Geometric Basis Research Articles

Limb Lengthening and Reconstruction Surgery

S. Robert Rozbruch and Svetlana Ilizarov, editors. New York: Informa Healthcare; 2007. 695 pages. $299.95. ISBN: 978-0-8493-4051-2. Based on the principles of distraction osteogenesis originally elucidated by Gavriil A. Ilizarov, Limb Lengthening and Reconstruction Surgery successfully covers the entire current spectrum of Ilizarov-based surgery. It was not long ago that Ilizarov, who is now universally known as the father of deformity correction, was working in Siberia and was unknown in the West. Fortunately, a group of Italians “discovered” the Ilizarov circular fixator in the early 1980s and brought a completely new branch of orthopaedic medicine to the rest of the world. Thereafter, awareness of Ilizarov and his contributions quickly increased, especially after the 1992 English-language publication of his major work, Transosseous Osteosynthesis, which detailed the principles he had spent nearly his entire professional life developing and applying clinically. As “Ilizarov fever” began to hit the United States in the mid-1980s, a number of American orthopaedic surgeons became dedicated Ilizarovians, including John Herzenberg and Dror Paley. Both were instrumental in solving and outlining the geometric basis of deformity correction, which later became the foundation of the center of rotation of angulation (CORA) method. …

Quantitative Description of Mitral Valve Geometry Using Real-Time Three-Dimensional Echocardiography

: Leaflet and annular geometry are important determinants of mitral valve (MV) stress. Repair techniques which optimize valvular geometry will reduce stress and potentially increase repair durability. The development of such procedures will require image processing methodologies that provide a quantitative description of three-dimensional valvular geometry. Using three-dimensional echocardiography in conjunction with novel geometric modeling and rendering techniques, we have developed a high-resolution, quantitative, three-dimensional methodology for imaging the human MV. : Five normal adults underwent MV imaging using real-time three-dimensional echocardiography. Using specially designed image analysis software, multiple valvular geometric parameters, including the magnitude and orientation of leaflet curvature, leaflet surface area, annular height, intercommissural width, septolateral annular diameter, and annular area were determined for each subject. Image rendering techniques that allow for the clear and concise presentation of this detailed information are also presented. : Although three-dimensional annular and leaflet geometry were found to be highly conserved among normal human subjects, substantial regional variation in leaflet geometry was observed. Interestingly, leaflet geometric heterogeneity was most pronounced in the midposterior leaflet, the region most commonly involved in leaflet flail in subjects with myxomatous disease. : The image processing and graphical rendering techniques that we have developed can be used to provide a complete description of three-dimensional MV geometry in human subjects. Widespread application of these techniques to normal subjects and patients with MV disease will provide insight into the geometric basis of both valvular pathology and repair durability.

Design and QSAR study of analogs of α-tocopherol with enhanced antiproliferative activity against human breast adenocarcinoma cells

Quantitative structure–activity relationships (QSAR) have been established for two sets of the antitumor drugs, α-tocopherol derivatives. Constitutional, geometrical, physico-chemical and electronic descriptors (using the B3LYP/6-31G (d, p) basis set) were computed and analyzed. The most relevant of these descriptors were grouped and multiple linear regressions have been carried out. QSAR model with four variables, R 2 = 0.98 and cross-validation parameter q pre 2 = 0.91 , was selected. Analogs of α-tocopherol (compounds D-1 and D-2) have been designed and their antiproliferative activities were evaluated using the proposed regression model. Calculated antiproliferative activities of the designed lysine/α-tocopherol/cholesterol conjugates, IC 50 (D-1) = 2.25 μM and IC 50 (D-2) = 3.42 μM, were significantly stronger than activities of the other analyzed compounds IC 50 > 4 μM.

Generic representation of 3D motion paths in dynamic animations of simulated construction processes

The presented research investigated generic and scalable techniques to accurately represent 3D motion paths in dynamic animations of operations simulated using Discrete-Event Simulation. The work designed and implemented methods that can be used to 1) define and manipulate arbitrarily-shaped trajectories to represent accurate 3D motion paths of virtual simulation objects, and 2) compute the precise three-dimensional spatial configuration of virtual simulation objects when they travel on defined paths. In order to address the problem of describing the accurate motion of simulation objects on realistic paths, the research investigated the requirements of two key technologies: 1) A mathematical representation for defining arbitrarily complex curves that can, by manipulating only high-level interaction parameters, be locally and/or globally edited to describe a realistic motion trajectory, and 2) A geometric basis to guide the computation of a simulation object's correct 3D orientation as it travels on an arbitrarily uneven virtual terrain surface. In order to achieve the first objective, the research investigated a technique of producing a general class of interpolating cubic splines whose shape can be locally or globally controlled by modifying three high-level control parameters. In addition, by implementing an innovative virtual terrain-following algorithm, a computation scheme was designed that correctly calculates and portrays the orientation of simulation objects along all three axes (yaw, pitch, and roll) as they travel inside animated 3D virtual worlds. The designed animation methods were implemented in a software tool called PathFinder that integrates as an add-on with the VITASCOPE visualization system.

