ALBERT

Notes: Abstract The theory of complementary variational principles is used to obtain maximum and minimum principles for diffusion problems with Michaelis-Menten kinetics. In an illustrative calculation we obtain an extremely accurate variational solution in good agreement with the numerical solution of McElwain (1978).

Notes: Abstract Necessary and sufficient conditions for primitivity of a product of two Leslie matrices are given. Such a product could be used in modeling the growth of a population governed alternately by two different sets of fertility and survival parameters.

Notes: Abstract Zadeh's transfer function method for linear time-variable systems is used to apply frequency-domain analysis to a periodically time-varying elastance model of the left ventricle. Left ventricular pressure computed from the system function of the time-varying elastance and the phasors of aortic flow shows a typical waveform of the measured ventricular pressure.

Notes: Abstract The entropy budget of a white-tailed deer (50kg) on a maintenance diet and a full-feed diet in a standing posture in an open field under clear nocturnal skies with an air temperature of −20°C is investigated based on the energetics given by Moen. Entropy inflow into a white-tailed deer due to infra-red radiation and entropy outflows from a deer due to infra-red radiation, convection, evaporation of water and conduction to ingested food are calculated. Also the entropy production due to metabolic heat production is estimated. Net entropy flow into a deer from its environment becomes negative. On the assumption that a white-tailed deer is in a steady state in entropy, the total entropy production in a deer on a maintenance diet becomes +0.46 J/sec/K. Positiveness of the entropy production shows that the Second Law of Thermodynamics certainly holds in a white-tailed deer. The entropy production per effective radiating surface area of a deer on a maintenance diet is 0.32×10−4 J/cm2/sec/K. On the other hand, the entropy production in a deer on a full-feed diet is 0.59 J/sec/K and that per effective surface area is 0.41×10−4 J/cm2/sec/K. Uptake of 1 g of food produces 22 J/K of entropy within the body of a white-tailed deer. Comparison is made with the results for entropy production in a lizard and in plant leaves.

Notes: Abstract Biological adaptability has been proved to be analysable by means of the Maximum Entropy Formalism (MAXENT) in some cases of non-interacting systems. This formalism is extended to the biomass statistical structures of populations exhibiting internal interactions (i.e. predatorprey effects).

Notes: Abstract The temporal behaviours of the nonlinear substructure of a self-organized compartmental model of calcium metabolism were investigated. The order-two autocatalytic process included in this simple two-dimensional model is compared to some secondary nucleation mechanisms which should take place at the extracellular fluid-bone interface. The model gives rise to complex dynamic behaviours, and multistability properties, involving up to two stable periodic regimes (birhythmicity), were established in different topological configurations. The bifurcations occurring on the boundaries between regions of different qualitative behaviour have been determined. These properties are discussed in relation to the dynamical behaviour of other two-variable models, especially those including the same nonlinearity.

Notes: Abstract A linearized oscillation theorem due to Kulenović, Ladas and Meimaridou (1987,Quart. appl. Math. XLV, 155–164) and an extension of it are applied to obtain the oscillation of solutions of several equations which have appeared in population dynamics. They include the logistic equation with several delays, Nicholson's blowflies model as described by Gurney, Blythe and Nisbet (1980,Nature, Lond. 287, 17–21) and the Lasota-Wazewska model of the red blood cell supply in an animal. We also developed a linearized oscillation result for difference equations and applied it to several equations taken from the biological literature.

Notes: Abstract A theoretical approach to the explanation of the structural design of metabolic pathway is presented. It is based on the hypothesis that due to natural selection during evolution the cellular metabolism of present-day organisms may be characterized by optimal properties. Two cardinal terms enter the theory: (i) the efficiency of a metabolic pathway and (ii) the evolutionary effort for the change of the kinetic parameters of enzymes by mutations of the corresponding genes. For both quantities simple mathematical expressions are proposed. While the efficiency is related to the reaction rates of the enzymes constituting the metabolic pathway, the evolutionary effort is considered to be a monotonically increasing function of the parameter values. By maximizing the efficiency under the constraint of a fixed evolutionary effort the theory allows the calculation of the optimal parameter distribution as the outcome of evolution processes. The methods developed are applied to the following systems: (a) linear reaction sequences with very low affinities of the enzymes towards substrates, (b) linear sequences consisting of saturable enzymatic reactions, (c) branched metabolic pathways consisting of segments of linear chains and (d) glycolysis of erythrocytes. The conclusion is derived that the optimal distribution of kinetic constants depends strongly on the equilibrium constants of the reactions as well as on the total osmolarity of the metabolic intermediates. Without osmotic constraints the evolutionary effort is mainly spent on the enzymes at the beginning of the chain. Using Michaelis-Menten equations the optimal state is characterized by a decrease of the maximal activities of the enzymes towards the end of the chain. These results are modified if osmotic constraints are taken into account. At the investigation of branched pathways the following results were obtained: firstly, if a certain end product may be synthesized along different pathways those which are thermodynamically more unfavourable (e.g. characterized by a small change of free energy) are eliminated in the course of evolution; secondly, if a branched pathway leads to several important end products those reaction segments which are thermodynamically unfavourable are characterized by a higher evolutionary effort. The application of the theory to a realistic model of glycolysis of erythrocytes leads to a correct description of various functionally important properties of the system, such as the ratio between fluxes through different branches and the ATP/ADP ratio, whereas the theory cannot predict the strong separation of time constants observed in the real glycolytic system. It is concluded that the improvement of the predictive power of the theory necessitates the use of more complex functionals for the efficiency which take into account not only the fluxes but also other system properties such as the stability of the pathway or homoeostatic effects.

Notes: Abstract Bayesian image processing formalisms which incorporatea priori information about valued-uncorrelated and valued-correlated (patterned) source distributions are introduced and the corresponding iterative algorithms are derived using the EM technique. Striking improvement in image processing is demonstrated when applying these algorithms to Poisson and Gaussian randomized data in one-dimensional cases.

Notes: Abstract For a broad class of interconnected nonlinear systems, this paper develops a complete design methodology for decentralized variable structure control. Specifically, the paper sets forth design schemes for local switching surfaces and the related local switched feedback gains which together force the original nonlinear interconnected system to behave as a reduced order interconnected equivalent system having a desired response such as stability, tracking, or prespecified eigenvalues. Also developed is a numerical algorithm for constructing the switched local feedback gains. A simple nonlinear example illustrates the control strategy.

