A CTRW-based model of time-resolved fluorescence lifetime imaging in a turbid medium.

We develop an analytic model of time-resolved fluorescent imaging of photons migrating through a semi-infinite turbid medium bounded by an infinite plane in the presence of a single stationary point fluorophore embedded in the medium. In contrast to earlier models of fluorescent imaging in which photon motion is assumed to be some form of continuous diffusion process, the present analysis is based on a continuous-time random walk (CTRW) on a simple cubic lattice, the object being to estimate the position and lifetime of the fluorophore. Such information can provide information related to local variations in pH and temperature with potential medical significance.

Effects of anisotropy on the depth of penetration of photons into turbid media.

Biomedical applications of near infrared radiation (NIR) techniques (i.e., based on light wavelengths roughly between 400 and 1100 nm) require that a preliminary estimate of the tissue volume being investigated be found.

Integral calculus problem solving: an fMRI investigation.

Only a subset of adults acquires specific advanced mathematical skills, such as integral calculus. The representation of more sophisticated mathematical concepts probably evolved from basic number systems; however its neuroanatomical basis is still unknown. Using fMRI, we investigated the neural basis of integral calculus while healthy participants were engaged in an integration verification task.

Propagators and related descriptors for non-Markovian asymmetric random walks with and without boundaries.

There are many current applications of the continuous-time random walk (CTRW), particularly in describing kinetic and transport processes in different chemical and biophysical phenomena. We derive exact solutions for the Laplace transforms of the propagators for non-Markovian asymmetric one-dimensional CTRW's in an infinite space and in the presence of an absorbing boundary. The former is used to produce exact results for the Laplace transforms of the first two moments of the displacement of the random walker, the asymptotic behavior of the moments as t-->infinity, and the effective diffusion constant.

Mean time-of-flight of photons in transillumination measurements of optically anisotropic tissue with an inclusion.

We study the effect of optical anisotropy on the mean time-of-flight of photons in a slab of turbid medium containing an inclusion whose optical properties differ from those of the bulk. For this analysis the difference in the mean time for a photon introduced into the slab to reach a specified target point with and without the inclusion is calculated. This difference is defined to be a measure of the contrast.

Time-dependent diffusion coefficients in periodic porous materials.

We derive an approximate solution for the Laplace transform of the time-dependent diffusion coefficient, D(t), of a molecule diffusing in a periodic porous material. In our model, the material is represented by a simple cubic lattice of identical cubic cavities filled with a solvent and connected by small circular apertures in otherwise reflecting cavity walls, the thickness of which can be neglected. The solution describes the decrease of D(t) from its initial value, D(0) = D, where D is the diffusion constant in the free solvent, to its asymptotic value, D(infinity) = D(eff), which is much smaller than D.

Diffusion in multilayer media: transient behavior of the lateral diffusion coefficient.

A general formalism for treating lateral diffusion in a multilayer medium is developed. The formalism is based on the relation between the lateral diffusion and the distribution of the cumulative residence time, which the diffusing particle spends in different layers. We exploit this fact to derive general expressions which give the global and local time-dependent diffusion coefficients in terms of the average cumulative times spent by the particle in different layers and the probabilities of finding the particle in different layers, respectively.

Monte Carlo simulations of increased/decreased scattering inclusions inside a turbid slab.

We analyse the effect on scattered photons of anomalous optical inclusions in a turbid slab with otherwise uniform properties. Our motivation for doing so is that inclusions affect scattering contrast used to quantify optical properties found from transmitted light intensity measured in transillumination experiments. The analysis is based on a lattice random walk formalism which takes into account effects of both positive and negative deviations of the scattering coefficient from that of the bulk.

Estimation of anisotropic optical parameters of tissue in a slab geometry.

The scattering and absorption coefficients of many homogeneous biological tissues such as muscle, skin, white matter in the brain, and dentin are often anisotropically oriented with respect to their bounding interface. In consequence the curves of equal intensity of re-emitted light on the surface of the slab will no longer be circular. We here consider the problem of determining the parameters allowing one to estimate the angles defining anisotropy, directional bias of diffusive spreading, and scattering and absorbing coefficients from data obtained from time-gated measurements of light intensity transmitted through a slab of the tissue.

Diffusion in the presence of periodically spaced permeable membranes.

The diffusion of molecules in biological tissues and some other microheterogeneous systems is affected by the presence of permeable barriers. This leads to the slowdown of diffusion at long times as compared to barrier-free diffusion. At short times the effect of barriers is weak.

Photon migration in turbid media with anisotropic optical properties.

