On the construction of a new generalization of Runge–Kutta methods

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On the chaotic nature of electro-discharge machining

The long-accepted thermoelectric model for electro-discharge machining is being brought into question. Several experimental facts prompt the proposal of a new theory based on the effect of gap pollution on the ignition of discharges. The first experimental proof comes from the recently reported observation of debris chains and clusters. In this view, each step of the process depends on the previous ones through a deterministic relation, even if the overall evolution is unpredictable. The paper establishes mathematical grounds for the abovementioned intuitions by setting up and solving a recursive equation for the machining energy employed at each discharge event. By means of numerical and algebraic tools, the above equation is studied and shows a chaotic evolution similar to that of the logistic map. Results reconcile the apparent paradox between deterministic nature and stochastic localization of the discharges and introduce a description of the chaotic dynamics of electro-discharge processes

A Computational Template for Three-Dimensional Modeling of the Vascular Scaffold of the Human Thyroid Gland

We recently designed an innovative scaffold-bioreactor unit for the bioengineering of a three-dimensional (3D) bioartificial human thyroid gland or its miniaturized replica as a part of a microfluidic chip test system.This device is based on the evidence that the 3D geometry of the intraglandular stromal/vascular scaffold (SVS; i.e., the fibrous and vascular matrix) of mammalian viscera plays a key role in guiding growth and differentiation of in vitro seeded cells. Therefore, we initiated a research program focused on computer-aided reconstruction of the 2nd to 4th order intralobar arterial network (IAN) of the human thyroid gland asa reliable surrogate for its 3D SVS, to be used as an input for rapid prototyping of a biomaterial replica. Tothis end, we developed a computational template that works within the Mathematica environment, giving rise to a quasi-fractal growth of the IAN distribution, constrained within an approximation of the thyroidl obe shape as a closed surface. Starting from edge detection of planar images of real human thyroid lobes acquired by in vivo real-time ultrasonography, we performed data approximation of the lobar profiles based on splines and Bezier curves, providing 3D lobar shapes as geometric boundaries for vessel growth by a diffusion-limited aggregation model. Our numerical procedures allowed for a robust connection between development of lobar arterial trees and thyroid lobe shape, led to a vascular self-similarity consistent with that of a cadaveric lobar arterial cast, and reproduced arterial vessels in a proportion not statistically different from that described for the real human thyroid gland. We conclude that our algorithmic template offers a reliable reproduction of the extremely complex IAN of the adult human thyroid lobe, potentially useful as a computational guidance for bioprinting of thyroid lobe matrix replicas. In addition, due to the simplicity and limited number of morphometrical parameters required by our system, we predict its application to the design of a number of patient-tailored human bioartificial organs and organs-on-chip,including parenchymal viscera and bones

Runtime complexity estimation for accurate solution of linear systems

An established idea for the accurate solution of linear systems is to use iterative refinement. More recently it has been shown that a modification of iterative refinement can be advantageous for high precision computation. In this work we describe a simplified complexity analysis that reliably shows when an iterative solution procedure is advantageous over a direct solution method. The analysis involves an estimate of the condition number of a matrix and is efficient enough to be used for automatic method selection at runtime in a linear solver. We also introduce a scaling technique that is advantageous when solving for the solution and correction steps using machine precision. Numerical experiments, using an implementation developed in the Mathematica kernel, are provided to confirm the theory that has been presented

Stiffness Detection Revisited

Many applied differential equations exhibit some form of stiffness, which restricts the step\u2013size and, hence, effectiveness of explicit solution methods. A number of implicit methods have been proposed to circumvent this problem. However, implicit methods can also be substantially less efficient, due to overhead associated with the intrinsic linear algebra. Several attempts have been made to provide user\u2013friendly codes, that would automatically attempt to detect stiffness, at run time, and switch between appropriate methods as necessary. In this work, we outline a new implementation to automatically equip a code with a stiffness detection device. Particular attention is given to the problem of estimation of the dominant eigenvalue of a matrix. We propose an efficient implementation, based on subspace and Krylov iteration, that is now part of the automated method selection used in the scientific problem solving environment of Mathematica. To demonstrate the effectiveness of our strategy, numerical experiments are given with a focus on stiff differential systems that arise in bio-medical applications

Glandular Morphology Biomodeling

Aim of this work is the modeling the parenchymal and vascular/stromal morphology of human glands, from their scintigraphic or echographic images; the model of interest, here, is the adult human thyroid [R. Toni et al, "Ex-situ bioengineering of bioartificial endocrine glands: a new frontier in regenerative medicine of soft tissue organs", Annals of Anatomy 193:381-394, 2011]. Simulation of the thyroid morphology is approached, here, via a hybrid technique, involving a numerical reconstruction of surfaces through edge-detection and data fitting, followed by a fractal modeling of the vascular/stromal skeleton (SSV), constrained by anatomical morphometric data information. A software is developed in Mathematica, constituted of a computational kernel of mathematical and visualization routines. The output of this software is exported in stereo-lithographic format, suitable for rapid prototyping. This work, funded by Grant PRIN 2008, stems from a cooperation of many researchers: M. Sofroniou, A. Gatto and the Research Units at Modena and Bologna Universities, R. Toni and the Research Units at Parma University and Hospital, Bologna CNR and IOR and Faenza CNR

A planar fractal analysis of the arterial tree of the human thyroid gland: implications for additive manufacturing of 3D ramified scaffolds

It is currently known that a number of human vascular systems have a fractal geometry. Since we have recently developed a technique to prototype single arterial branches of human soft tissue organs by additive layer manufacturing (AM), we have explored the possibility that auto-similarity in vessel branching represents a key variable for accurate computational modeling of the organ three-dimensional (3D), macro / microscopic anatomy, and its reproduction by inverse engineering. To this purpose, ramification features of the intra-lobar arteries of the human thyroid were studied using injection-corrosion casts of a cadaveric gland. Vessel diameters, ramification angles and branch lengths were measured by light microscopic, computer-aided optical metrology. Distribution of morphological variables was considered on a cumulative basis, and special focus was given to the branching laws. To reduce the bias of vascular distortion due to the pressure of intra-vascular resin injection, measures were made dimensionless through the use of a scaling parameter set on the vascular caliber of major afferent arteries. In addition, using high resolution micro-tomography (mCT Skyscan 1172, Bruker micro-CT) equipped with CTAn software and the Otsu algorithm for segmentation, spaces occupied by vascular branches (referred to as Volume of Interests, VOI) were selected, and their planar fractal dimension calculated (Mandelbrot 1982). Finally, a computational simulation of the vascular tree was achieved using a mixed, stochastic / deterministic algorithm, based on diffusion limited aggregation (DLA; Witten and Sander, 1981), constrained by mean values of vascular variables. Ratios among decreasing cast calibers, ramification angles and branch lengths, respectively, were found strictly interrelated, mCT-VOI depicted fractal dimensions, and DLA simulation led to a fractal-like organization consistent with real data morphometrics. In summary, thyroid arterial geometry reliably exhibited a degree of auto-similarity, suggesting that fractality is a key feature for computational modeling and eventual AM of 3D vascular networks of the human thyroid