MR image-based measurement of rates of change in volumes of brain structures. Part II: application to a study of Alzheimer's disease and normal aging

We present global and regional rates of brain atrophy measured on serially acquired T1-weighted brain MR images for a group of Alzheimer's disease (AD) patients and age-matched normal control (NC) subjects using the analysis procedure described in Part I. Three rates of brain atrophy: the rate of atrophy in the cerebrum, the rate of lateral ventricular enlargement and the rate of atrophy in the region of temporal lobes, were evaluated for 14 AD patients and 14 age-matched NC subjects. All three rates showed significant differences between the two groups. However, the greatest separation of the two groups was obtained when the regional rates were combined. This application has demonstrated that rates of brain atrophy, especially in specific regions of the brain, based on MR images can provide sensitive measures for evaluating the progression of AD. These measures will be useful for the evaluation of therapeutic effects of novel therapies for AD.     

Automatic phasing of MR images. Part I: linearly varying phase

In spin-echo and well shimmed gradient-echo images, the phase of the complex image often varies linearly in both the readout and phase-encode directions. Thus, in principle, it is possible to display an image in absorption mode. However, manually determining the two first-order and one zero-order phase parameters needed to display an absorption-mode image is a formidable task. In this paper, the Bayesian calculations needed to automatically determine these parameters are presented, and the calculations are illustrated using spin-echo images.     

Modeling brain tissue volumes over the lifespan: quantitative analysis of postmortem weights and in vivo MR images

Normative measurements of brain gray matter and white matter tissue volumes across the lifespan have not yet been established. The purpose of this article was to use mathematical modeling and analytical functions to demonstrate the growth trajectory of gray matter and white matter from age 0 to age 90. For each gender, brain weight functions were generated by utilizing existing autopsy data from 4400 subjects. Brain gray matter, white matter and lateral ventricular volumes were measured from 39 MR volumes of normal individuals. These were converted to weight by multiplying the tissue volumes by the specific gravity of that tissue. White matter volumes were described by a saturating exponential function, and the gray matter volume function was calculated by subtracting the white matter weight function from the brain weight function. For each gender, equations were generated for white matter and gray matter volumes as a function of age over the lifespan.     

Dirac structures in Lagrangian mechanics Part I: Implicit Lagrangian systems

This paper develops the notion of implicit Lagrangian systems and presents some of their basic properties in the context of Dirac structures. This setting includes degenerate Lagrangian systems and systems with both holonomic and nonholonomic constraints, as well as networks of Lagrangian mechanical systems. The definition of implicit Lagrangian systems with a configuration space $Q$ makes use of Dirac structures on $T^{1}Q$ that are induced from a constraint distribution on $Q$ as well as natural symplectomorphisms between the spaces $T^{1}TQ$, $T{T}^{1}Q$, and $T^1{T}^{1}Q$. Two illustrative examples are presented; the first is a nonholonomic system, namely a vertical disk rolling on a plane, and the second is an L–C circuit, a degenerate Lagrangian system with holonomic constraints.     

High order multi-moment constrained finite volume method. Part I: Basic formulation

A new high-order finite volume method based on local reconstruction is presented in this paper. The method, so-called the multi-moment constrained finite volume (MCV) method, uses the point values defined within single cell at equally spaced points as the model variables (or unknowns). The time evolution equations used to update the unknowns are derived from a set of constraint conditions imposed on multi kinds of moments, i.e. the cell-averaged value and the point-wise value of the state variable and its derivatives. The finite volume constraint on the cell-average guarantees the numerical conservativeness of the method. Most constraint conditions are imposed on the cell boundaries, where the numerical flux and its derivatives are solved as general Riemann problems. A multi-moment constrained Lagrange interpolation reconstruction for the demanded order of accuracy is constructed over single cell and converts the evolution equations of the moments to those of the unknowns. The presented method provides a general framework to construct efficient schemes of high orders. The basic formulations for hyperbolic conservation laws in 1- and 2D structured grids are detailed with the numerical results of widely used benchmark tests.     

