One-dimensional model for propagation of a pressure wave in a model of the human arterial network: comparison of theoretical and experimental results

Pulse wave evaluation is an effective method for arteriosclerosis screening. In a previous study, we verified that pulse waveforms change markedly due to arterial stiffness. However, a pulse wave consists of two components, the incident wave and multireflected waves. Clarification of the complicated propagation of these waves is necessary to gain an understanding of the nature of pulse waves in vivo. In this study, we built a one-dimensional theoretical model of a pressure wave propagating in a flexible tube. To evaluate the applicability of the model, we compared theoretical estimations with measured data obtained from basic tube models and a simple arterial model. We constructed different viscoelastic tube set-ups: two straight tubes; one tube connected to two tubes of different elasticity; a single bifurcation tube; and a simple arterial network with four bifurcations. Soft polyurethane tubes were used and the configuration was based on a realistic human arterial network. The tensile modulus of the material was similar to the elasticity of arteries. A pulsatile flow with ejection time 0.3 s was applied using a controlled pump. Inner pressure waves and flow velocity were then measured using a pressure sensor and an ultrasonic diagnostic system. We formulated a 1D model derived from the Navier-Stokes equations and a continuity equation to characterize pressure propagation in flexible tubes. The theoretical model includes nonlinearity and attenuation terms due to the tube wall, and flow viscosity derived from a steady Hagen-Poiseuille profile. Under the same configuration as for experiments, the governing equations were computed using the MacCormack scheme. The theoretical pressure waves for each case showed a good fit to the experimental waves. The square sum of residuals (difference between theoretical and experimental wave-forms) for each case was <10.0%. A possible explanation for the increase in the square sum of residuals is the approximation error for flow viscosity. However, the comparatively small values prove the validity of the approach and indicate the usefulness of the model for understanding pressure propagation in the human arterial network.     

A simple model for the two dimensional blood flow in the collapse of veins

Veins in the cardiovascular system may collapse if the internal pressure is less than the external pressure. Such collapse or buckling will have important consequence in terms of the rate of blood flow. Here a steady, parallel unidirectional flow as an exact solution of the continuity and the Navier Stokes equations is constructed. Various stages of the deformation process of the elastic tube (before contact of opposite sides occurring), from an ellipse to a `strongly buckled' configuration, are obtained in analytical forms as a by-product of the calculations. The pressure – area and the pressure – flow rate diagrams computed numerically from the model agree with the trends measured experimentally. Partial Financial Support has been provided by the Research Grants Council Contract HKU 7184/04E.     

A study of the bifurcation behaviour of a model of flow through a collapsible tube

Most of the elastic tubes found in the mammalian body will collapse from a distended circular cross section and when collapsed may undergo flow-induced oscillations. A mathematical model describing fluid flow in a collapsible tube is analysed using the software package AUTO-86. AUTO-86 is used for continuation and bifurcation problems in systems of non-linear ordinary differential equations. The model is a third-order lumped-parameter type and is based on the classical “Starling resistor”; it describes the unsteady flow behaviour and, in particular, the experimentally observed self-excited oscillations, in a way which is simple enough to give physical understanding, yet still firmly based on fluid mechanical principles. Some of the bifurcation types found in this model bear close resemblance to the types suggested by experimental observations of self-excited oscillations in collapsible tubes; they thus shed some light on the various topological changes which occur in practice, particularly in view of the fact that some of the points found numerically are diffcult to achieve experimentally, while the existence of others can only be inferred indirectly and uncertainly from experiment.     

Mathematical modelling of pulmonary gas transport

 Equations governing the transport of the gases oxygen and carbon dioxide inside the pulmonary capillaries are written down. By analysing these equations it is predicted that there will be negligible limitation to the transport of oxygen when oxygen concentration takes a normal physiological or higher value. For low values of oxygen concentration, there may be limitation to oxygen transport. It is predicted further that the quantity of carbon dioxide excreted from blood into alveolar gas is dependent on oxygen concentration, with low oxygen concentrations inhibiting the carbon dioxide transport process. The relatively slow reaction involving carbon dioxide in plasma also inhibits the excretion of carbon dioxide. These predictions are verified by solving the whole system of governing equations numerically. Received: 1 May 2002 / Revised version: 20 October 2002 / Published online: 19 March 2003 JPW was supported by a grant from the Engineering and Physical Sciences Research Council of Great Britain. Key words or phrases: Pulmonary gas transport – Haemoglobin – Saturation     