Notes: Abstract A new stabilization method of a large-scale dynamic system, consisting of a set of interconnected subsystems, is presented in this paper. The topology of the interconnected subsystems is given as a network containing nodes with only one ingoing link, and none, one, or more outgoing links. Here, when the notion “node” is used a subsystem is assumed, and the links stand for the subsystem interconnections. The stabilization method is made only by the use of local linear state feedback around each subsystem, in order to satisfy constraints given in the problem. The interconnections among the subsystems are assumed to be nonlinear, time-varying. According to the topology of the large-scale system, the method of stabilization is hierarchic, one proceeds from node to node, and is applicable from a computer standpoint. A design algorithm follows directly, and can be made using the Generate and Test method for each subsystem independently, thus enabling designers to use a computer which has a video terminal as a peripheral unit and providing a possibility for interactive applications.

Notes: Abstract The estimation of covariance matrices which are structured, for example, of Toeplitz type, from measurement data is considered. The problem is considered in the context of array beamforming, and various methods of estimation are derived and compared, such comparison including consideration of the behavior of the estimate in beamforming applications.

Notes: Abstract The planar least-squares inverse (PLSI) polynomials are used for stabilization of two-dimensional unstable recursive filters. In order to obtain the PLSI polynomials, the main work involved consists in forming a set of linear equations and then solving them. In this paper we present an efficient and simple method to form the necessary set of linear equations (i.e., the required coefficient matrix) for a chosen pattern and order of the desired PLSI polynomial, starting from the denominator polynomial of a two-dimensional unstable recursive filter.

Notes: Abstract This paper shows how to use orthogonal functions to invert singular (i.e., generalized state-space) systems. The approach is to express the inverse system itself as a singular system, and then to apply the theory of orthogonal functions to convert that differential-algebraic system to a purely algebraicgeneralized Lyapunov equation whose solution yields the input of the original system given its output. Both left and right inversion are treated. Necessary and sufficient conditions for the existence and uniqueness of the generalized Lyapunov equation are derived, and a generalizedQZ algorithm is given for its efficient solution. It is also shown that the coefficients in the Walsh function expansion may be approximately found using an FFT-type butterfly network. These results provide both an extension in theory, by investigating the properties of a new Lyapunov equation, and an extension in the implementation of system inversion, by providing a scheme which applies to generalized state-space systems and uses an unconventional approach which may prove to be a useful contribution.

Notes: Abstract In digital communication networks, a special class of complex biquad recursive digital filters called orthogonal filters is increasingly being used. The separate effects of the overflow and quantization nonlinearities on these orthogonal filters' responses have been investigated [5], [6]. In this paper we examine the zero-input stability properties of the actual orthogonal filter having both overflow and quantization nonlinearities. The overflow nonlinearities considered include saturation, bit-by-bit inversion, zeroing, and modulo 2 arithmetic. The quantization techniques used may be roundoff, magnitude, or value truncation. An example demonstrates the adverse coupling effect between the overflow and quantization nonlinearities. Two criteria are therefore derived to ensure asymptotic overflowstability of the filter in the presence of quantization. These criteria have been translated to the coefficient plane; various regions corresponding to different minimum wordlengths required to ensure decoupling of the overflow and quantization phenomena have been derived.

Notes: Abstract The identification problem for electromagnetic objects excited by transients is discussed. Several classes of models are reviewed, and an output error model is selected. An algorithm for solving the transient identification problem using this model is presented, and some of the issues connected with its use are considered. Examples of the application of this algorithm to electromagnetic data are given.

Notes: Abstract A theory of noise fluctuations is developed which is applicable to systems of any size in which unimolecular or bimolecular reactions are occurring. The main difference between small and large reacting systems is that in the former the probability of finding a particle in a particular state does not obey a Gaussian distribution, but satisfies a distribution which reflects the mechanism of the chemical reaction. This difference is reflected in the main result of the theory: an autocorrelation function that is expressible as a sum of exponentials, the amplitudes of which are explicit functions of the moments of the distribution. Thus, by using small systems, the autocorrelation function,in principle, allows the elucidation of reaction mechanisms. Numerical simulations indicate that for reacting systems having ten or fewer particles, the deviation of the autocorrelation function from a single exponential should be easily detectable, and that estimates of the first four moments of the distribution should be possible. Accurate inference of the distribution, however, will require further mathematical and experimental advances.

Notes: Abstract The recent mathematical formalization of the concepts of matter and extrinsical energy, which are used for the relational representation of biological systems, is employed in the analysis of the important experimental discoveries of Comorosanet al. related to low energy electromagnetic irradiations on enzyme substrates. By means of the present analysis one of the properties inherent to the experimental phenomena is more precisely exposed, and theoretical developments corresponding to “energetical evolutions” in a biological system (Leguizamón, 1976) may now have an experimental basis. Important limitations are introduced for the validity of the commutativity and associativity of cartesian product of sets, when they represent matter and its linked extrinsical energy. In connection with this last aspect, new important knowledge is obtained for the relational mathematical representation of biological systems.

Notes: Abstract The structure of solutions to a simple spatially dependent population model involving growth and death is investigated. Two forms of motility of the population are considered: (1) random motion only modeled by a Fickian law, and (2) a directed component of motion (chemotaxis), included in addition to the random motion. Under certain growth conditions a traveling wave of constant speed is approached. This speed can be increased by the addition of the chemotaxis with a corresponding increase in the asymptotic population. Development of initial conditions into a wave is illustrated numerically.

Notes: Abstract This paper describes mechanisms of intracellular and intercellular adaptation that are due to spatial or temporal factors. The spatial mechanisms support self-regulating pattern formation that is capable of directing self-organization in a large class of systems, including examples of directed intercellular growth, transmitter production, and intracellular conductance changes. A balance between intracellular flows and counterflows causes adaptation. This balance can be shifted by environmental inputs. The decrease in Ca2+-modulated outward K+ conductance in certain molluscan nerve cells is a likely example. Examples wherein Ca2+ acts as a second messenger that shunts receptor sensitivity can also be discussed from this perspective. The systems differ in basic ways from recent diffusion models. Chemical transducers driven by membrane-bound intracellular signals can establish long-range intercellular interactions that compensate for variable intercellular distances and are invariant under developmental size changes; diffusional signals do not. The intracellular adaptational mechanisms are formally analogous to intercellular mechanisms that include cellular properties which are omitted in recent reaction-diffusion models of pattern formation. The cellular models use these properties to compute size-invariant properties despite wide variations in their intercellular signals. Mechanisms of temporal adaptation can be derived from the simplest laws of chemical transduction by using a correspondence principle. These mechanisms lead to such properties of intercellular signals as transient overshoot, antagonistic rebound, and an inverted U in sensitivity as intracellular signals or adaptation levels shift. Such effects are implicated in studies of behavioral, reinforcement, motor control, and cognitive coding.

