Kalvis M. Jansons

Persistent angular structure: new insights from diffusion magnetic resonance imaging data

We determine a statistic called the (radially) persistent angular structure (PAS) from samples of the Fourier transform of a three-dimensional function. The method has applications in diffusion magnetic resonance imaging (MRI), which samples the Fourier transform of the probability density function of particle displacements. The PAS is then a representation of the relative mobility of particles in each direction. In PAS-MRI, we compute the PAS in each voxel of an image. This technique has biomedical applications, where it reveals the orientations of microstructural fibres, such as white-matter fibres in the brain. Scanner time is a significant factor in determining the amount of data available in clinical brain scans. Here, we use measurements acquired for diffusion-tensor MRI, which is a routine diffusion imaging technique, but extract richer information. In particular, PAS-MRI can resolve the orientations of crossing fibres. We test PAS-MRI on human brain data and on synthetic data. The human brain data set comes from a standard acquisition scheme for diffusion-tensor MRI in which the samples in each voxel lie on a sphere in Fourier space.

Persistent Angular Structure: New Insights from Diffusion MRI Data. Dummy Version

We determine a statistic called the radially Persistent Angular Structure (PAS) from samples of the Fourier transform of a three-dimensional function. The method has applications in diffusion magnetic resonance imaging (MRI), which samples the Fourier transform of the probability density function of particle displacements. The persistent angular structure is then a representation of the relative mobility of particles in each direction. In combination, PAS-MRI computes the persistent angular structure at each voxel of an image. This technique has biomedical applications, where it reveals the orientations of microstructural fibres, such as white-matter fibres in the brain. We test PAS-MRI on synthetic and human brain data. The data come from a standard acquisition scheme for diffusion-tensor MRI in which the samples in each voxel lie on a sphere in Fourier space.

Non-linear Registration with the Variable Viscosity Fluid Algorithm

In this paper we classify inhomogeneous non-linear registration algorithms into those of variable data influence, of variable deformability and of variable model type. As examples we introduce three modifications of the viscous fluid registration algorithm: passing a filter over the computed force field, adding boundary conditions onto the velocity field, and re-writing the viscous fluid PDE to accommodate a spatially-varying viscosity field. We demonstrate their application on artificial test data, on pre-/post-operative MR head slices and on MR neck volumes.

Determination of the macroscopic (partial) slip boundary condition for a viscous flow over a randomly rough surface with a perfect slip microscopic boundary condition

Consider a viscous fluid, at zero Reynolds number, moving over a solid surface flat except for a random array of microscopic defects having a small area fraction c. Assuming a microscopic boundary condition of perfect slip, the macroscopic boundary condition is determined from first principles. The asymptotic structure of the solution for a random surface with finite slope is quite different from those of earlier studies in the limit of an ‘‘almost flat’’ surface. The results of this study show that very small amounts of roughness can well approximate a no‐slip boundary condition macroscopically, for example, one defect of the order of 10−9 m per (10−7 m)2 gives a slip length of only 10−5 m.

On the application of geometric probability theory to polymer networks and suspensions, I

The theory of stochastic processes and integral geometry is applied to polymer networks and suspensions. Averaged monomer density is calculated for adsorbed and depleted layers of single-chain polymers modelled as brownian paths near plane and spherical surfaces, and exact solutions are given for an adsorbed layer with a force field. Asymptotic expressions are deduced for the available space fraction for a small solute in such layers. A measure of the typical asymmetry of a single-chain polymer modelled as a brownian path is obtained by calculating the averaged integral of the mean curvature, surface area and volume of its convex hull. The averaged integral of the mean curvature and surface area also give the first two terms of an asymptotic expression for the depletion near a smooth, slightly curved surface and an upper bound on the averaged capacitance. These quantities are also calculated for, model polymer aggregates, brownian rings and brownian paths with drift. From the latter, we deduce averages of powers of the end-to-end distance of a brownian path weighted by the integral of the mean curvature and surface area of its convex hull. The depletion due to a lamina with a smooth, slightly curved boundary in a solution of single-chain polymers is calculated, and the result is interpreted in terms of averaged geometrical properties of brownian paths. In a previous paper (Jansons & Phillips 1990) we calculated available space fractions for solutes and the polymer depletion near surfaces in homogeneous solutions of single-chain polymers, modelled as brownian paths. Here the treatment is extended to surface layers of polymers (e.g. the glycocalyx), for which averaged monomer densities and available space fractions are calculated, and to other types of polymers, such as ring polymers and polymer aggregates (e.g. proteoglycans), for which the polymer depletion near a smooth boundary and bounds on the averaged capacitance are obtained. The averaged volume of the convex hull of a brownian path is calculated, and used to derive a measure of typical polymer asymmetry that agrees well with earlier numerical calculations. We also investigate the relation between the end-to-end distance of a brownian path and the integral of the mean curvature and surface area of its convex hull. Further information on typical polymer shape is obtained from a geometrical interpretation of the polymer depletion due to a lamina. In ?3, we consider adsorbed layers of polymers and depleted layers near surfaces in polymer solutions, calculating the averaged monomer density and the available space fraction for small solutes. Many results for adsorbed layers follow simply from well known results in the theory of stochastic processes, such as the Ray-Knight

Exponential Timestepping with Boundary Test for Stochastic Differential Equations

We present new numerical methods for scalar stochastic differential equations. Successive time increments are independent random variables with an exponential distribution. We perform numerical experiments using a double-well potential. Exponential timestepping algorithms are efficient for escape-time problems because a simple boundary test can be performed at the end of each step.

The short-time transient of diffusion outside a conducting body

We consider the short-time transient of the diffusion of mass (or heat) to a body whose surface is held at zero concentration, with the space outside initially at unit concentration. The problem is expressed in terms of the probability that a brownian path of short duration intersects the body. A three-term asymptotic series for the absorption rate is derived for an arbitrary smooth body, together with the leading corrections due to edges and lines of contact with insulating surfaces. A three-term series is also derived for a plane laminar conductor with a smooth boundary, or equivalently a conducting film mounted on an insulating plane. These results are used to derive short-time absorption rates for shapes such as discs, rings and cylinders, commonly used for microelectrodes and hot-film devices.

Efficient Numerical Solution of Stochastic Differential Equations Using Exponential Timestepping

We present an exact timestepping method for Brownian motion that does not require Gaussian random variables to be generated. Time is incremented in steps that are exponentially-distributed random variables; boundaries can be explicitly accounted for at each timestep. The method is illustrated by numerical solution of a system of diffusing particles.

On polymer conformations in elongational flows

We consider various models of polymer conformations using paths of Gaussian processes such as Brownian motion. In each case, the calculation of the law of the moment of inertia of a random polymer structure (which is equivalent to the calculation of the partition function) is reduced to the problem of finding the law of a certain quadratic functional of a Gaussian process. We present a new method for computing the Laplace transforms of these quadratic functionals which exploit their special form via the Ray-Knight Theorem and which does not involve the classical method of eigenvalue expansions. We apply the method to several simple examples, then show how the same techniques can be applied to more complicated cases with the aid of a little excursion theory.

Stochastic Calculus: Application to Dynamic Bifurcations and Threshold Crossings

For the dynamic pitchfork bifurcation in the presence of white noise, the statistics of the last time at zero are calculated as a function of the noise level ∈ and the rate of change of the parameter μ. The threshold crossing problem used, for example, to model the firing of a single cortical neuron is considered, concentrating on quantities that may be experimentally measurable but have so far received little attention. Expressions for the statistics of pre-threshold excursions, occupation density, and last crossing time of zero are compared with results from numerical generation of paths.