Mathematical modelling of oxygen transport to tissue

 The equations governing oxygen transport from blood to tissue are presented for a cylindrical tissue compartment, with blood flowing along a co–axial cylindrical capillary inside the tissue. These governing equations take account of: (i) the non–linear reactions between oxygen and haemoglobin in blood and between oxygen and myoglobin in tissue; (ii) diffusion of oxygen in both the axial and radial directions; and (iii) convection of haemoglobin and plasma in the capillary. A non–dimensional analysis is carried out to assess some assumptions made in previous studies. It is predicted that: (i) there is a boundary layer for oxygen partial pressure but not for haemoglobin or myoglobin oxygen saturation close to the inflow boundary in the capillary; (ii) axial diffusion may not be neglected everywhere in the model; (iii) the reaction between oxygen and both haemoglobin and myoglobin may be assumed to be instantaneous in nearly all cases; and (iv) the effect of myoglobin is only significant for tissue with a low oxygen partial pressure. These predictions are validated by solving the full equations numerically and are then interpreted physically. Received: 13 October 2000 / Revised version: 12 June 2001 / Published online: 17 May 2002     

Observations of pollen tube growth in Nicotiana alata and their implications for the mechanism of self-incompatibility

 Style squashes and stylar grafts were used to examine the growth of Nicotiana alata pollen tubes in self-compatible and self-incompatible styles. Compatible tubes typically showed a uniform layer of callose deposition in the walls and in small plugs spaced at regular intervals within the tube. Incompatible tubes were characterised by the variability of callose deposition in the walls and by larger, closer and more irregularly spaced plugs. There was no difference in the growth rate of compatible and incompatible tubes during growth through the stigma, but within the style most compatible tubes grew 20–25 mm day^-1 (maximum 30 mm day^–1), whereas incompatible tubes grew 1.0–1.5 mm day^-1 (maximum 5 mm day^–1). Many incompatible tubes continued to grow until flowers senesced, and only a small proportion died as a consequence of tip bursting. Grafting compatibly pollinated styles onto incompatible styles showed that the incompatible reaction could occur in pollen tubes between 2 and 50 mm long, and that inhibition of pollen tube growth occurred in both the upper and lower parts of the transmitting tract. Grafting incompatibly pollinated styles onto compatible styles showed that the incompatible reaction was fully reversible in at least a proportion of the pollen tubes. The findings are not consistent with the cytotoxic model of inhibition of self-pollen tubes in solanaceous plants, which assumes that the incompatible response results from the degradation of a finite amount of rRNA present in the pollen tube. However, if pollen tubes do in fact synthesise rRNA, the findings become consistent with this model. Received: 23 May 1996 / Revision accepted: 22 August 1996     

A numerical method for renal models that represent tubules with abrupt changes in membrane properties

 The urine concentrating mechanism of mammals and birds depends on a counterflow configuration of thousands of nearly parallel tubules in the medulla of the kidney. Along the course of a renal tubule, cell type may change abruptly, resulting in abrupt changes in the physical characteristics and transmural transport properties of the tubule. A mathematical model that faithfully represents these abrupt changes will have jump discontinuities in model parameters. Without proper treatment, such discontinuities may cause unrealistic transmural fluxes and introduce suboptimal spatial convergence in the numerical solution to the model equations. In this study, we show how to treat discontinuous parameters in the context of a previously developed numerical method that is based on the semi-Lagrangian semi-implicit method and Newton's method. The numerical solutions have physically plausible fluxes at the discontinuities and the solutions converge at second order, as is appropriate for the method. Received: 13 November 2001 / Revised version: 28 June 2002 / Published online: 26 September 2002 This work was supported in part by the National Institutes of Health (National Institute of Diabetes and Digestive and Kidney Diseases, grant DK-42091.) Mathematics Subject Classification (2000): 65-04, 65M12, 65M25, 92-04, 92C35, 35-04, 35L45 Keywords or phrases: Mathematical models – Differential equations – Mathematical biology – Kidney – Renal medulla – Semi-Lagrangian semi-implicit     

Localization of pectins in the pollen tube wall of Ornithogalum virens L. Does the pattern of pectin distribution depend on the growth rate of the pollen tube?

