A model of extravascular bubble evolution: effect of changes in breathing gas composition.

Observations of bubble evolution in rats after decompression from air dives (O. Hyldegaard and J. Madsen. Undersea Biomed. Res. 16: 185-193, 1989; O. Hyldegaard and J. Madsen. Undersea Hyperbaric Med. 21: 413-424, 1994; O. Hyldegaard, M. Moller, and J. Madsen. Undersea Biomed. Res. 18: 361-371, 1991) suggest that bubbles may resolve more safely when the breathing gas is a heliox mixture than when it is pure O(2). This is due to a transient period of bubble growth seen during switches to O(2) breathing. In an attempt to understand these experimental results, we have developed a multigas-multipressure mathematical model of bubble evolution, which consists of a bubble in a well-stirred liquid. The liquid exchanges gas with the bubble via diffusion, and the exchange between liquid and blood is described by a single-exponential time constant for each inert gas. The model indicates that bubbles resolve most rapidly in spinal tissue, in adipose tissue, and in aqueous tissues when the breathing gas is switched to O(2) after surfacing. In addition, the model suggests that switching to heliox breathing may prolong the existence of the bubble relative to breathing air for bubbles in spinal and adipose tissues. Some possible explanations for the discrepancy between model and experiment are discussed.     

Mathematical models of diffusion-limited gas bubble dynamics in tissue

Mathematical models of bubble evolution in tissue have recentlybeen incorporated into risk functions for predicting the incidence ofdecompression sickness (DCS) in human subjects after diving and/or flying exposures. Bubble dynamics models suitable forthese applications assume the bubble to be either contained in anunstirred tissue (two-region model) or surrounded by a boundary layerwithin a well-stirred tissue (three-region model). The contrastingpremises regarding the bubble-tissue system lead to differentexpressions for bubble dynamics described in terms of ordinarydifferential equations. However, the expressions are shown to bestructurally similar with differences only in the definitions ofcertain parameters that can be transformed to make the modelsequivalent at large tissue volumes. It is also shown that thetwo-region model is applicable only to bubble evolution in tissues ofinfinite extent and cannot be readily applied to bubble evolution infinite tissue volumes to simulate how such evolution is influenced byinteractions among multiple bubbles in a given tissue. Two-regionmodels that are incorrectly applied in such cases yield results thatmay be reinterpreted in terms of their three-region model equivalents but only if the parameters in the two-region model transform into consistent values in the three-region model. When such transforms yieldinconsistent parameter values for the three-region model, results maybe qualitatively correct but are in substantial quantitative error.Obviation of these errors through appropriate use of the differentmodels may improve performance of probabilistic models of DCSoccurrence that express DCS risk in terms of simulated in vivo gas andbubble dynamics.     

Mathematical analysis of a model describing evolution of an asexual population in a changing environment

We investigate a mathematical model for an asexual population with non-overlapping (discrete) generations, that exists in a changing environment. Sexual populations are also briefly discussed at the end of the paper. It is assumed that selection occurs on the value of a single polygenic trait, which is controlled by a finite number of loci with discrete-effect alleles. The environmental change results in a moving fitness optimum, causing the trait to be subject to a combination of stabilising and directional selection.This model is different from that investigated by Waxman and Peck [Genetics 153 (1999) 1041] where overlapping generations and continuous effect alleles were considered. In this paper, we consider non-overlapping generations and discrete effect alleles. However in [Genetics 153 (1999) 1041] and the present work, there is the same pattern of environmental change, namely a constant rate of change of the optimum.From [Genetics 153 (1999) 1041], no rigorous theoretical conclusion can be drawn about the form of the solutions as t grows large. Numerical work carried out in [Genetics 153 (1999) 1041] suggests that the solution is a lagged travelling wave solution, but no mathematical proof exists for the continuous model. Only partial results, regarding existence of travelling wave solutions and perturbed solutions, have been established (see [Nonlin. Anal. 53 (2003) 683; An integral equation describing an asexual population in a changing environment, Preprint]). For the discrete case of this paper, under the assumption that the ratio between the unit of genotypic value and the speed of environment change is a rational number, we are able to give rigorous proof of the following conclusion: the population follows the environmental change with a small lag behind, moreover, the lag is represented using a calculable quantity.     

Mathematical model for adaptive evolution of populations based on a complex domain

A mutation is ultimately essential for adaptive evolution in all populations. It arises all the time, but is mostly fixed by enzymes. Further, most do consider that the evolution mechanism is by a natural assortment of variations in organisms in line for random variations in their DNA, and the suggestions for this are overwhelming. The altering of the construction of a gene, causing a different form that may be communicated to succeeding generations, produced by the modification of single base units in DNA, or the deletion, insertion, or rearrangement of larger units of chromosomes or genes. This altering is called a mutation. In this paper, a mathematical model is introduced to this reality. The model describes the time and space for the evolution. The tool is based on a complex domain for the space. We show that the evolution is distributed with the hypergeometric function. The Boundedness of the evolution is imposed by utilizing the Koebe function.     

