ALBERT

Notes: Abstract In order to evaluate the effect of anatomic asymmetries on the gas concentration distribution in the pulmonary airways, a Monte Carlo simulation of combined bulk flow and molecular diffusion was carried out in a realistic distal airway model (Parkeret al., 1971). This airway model, composed of branches distal to the 0.5-ram diameter airways, contained an upper symmetric segment consisting of four generations of conducting airways and a lower asymmetric segment of alveolar ducts and sacs arranged in five transport paths of varying lengths. In accounting for the volume increases of these ducts and sacs occurring during normal respiration, uniform alveolar filling rates and a fixed length-to-diameter ratio of all airways were assumed. For a pulse injection of inert tracer gas, the simulation was employed to determine the longitudinal concentration profiles in the conducting airways. In the alveolated airways, not only were the longitudinal profiles determined along each path, but radial transport from the core to the periphery of the airways was considered. The results of the simulations indicate that geometric asymmetries alone contribute substantially to regional concentration variations in the distal airways. For example, when a gas bolus is injected at mid*inspiration, there are concentration differences as great as 40% between two points along different transport paths located equi-distant from the proximal end of the model. As viewed from the terminal end of the model (acinus), average concentration differences as large as 6-to-1 exist between the longest and shortest transport paths respectively for gas boli introduced near the end of inspiration. The results further indicate because of large radial diffusion rates, no significant concentration differences exist between the periphery a-ld the central core of alveolated airways. Simulation of the expired concentration profiles indicate that boll injected very late during inspiration exhibit a sloping tail, unlike the earlier injected boll whose tails are virtually horizontal. Through the use of superposition teehniqnes, it was found that these sloping tails correspond to an alveolar slope of 1.5 vol% between 750 and 1250 ml expired for a continuous washing of tracer. This result is in disagreement with other transport analyses which did not directly account for the effect of geometric asymmetries.

Notes: Abstract Assuming a spherical geometry for the left ventricle, passive elastic stiffness-stress relations have been obtained on the basis of linear elasticity theory and large deformation theory. Employing pressure-volume aata taken from rat hearts of various age groups, it is shown that young rat heart muscle (1 month) is stiffer than either adult (7 months) or old rat heart muscle (17 months). Although the qualitative results are similar for both elasticity theories, the large deformation theory gave results in closer agreement with those obtained from papillary muscle studies. These results imply that stiffness of muscleper se can be assessed from left ventricular pressure-volume data.

Notes: Abstract Deliberate evaluation of the quantum theory of nerve excitation is made by comparing it with Hill's theory in fitting the experimental data on threshold-frequency relation, optimum frequency (v0) for nerve excitation and strength-duration relation. Decrease of v0 and increase of all the time constants (Hill's λ andk, Wei'sT 2 and spike durationw) with decreasing temperature are interpreted on the basis of the dipole relaxation timeT 2 but inexplicable from Hill's theory or any other existing theory. The closeness ofk,T 2 andw values is explained. A variety of experimental results obtained by others is discussed. Finally, a comparison is made between the Hodgkin-Huxley equations and the quantum theory. Most of the facts (electrical and non-electrical) tend to support the thesis that nerve excitation is a macroscopic expression of quantum transitions of dipoles between energy states.

Notes: Abstract In the periodic Leslie model the asymptotic period of total population is a divisor of the asymptotic period of the population vector. Under reasonable circumstances these periods are identical.

Notes: Abstract A number of biological branching systems, such as the bronchial and pulmonary arterial trees, are being investigated in an ongoing study in order to define their physiological properties. The technique involves the description of branching trees by the use of hierarchical systems of ordering, especially those described by Horsfield and by Strahler. During this work some mathematical properties of branching trees were demonstrated and these are described in this paper.

Notes: Abstract Luteinizing hormone (LH) is secreted continuously from the anterior pituitary gland. The concentration in the blood of this gonadotropic hormone plays a regulatory role in the development of puberty in both sexes, in the induction of ovulation in females, and in the production of testosterone in males. The secretion of LH is in turn controlled by luteinizing hormone releasing hormone (LHRH) secreted by the hypothalamus. LH and LHRH are removed from the blood by degradation and excretion. This hormonal system is modelled by a system of ordinary differential equations based upon specific physiological and biochemical assumptions current among experimentalists in this field. The one exception is the assumption that LHRH may bind reversibly to a serum protein; an analysis of the data shows that this or a similar mechanism is a crucial specification. Data on the serum levels of LH and LHRH in two human subjects were fitted using the model. The data consist of the transients and subsequent decays created by a bolus intravenous injection of LHRH.