Monoclonal antibodies that recognize pectins were used for the localization of esterified (JIM7) and acidic, unesterified (JIM5) forms of pectin in pollen tube walls of Ornithogalum virens L. (x = n = 3). The results indicated that the distribution of the two forms of pectin in the pollen tube wall depended on the medium (liquid or solid) used for pollen germination. In pollen tubes grown in the liquid medium, the localization of JIM7 was limited to the very tip of the pollen tube, whereas the localization of JIM5 indicated a uniform distribution of unesterified pectins in the very tip of the tube and along the subapical parts of the tube wall. In tubes germinated on the medium stabilized with agar (1–2%) the localization of JIM7 and JIM5 indicated the presence of both forms of pectin in the tube tip and along the whole length of the pollen tube wall in a ring-like pattern. Thus, the localization of esterified pectins in the sub-apical part of the pollen tube wall, below the apex of the tube, is described for the first time. Measurements of the growth rates of pollen tubes growing on the two types of medium indicated that oscillations in tube growth rate occur but these do not coincide with the pattern of pectin distribution in the tube wall. Our results complement the previous data obtained for the localization of JIM5 and JIM7 in pollen tube walls of other plant species. (Y.-Q. Li et al. 1994, Sex Plant Reprod 7: 145–150) and provide new insight into an understanding of the construction of the pollen tube wall and the physiology of pollen grain germination. Received: 25 January 1999 / Accepted: 23 June 1999     

Early development and quorum sensing in bacterial biofilms

 We develop mathematical models to examine the formation, growth and quorum sensing activity of bacterial biofilms. The growth aspects of the model are based on the assumption of a continuum of bacterial cells whose growth generates movement, within the developing biofilm, described by a velocity field. A model proposed in Ward et al. (2001) to describe quorum sensing, a process by which bacteria monitor their own population density by the use of quorum sensing molecules (QSMs), is coupled with the growth model. The resulting system of nonlinear partial differential equations is solved numerically, revealing results which are qualitatively consistent with experimental ones. Analytical solutions derived by assuming uniform initial conditions demonstrate that, for large time, a biofilm grows algebraically with time; criteria for linear growth of the biofilm biomass, consistent with experimental data, are established. The analysis reveals, for a biologically realistic limit, the existence of a bifurcation between non-active and active quorum sensing in the biofilm. The model also predicts that travelling waves of quorum sensing behaviour can occur within a certain time frame; while the travelling wave analysis reveals a range of possible travelling wave speeds, numerical solutions suggest that the minimum wave speed, determined by linearisation, is realised for a wide class of initial conditions. Received: 10 February 2002 / Revised version: 29 October 2002 / Published online: 19 March 2003 Key words or phrases: Bacterial biofilm – Quorum sensing – Mathematical modelling – Numerical solution – Asymptotic analysis – Travelling wave analysis     

Ion dynamics and hydrodynamics in the regulation of pollen tube growth

A coherent picture of pollen tube growth is beginning to emerge that couples ion dynamics with biochemical, biophysical and cytological processes in ordered and controlled feedback circuits that define the nature of polarized apical growth. It is a paradox, however, that complete understanding of the mechanical forces that drive cell elongation in this system still remains to be fully achieved. The results of our recent studies to characterize Cl^– ion dynamics during apical growth in tobacco pollen tubes led us to re-examine this question in the light of a possible force-generating role provided by hydrodynamic flow. Previously we found that oscillatory Cl^– efflux from the apex is closely coupled to oscillatory growth and the cell volume of the apical domain. Cl^– influx occurs in a region of the tube that is distal to the clear zone; hence, a vectorial flow of anion traverses the apical domain and fluxes out of the tip with oscillatory dynamics. Because of the effects that this could induce on charge and osmotic potentials, water could potentially flow through the apical domain, linked to the flux of Cl^–. This conjecture is consistent with studies in other plant cells that demonstrate a pivotal role for flux through anion channels in the control or normalization of osmotic status. In the current report, the relationship between Cl^– efflux oscillations and the physical characteristics of the apical dome during oscillatory growth is examined in closer detail. Evidence is presented that shows a cyclic deformation of the extreme apex occurs during the growth pulse and is correlated with cyclic Cl^– efflux. In addition, there is a dramatic increase in the number and density of clear thread-like zones traversing the apical plasma membrane during the process of tip elongation. Possible functional roles of Cl^– flux and hydrodynamics are discussed in the context of what drives tip elongation during cycles of pollen tube growth. Received: 23 November 2000 / Revision accepted: 19 June 2001     

Development of a biologically-based controlled growth and differentiation model for developmental toxicology