Concentration distribution around a growing gas bubble in tissue

This paper presents the concentration distribution around a growing nitrogen gas bubble in the blood and other tissues of divers who surface too quickly, when the ambient pressure through the decompression process is variable and constant. This effort is a modification of Sirinivasan et al. model (1999) [9]. The mathematical model is solved analytically to find the growth rate of a gas bubble in a tissue after decompression in the ambient pressure. Moreover, the concentration distribution around the growing bubble is introduced. The growth process is affected by ascent rate , tissue diffusivity D_T, initial concentration difference ΔC₀, surface tension σ and void fraction ?₀.     

Numerical simulation of sinusoidal flow in a straight elastic tube: effects of phase angles

Numerical simulations of flow in straight elastic (moving wall) tubes subjected to a sinusoidal pressure gradient were performed for conditions prevailing in large and medium sized arteries. The effects of varying the phase angle between the pressure gradient and the tube radius, the amplitude of wall motion, and the unsteadiness parameter (alpha) on flow rate and wall shear stress were investigated. Mean and peak flow rates and shear stresses were found to be strongly affected by the phase angle between the pressure gradient and the tube radius with greater sensitivity at higher diameter variation and higher alpha. In large artery simulations (alpha = 12), means flow rate was found to be 60% higher and peak flow rate to be 73% higher than corresponding rigid tube values for certain phase angles, while a threefold increase in mean wall shear stress and sevenfold increase in peak wall shear stress were observed in a sensitive phase angle range. Significant reversal in the wall shear stress direction occurred in the sensitive phase angle range even when there was negligible flow rate reversal. All effects were greatly diminished in simulations of medium sized vessels (alpha = 4). Some experimental evidence to support the predictions of a strong effect of phase angle on wall shear stress in large vessels is presented. Finally, physiological implications of the present work are discussed from a basis of aortic input impedance data, and a physical explanation for the extreme sensitivity of the flow field to small amplitude wall motion at high alpha is given.     

Mathematical modeling of evolution. Solved and open problems

Evolution is a highly complex multilevel process and mathematical modeling of evolutionary phenomenon requires proper abstraction and radical reduction to essential features. Examples are natural selection, Mendel’s laws of inheritance, optimization by mutation and selection, and neutral evolution. An attempt is made to describe the roots of evolutionary theory in mathematical terms. Evolution can be studied in vitro outside cells with polynucleotide molecules. Replication and mutation are visualized as chemical reactions that can be resolved, analyzed, and modeled at the molecular level, and straightforward extension eventually results in a theory of evolution based upon biochemical kinetics. Error propagation in replication commonly results in an error threshold that provides an upper bound for mutation rates. Appearance and sharpness of the error threshold depend on the fitness landscape, being the distribution of fitness values in genotype or sequence space. In molecular terms, fitness landscapes are the results of two consecutive mappings from sequences into structures and from structures into the (nonnegative) real numbers. Some properties of genotype–phenotype maps are illustrated well by means of sequence–structure relations of RNA molecules. Neutrality in the sense that many RNA sequences form the same (coarse grained) structure is one of these properties, and characteristic for such mappings. Evolution cannot be fully understood without considering fluctuations—each mutant originates form a single copy, after all. The existence of neutral sets of genotypes called neutral networks, in particular, necessitates stochastic modeling, which is introduced here by simulation of molecular evolution in a kind of flowreactor.     

Optical monitoring of bubble size and shape in a pulsating bubble surfactometer.

The pulsating bubble surfactometer (PBS) is often used for in vitro characterization of exogenous lung surfactant replacements and lung surfactant components. However, the commercially available PBS is not able to dynamically track bubble size and shape. The PBS therefore does not account for bubble growth or elliptical bubble shape that frequently occur during device use. More importantly, the oscillatory volume changes of the pulsating bubble are different than those assumed by the software of the commercial unit. This leads to errors in both surface area and surface tension measurements. We have modified a commercial PBS through the addition of an image-acquisition system, allowing real-time determination of bubble size and shape and hence the accurate tracking of surface area and surface tension. Compression-expansion loops obtained with the commercially available PBS software were compared with those provided by the image-analysis system for dipalmitoylphosphatidylcholine, Infasurf, and Tanaka lipids (dipalmitoylphosphatidylcholine-palmitoyloleoylphosphatidyl-glycerol-palmitic acid, 68:22:9) at concentrations of 0.1 and 1.0 mg/ml and at frequencies of 1 and 20 cycles/min. Whereas minimum surface tension as determined by the image-analysis system is similar to that measured by the commercially available software, the maximum surface tension and the shapes of the interfacial area-surface tension loops are quite different. Differences are attributable to bubble drift, nonsinusoidal volume changes, and variable volume excursions seen with the modified system but neglected by the original system. Image analysis reveals that the extent of loop hysteresis is greatly overestimated by the commercial device and that an apparent, rapid increase in surface tension upon film expansion seen in PBS loops is not observed with the image-analysis system. The modified PBS system reveals new dynamic characteristics of lung surfactant preparations that have not previously been reported.     

