Bistable mathematical and mechanical models related to the urethra

Normal micturition is controlled primarily by a neural system. Certain physical effects become evident when neural control is destroyed, and the automatic or autonomous bladder phenomena occur. It is shown in this paper that a physical system simulating the alternating periods of continence and voiding of the automatic bladder may comprise only passive elastic components, and that periodic voiding does not per se imply neural control.     

Comparison of mathematical models for the maternal age dependence of Down's syndrome rates

Summary The maternal age dependence of Down's syndrome rates was analyzed by two mathematical models, a discontinuous (DS) slope model which fits different exponential equations to different parts of the 20–49 age interval and a CPE model which fits a function that is the sum of a constant and exponential term over this whole 20–49 range. The CPE model had been considered but rejected by Penrose, who preferred models postulating changes with age assuming either a power function X¹⁰, where X is age or a Poisson model in which accumulation of 17 events was the assumed threshold for the occurrence of Down's syndrome. However, subsequent analyses indicated that the two models preferred by Penrose did not fit recent data sets as well as the DS or CPE model. Here we report analyses of broadened power and Poisson models in which n (the postulated number of independent events) can vary. Five data sets are analyzed. For the power models the range of the optimal n is 11 to 13; for the Poisson it is 17 to 25. The DS, Poisson, and power models each give the best fit to one data set; the CPE, to two sets. No particular model is clearly preferable. It appears unlikely that, with a data set from any single available source, a specific etiologic hypothesis for the maternal age dependence of Down's syndrome can be clearly inferred by the use of these or similar regression models.     

Regional ventilation during spontaneous breathing and mechanical ventilation in dogs

We evaluated the effects of the different patterns of chest wall deformation that occur with different body positions and modes of breathing on regional lung deformation and ventilation. Using the parenchymal marker technique, we determined regional lung behavior during mechanical ventilation and spontaneous breathing in five anesthetized recumbent dogs. Regional lung behavior was related to the patterns of diaphragm motion estimated from X-ray projection images obtained at functional residual capacity (FRC) and end inspiration. Our results indicate that 1) in the prone and supine positions, FRC was larger during mechanical ventilation than during spontaneous breathing; 2) there were significant differences in the patterns of diaphragm motion and regional ventilation between mechanical ventilation and spontaneous breathing in both body positions; 3) in the supine position only, there was a vertical gradient in lung volume at FRC; 4) in both positions and for both modes of breathing, regional ventilation was nonlinearly related to changes in lobar and overall lung volumes; and 5) different patterns of diaphragm motion caused different sliding motions and differential rotations of upper and lower lobes. Our results are inconsistent with the classic model of regional ventilation, and we conclude that the distribution of ventilation is determined by a complex interaction of lung and chest wall shapes and by the motion of the lobes relative to each other, all of which help to minimize distortion of the lung parenchyma.     

Comparison of several mathematical models for describing the joint effect of temperature and ph on glucanex activity

The aim of the present work was to evaluate with different statistical criteria the suitability of nine equations for describing and optimizing the simultaneous effect of temperature and pH on glucanex activity using two characteristic polysaccharides (curdlan and laminarin) as substrates. The most satisfactory solutions were found with an empirical equation constituted with parameters of practical interest (Rosso model), and a hybrid model between the Arrhenius equation and the mathematical expression generated by the protonation-hydroxylation mechanism (Tijskens model). The joint optimal values of pH and temperature calculated with the Rosso model were obtained at 4.64 and 50°C with curdlan and 4.64 and 48°C using laminarin as substrate.     

Continuous and discrete mathematical models of tumor-induced angiogenesis

Angiogenesis, the formation of blood vessels from a pre-existing vasculature, is a process whereby capillary sprouts are formed in response to externally supplied chemical stimuli. The sprouts then grow and develop, driven initially by endothelial-cell migration, and organize themselves into a dendritic structure. Subsequent cell proliferation near the sprout tip permits further extension of the capillary and ultimately completes the process. Angiogenesis occurs during embryogenesis, wound healing, arthritis and during the growth of solid tumors. In this paper we present both continuous and discrete mathematical models which describe the formation of the capillary sprout network in response to chemical stimuli (tumor angiogenic factors, TAF) supplied by a solid tumor. The models also take into account essential endothelial cell-extracellular matrix interactions via the inclusion of the matrix macromolecule fibronectin. The continuous model consists of a system of nonlinear partial differential equations describing the initial migratory response of endothelial cells to the TAF and the fibronectin. Numerical simulations of the system, using parameter values based on experimental data, are presented and compared qualitatively with in vivo experiments. We then use a discretized form of the partial differential equations to develop a biased random-walk model which enables us to track individual endothelial cells at the sprout tips and incorporate anastomosis, mitosis and branching explicitly into the model. The theoretical capillary networks generated by computer simulations of the discrete model are compared with the morphology of capillary networks observed in in vivo experiments.     

根瘤菌生态学研究中两个数学模型的探讨

以自然重壤土和无菌砂为培养基质,栽种大豆,采用MPN法测数,证明了大豆根瘤菌数量的增长符合逻辑斯谛模型并编写了相应的BASIC程序。通过6组试验数据的计算,发现根瘤菌接种量和其占瘤率之间存在一定的数学关系,并编写了相应的BASIC程序。     

A note on equivalent mathematical models

A mathematical model of a process contains parameters supposedly characterizing the system which manifests the process. If the parameters are statistically distributed in a population of such systems, the process manifested by the entire population will in general be described by a different mathematical model. Thus a choice is always at hand between two or more mathematical models, depending on which parameters (if any) are assumed to be distributed and, if so, how. Examples of such alternative interpretations are given for mathematical models of some behavioral processes.     

Synthesis of mathematical models of branching axons and dendrites

A synthesis was made of models of branching neuronal cable structures from a full set of standard basic models. The study aimed to produce an instrument of mathematical modelling making it possible to reflect true life morphological and electrophysiological characteristics of axons and dendrites, discarding some of the restrictions and simplifications characterizing existing models of the structures mentioned. Equivalent electrical circuits of branching axons and dendrites were set up with in-series and node connections of standard four-terminal networks corresponding to basic segments with active or passive membrane. Equations were obtained for electrical processes in branching neuronal neurites, generalized in the case of multiple binary branching with arbitrary symmetry and branching structure. A difference scheme common to the whole class of models contemplated was produced and the algorithm of a numerical solution to the difference equations thus obtained was elaborated. The instrument described makes it possible to synthesize diverse models of branching axons and dendrites, offering considerably greater opportunities for modelling the main electrophysiological processes developing in these structures of electrotonus, propagation of excitation, and interaction between these two factors.State University Commemorating Tricentenary of Russo-Ukrainian Union. Dnepropetrovsk. Translated from Neirofiziologiya, Vol. 20, No. 4, pp. 471–479, July–August, 1988.     

An application of mathematical models in posturography

The measurement of the horizontal displacements (HD) of the centre of mass of the human body when standing still, is often performed by means of a platform. The resulting measured HD as a function of time, both in the left-right or sinister-dexter and the posterior-anterior direction, are known as stabilograms. Such stabilograms are being more widely used to obtain the so called measured statokinesigram, similar to a Lissajous-figure, by eliminating time from the sinister-dexter stabilogram and the posterior-anterior stabilogram. It is shown that in using mathematical models of the standing person to obtain the stabilograms, the resulting statokinesigram is very different from the measured statokinesigram. It is also shown that the quantity line-integral calculated from a statokinesigram is much smaller when determined from a statokinesigram corrected by means of the mathematical models.