 A mathematical model is developed with a highly controlled birth and death process for precursor cells. This model is both biologically- and statistically-based. The controlled growth and differentiation (CGD) model limits the number of replications allowed in the development of a tissue or organ and thus, more closely reflects the presence of a true stem cell population. Leroux et al. (1996) presented a biologically-based dose-response model for developmental toxicology that was derived from a partial differential equation for the generating function. This formulation limits further expansion into more realistic models of mammalian development. The same formulae for the probability of a defect (a system of ordinary differential equations) can be derived through the Kolmogorov forward equations due to the nature of this Markov process. This modified approach is easily amenable to the expansion of more complicated models of the developmental process such as the one presented here. Comparisons between the Leroux et al. (1996) model and the controlled growth and differentiation (CGD) model as developed in this paper are also discussed. Received: 8 June 2001 / Revised version: 15 June 2002 / Published online: 26 September 2002 Keywords or phrases: Teratology – Multistate process – Cellular kinetics – Numerical simulation     

Inertial migration based concentration factors for suspensions of Chlorella micro-algae in branched tubes

When a dilute suspension flows in the laminar regime through a tube, under certain conditions the suspended particles migrate radially to an equilibrium radial position. Branched tubes can use this radial concentration distribution to concentrate dilute suspensions. Suspensions of microalgae, Chlorella vulgaris, were pumped through tubes of various diameters for tube Reynolds number ranging from 47–1839 and photographed. Upstream particle concentration profdes were obtained by image analysis of the photographs. The dividing stream surfaces in branched tubes were obtained from the three-dimensional numerical solutions of the Navier-Stokes equations for steady, laminar, and homogeneous flow through tubes having one and two orthogonal branches. Concentration factors for Chlorella suspensions in branched tubes, predicted by a general method, fall in the range of 1.0–1.3     

Volatile constituents of glutathione--ribose model system and its antioxidant activity

Summary.  Reaction between glutathione and ribose was carried out to study the volatiles formed via Maillard reaction and their antioxidant activity as well as their role in inhibition of LDL oxidation. The simultaneous distillation – extraction technique was used for trapping the volatile components followed by GC – MS analysis. Thirty six compounds were identified with the predominance of carbonyls and sulfur – containing compounds in the volatiles of this model system. Sensory evaluation was performed for the model system product according to the International Standard Methods (ISO). The results showed a high decrease in roasted and burnt attributes and remarkable increase in the like – boiled and roasted meat attributes. The sensory results of the model system product were confirmed by the presence of high concentrations of some volatile compounds having meat – like aroma such as 2-methyl-3-furanthiol and 2-furylmethanethiol. The radical scavenging activity of glutathione – ribose model system was quantified spectrophotometrically, using DPPH radical. The activity of the model system product was found to be slightly lower than that of gallic acid and BHA, but it was much higher than that of cinnamic acid (200 ppm. for each). A highly antioxidative activity was recorded by the model system product during the inhibition of LDL – oxidation in comparison with L-ascorbic acid as well as reduced glutathione (as a concentration of 0.5 μmol/L, for each) which may be due to the presence of some compounds such as 2-furylmethanethiol, 2-acetyl thiazole, 4-hydroxy-5-methyl-3(2H)-furanone. Received October 15, 2001 Accepted April 3, 2002 Published online September 4, 2002 Authors' address:  Khaled Farouk El-massry, Flavour and Aromatic Chemistry Department, National Research Center, Tahrir st., Dokki, Cairo, Egypt, Fax: 002 02 337 0 931, E-mail: kfarouk@yahoo.com     

Ultrastructure of feeding tubes formed in giant-cells induced in plants by the root-knot nematodeMeloidogyne incognita

Summary The plant pathogenic nematodeMeloidogyne incognita forms conspicuous tubular structures referred to as feeding tubes in special food cells, called giant-cells, induced and maintained in susceptible host roots by feeding nematodes. Feeding tubes are formed by nematode secretions injected into giant-cells via a stylet and apparently function to facilitate withdrawal of soluble assimilates by the parasite. In giant-cells in roots of the four host species examined in this study, feeding tube morphology was identical. Tubes were straight to slightly curved structures just less than 1 μm wide and up to slightly more than 70 μm long. At the ultrastructural level, each tube consisted of a 190–290 nm thick, electron-dense, crystalline wall surrounding an electron-transparent lumen with a diameter of 340–510 nm. The distal end of the tube was sealed with wall material. Older tubes were found free in the host cytoplasm while the proximal ends of young tubes were attached to the host cell wall via short wall ingrowths through which the nematode's stylet was inserted. An elaborate membrane system was associated with the feeding tubes and was most extensive around newly formed tubes. Contiguous to the feeding tube wall, this membrane system consisted of strands of smooth endoplasmic reticulum while rough endoplasmic reticulum predominated toward the outer margin of the membrane system. Vacuoles and mitochondria were excluded from a zone of cytoplasm surrounding feeding tubes. This zone of exclusion, as well as the membrane system noted above, tended to be less pronounced or absent around older tubes no longer being used by the nematode.     