Protein-fold evolution in the test tube

Currently, the combination of library selection and directed evolution is the most powerful approach for finding proteins with novel folds or functions. In the past, most studies concentrated either on protein scaffolds with a given fold or on short peptides. With the recent development of potent in vitro selection and evolution techniques, the screening of much larger sequence space is possible, allowing for the de novo generation of proteins.     

Mathematical model of a batch acetone-butanol fermentation

A mathematical model for the batch culture of Clostridium acetobutylicum was formulated using experimental data for anaerobic solvent production. The model summarizes biochemical as well as physiological aspects of growth and metabolite synthesis by the production strain. The key fermentation rates are expressed and evaluated with regard to substrate consumption and butanol end-product inhibitory effects. Parametric sensitivity analysis of the batch process model was carried out, indicating the importance of the key process parameters.     

Mathematical model of a cell size checkpoint

How cells regulate their size from one generation to the next has remained an enigma for decades. Recently, a molecular mechanism that links cell size and cell cycle was proposed in fission yeast. This mechanism involves changes in the spatial cellular distribution of two proteins, Pom1 and Cdr2, as the cell grows. Pom1 inhibits Cdr2 while Cdr2 promotes the G2 → M transition. Cdr2 is localized in the middle cell region (midcell) whereas the concentration of Pom1 is highest at the cell tips and declines towards the midcell. In short cells, Pom1 efficiently inhibits Cdr2. However, as cells grow, the Pom1 concentration at midcell decreases such that Cdr2 becomes activated at some critical size. In this study, the chemistry of Pom1 and Cdr2 was modeled using a deterministic reaction-diffusion-convection system interacting with a deterministic model describing microtubule dynamics. Simulations mimicked experimental data from wild-type (WT) fission yeast growing at normal and reduced rates; they also mimicked the behavior of a Pom1 overexpression mutant and WT yeast exposed to a microtubule depolymerizing drug. A mechanism linking cell size and cell cycle, involving the downstream action of Cdr2 on Wee1 phosphorylation, is proposed.     

Observation and quantification of gas bubble formation on a mechanical heart valve

Clinical studies using transcranial Doppler ultrasonography in patients with mechanical heart valves (MHV) have detected gaseous emboli. The relationship of gaseous emboli release and cavitation on MHV has been a subject of debate in the literature. To study the influence of cavitation and gas content on the formation and growth of stable gas bubbles, a mock circulatory loop, which employed a Medtronic-Hall pyrolytic carbon disk valve in the mitral position, was used. A high-speed video camera allowed observation of cavitation and gas bubble release on the inflow valve surfaces as a function of cavitation intensity and carbon dioxide (CO2) concentration, while an ultrasonic monitoring system scanned the aortic outflow tract to quantify gas bubble production by calculating the gray scale levels of the images. In the absence of cavitation, no stable gas bubbles were formed. When gas bubbles were formed, they were first seen a few milliseconds after and in the vicinity of cavitation collapse. The volume of the gas bubbles detected in the aortic track increased with both increased CO2 and increased cavitation intensity. No correlation was observed between O2 concentration and bubble volume. We conclude that cavitation is an essential precursor to stable gas bubble formation, and CO2, the most soluble blood gas, is the major component of stable gas bubbles.     

Effects of gas properties and waveform asymmetry on gas transport in a branching tube network

Local gas transport coefficients, quantifying longitudinal dispersion through a symmetrical constant-diameter tube network, have been measured during oscillation with both symmetrical and nonsymmetrical waveforms. Experiments were carried out over a range of conditions that would prevail in the central to lower airways during high-frequency ventilation at moderate frequency (5 Hz) and tidal volume (15-80 ml). Gas transport coefficients resulting from oscillation of three different resident-trace gas pairs were measured using a new analytic technique. This technique allowed rapid determination of the transport coefficient distribution along the entire network. Results demonstrate a small but significant influence attributable to changes in gas properties that is similar to that found in a straight tube and indicate that augmented dispersion is an important mechanism of axial transport. Gas transport coefficients were found to be unaffected by changes in flow waveform symmetry, suggesting that previously reported improvements in gas exchange associated with decreasing inspiratory to expiratory time ratios are not due to a change in local conditions such as asymmetry in the velocity profile.     