Stretching-based diagnostics and reduction of chemical kinetic models with diffusion

A new method for diagnostics and reduction of dynamical systems and chemical kinetic models is proposed. The method makes use of the local structure of the normal stretching rates by projecting the dynamics onto the local directions of maximal stretching. The approach is computationally very simple as it implies the spectral analysis of a symmetric matrix. Notwithstanding its simplicity, stretching-based analysis derives from a geometric basis grounded on the pointwise applications of concepts of normal hyperbolicity theory. As a byproduct, a simple reduction method is derived, equivalent to a “local embedding algorithm”, which is based on the local projection of the dynamics onto the “most unstable and/or slow modes” compared to the time scale dictated by the local tangential dynamics. This method provides excellent results in the analysis and reduction of dynamical systems displaying relaxation towards an equilibrium point, limit cycles and chaotic attractors. Several numerical examples deriving from typical models of reaction/diffusion kinetics exhibiting complex dynamics are thoroughly addressed. The application to typical combustion models is also analyzed.

Analysis of Profile Functions for General Linear Regularization Methods

The stable approximate solution of ill‐posed linear operator equations in Hilbert spaces requires regularization. Tight bounds for the noise‐free part of the regularization error are constitutive for bounding the overall error. Norm bounds of the noise‐free part which decrease to zero along with the regularization parameter are called profile functions and are the subject of our analysis. The interplay between properties of the regularization and certain smoothness properties of solution sets, which we shall describe in terms of sourcewise representations, is crucial for the decay of associated profile functions. On the one hand, we show that a given decay rate is possible only if the underlying true solution has appropriate smoothness. On the other hand, if smoothness fits the regularization, then decay rates are easily obtained. If smoothness does not fit, then we will measure this in terms of some distance function. Tight bounds for these allow us to obtain profile functions. Finally we study the most realistic case when smoothness is measured with respect to some operator which is related to the one governing the original equation only through a link condition. In many parts the analysis is done on geometric basis, extending classical concepts of linear regularization theory in Hilbert spaces. We emphasize intrinsic features of linear ill‐posed problems which are frequently hidden in the classical analysis of such problems.

Altered Myocardial Shear Strains Are Associated With Chronic Ischemic Mitral Regurgitation

Ischemic mitral regurgitation (IMR) limits life expectancy and can lead to postinfarction global left ventricular (LV) dilatation and remodeling, the pathogenesis of which is not completely known. We tested the hypothesis that IMR perturbs adjacent myocardial LV systolic strains. Thirteen sheep had three columns of miniature beads inserted across the lateral LV wall, with additional epicardial markers silhouetting the ventricle. One week later posterolateral infarction was created. Seven weeks thereafter, the animals were divided into two groups according to severity of IMR (< or = 1+, n = 7, IMR[-] vs > or = 2+, n = 6, IMR[+]). Four dimensional marker coordinates and quantitative histology were used to calculate ventricular volumes, transmural myocardial systolic strains, and systolic fiber shortening. Seven weeks after infarction, end-diastolic (ED) volume increased similarly in both groups, end-systolic (ES) E13 (circumferential-radial) shear increased in both groups, but more so in IMR(+) than IMR(-) (+0.12 vs 0.04, p < 0.005), and E12 (circumferential-longitudinal) shear increased in IMR(-) but not IMR(+) (+0.04 vs -0.01, p < 0.005). There were no significant differences in ED or ES remodeling strains or systolic fiber shortening between IMR(-) and IMR(+). An equivalent increase in LV end-diastolic (ED) volume in both groups, coupled with unchanged ED and end-systolic remodeling strains as well as systolic circumferential, longitudinal, and radial strains, argue against a global LV or regional myocardial geometric basis for the cardiomyopathy associated with IMR. Further, similar systolic fiber shortening in both groups militates against an intracellular (cardiomyocyte) mechanism. The differences in subepicardial E12 and E13 shears, however, suggest a causal role of altered interfiber (cytoskeleton and extracellular-matrix) interactions.