A new silverside, <Emphasis Type="Italic">Atherinomorus aetholepis</Emphasis> sp. nov., from the West Pacific (Atheriniformes: Atherinidae)

Atherinomorus aetholepis sp. nov. is described from the holotype and 51 paratypes, 44–72 mm in standard length, collected from Indonesian and Philippine waters. The species is similar to other congeners in general body appearance, especially A. duodecimalis and A. regina, in having a slender body, a tubercle on the posterior end of the dentary, and a narrow midlateral band, but clearly differing from them in having a long spatular outgrowth on the posterior margins of most of the predorsal and interdorsal scales. Additionally, the species differs from A. duodecimalis in having a more slender body [body depth 17–22 (mean 19) % SL vs. 19–25 (mean 22) % SL], more midlateral scales [37–40 (mean 38.4) vs. 35–38 (mean 36.6)], more total vertebrae [38–42 (mean 39.9) vs. 36–40 (mean 38.0)], and fewer lower gill rakers [18–22 (mean 20.2) vs. 20–25 (mean 22.3)], and from A. regina in having more anal fin soft rays (12–14 vs. 9–10). Electronic supplementary material to this article is available at and accessible to authorized users. Received: October 22, 2001 / Revised: March 14, 2002 / Accepted: March 26, 2002 An erratum to this article is available at .     

An integral method for the analysis of blood flow

The existing methods to solve the problems of pulsatile flow in the cardiovascular system are based on either linear axisymmetric equations or non-linear one-dimensional equations. The solutions thus obtained give only a mediocre comparison with measurements. In this paper, a non-linear axisymmetric theory is proposed. The starting point of the present theory is a third degree polynomial representation of the velocity profile. Integral methods are then applied to obtain the governing equations. To ascertain the accuracy of the theory proposed above, the calculations for a simple case involving pulsatile flow in a long rigid tube were performed. The results are: (a) the average velocities compare very well with exact solutions and (b) the velocity profiles for a given frequency agree very well with exact solutions for flow in small tubes, but tend to differ as tube size is increased.     

Travelling Waves in Hyperbolic Chemotaxis Equations

Mathematical models of bacterial populations are often written as systems of partial differential equations for the densities of bacteria and concentrations of extracellular (signal) chemicals. This approach has been employed since the seminal work of Keller and Segel in the 1970s (Keller and Segel, J. Theor. Biol. 30:235–248, 1971). The system has been shown to permit travelling wave solutions which correspond to travelling band formation in bacterial colonies, yet only under specific criteria, such as a singularity in the chemotactic sensitivity function as the signal approaches zero. Such a singularity generates infinite macroscopic velocities which are biologically unrealistic. In this paper, we formulate a model that takes into consideration relevant details of the intracellular processes while avoiding the singularity in the chemotactic sensitivity. We prove the global existence of solutions and then show the existence of travelling wave solutions both numerically and analytically.     

Mathematical analysis of oxygen concentration in a two dimensional array of capillaries

 An approach is presented for modeling transport and exchange in skeletal muscle that can be used to analyze vascular beds consisting of a large number of interacting capillaries. First the oxygen concentration is determined in a functional unit consisting of a single capillary surrounded by a region of tissue in which a flux is prescribed on the outer boundary of the region. This flux, which is a result of the interaction among all of the capillaries comprising the vascular bed, is then found by matching the concentration along the borders between adjacent units. This leads to a system of ordinary differential equations for the oxygen concentration in the capillaries coupled with a system of algebraic equations for the fluxes. The method is illustrated by obtaining the oxygen concentration within an array of capillaries for the case when each capillary has a different initial concentration and for the case when each capillary has a different flow rate. Received: 12 June 2001 / Revised version: 18 April 2002 / Published online: 17 January 2003 Key words or phrases: Skeletal muscle – Transport – Microcirculation     

Further properties of the two-membrane model

The properties of nonlinear equations describing the solute and solvent transport across a simplified Patlak-Goldstein-Hoffman model (two membranes in series without unstirred layers) are investigated both analytically and numerically. The analysis shows that the principal coefficients measured in transport experiments in the presence of active transport are dependent on the experimental conditions. These ‘apparent’ system parameters are extensions of the corresponding parameters determined both in passive systems and in the linear Kedem-Katchalsky theory. Moreover, they are related to the local phenomenological coefficients of the single membranes of the array. Several relationships between measurable quantities and the local system parameters are indicated, allowing the planning of experiments aimed at the measurement of the latter. Data in the literature have been used to check the proposed volume flow equation.