptor interaction, and estimated the size of the binding site as b = 0.5 nm. The latter allows one to recalculate the rate constants for a different ligand, fluorescent label, medium viscosity, and/or temperature. The knowledge of the rate constants is essential for the calibration-free determination of the number of receptors per cell from a single kinetic curve of the cells mean fluorescence intensity.

Nuclear apoptotic volume decrease in individual cells: Confocal microscopy imaging and kinetic modeling.

The dynamics of nuclear morphology changes during apoptosis remains poorly investigated and understood. Using 3D time-lapse confocal microscopy we performed a study of early-stage apoptotic nuclear morphological changes induced by etoposide in single living HepG2 cells. These observations provide a definitive evidence that nuclear apoptotic volume decrease (AVD) is occurring simultaneously with peripheral chromatin condensation (so called "apoptotic ring"). In order to describe quantitatively the dynamics of nuclear morphological changes in the early stage of apoptosis we suggest a general molecular kinetic model, which fits well the obtained experimental data in our study. Results of this work may clarify molecular mechanisms of nuclear morphology changes during apoptosis.

Impressed sources and fields in the volume-integral-equation formulation of electromagnetic scattering by a finite object: a tutorial.

Although free space cannot generate electromagnetic waves, the majority of existing accounts of frequency-domain electromagnetic scattering by particles and particle groups are based on the postulate of existence of an impressed incident field, usually in the form of a plane wave. In this tutorial we discuss how to account for the actual existence of impressed source currents rather than impressed incident fields. Specifically, we outline a self-consistent theoretical formalism describing electromagnetic scattering by an arbitrary finite object in the presence of arbitrarily distributed impressed currents, some of which can be far removed from the object and some can reside in its vicinity, including inside the object. To make the resulting formalism applicable to a wide range of scattering-object morphologies, we use the framework of the volume integral equation formulation of electromagnetic scattering, couple it with the notion of the transition operator, and exploit the fundamental symmetry property of this operator. Among novel results, this tutorial includes a streamlined proof of fundamental symmetry (reciprocity) relations, a simplified derivation of the Foldy equations, and an explicit analytical expression for the transition operator of a multi-component scattering object.

Far-field Lorenz-Mie scattering in an absorbing host medium. II: Improved stability of the numerical algorithm.

A recently developed FORTRAN program computing far-field optical observables for spherical particles in an absorbing medium has exhibited numerical instability arising when the product of the particle vacuum size parameter and the imaginary part of the refractive index of the host becomes sufficiently large. We offer a simple analytical explanation of this instability and propose a compact numerical algorithm for the stable computation of Lorenz-Mie coefficients based on an upward recursion formula for spherical Hankel functions of a complex argument. Extensive tests confirm an excellent accuracy of this algorithm approaching machine precision. The improved public-domain FORTRAN program is available at https://www.giss.nasa.gov/staff/mmishchenko/Lorenz-Mie.html.

On the concept of random orientation in far-field electromagnetic scattering by nonspherical particles.

Although the model of randomly oriented nonspherical particles has been used in a great variety of applications of far-field electromagnetic scattering, it has never been defined in strict mathematical terms. In this Letter, we use the formalism of Euler rigid-body rotations to clarify the concept of statistically random particle orientations and derive its immediate corollaries in the form of the most general mathematical properties of the orientation-averaged extinction and scattering matrices. Our results serve to provide a rigorous mathematical foundation for numerous publications in which the notion of randomly oriented particles and its light-scattering implications have been considered intuitively obvious.

First-principles modeling of electromagnetic scattering by discrete and discretely heterogeneous random media.

A discrete random medium is an object in the form of a finite volume of a vacuum or a homogeneous material medium filled with quasi-randomly and quasi-uniformly distributed discrete macroscopic impurities called small particles. Such objects are ubiquitous in natural and artificial environments. They are often characterized by analyzing theoretically the results of laboratory, in situ, or remote-sensing measurements of the scattering of light and other electromagnetic radiation. Electromagnetic scattering and absorption by particles can also affect the energy budget of a discrete random medium and hence various ambient physical and chemical processes. In either case electromagnetic scattering must be modeled in terms of appropriate optical observables, i.e., quadratic or bilinear forms in the field that quantify the reading of a relevant optical instrument or the electromagnetic energy budget. It is generally believed that time-harmonic Maxwell's equations can accurately describe elastic electromagnetic scattering by macroscopic particulate media that change in time much more slowly than the incident electromagnetic field. However, direct solutions of these equations for discrete random media had been impracticable until quite recently. This has led to a widespread use of various phenomenological approaches in situations when their very applicability can be questioned. Recently, however, a new branch of physical optics has emerged wherein electromagnetic scattering by discrete and discretely heterogeneous random media is modeled directly by using analytical or numerically exact computer solutions of the Maxwell equations. Therefore, the main objective of this Report is to formulate the general theoretical framework of electromagnetic scattering by discrete random media rooted in the Maxwell-Lorentz electromagnetics and discuss its immediate analytical and numerical consequences. Starting from the microscopic Maxwell-Lorentz equations, we trace the development of the first-principles formalism enabling accurate calculations of monochromatic and quasi-monochromatic scattering by static and randomly varying multiparticle groups. We illustrate how this general framework can be coupled with state-of-the-art computer solvers of the Maxwell equations and applied to direct modeling of electromagnetic scattering by representative random multi-particle groups with arbitrary packing densities. This first-principles modeling yields general physical insights unavailable with phenomenological approaches. We discuss how the first-order-scattering approximation, the radiative transfer theory, and the theory of weak localization of electromagnetic waves can be derived as immediate corollaries of the Maxwell equations for very specific and well-defined kinds of particulate medium. These recent developments confirm the mesoscopic origin of the radiative transfer, weak localization, and effective-medium regimes and help evaluate the numerical accuracy of widely used approximate modeling methodologies.