Wavefront sensorless adaptive optics for large aberrations

In some adaptive optics systems the aberration is determined not by using a wavefront sensor but by sequential optimization of the adaptive correction element. Efficient schemes for the control of such systems are essential if they are to be effective. A scheme is introduced that permits the efficient measurement of large amplitude wavefront aberrations that are represented by an appropriate series of modes. This scheme uses an optimization metric based on the root-mean-square spot radius (or focal spot second moment) and an aberration expansion using polynomials suited to the representation of lateral aberrations. Experimental correction of N aberration modes is demonstrated with a minimum of N+1 photodetector measurements. The geometrical optics basis means that the scheme can be extended to arbitrarily large aberrations.

Geometrical basis of perception of gaze direction

Perception of gaze direction depends not only on the position of the irises within the looker’s eyes but also on the orientation of the looker’s head. A simple analysis of the geometry of gaze direction predicts this dependence. This analysis is applied to explain the Wollaston effect, the Mona Lisa effect, and the newly presented Mirror gaze effect. In an experiment synthetic faces were used in which the position of the iris and the angle of head rotation were varied. Different groups of subjects judged iris position, head rotation, and gaze direction of the same stimuli. The results illustrate how cues of iris location and head orientation interact to determine perceived gaze direction.

The geometry of the forisome-sieve element-sieve plate complex in the phloem of Vicia faba L. leaflets

Forisomes are contractile protein bodies that appear to control flux rates in the phloem of faboid legumes by reversibly plugging the sieve tubes. Plugging is triggered by Ca(2+) which induces an anisotropic deformation of forisomes, consisting of a longitudinal contraction and a radial expansion. By conventional light microscopy and confocal laser-scanning microscopy, the three-dimensional geometry of the forisome-sieve element-sieve plate complex in intact sieve tubes of leaflets of Vicia faba L. was reconstructed. Forisomes were mostly located close to sieve plates, and occasionally were observed drifting unrestrainedly along the sieve element, suggesting that they might be utilized as internal markers of flow direction. The diameter of forisomes in the resting state correlated with the diameter of their sieve elements, supporting the idea that radial expansion of forisomes is the geometric basis of reversible sieve tube plugging. Comparison of the present results regarding forisome geometry in situ with previously published data on forisome reactivity in vitro makes it questionable, however, whether forisomes are capable of completely sealing sieve tubes in V. faba leaves.

The principle of least action and the geometric basis of D-branes

We analyze thoroughly the boundary conditions allowed in classical non-linear sigma models and derive from first principle the corresponding geometric objects, i.e. D-branes. In addition to giving classical D-branes an intrinsic and geometric foundation, D-branes in nontrivial H flux and D-branes embedded within D-branes are precisely defined. A well known topological condition on D-branes is replaced.

Evolutionary approach to inverse planning in coplanar radiotherapy

A unified evolutionary approach to coplanar radiotherapy inverse planning is proposed. It consists of a genetic algorithm-based framework that solves with little modification treatment planning for three different kinds of radiation therapy: conformal, aperture-based and intensity modulated. Thanks to evolutionary optimisation techniques we have been able to search for full beam configurations, that is beam intensity, beam shape and especially beam orientation. Unlike some previous works found in literature, our proposed solution automatically determines exact beam angles not relaying solely on a geometrical basis but involving beam intensity profiles, thus considering the effective delivered dose. Our dose distribution model has been validated through comparison with commercial system: fixed the same beam configuration, both calculated beam shapes and DVH have been compared. Then we have tested the optimisation algorithm with real clinical cases: these involved both simple (convex target, far organs at risk) and complex (concave target, close organs at risk) ones. As stated by physician and by simulation with the same commercial system, our tools found good solutions in both cases using corresponding correct therapy.

The Worldwide Equity Premium: A Smaller Puzzle

We use a new database of long-run stock, bond, bill, inflation, and currency returns to estimate the equity risk premium for 17 countries and a world index over a 106-year interval. Taking U.S. Treasury bills (government bonds) as the risk-free asset, the annualised equity premium for the world index was 4.7% (4.0%). We report the historical equity premium for each market in local currency and US dollars, and decompose the premium into dividend growth, multiple expansion, the dividend yield, and changes in the real exchange rate. We infer that investors expect a premium on the world index of around 3-3 1/2% on a geometric mean basis, or approximately 4 1/2-5% on an arithmetic basis.