Notes: Abstract Large radiation doses to the lung can cause early death from cardiopulmonary insufficiency resulting from radiation pneumonitis and pulmonary fibrosis. A model for early death following inhalation of insoluble radioactive particles is propose. The model is based on three assumptions: (1) early death results from damage to a cluster of cells from a large number of cell clusters at risk, (2) the dose that causes early death depends on how the radiation is delivered in time and (3) the cell clusters at risk to damage are equally sensitive ro radiation. Results from asymptotic theory of extreme values, along with biophysical considerations, suggest that the cumultive distribution function for the absorbed radiation dose to the production of pulmonary injury sufficient to cause early death is best estimated by the third asymptotic distribution without a threshold. This distribution function is identical to the Weibull cumulative distribution function. Data for Beagle dogs after inhaling relatively insoluble forms of alpha- or beta-gamma-emitting particles are shown to support the Weibull model.

Notes: Abstract Models of the human respiratory tract were developed based on detailed morphometric measurements of a silicone rubber cast of the human tracheobronchial airways. Emphasis was placed on the “Typical Path Lung Model” which used one typical pathway to represent a portion of the lung, such as a lobe, or to represent the whole lung. The models contain geometrical parameters, including airway segment diameters, lengths, branching angles and angles of inclination to gravity, which are needed for estimating inhaled particle deposition. Aerosol depositions for various breathing patterns and particle sizes were calculated using these lung models and the modified Findeisen-Landahl computational scheme. The results agree reasonably well with recent experimental data. Regional deposition, including lobar deposition fractions, are also calculated and compared with results based on the ICRP lung deposition model.

Notes: Abstract The completely symmetrical system is defined as having identical transfer coefficients between pairs of compartments and the same loss coefficient for each compartment. The eigenvalues and eigenvector are explicitly found along with the inverses of the system matrix and the matrix of eigenvectors. Many properties, special instances of more general theorems, can be seen at once from the explicit analytic solution of the initial value, washout and washin problems. The system serves as a known case for testing estimation procedures, algorithms for solutions of linear systems, eigenvalue-eigenvector and inversion routines and is of considerable tutorial value.

Notes: Abstract The mathematical structures underlying the theories of organismic sets, (M, R)-systems and molecular sets are shown to be transformed naturally within the theory of categories and functors. Their natural transformations allow the comparison of distinct entities, as well as the modelling of dynamics in “organismic” structures.

Notes: Abstract To explain the sodium conductance change using Wei's dipole model (Wei, 1969), we may expect that during depolarization the dipole's population difference, ΔN, is first reduced and then returns more slowly to its resting value. This paper shows that the experimental results of gating currents support this idea. Such time course of ΔN, however, is not a usual relaxation process. To account for the unusual behavior of ΔN, we propose two additional assumptions: (1) there exists a special coupling system (probably the intramolecular vibrations) whose coupling strength with the dipoles is much stronger than with the thermal bath (intermolecular vibrations), and (2) there also exist “traps” for the dipole's excitation energy so that this energy is transformed into other energy forms at a rate increasing with the increase of depolarization. Experiments suggest that the traps are proteins located at the inner membrane surface.

Notes: Abstract Current research into the dynamics of iterative ecological and biological models has lead to a number of theorems concerning the existence of various types of iterative dynamical behavior. In particular, much study has been done on the dynamical behavior of the “simplest dynamical system”f b(x)=bx(1−x), which is just the canonical discrete form of logistic growth equations found in ecology, sociobiology, and population biology. In this paper, we make use of some of the techniques and concepts of topological dynamics to construct a number of generalized conjugacy theorems. These theorems are then used to demonstrate that the mappingf b has a number of conjugacy classes in which the dynamics of the iterates is equivalent to within a change of variables. The concepts of fitness and survival in logistic equations are then shown to be independent, if we follow certain intuitive definitions for these concepts. This conclusion follows from a comparison of the conjugacy classes of the functionf b and the extinction sets off b.

Notes: Abstract For chemical reactions not at equilibrium but proceeding in the forward direction in the steady state, a result found by a method first introduced by H. G. Britton (1963, 1965) is generalized to prove that if $${{\vec J} \mathord{\left/ {\vphantom {{\vec J} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\leftarrow}$}}{J} }}} \right. \kern-\nulldelimiterspace} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\leftarrow}$}}{J} }}$$ is the unidirectional flux ratio, $${{\vec J} \mathord{\left/ {\vphantom {{\vec J} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\leftarrow}$}}{J} }}} \right. \kern-\nulldelimiterspace} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\leftarrow}$}}{J} }}$$ exp (−ΔG/RT). The conditions under which the equality or inequality applies are discussed. If the unidirectional fluxes are not in the steady state, the unidirectional flux ratio is time invariant in certain specific situations. One such important case is for chemical reaction systems with an ordered sequence of reactions. For systems with more than one pathway, $${{\vec J} \mathord{\left/ {\vphantom {{\vec J} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\leftarrow}$}}{J} }}} \right. \kern-\nulldelimiterspace} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\leftarrow}$}}{J} }}$$ is not constant except for special cases. These results also apply to diffusional and active transport systems.