We analyse properties of photon migration in reflectance measurements made on a semi-infinite medium bounded by a plane, in which optical parameters may vary in directions neither parallel to, nor perpendicular to the bounding plane. Our aim in doing this is to develop the formulae necessary to deduce parameters of directionality from both time-gated and continuous wave measurements. The mathematical development is based on a diffusion picture, in which the bounding plane is regarded as being totally absorbing so that all photons reaching the surface contribute to the reflectance.

Rate constant for diffusion-influenced ligand binding to receptors of arbitrary shape on a cell surface.

The theory of ligand binding to receptors on a cell surface suggested by Berg and Purcell and generalized by Zwanzig and Szabo uses the assumption that receptors are circular absorbing disks on an otherwise reflecting sphere. One of the key ingredients of this theory is a solution for the rate constant for ligand binding to a single circular receptor on a reflecting plane. We give an exact solution for the rate constant for binding to a single elliptic receptor and an approximate solution for binding to a single receptor of more general shape.

Kinetics of ligand equilibration between tubular and vesicular parts of the endosome.

The kinetics of ligand equilibration between the tubular and vesicular parts of the endosome are studied for ligands diffusing in the vesicle and in a narrow cylindrical tubule attached to it. The key quantity in our analysis is the fraction of ligands in the vesicle at time t, P(ves)(t). We derive an expression for the Laplace transform of P(ves)(t) as a function of the vesicle volume and the length and radius of the tubule as well as the ligand diffusion coefficients in the vesicle and in the tubule.

Effects of anisotropic optical properties on photon migration in structured tissues.

It is often adequate to model photon migration in human tissue in terms of isotropic diffusion or random walk models. A nearly universal assumption in earlier analyses is that anisotropic tissue optical properties are satisfactorily modelled by using a transport-corrected scattering coefficient which then allows one to use isotropic diffusion-like models. In the present paper we introduce a formalism, based on the continuous-time random walk, which explicitly allows the diffusion coefficients to differ along the three axes.

Continuous-time random-walk model for financial distributions.

We apply the formalism of the continuous-time random walk to the study of financial data. The entire distribution of prices can be obtained once two auxiliary densities are known. These are the probability densities for the pausing time between successive jumps and the corresponding probability density for the magnitude of a jump.

Number of distinct sites visited by a random walker trapped by an absorbing boundary.

The number of distinct sites visited by a lattice random walker is a subject of continuing interest in both mathematics and physics. All previous investigations have used the assumption that the lattice is unbounded. An assessment of the amount of tissue interrogated by a photon in reflectance measurements for diagnostic purposes suggests analyzing properties of the average number of distinct sites visited by a random walker trapped by an absorbing plane at time t.

Tables of the Inverse Laplace Transform of the Function [Formula: see text].

The inverse transform, [Formula: see text], 0 < < 1, is a stable law that arises in a number of different applications in chemical physics, polymer physics, solid-state physics, and applied mathematics. Because of its important applications, a number of investigators have suggested approximations to (). However, there have so far been no accurately calculated values available for checking or other purposes.

Fourier Representations of Pdf's Arising in Crystallography.

A survey is given of some recent calculations of univariate and multivariate probability density functions (pdfs) of structure factors used to interpret crystallographic data. We have found that in the presence of sufficient atomic heterogeneity the frequently used approximations derived from the central limit theorem in the form of Edgeworth or Gram-Charlier series can be quite unreliable, and in these cases the more exact, but lengthier, Fourier calculations must be made.

Stable Law Densities and Linear Relaxation Phenomena.

Stable law distributions occur in the description of the linear dielectric behavior of polymers, the motion of carriers in semi-conductors, the statistical behavior of neurons, and many other phenomena. No accurate tables of these distributions or algorithms for estimating the parameters in these relaxation models exist. In this paper we present tables of the functions together with related functional properties of ().

On the Calculation of Moments of Molecular Weight Distribution from Sedimentation Equilibrium Data.

In this paper we discuss a technique for calculating moments of polydisperse materials in terms of concentration readings along the cell. The proposed method minimizes dependence on data from the end points where they may be unreliable. An analysis is given of the errors involved in the use of the proposed method when the underlying molecular weight distribution is the Schulz distribution or the lognormal.

The Solution to a Nonlinear Lamm Equation in the Faxén Approximation.

An exact solution in the Faxén approximation is given for the Lamm equation in which the sedimentation coefficient is related to concentration as (1-). It is shown that the solution in this case can be expressed in terms of the solution to the linear case ( =0) with a modified argument. The boundary sharpening phenomenon expresses itself very clearly in the solution presented here.