Systematic Sequences of Geometric Relativistic Basis Sets. I:s- and p-Block Elements up to Xe

In this work a scheme for constructing systematic sequences of relativistic SCF basis sets at a reasonable computational cost is presented and applied to atoms of the s- and p-block up to Xe. This scheme, which couples simplex optimization and the use of geometric series given by four-term polynomial expressions for the logarithm of the exponents, allows for the construction of basis sets that exhibit very regular patterns of convergence to the numerical reference values of atomic total energies, spinor energies and radial expectation values. This regularity, together with the broad range of basis set sizes presented, enables these sets to be used as building blocks for basis sets applicable in both routine and benchmark relativistic calculations on atomic and molecular systems.

Proximal femur geometric basis for the choice of femoral implant in cemented hip replacement

Proximal femur geometric basis for the choice of femoral implant in cemented hip replacement

QSAR studies on 1-phenylbenzimidazoles as inhibitors of the platelet-derived growth factor

This work is devoted to the development of quantitative structure–activity relationship (QSAR) models of the biological activity of 123 1-phenylbenzimidazoles as inhibitors of the PDGF receptor. The molecular features are represented by chemical descriptors that have been calculated on geometrical, topological, quantum mechanical, and electronic basis by using CODESSA PRO. The obtained models, linear (multilinear regression) and nonlinear (artificial neural network), are aimed to link the structures to their reported activity log 1/IC 50. The former model can be used for physico-chemical interpretation, while the latter possesses a superior predictive ability.

Highly accurate relativistic universal Gaussian basis set for Dirac–Fock–Breit calculations

AbstractA universal Gaussian basis set is presented to perform highly accurate Dirac–Fock–Breit calculations. The Gaussian‐type functions of our universal basis set satisfy the relativistic boundary conditions associated with the finite nucleus model, for a finite velocity of light, and conform to the so‐called kinetic balance at the nonrelativistic limit. The high accuracy obtained in our Dirac–Fock–Breit calculations is attributed to the fact that with our universal Gaussian basis set we are able to better represent the relativistic kinematics of an electron inside the nucleus. Results for the Dirac–Fock–Breit energies for a number of neutral atoms up to nobelium and some of their negative ions are presented and compared with the results of Parpia et al., obtained by employing a geometrical Gaussian basis set and a more accurate value for the velocity of light. The Dirac–Fock–Breit energy results presented in this work, using our universal Gaussian basis set, are among the best obtained with Gaussian‐type functions in the literature. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2005

Reference frame, angular momentum, and porphyroblast rotation

Rotation of small rigid objects in a deforming ductile matrix can produce two different types of microstructure: a shape fabric due to alignment of the principal axes of a population of elongate objects and the inclusion trail microstructure preserved in syntectonic porphyroblasts. We use numerical modeling to show that inclusion trails of elongate porphyroblasts are expected to be extremely complex. In contrast, snowball garnets are readily interpretable. But misuse of reference frame and kinematic misconceptions have obfuscated the discussion on the formation of porphyroblast inclusion trails in general and snowball garnet inclusion trails in particular. We clarify this point. Models for snowball garnet formation that are based on the notion of garnets being irrotational with respect to the earth can be rejected on a geometrical and kinematic basis. Further, the notion that rigid objects embedded in a deforming ductile matrix generally do not rotate is unsound—it violates the fundamental physical law of balance of angular momentum.

Single-walled Carbon Nanotubes Are a New Class of Ion Channel Blockers

Here we identify a novel class of biological membrane ion channel blockers called single-walled carbon nanotubes (SWNTs). SWNTs with diameter distributions peaked at approximately 0.9 and 1.3 nm, C60 fullerenes, multi wall nanotubes (MWNTs), and hyperfullerenes (nano-"onions") were synthesized by several techniques and applied to diverse channel types heterologously expressed in mammalian cells. External as-fabricated and purified SWNTs blocked K+ channel subunits in a dose-dependent manner. Blockage was dependent on the shape and dimensions of the nanoparticles used and did not require any electrochemical interaction. SWNTs were more effective than the spherical fullerenes and, for both, diameter was the determining factor. These findings postulate new uses for SWNTs in biological applications and provide unexpected insights into the current view of mechanisms governing the interaction of ion channels with blocking molecules.

Einstein–Cartan theory as a theory of defects in space–time

The Einstein–Cartan theory of gravitation and the classical theory of defects in an elastic medium are presented and compared. The former is an extension of general relativity and refers to four-dimensional space–time, while we introduce the latter as a description of the equilibrium state of a three-dimensional continuum. Despite these important differences, an analogy is built on their common geometrical foundations, and it is shown that a space–time with curvature and torsion can be considered as a state of a four-dimensional continuum containing defects. This formal analogy is useful for illustrating the geometrical concept of torsion by applying it to concrete physical problems. Moreover, the presentation of these theories using a common geometrical basis allows a deeper understanding of their foundations.