Notes: Abstract The nonlinear second-order difference equationx n+1=axn(1-xn−1), where 0≦x nX≦1 anda ≧1, is examined from varying points of view, analytical, numerical and geometrical. An analytic expression is obtained for an invariant attracting curveC ∞ (a) in phase space, which becomes the central object of study. This basic curve, which replaces the simple parabolic shape typical of many analogous first-order models, may have a complicated geometrical structure. As the parametera increases,C ∞(a) undergoes transformations characterized by the dynamical descriptions: stable node→stable focus→stable limit cycle →chaotic attractor. Although the limited characterization ofchaos by the appearance of nonperiodic solutions and solutions of arbitrarily large period is relied upon, this appears to be only a simplified approximation of the real behavior of solutions. Trajectories (x n, xn+1),n=0,1,…, are calculated using the related nonlinear planar mapT a(x,y)=(y,ay(1−x)), and regions of persistence and escape are described for characteristic values ofa. The study of persistence, of even more fundamental interest than the associated problems of periodicity and stability, receives special attention. We introduce a geometrical model, similar in many respects to that for the well-known analoguex n+1=axn(1−x n), but having several new and important features. It appears that as the parametera increases in the chaotic regime there are infinitely many intermittent bursts of increase in the probability that any initial point (x 0, x1) will persist in the unit square under successive iterations of the mappingT a, an unexpected property that should be of interest for applications. A discussion of the applicability of these results to population dynamics theory is given, and it is suggested that such equations might find useful application to problems in developmental biology as well.

Notes: Abstract Based on the principle of minimum power, a mathematical model of the functional state of the oxygen transport system is presented. The optimization model minimizes the power expenditure of the heart, bone marrow, lung and other tissues. The model is used to determine the functional parameters of the oxygen transport system in man under both normal and varying barometric pressures. Theoretical results are compared with experimental data.

Notes: Abstract A quantitative model of ion binding and molecular interactions in the lipid bilayer membrane is proposed and found to be useful in examining the factors underlying such membrane characteristics as shape, sidedness, stability and vesicle size at various cation concentrations. The lipid membrane behaves as a bilayer couple whose preferential radius of curvature depends on the expansion or contraction of one monolayer relative to the other. It is proposed that molecular packing may be altered by electrostatic repulsion of adjacent like-charged phospholipid headgroups, or by bringing two headgroups closer together by divalent cation crossbridging. The surface concentrations of each type of cation-phospholipid complex can be described by simple binding equilibria and the Gouy-Chapman-Stern formulation for the surface potential in a diffuse double layer. The asymmetric distribution of acidic phospholipids in most biological membranes can account for the differential effects of identical ionic environments on either side of the bilayer. The fraction of vesicle material which tends to have a right-side-out orientation may be approximated by a normal distribution about the mean curvature. The theory generates vesicle sidedness distributions that, when fitted to experimental results from human erythrocyte membranes, provide an alternative method of estimating intrinsic cationphospholipid dissociation constants and other molecular parameters of the bilayer. The results also corroborate earlier suggestions that the Gouy-Chapman theory tends to overestimate free counter-ion concentrations at the surface under large surface potentials.

Notes: Abstract The “yellow strips” on the cuticle of the Oriental Hornet (Vespa orientalis, Hymenoptera, Vespinae), present photoelectric properties. A mathematical model for the relative changes in resistance as a photoconductive process conforms to the general model for a semiconductor with traps.

Notes: Abstract Damped nonlinear oscillations in biological and biochemical systems are investigated by the extended Krylov-Bogoliubov-Mitropolskii (KBM) method. A review on the extension made by Popov to the KBM method is given and also further improvements are presented. Applications are made to models of oscillating chemical reactions (Lefever and Nicolis, 1971), FitzHugh (1961) equations, and population dynamics (Gatto and Rinaldi, 1977). Comparison to damped oscillating physical and engineering systems is made.

Notes: Abstract The conditions that will allow the lumping together of several age classes in the Leslie model are investigated. We show that if the lumping is to be valid for all population distributions, then the parameters of the model must be periodic. Lumping is valid when the population is in equilibrium, but equilibrium should be tested before the model is lumped.

Notes: Abstract Catastrophe theory is a mathematical theory which, allied with a new and controversial methodology, has claimed wide application, particularly in the biological and the social sciences. These claims have recently been heatedly opposed. This article describes the debate and assesses the merits of the different arguments advanced.

Notes: Abstract Estimates of capillary tracer permeability calculated using multiple indicator data depend upon the particular model adopted to describe blood tissue exchange. The model proposed by Crone (1963) is appropriate when some of the injected tracer diffuses into the tissue but does not return appreciably to the bloodstream before data collection is terminated. Under these conditions extraction of tracer by the tissue depends on a single dimensionless parameter, αcap, defined as the ratio of capillary permeability surface area to water flow. The effects of finite red cell tracer permeability on the Crone model estimate of capillary permeability are examined in the present study. The results indicate that even when back diffusion from the extravascular space is negligible, significant errors in the Crone model estimate can be expected when capillary permeability is relatively high and the ratio of red cell to capillary permeability is less than unity. However, when an aliquot of blood is equilibrated with tracer prior to injection and the dimensionless capillary permeability is relatively low (i.e. αcap ≦ 0.25 for a haematocrit≦50%), the whole blood Crone model estimate of αcap will be within 10% of the actual value, irrespective of red cell permeability. Red cell-plasma exchange for commonly used tracer-organ combinations should not significantly affect Crone estimates of capillary permeability under normal physiological conditions, but may be important in low flow situations.

Notes: Abstract Assuming truncated ellipsoidal geometry for the right and left ventricles, a model is developed for the myocardium enabling biventricular mechanical behavior to be studied. Employing pressure-volume data taken from normal dog hearts and from hearts in which the pulmonary artery has been banded over periods of 2–40 weeks, it is shown that: (a) right ventricular wall stresses are higher than left ventricular stresses; (b) right ventricular wall stress increases initially to a maximum after 3–4 weeks followed by a decline to normal and even subnormal levels, attaining a minimum value at 32–33 weeks; (c) left ventricular stresses behave in a similar manner, attaining their maximum and minimum levels after 7–8 weeks and 32–33 weeks respectively. These results suggest that surgical or medical therapy in patients with hypertrophied ventricles might be more appropriate during the period of wall stress reduction.

Notes: Abstract In this paper we describe a mathematical model of the oscillations of the diaphragm which limits the vitreous body from the anterior segment of the human eye after the lens has been removed in a cataract operation. We study the motion of this diaphragm driven by movements of the eye. Firstly, a mathematical statement of the problem is given and then we solve the problem exactly for a given class of eye movements. From the analysis we deduce that significant oscillations of the membrane are driven by saccades and that it is the angular acceleration of the eye which causes these types of oscillations. A numerical example is given.

Notes: Abstract The Lotka-Volterra system of prey-predator equations is considered with a special type of continuous time delay. In the case of equal diffusion coefficients Hopf’s bifurcation technique is used to show the existence of travelling wave train solutions for the prey-predator system.