Super-resolved calibration-free flow cytometric characterization of platelets and cell-derived microparticles in platelet-rich plasma.

Importance of microparticles (MPs), also regarded as extracellular vesicles, in many physiological processes and clinical conditions motivates one to use the most informative and precise methods for their characterization. Methods based on individual particle analysis provide statistically reliable distributions of MP population over characteristics. Although flow cytometry is one of the most powerful technologies of this type, the standard forward-versus-side-scattering plots of MPs and platelets (PLTs) overlap considerably because of similarity of their morphological characteristics. Moreover, ordinary flow cytometry is not capable of measurement of size and refractive index (RI) of MPs. In this study, we 1) employed the potential of the scanning flow cytometer (SFC) for identification and characterization of MPs from light scattering; 2) suggested the reference method to characterize MP morphology (size and RI) with high precision; and 3) determined the lowest size of a MP that can be characterized from light scattering with the SFC. We equipped the SFC with 405 and 488 nm lasers to measure the light-scattering profiles and side scattering from MPs, respectively. The developed two-stage method allowed accurate separation of PLTs and MPs in platelet-rich plasma. We used two optical models for MPs, a sphere and a bisphere, in the solution of the inverse light-scattering problem. This solution provides unprecedented precision in determination of size and RI of individual spherical MPs-median uncertainties (standard deviations) were 6 nm and 0.003, respectively. The developed method provides instrument-independent quantitative information on MPs, which can be used in studies of various factors affecting MP population.

Influence of magnesium sulfate on HCO3/Cl transmembrane exchange rate in human erythrocytes.

Magnesium sulfate (MgSO4) is widely used in medicine but molecular mechanisms of its protection through influence on erythrocytes are not fully understood and are considerably controversial. Using scanning flow cytometry, in this work for the first time we observed experimentally (both in situ and in vitro) a significant increase of HCO3(-)/Cl(-) transmembrane exchange rate of human erythrocytes in the presence of MgSO4 in blood. For a quantitative analysis of the obtained experimental data, we introduced and verified a molecular kinetic model, which describes activation of major anion exchanger Band 3 (or AE1) by its complexation with free intracellular Mg(2+) (taking into account Mg(2+) membrane transport and intracellular buffering). Fitting the model to our in vitro experimental data, we observed a good correspondence between theoretical and experimental kinetic curves that allowed us to evaluate the model parameters and to estimate for the first time the association constant of Mg(2+) with Band 3 as KB~0.07mM, which is in agreement with known values of the apparent Mg(2+) dissociation constant (from 0.01 to 0.1mM) that reflects experiments on enrichment of Mg(2+) at the inner erythrocyte membrane (Gunther, 2007). Results of this work partly clarify the molecular mechanisms of MgSO4 action in human erythrocytes. The method developed allows one to estimate quantitatively a perspective of MgSO4 treatment for a patient. It should be particularly helpful in prenatal medicine for early detection of pathologies associated with the risk of fetal hypoxia.

Time-domain discrete-dipole approximation for simulation of temporal response of plasmonic nanoparticles.

We develop the time-domain discrete dipole approximation (DDA), describing the temporal evolution of electric field in plasmonic nanostructures. The main equation is obtained by taking the inverse Fourier transform of the Taylor expansion of the frequency-domain DDA in terms of frequency deviation from the central frequency. Thus we assume that incident wavefronts of different frequencies accumulate relatively small phase difference when passing the particle. This assumption is always valid for nanoparticles much smaller than the wavelength. Being the time-domain method, the proposed approach also requires an analytic frequency dependence of electric permittivity, e.g. the Drude model. We present numerical results of application of the time-domain DDA to silver nanosphere, rod, and disk, which agree well with that obtained with its frequency-domain counterpart and the finite-difference time-domain method. Moreover, the time-domain DDA is the fastest of the three methods for incident pulses of several-femtoseconds width. Thus, it can effectively be applied for modeling the temporal responses of plasmonic nanostructures.