A careful physical analysis of gas bubble dynamics in xylem

Many studies have confirmed that cavitation in xylem is caused by air bubbles. Recently Shen et al. (Tree Physiol. 22 (2002) R655), analysed the expansion of a pre-existent bubble in xylem and one formed by air seeding. The present paper makes a further analysis of bubble expansion by the equilibrium criterion of the Helmholtz function. It has been proved that when xylem pressure P'l decreases to a special value P'l* from a value higher than, or equal to, or lower than -Po (Po is atmospheric pressure), an air bubble in xylem can grow up steadily, corresponding to minimums of the Helmholtz function F(r). As soon as P'l is lower than P'l*, since F(r) will be a decreasing function when P'l < P'l*, resulting in non-equilibrium of the bubble, it will break inducing a cavitation event. The analysis is consistent with the results of mechanism. Given P'l > or = -3Po, if an air bubble could enter a conduit, it would be in a stable equilibrium. When P'l < -3Po an air bubble entering a conduit will be in an unstable equilibrium. As the water further vaporizes, it will break at once. This is the case to which the former published formula P'l = -2sigma/rp is applicable.     

Characterization of gas transfer and mixing in a bubble column equipped with a rubber membrane diffuser

Gas transfer and mixing were characterized in a 32-L bubble column reactor equipped with a commercially available rubber membrane diffuser. The performance of the membrane diffuser indicates that the slits in the membrane are best described as holes with elastic lids, acting as valves cutting off bubbles from the gas stream. The membrane diffuser thus functions as a one-way valve preventing backflow of liquid. Our design of the bottom plate of the reactor enabled us to optimize the aeration by changing the tension of the membrane. We thereby achieved mass transfer coefficients higher than those previously reported in bubble columns. A strong dependence of mass transfer on gas holdup and bubble size was indicated by estimates based on these two variables. The microalga, Rhodomonas sp. , sensitive to chemical and physical stress, was maintained for 8 months in continuous culture with a productivity identical to cultures grown in stirred tank reactors. Copyright 1998 John Wiley & Sons, Inc.     

Mathematical analysis of a model for a plant-herbivore system

The apple twig borer (Amphicerus bicaudatus) is an insect pest of the grape vine, causing considerable damage to the grape vine in early spring. A simple difference equation model is formulated and analysed for this plant-herbivore system based on two control strategies, cane removal and pesticide application. The system has two equilibria, one where the pest is present and one where the pest is absent. Regions are found in parameter space for global stability of the equilibria and in the absence of global stability it is shown that there exist periodic or quasiperiodic solutions.     

生态脆弱区生态系统状态演变分析的若干数学方法

生态脆弱区往往存在多个生态系统(草原、荒漠和灌木等)共存的现象.由于外部环境和人类活动等因素的影响,生态脆弱区会发生从一种生态系统转变为另一种生态系统的现象,即突变现象.分析生态脆弱区多生态系统共存情况下生态系统的稳定性对了解生态脆弱区生态系统的变化具有重要意义.本文回顾了目前能够描述生态脆弱区多生态系统的动力系统及其...     

Mathematical properties of responses of a neuron model

Recently, Nagumo and Sato proposed a mathematical neuron model in the form of a nonlinear difference equation and investigated its response characteristic. The result showed that the input-output relationship of the neuron model is quite complicated and takes the form of an extended Cantor's function. It also explained the unusual and unsuspected phenomenon found by Harmon in experimental studies with his transistor neuron model. — In this paper, a fraction representation of a sequence of pulses is proposed. A mathematical treatment of the same neuron model based on the representation gives the same result as in the previous paper. Moreover, many mathematical properties, including the one where the ratio of the number of 1's contained in a cycle of a sequence to the length of the cycle gives any rational number between 0 and 1, were obtained by investigating sequences generated by the model.     

Oxygen mass transfer and gas holdup in a bubble column bioreactor with biosynthesis liquids

The influence of the rheology of some antibiotic biosynthesis liquids produced by Streptomyces aureofaciens, Nocardia mediterranei and Penicillium chrysogenum on the volumetric liquid phase oxygen transfer coefficient, k_La, and gas holdup, ε_G, together with the influence of superficial gas velocity, were studied in a bubble column bioreactor, using samples of fermentation liquids taken from industrial stirred tank fermenters, at 30-hour intervals during fermentation batch. The results were compared to those of previous studies from literature on non-Newtonian homogeneous fluids, such as CMC-Na, xanthan and starch solutions, respectively. In the heterogeneous broths, ε_G and k_La decreased with increasing apparent viscosity of the broth and increased with increasing superficial velocity. The experimental data were correlated using non-linear regression with correlation coefficients above 0.85.