Notes: Abstract A mathematical model of prothrombin activation is being proposed which includes the feedback mechanism of thrombin and the alteration of factor V by thrombin. This model is in good agreement with experimental data for the dependence of the rate of thrombin formation on the concentrations of factors V and X a . In particular, it correctly predicts the existence and location of a maximum in both of these cases.

Notes: Abstract A new model of the upper tracheobronchial tree is proposed to account for the three-dimensional nature of the airway system. In addition to the tube length, the tube diameter, and the branching angle, the model includes information on the orientation angle of each tube relative to its parent tube. The orientation angle, defined as the angle between two successive bifurcations, is useful for calculating the gravitational inclination of each tube. The information on orientation angle is further used to construct a binary coding system for identifying individual tubes in the airway tree. The proposed model is asymmetrical, but the same principles can be readily used to construct a symmetrical one.

Notes: Abstract In any control system for which the number of independent controls is smaller than the number of degrees of freedom to be controlled, our choice of control in any state is restricted to a submanifold of smaller dimension than the tangent space. This simple fact has a number of important consequences for questions of biological import; we consider its implications for adaptation, for senescent phenomena and for the determination of tertiary structures of polypeptides through control of certain average properties. We also formulate the Pontryagin Maximum Principle of Optimal control theory in such a way as to inquire whether specific biodynamic systems can be regarded as optimal with respect to rate of accumulation of particular quantities of the system. We find that if this is possible, the quantity in question must play the role of a clock.

Notes: Abstract A stochastic model for the population regulated by logistic growth and spreading in a given region of two-or three-dimensional space has been introduced. For many-species population the interactions among the species have also been icorporated in this model. From the random variables that describe stochastic processes of a Wiener type the space-dependent random population densities have been formed and shown to satisfy the Langevin equations. The Fokker-Planck equation corresponding to these Langevin equations has been approximately solved for the transition probability of the population spreading and it has been found that such approximate expressions of the transition probability depend on the solutions of the deterministic equations of the diffusion model with logistic growth and interactions. Also, the stationary or equilibrium solutions of the Fokker-Planck equation together with the special discussion on the pattern of single-species population spreading have been made.

Notes: Abstract The notion of an evolutive hierarchical system proposed in this paper is a mathematical model for systems, like organisms, with more or less complex objects. This model, based on category theory, retains the following characteristics of natural systems: they have an internal organization consisting of components with interrelations; they maintain their organization in time though their components are changing; their components are divided into several levels corresponding to the increasing complexity of their own organization, and the system may be studied at any of these levels (e.g. molecular, cellular...). The state of the system at a given instant is modeled by a category whose objects are its components, the state transition by a functor, a complex object by the (direct) limit of a pattern of linked objects (which describes its internal organization). The properties of limits in a category make it possible to ‘measure’ the emergence of properties for a complex object with respect to its components, and to reduce the study of a hierarchical system to that of its components of the lowest degree and their links. Categorical constructions describe the formation of a hierarchical evolutive system stepwise, by means of the operations: absorption of external objects, destruction of some components, formation of new complex objects.

Notes: Abstract An algorithmic formulation is presented for the inference procedure concerning lineage models. The problem is to find lineage rules from observed sequences of tree structures under the assumption that no interactions take place in the course of development and that sufficiently frequent observations are available at equal time intervals. The underlying structural pattern is taken to be a OL system, and the goal is to find propagating and deterministic OL schemes with minimal properties satifsying certain biological reliance criteria. Upper bounds have been found for the complexity of the inference algorithms.

Notes: Abstract A statistical theory of non-equilibrium fluctuation in Volterra-Lotka systems has been presented on the basis of the technique of statistical linearization of non-linear coupled stochastic differential equations.

Notes: Abstract Under certain basic assumptions the branching pattern of dendrites can be modeled as a Galton-Watson process in varying environment. Using results from graph theory we compute the probability distributions, expectations and variances for biologically significant variables such as the number of (intermediate and terminal) branches, the maximum number of orders, etc., together with the limit behavior of these quantities. Furthermore, the probability measure induced by the Galton-Watson process on the set of all trees is calculated. The measure assigns to any set of branching patterns the probability that it is realized by a certain process, which is completely described through the bifurcation probabilities.

Notes: Abstract Theoretical models for DNA repair mechanisms are constructed. Reliability studies considering the living cell as a repairable system are done. The DNA repair process is discussed along with applications and comparison with available experimental data.

Notes: Abstract A mathematical model has been developed to simulatein vivo transmural accumulation of an intravenously injected tracer in the aortic wall of experimental animals. Parameters have been included to represent the following processes that affect tracer distribution: permeation of the blood-tissue interface, diffusion through the layers of the artery wall,convective solute drag through the same, and degradation. Of particular interest for thein vivo situation situation is the inclusion of boundary conditions that account for the variation in the plasma concentration of injected tracer as a function of time. Two analytical solutions are presented. The first describes a system in which two boundaries must be delineated; it pertains if the tracer is allowed to circulate until it enters the avascular media of the artery wall both across its luminal boundary and from the capillaries in its outer layer. The second applies to shorter duration experiments in which entry across only the luminal boundary is considered. A limiting case of the solution for short circulation times is presented, compared with a previously published solution, and examined for its potential utility in parameter estimation. Because of its treatment of time-dependent boundary conditions, the model has unique application toin vivo experiments related to macromolecular transport in atherosclerosis that may otherwise elude proper interpretation.

Notes: Abstract Several current reaction-diffusion mechanisms have been proposed as models for morphogenesis in the Turing (1952,Phil. Trans. R. Soc. Lond. B 237, 37–72) sense. We introduce and exploit a quantity, we have termed heterogeneity, which allows us to elaborate the differences between the various models with regard to spatial pattern formation. It is shown that this quantity provides a concise view for the comparison of theoretical models with experimental observations. Two model mechanisms are treated explicitly both for linear and for biased diffusion.