Convergence of the discrete dipole approximation. I. Theoretical analysis: erratum.

We report and address the errors in the analysis of the weighted discretization in Section 2.F of our published paper [J. Opt. Soc. Am. A23, 2578 (2006)JOAOD61084-752910.1364/JOSAA.23.002578].

Brownian aggregation rate of colloid particles with several active sites.

We theoretically analyze the aggregation kinetics of colloid particles with several active sites. Such particles (so-called "patchy particles") are well known as chemically anisotropic reactants, but the corresponding rate constant of their aggregation has not yet been established in a convenient analytical form. Using kinematic approximation for the diffusion problem, we derived an analytical formula for the diffusion-controlled reaction rate constant between two colloid particles (or clusters) with several small active sites under the following assumptions: the relative translational motion is Brownian diffusion, and the isotropic stochastic reorientation of each particle is Markovian and arbitrarily correlated. This formula was shown to produce accurate results in comparison with more sophisticated approaches. Also, to account for the case of a low number of active sites per particle we used Monte Carlo stochastic algorithm based on Gillespie method. Simulations showed that such discrete model is required when this number is less than 10. Finally, we applied the developed approach to the simulation of immunoagglutination, assuming that the formed clusters have fractal structure.

High-precision characterization of individual E. coli cell morphology by scanning flow cytometry.

We demonstrate a flow-cytometric method to measure length and diameter of single Escherichia coli cells with sub-diffraction precision. The method is based on the original scanning flow cytometer that measures angle-resolved light-scattering patterns (LSPs) of individual particles. We modeled the shape of E. coli cells as a cylinder capped with hemispheres of the same radius, and simulated light scattering by the models using the discrete dipole approximation. We computed a database of the LSPs of individual bacteria in a wide range of model parameters and used it to solve the inverse light-scattering problem by the nearest-neighbor interpolation. The solution allows us to determine length and diameter of each individual bacterium, including uncertainties of these estimates. The developed method was tested on two strains of E. coli. The resulting precision of bacteria length and diameter measurements varied from 50 nm to 250 nm and from 5 nm to 25 nm, respectively. The measured distributions of samples over length and diameter were in good agreement with measurements performed by optical microscopy and literature data. The described approach can be applied for rapid morphological characterization of any rod-shaped bacteria.

A case study on the reciprocity in light scattering computations.

The fulfillment of the reciprocity by five publicly available scattering programs is investigated for a number of different particles. Reciprocity means that the source and the observation point of a given scattering configuration can be interchanged without changing the result. The programs under consideration are either implementations of T-matrix methods or of the discrete dipole approximation. Similarities and differences concerning their reciprocity behavior are discussed. In particular, it is investigated whether and under which conditions reciprocity tests can be used to evaluate the scattering results obtained by the different programs for the given particles.

Comparison of the discrete dipole approximation and the discrete source method for simulation of light scattering by red blood cells.

The discrete sources method (DSM) and the discrete dipole approximation (DDA) were compared for simulation of light scattering by a red blood cell (RBC) model. We considered RBCs with diameters up to 8 mum (size parameter up to 38), relative refractive indices 1.03 and 1.06, and two different orientations. The agreement in the angle-resolved S(11) element of the Mueller matrix obtained by these methods is generally good, but it deteriorates with increasing scattering angle, diameter and refractive index of a RBC. Based on the DDA simulations with very fine discretization (up to 93 dipoles per wavelength) for a single RBC, we attributed most of the disagreement to the DSM, which results contain high-frequency ripples. For a single orientation of a RBC the DDA is comparable to or faster than the DSM. However, the relation is reversed when a set of particle orientations need to be simulated at once. Moreover, the DSM requires about an order of magnitude less computer memory. At present, application of the DSM for massive calculation of light scattering patterns of RBCs is hampered by its limitations in size parameter of a RBC due to the high number of harmonics used for calculations.

Scattering of a damped inhomogeneous plane wave by a particle in a weakly absorbing medium.

We use the volume integral equation formulation to consider frequency-domain electromagnetic scattering of a damped inhomogeneous plane wave by a particle immersed in an absorbing medium. We show that if absorption in the host medium is sufficiently weak and the particle size parameter is sufficiently small, then (i) the resulting formalism (including the far-field and radiative-transfer regimes) is largely the same as in the case of a nonabsorbing host medium, and (ii) one can bypass explicit use of sophisticated general solvers of the Maxwell equations applicable to inhomogeneous-wave illumination. These results offer dramatic simplifications for solving the scattering problem in a wide range of practical applications involving absorbing host media.