Photoresist reflow method of microlens production Part I: Background and experiments

We present a two part study of melted microlens arrays. This first part concentrates on the production and measurement of microlens arrays while the second part examines attempts to model the microlens profiles. In this paper we first review some of the fabrication techniques used over the past twenty years to produce lens arrays. Some applications of microlens arrays are then discussed. Particular emphasis is placed on the photoresist reflow method of microlens production that was suggested by Popovic et al., as this was the method used to produce the microlens examined in this study. Lenses produced using this method can have large deviations from the spherical case, i.e. the profile that would be expected from a simple minimisation of the surface energy. These deviations have not been explained to date in the literature, however a number of possible causes for this deviation are given in this paper.Therefore the fundamental questions we wish to explore here are: (1) Why physically do dips occur? and (2) Can the resulting surface profile be predicted? Any model developed to quantitatively estimate the optical effects of surface shape will depend on the physical assumptions made regarding the surface formation mechanism. However as we shall indicate at this point only an informed guess regarding the relative importance of a number of possible mechanisms can be made.     

Stochastic radiative transfer in multilayer broken clouds. Part II: validation tests

In the second part of our two-part paper, we estimated the accuracy and robustness of the approximated equations for the mean radiance that were derived in Part I. In our analysis we used the three-dimensional (3D) cloud fields provided by (i) the stochastic Boolean model, (ii) large-eddy simulation model and (iii) satellite cloud retrieval. The accuracy of the obtained equations was evaluated by comparing the ensemble-averaged radiative properties that were obtained by the numerical averaging method (reference) and the analytical averaging method (approximation). The robustness of these equations was estimated by comparing the domain-averaged radiative properties obtained by using (i) the full 3D cloud structure (reference) and (ii) the bulk cloud statistics (approximation). It was shown that the approximated equations could provide reasonable accuracy (∼15%) for both the ensemble- and domain-averaged radiative properties.     

Automatic phasing of MR images. Part II: voxel-wise phase estimation

Magnetic resonance images are typically displayed as the absolute value of the discrete Fourier transform of the k-space data. However, absorption-mode images, the real part of the discrete Fourier transform of the data after applying an appropriate phase correction, have significant advantages over absolute-value images. In a companion paper, the problem of estimating the phase parameters needed to produce an absorption-mode image when the phase of the complex image varies linearly as a function of position, a situation common in magnetic resonance images, was addressed. However, some magnetic resonance images have phases that can vary in a complicated, nonlinear, positionally dependent fashion. To produce an absorption-mode image from these data, one must first estimate the positionally dependent phase, and then use that phase estimate to produce an absorption-mode image. This paper addresses both of these problems by first using Bayesian probability theory to estimate the constant or zero-order phase as a function of image position, and then the calculations are illustrated by using them to generate absorption-mode images from data where the phase of the image is a nonlinear function of position.     

A high resolution spatially adaptive vortex method for separating flows. Part I: Two-dimensional domains

A grid-free high-resolution spatially-adaptive vortex method for two-dimensional incompressible flow in bounded domains is presented. The computational algorithm is based on operator splitting in which convection and diffusion are handled separately every time step. In the convection step, computational elements are convected with velocities obtained by fast approximations of the Biot–Savart superposition with second-order Runge–Kutta time integration scheme. Diffusion is performed using the smooth redistribution method that employs a Gaussian basis function for vorticity in the interior. Near solid walls, the core functions are modified to conserve circulation. The no-slip boundary condition is enforced by creating of a vortex sheet that is redistributed to neighboring elements using the redistribution method. The proposed method enables accurate and smooth recovery of the vorticity and does not require explicit use of vortex images or occasional re-meshing. Algorithms for reduction in computational cost by accurately removing elements in overcrowded regions and for spatial adaptivity that allows for variable core sizes and variable element spacing are presented. Computations of flow around an impulsively started cylinder for Reynolds number values of 1000, 3000, and 9500 are preformed to investigate various aspects of the proposed method.     

Multiwave non-linear couplings in elastic structures. Part II: two-dimensional example

The non-linear resonance coupling in a thin plate in the Kirchhoff-Love approximation is selected as a two-dimensional example of mechanical systems exhibiting a rich range of resonant wave-like phenomena. This is originally examined by use of Whitham's average-Lagrangian method. In particular, the existence of three basic resonant triads between longitudinal, shear and bending modes is shown. Some of these necessarily enter cascade wave processes related to the instability of some of the mode components of the triad under small perturbations. A short comparison with Kolmogorov's cascades of turbulence is given.     

Identification of a continuous structure with a geometrical non-linearity. Part I: Conditioned reverse path method

Particular effort has been spent in the field of identification of multi-degree-of-freedom non-linear systems. The newly developed methods permit the structural analyst to consider increasingly complex systems. The aim of this paper and a companion paper is to study, by means of two methods, a continuous non-linear system consisting of an experimental cantilever beam with a geometrical non-linearity. In this paper (Part I), the ability of the conditioned reverse path method, which is a frequency domain technique, to identify the behaviour of this structure is assessed. The companion paper (Part II) is devoted to the application of proper orthogonal decomposition, which is an updating technique, to the test example.     