Notes: Abstract The optimum number of total capillaries in the whole human body was estimated from the analysis of the efficiency for oxygen (O2) transport in the vascular-tissue system. We used a tissue model composed of uniform spheres in which O2 diffuses from the capillary located at the centre of each sphere towards the surrounding tissue consuming O2 at a constant rate. The tissue mass supplied by a single capillary was estimated as the area of positive O2 concentration under a given condition of capillary flow and O2 consumption rate. Total tissue mass was determined as the function of the capillary numbern and the total blood flow. On the other hand, the energy cost required to maintain the vascular system withn terminals was assessed by using the minimum volume model (Kamiya and Togawa,Bull. math. Biophys. 34, 431–438, 1972). The efficiency of the entire vascular-tissue system was evaluated by calculating the ratio of total tissue mass/cost function. The result of the calculation using physiological data of cardiac output and O2 consumption for an average human adult during a heavy exercise revealed the maximum efficiency occurring at the capillary number 3.7×1010 which coincided well with its normal range of physiological estimates (3.2×1010–4.2×1010). We concluded that the entire vascular-tissue system is constructed so as to attain the highest efficiency in O2 transport at its maximum activity.

Notes: Abstract A kinetic model involving synthesis of proinsulin in the rough endoplasmic reticulum, maturation through the Golgi apparatus and granules, with conversion to insulin, is proposed to account for data on the amount of insulin and of proinsulin both secreted during various time intervals and remaining in islets. Introducing three compartments for granules makes it possible to account for the measurement of both hot (pulse labeled with tritiated leucine) and cold proinsulin and insulin over a period of 21/2 hr under constant glucose. Data from islets from animals pretreated with tolbutamide are also presented and modeled. The model is then expanded so that it can be successfully applied to available data on the effects of a period of glucose deprivation on secretion of both hot and cold hormone. Parameters have essentially the same values, where they overlap, as were obtained (Landahl and Grodsky, 1982Bull. math. Biol. 44, 399–410) from insulin secretion by perfused rat pancreas stimulated by a variety of temporal patterns of glucose concentration.

Notes: Abstract Two models of binary tree growth are examined in terms of the Strahler order branching ratio (Rb) and the types of vertex produced during growth, and their inter-relationship. The sequential growth model is that described by Van Pelt and Verwer (1985,Bull. math. Biol. 47, 323–336) in which random growth occurs according to attributed probabilities on terminal or internal segments, one branch at a time. This model generates values ofRb≥3. The synchronous growth model is new and permits more than one segment to branch at a time, again randomly with attributed probabilities. This model generates values ofRb≥2 and in particular, when only terminal branching is permitted, gives 2≤Rb〈3. Such a model might explain the branching in the human bronchial tree, in which 2.5≤Rb≤2.8. Our synchronous model is an alternative to the centrifugal-order-dependent sequential model of Van Pelt and Verwer.

Notes: Abstract From an energy budget of a deciduous plant leaf in moderate conditions, entropy fluxes into or out of the leaf due to solar radiation, infrared radiation, evaporation of water and heat conduction are calculated. Net entropy flow into the leaf is negative. On the assumption that the entropy in the leaf is in a steady state, the entropy production in the typical deciduous leaf in moderate conditions [the solar energy absorbed by both sides of the leaf isE solar=0.0602 (J cm−2 s−1)] becomesS prod=1.8×10−4 (J cm−2 s−1 K−1). The positiveness of the entropy production shows that the Second Law of Thermodynamics certainly holds in the plant leaf. Entropy productions in other conditions are also calculated. The entropy production in the leafS prod becomes a linear function of the solar energy absorbed by the leafE solar:S prod≈-(29.5E solar)×10−4. A theorem is presented: the entropy production in plant leaves oscillates during the period of one day, paralleling the daily solar energy absorbed by leaves.

Notes: Abstract Molecular biologists strive to infer evolutionary relationships from quantitative macromolecular comparisons obtained by immunological, DNA hybridization, electrophoretic or amino acid sequencing techniques. The problem is to find unrooted phylogenies that best approximate a given dissimilarity matrix according to a goodness-of-fit measure, for example the least-squares-fit criterion or Farris'sf statistic. Computational costs of known algorithms guaranteeing optimal solutions to these problems increase exponentially with problem size; practical computational considerations limit the algorithms to analyzing small problems. It is established here that problems of phylogenetic inference based on the least-squares-fit criterion and thef statistic are NP-complete and thus are so difficult computationally that efficient optimal algorithms are unlikely to exist for them.

Notes: Abstract A computational model to predict deposition of a wide variety of particulate pollutants in several species of mammals is presented. The model incorporates breathing pattern and detailed anatomical models of the respiratory tract based on extensive morphometric measurements of individual airways. The predicted deposition from this general model is in close agreement with observed deposition of monodisperse aerosols in rats. Particle size and density and respiratory breathing patterns are the critical parameters affecting regional deposition.

Notes: Abstract A theory of antigen-antibody induced particulate aggregation is developed by investigating the stability of model systems of particles. Conditions for the formation of large aggregates are derived by imposing the requirement that at equilibrium a statistically significant number of redundant bonds would occur in a reduced monomer-dimer model system. A relationship is obtained which predicts the fractional agglutination in the reduced dimer system as a function of the antigen, antibody and particulate concentrations: $$\frac{g}{{2f c_0 (1 - g)^{2^ - } }} = \frac{{s_1 }}{r} + \frac{{s_1 s_2 }}{{2!r^2 }} + ... + \frac{{s_1 s_2 ...s_j }}{{j!r^j }},$$ wherec 0 is the initial concentration of monomer,f is a proximity factor,g is the fractional agglutination,s i is the average rate of formation of theith bond from an (i−1)th bound dimer, andr is the average rate of dissociation of a single antibody-antigen bond.

Notes: Abstract The roles of the concentrations of the three interacting constituents in the aggregation process (antibodies, antigens and particulates) are analyzed in detail. It is shown that the basic equation derived in Part I is consistent over a broad range of conditions with experimental findings previously reported.

Notes: Abstract A general mathematical model describing the biochemical interactions of the hormones luteinizing hormone releasing hormone (LHRH), luteinizing hormone (LH) and testosterone (T) in the male is presented. The model structure consists of a negative feedback system of three ordinary differential equations, in which the qualitative behavior is either a stable constant equilibrium solution or oscillatory solutions. A specific realization of the model is used to describe the experimental observations of pulsatile hormone release, its experimental suppression, the onset of puberty, the effects of castration, and several other qualitative and quantitative results. This model is presented as a first step in understanding the physicochemical interactions of the hypothalamic-pituitary-gonadal axis.

Notes: Abstract Based upon the transition rate equation of dipoles in the membrane, we deal with two important aspects of interaction of nerve signals: (1) conditions for nerve excitation and (2) frequency spectrum analysis of nerve impulse. Interrelations between signal amplitudes and frequencies are formulated in detail. There are several important conclusions which can be drawn from our calculations. First, toexcite the nerve, low frequencies are generally more effective than high frequencies. Second, tosedate the nerve (i.e. to suppress undesired activities), high frequencies would suit better. Third, harmonics produced through interactions of nerve signals are not necessarily weaker than the fundamental frequencies. The great significance of our theory is that it indicates in principle the feasibility to alter or rewrite the information contents of a nerve message in our body by applying stimulations of appropriate strengths and frequencies. Thus, the theory provides a physical basis and hence some understanding for a new branch of medicine—neuro therapy such as Nogier's auriculotherapy, Lamy's phonophoresis, Voll's electroacupuncture and the fast rising TENS (transcutaneous electro-neuro stimulation).

Notes: Abstract A new physical property, called resonance of the B-type is hypothetically attached to the λ =546 nm irradiated crystalline (small) molecules. In this respect an up or down configuration is assumed for those states obtained through irradiation times that are multiples of 5 sec. With these assumptions, the cellular receptors that may detect these states appear to belong to three classes: the up, down and alternatively mixed up-down. Using the classic formalism of eigenvectors and eigenvalues, a simple spectroscopic type of formula is derived, through which all the possible states of the above characteristic may be obtained.

Notes: Abstract A model of ecosystems with migration is proposed from the viewpoint of flow. This model explains the following two points: (1) How the density-dependent terms in population dynamics arise as a consequence of migration. (2) How the ecosystem exhibits a hierarchy in energy per unit biomass.

Notes: Abstract For precise boundary conditions of biological relevance, it is proved that the steadily propagating plane-wave solution to the Fisher equation requires the unique (eigenvalue) velocity of advance 2(Df)1/2, whereD is the diffusivity of the mutant species andf is the frequency of selection in favor of the mutant. This rigorous result shows that a so-called “wrong equation”, i.e. one which differs from Fisher's by a term that is seemingly inconsequential for certain initial conditions, cannot be employed readily to obtain approximate solutions to Fisher's, for the two equations will often have qualitatively different manifolds of exact solutions. It is noted that the Fisher equation itself may be inappropriate in certain biological contexts owing to the manifest instability of the lowerconcentration uniform equilibrium state (UES). Depicting the persistence of a mutantdeficient spatial pocket, an exact steady-state solution to the Fisher equation is presented. As an alternative and perhaps more faithful model equation for the propagation of certain species properties through a homogeneous population, we consider a reaction-diffusion equation that features a cubic-polynomial rate expression in the species concentration, with two stable UES and one intermediate unstable UES. This equation admits a remarkably simple exact analytical solution to the steadily propagating plane-wave eigenvalue problem. In the latter solution, the sign of the eigenvelocity is such that the wave propagates to yield the “preferred” stable UES (namely, the one further removed from the unstable intermediate UES) at all spatial points ast→∞. The cubic-polynomial equation also admits an exact steady-state solution for a mutant-deficient or mutant-isolated spatial pocket. Finally, the perpetuating growth of a mutant population from an arbitrary localized initial distribution, a mathematical problem analogous to that for ignition in laminar flame theory, is studied by applying differential inequality analysis, and rigorous sufficient conditions for extinction are derived here.

Notes: Abstract The binding of mono-, di- and trivalent cations to negatively charged surfaces is studied within the framework of a modified Gouy-Chapman equation. For any given combination of ions of the above valences, the existence and uniqueness of the solution for the surface potential is shown. The treatment provides the surface potential and charge density. For a system containing only monovalent and divalent ions, analytical solutions are given. When trivalent ions are also present, a procedure based on numerical integration is described. The distance dependence of the electrostatic potential for planar surfaces is given. The calculations provide the amount of cations tightly bound and the amount trapped in the double layer region. The competition between cations for binding to surfaces is elucidated.

Notes: Abstract This paper deals with a model describing the behavior of barium-treatedApalysia neurons. The model is represented by a dynamical system, so-called “complete system”, defined in R4 and depending on a small parameter. The study of this system under zero membrane current conditions was performed with the use of the qualitative theory of singular perturbations. We show that this system has a stable periodic solution of the discontinuous type when the small parameter tends to 0+. A reduced system defined in R3, associated to the complete system was also studied: it corresponds to a constant activation of the inward current. We demonstrate that the corresponding hypothetical cell remains silent under zero current conditions.

Notes: Abstract In the study of chemical modification of enzymes and other biologically active proteins, plots of fractional residual activity as a function of number of groups modified per enzyme molecule are often used to establish a correlation between the chemical modification and enzyme inactivation reactions and to determine the stoichiometry of the modification reaction. This paper presents a critical examination of the underlying theoretical framework of such graphs. Whereas these plots are usually presented as linear functions, it is shown here that the general equation describing the relationship between inactivation and modification contains an exponential term; therefore, in the general case, the plot is actually a curve. It is suggested that caution be exercised in the interpretation of such plots and that equations such as those derived in the text be used to fit theoretical curves to the data, in order to maximize the information gained from chemical modification experiments.

Notes: Abstract This communiction argues that so-called “hermaphroditic” tracer systems, which are neither open nor closed, do not exist physically. The argument is based on the assumption that any observable (possibly nonhomogeneous) macroscopic compartment can be approximated by a compartmentC with a finite number of entry points for the tracer, each associated with an abstract subcompartment ofC. It is shown that the “hermaphroditic” property requires that the mean waiting time be infinite in at least one of the subcompartments, or in a subcompartment elsewhere in the system. A subcompartment with infinite mean waiting time must have some sort of memory, of infinite duration, which knows how long a given particle has been retained, however long that is, and thereby determines its probability of departure. Assuming, as seems likely, that no physical basis exists for such an infinite memory, it follows that “hermaphroditic” systems do not exist.

Notes: Abstract The implication of state space structure on the existence of a repeatable experimentE designed to determine if a states∈L has propertyP or notP is investigated. It is shown that if a state spaceL is connected, then no experimentE is repeatable. This formalism is used to demonstrate that if a propertyP has an associated set of points inL which is dense with dense complement inL, then there exists no repeatable experimentE which can be used to test whethers has propertyP or notP. Other consequences of this formalization are discussed.