A rapid and reliable semiautomated method for measurement of total abdominal fat volumes using magnetic resonance imaging

The aim of this study was the development of a reliable and fast method to estimate total abdominal fat volumes (TAF) in diabetic subjects on the basis of T1-weighted MR images. Thirty-seven patients with diabetes were examined (age 48 +/- 13 y mean +/- SD). A semiautomated computer assisted software program was developed to quantify intraabdominal (IAF), subcutaneous (SCF), and total abdominal fat volumes (TAF). The variability of image analysis for fat measurements between two observers and within observers was assessed. Mean volumes (+/- SD) for IAF, SCF and TAF were 10.5 1 (+/- 5.0 1), 15.1 1 (+/-7.3 1) and 25.7 1 (+/-11.5 1), respectively. Inter- and intraobserver reliability was excellent (r = 0.999 to r = 1.0). Per patient, the analysis required nine minutes in addition to a scan duration of seven minutes. As this analytic method using T1-weighted MR images allows a fast and reliable quantification of TAF, IAF and SCF, it may serve as a valuable tool for respective studies in diabetic subjects.     

Automatic measurement of changes in brain volume on consecutive 3D MR images by segmentation propagation

This article presents a technique to automatically measure changes in the volume of a structure of interest in successive 3D magnetic resonance (MR) images and its application in the study of the brain and lateral cerebral ventricles. The only manual step is a segmentation of the structure of interest in the first image. The analysis comprises, first, precise rigid co-registration of the time series of images; second, computation of residual deformations between pairs of images; third, automatic quantification of the volume change, obtained by propagation of the segmentation of the structure of interest through the series of MR images. This approach has been applied to monitor changes in the volume of the brain and lateral cerebral ventricles in a healthy subject and a patient with primary progressive aphasia (PPA). Results are consistent with those obtained by application of the boundary shift integral (BSI) and by stereology in the same subjects.     

Recursion operators and bi-Hamiltonian structures in multidimensions. I

The algebraic properties of exactly solvable evolution equations in one spatial and one temporal dimensions have been well studied. In particular, the factorization of certain operators, called recursion operators, establishes the bi-Hamiltonian nature of all these equations. Recently, we have presented the recursion operator and the bi-Hamiltonian formulation of the Kadomtsev-Petviashvili equation, a two spatial dimensional analogue of the Korteweg-deVries equation. Here we present the general theory associated with recursion operators for bi-Hamiltonian equations in two spatial and one temporal dimensions. As an application we show that general classes of equations, which include the Kadomtsev-Petviashvili and the Davey-Stewartson equations, possess infinitely many commuting symmetries and infinitely many constants of motion in involution under two distinct Poisson brackets. Furthermore, we show that the relevant recursion operators naturally follow from the underlying isospectral eigenvalue problems.     

An MR imaging method for simultaneous measurement of gaseous diffusion constant and longitudinal relaxation time

A magnetic resonance imaging method for simultaneous and accurate determination of gaseous diffusion constant and longitudinal relaxation time is presented. The method is based on direct observation of diffusive motion. Initially, a slice-selective saturation of helium-3 (3He) spins was performed on a 3He/O2 phantom (9 atm/2 atm). A time-delay interval was introduced after saturation, allowing spins to diffuse in and out of the labeled slice. Following the delay interval a one-dimensional (1-D) projection image of the phantom was acquired. A series of 21 images was collected, each subsequent image having been acquired with an increased delay interval. Gradual spreading of the slice boundaries due to diffusion was thus observed. The projection profiles were fit to a solution of the Bloch equation corrected for diffusive motion. The fitting procedure yielded a value of D3He = 0.1562+/-0.0013 cm2/s, in good agreement with a measurement obtained with a modified version of the standard pulsed-field gradient technique. The method also enabled us to accurately measure the longitudinal relaxation of 3He spins by fitting the change of the total area under the projection profiles to an exponential. A value of T1 = 1.67 s (2 T field) was recorded, in excellent agreement with an inversion recovery measurement.     

A hybrid method for hydrodynamic-kinetic flow Part I: A particle-grid method for reducing stochastic noise in kinetic regimes

In this work we present a hybrid particle-grid Monte Carlo method for the Boltzmann equation, which is characterized by a significant reduction of the stochastic noise in the kinetic regime.