Notes: Abstract It has been shown that the resistance of flow and the wall shear increase with the size of the stenosis but these increases are comparatively small due to non-Newtonian behaviour of the blood indicating the usefulness of its rheological character in the functioning of the diseased arterial circulation.

Notes: Abstract The modern theory of generalized Hamiltonian systems is used to construct a unified canonical description of the linear Lagrangian biodynamics introduced by Kerner.

Notes: Abstract The purpose of this paper is to justify an asymptotic method developed for the study of peristaltic transport in a tube of arbitrary cross section. Within the framework of long wave approximation, the three-dimensional nonlinear Navier-Stokes equations are reduced to a sequence of two-dimensional linear boundary value problems of Laplace and biharmonic operators. It is shown that, if a Reynolds number is less than some constant, the solution of the approximate equations is indeed an asymptotic approximation to the exact solution of the problem as the ratio of the maximum radius of the tube to the wave length of the peristaltic motion of the wall tends to zero, and the error estimates are expressed inL 2 norms. Furthermore, under the same condition the exact solution is shown to be unique and stable under arbitrary perturbation of spatially periodic disturbance. Application of the stability condition to peristaltic transport in a tube of circular cross section is given.

Notes: Abstract A mathematical analysis, including existence and uniqueness, is given for some boundary value problems which model the flow of a fluid-solute mixture in a tube which is placed in an interstitium. The model permits an interchange of fluid and solute across the tube walls.

Notes: Abstract The vertebrate nervous system has topographic interconnections in many parts, known for example as retinotopy, somatotopy, etc. It is plausible that modifiable synapses play an important role in forming and refining these connections together with the sensory experiences. To elucidate the mechanism of topographic organization, we propose a simple model consisting of two nerve fields connected by modifiable excitatory synapses. The model also includes modifiable inhibitory synapses. The behavior of the model is described by a set of simultaneous non-linear integro-differential equations. By analyzing the equations, we obtain the equilibrium solution of topographic connections. It is also proved that a part of the presynaptic field which is frequently stimulated comes to be mapped on a large area of the postsynaptic field so that it has a good resolution.

Notes: Abstract The steady-state solution of the equations governing substrate exchange between vascular and extravascular compartments separated by a membrane with finite, symmetrical substrate permeability is presented. Substrate removal from the extravascular compartment by Michaelis-Menten saturation type kinetics with negligible diffusion in the axial and instantaneous diffusion in the transverse directions in both compartments are assumed. It is shown that the solution degenerates into known expressions for special linearized and asymptotic cases. The method of solution is also applied to an extension of the original model incorporating autoregulatory feedback effects upon the process responsible for substrate removal.

Notes: Abstract Global asymptotic stability and equilibrium coexistence is established in two species Lotka-Volterra-type competition when there are time delays in interspecific interaction terms and the intraspecies competition is stronger than the interspecies competition.

Notes: Abstract The realized (observed) value of Landau’s dominance hierarchy index is examined. Under a model of constant pairwise dominance probabilities, the observed index is shown to be a strongly consistent estimator of the underlying (true) index. However, a large number of encounters between animals is shown to be required in order to reduce bias and variance to practical levels except when the pairwise dominance probabilities are near one.

Notes: Abstract The steady state potassium conductance as a function of measured membrane potential difference (p.d.) ϕ of the squid giant axon is corrected for the effect of accumulation of potassium in the periaxonal space. This correction is made on the assumption that several mathematical models of the axon are valid. These are (i) the McIlroy (1975), McIlroy-Hahn (1978) model of membrane conductanceg i(i=K, Na) which is a detailed model of passive transport of ions across the axonal membrane with the aid of mobile, negatively-charged carriers, (ii) the Adelmanet al. (1973) compartmental model of the periaxonal and external bathing-solution spaces, (iii) the enzymatic theory of nervous conduction due to McIlroy (1970 a, b, c), (iv) the Wien dissociative effect of the axolemmic electric field on the weak membrane buffer proposed by Bass and Moore (1968) as a trigger mechanism in nervous excitation and (v) the model (McIlroy, 1979) of the interfacial double-layer p.d.s. which are assumed to exist at the membrane’s surfaces because of the presence of a fixed surface charge. From the correctedg k (ϕ) curves the values of the double-layer p.d.s. of model (v) are deduced and these are shown to lead to a consistent, physically reasonable solution for the distance (approx. 6.8Å) between the fixed surface charges and for the dissociation constants of these sites in their interactions with the ions of the extra-membrane electrolytes. Assuming that the selectivity coefficint of the potassium conducting system for the squid giant axon is approx. 52 it is deduced that the potassium permeability,P ks , of the periaxonal barrier ≈1.37(±0.5)×10−4 cm sec−1 and the thickness of the periaxonal space ≈451±159Å.

Notes: Abstract The effects of peripheral layer viscosity on physiological characteristics of blood flow through the artery with mild stenosis have been studied. It has been shown that the resistance to flow and the wall shear decrease as the peripheral layer viscosity decreases.

Notes: Abstract A study was made of Higgins’ model of glycolysis incorporating molecular diffusion of intermediates, utilizing an earlier conjecture due to Landau. Conditions for the existence of asymptotically stable spatio-temporal periodic solutions are obtained.

Notes: Abstract In order to understand the abnormal flow conditions of blood in a locally constricted blood vessel, the analytical results are obtained for the oscillatory flow of blood which behaves as a Newtonian fluid. It is here assumed that the surface roughness is cosine-shaped and the maximum height of the roughness is very small compared with the radius of the unconstricted tube. Numerical solutions are presented for the instantaneous flow rate, resistive impedance, wall shear stress and phase lag.

Notes: Abstract In order to better understand the effect of initial stress in blood flow in arteries, a theoretical analysis of wave propagation in an initially inflated and axially stretched cylindrical thick shell is investigated. For simplicity in the mathematical analysis, the blood is assumed to be an incompressible inviscid fluid while the arterial wall is taken to be an isotropic, homogeneous and incompressible elastic material. Employing the theory of small deformations superimposed on a large initial field the governing differential equations of perturbed solid motions are obtained in cylindrical polar coordinates. Considering the difficulty in obtaining a closed form solution for the field equations, an approximate power series method is utilized. The dispersion relations for the most general case of this approximation and for the thin tube case are thoroughly discussed. The speeds of waves propagating in an unstressed tube are obtained as a special case of our general treatment. It is observed that the speeds of both waves increase with increasing inner pressure and axial stretch.