Notes: Abstract A thick-wall incompressible, elastic sphere was used as a model for the diastolic rat left ventricle. A model for myocardial nonhomogeneity was derived assuming that fiber (circumferential) stress was independent of position in the ventricular wall. The theoretical implications of the resulting constitutive relations together with the spherical model were analyzed in the context of large deformation elasticity theory. It was found that muscle stiffness at a given level of uniaxial stress increased monotonically from the endocardium to the epicardium. In addition, fiber stress was found to be essentially a linear function of transmural pressure above a pressure of 6 g/cm2. It was also shown theoretically that neglecting the nonhomogeneity of the myocardium resulted in a state of stress which differed significantly from that predicted by the nonhomogeneous model. For example, at a transmural pressure of 14 g/cm2, fiber stress in the nonhomogenous model was equal to 17 g/cm2 while fiber stress in the homogeneous model varied between 100 g/cm2 at the endocardial surface and 2 g/cm2 at the epicardial surface. The change in muscle stiffness with position which characterized the nonhomogeneous model also tended to linearize the highly curvilinear radial stress distribution predicted by the homogeneous model at a given transmural pressure.

Notes: Abstract The phenomenon of axonal transport has been well documented (Ochs, 971; Lasek, 1970; and Grafstein, 1967). In a previous paper, we showed how diffusion alone could not account for this process. In this report we show that convection or convection with diffusion can account for the observed build-up of material. By including a first-order catabolic sequestration term, we are able to offer an understanding of the several apparent rates of transport with the same underlying velocity and variable sequestration.

Notes: Abstract It is known that the Lotka-Volterra coupled nonlinear differential equations for a two-species prey-predator ecosystem possess a periodic solution, although its exact form is not yet obtained analytically. The conventional linearization approximation for solving these nonlinear equations leads to a harmonic oscillator whose frequency depends only on the intraspecific coefficients. We propose here a prescription for obtaining nonlinear correction to the linear frequency by using the Hamilton-Jacobi canonical formalism of classical mechanics. It is found that the first-order correction, which also involves interspecific parameters, exhibits the basic qualitative features of the nonlinearity.

Notes: Abstract Environmental safety testing typically requires procedures for extrapolating from the relatively high experimental to the very low use doses of potentially harmful substances. In the present paper, a stochastic mammillary compartmental model for environmental safety testing is proposed and extrapolation procedures based on its dose-response relationship are developed. The proposed model is a direct generalization of one of the basic safety models, the one-hit model, in that a harmful reaction is assumed to occur if at any time any of the peripheral compartments attains a specified threshold of particles. Consideration of a closed model yields an upper bound on the probability of attaining a certain threshold level, thus providing a conservative procedure for extrapolating to a low dose, while a lower bound obtained from a related open model provides a useful monitoring device as to the sharpness of the upper, bound. The extrapolation procedure is illustrated with simulated data and approximations for initial values are developed.

Notes: Abstract By using chromosome images as a framework, algorithms for finding most dissimilar images are presented and illustrated by examples. In terms of angles, a chromosome image consists of two exterior biangles and two interior biangles. Biangles are defined and classified into 180° biangles, 〉180° biangles and 〈180° biangles. The dissimilarity of biangles and its geometric interpretation together with various properties of biangles are also presented. The results may have useful applications in pattern recognition, scene analysis, information storage and retrieval, artificial intelligence and fuzzy set theory.

Notes: Abstract The Volterra equations which represent competitions between two species are utilized to examine the phenomenon of boundary formation between two species of plants. The set of stable stationary points for these equations is determined and is illustrated in a product space of parameters and dynamical variables. The stages of boundary appearance and succession are visualized by considering slow changes of the parameters as functions of time and space.

Notes: Abstract For certain environments, the Darwinian model allows unique prediction of a function that any surviving system adapted to such an environment has to perform. This is the case for those environments that determine a “survival functional” of position in space-time of known shape. Purely temporal survival functionals can be distinguished from spatial and mixed ones. In each case, there exists an optimum path in combined physical and (reduced) metabolic space. Dependent on the admissible error, approximate solutions of different complexity are sufficient. All solutions possess an afferent, a central, and an efferent part. Within this general frame, specific, “probably simplest”, solutions are proposed for adaptive chemotaxis, insect locomotion, lower vertebrates locomotion, higher vertebrates locomotion, chronobiological systems, and immune systems, respectively—or rather, for the underlying functionals.

Notes: Abstract Mathematical models afford a procedure of unifying concepts and hypotheses by expressing quantitative relationships between observables. The model presented indicates the roles of both insulin and glucagon as regulators of blood glucose, albeit in different ranges of the blood glucose concentrations. Insulin secretion is induced during hyperglycemia, while glucagon secretion results during hypoglycemia. These are demonstrated by simulations of a mathematical model conformed to data from the oral glucose tolerance test and the insulin infusion test in normal control subjects and stable and unstable diabetic patients. The model studies suggest the parameters could prove of value in quantifying the diabetic condition by indicating the degree of instability.

Notes: Abstract A model for the dynamics of a single-species population whose birth rate depends on densities of previous generations is introduced. A difference equation formulation is proposed and the solutions classified for the various parameter values. Data from an experimental population of mice growing in limited space is cited and compared with the model predictions.

Notes: Abstract All soft tissues are modeled as either one-dimensionalstrings, two-dimensionalmembranes, or three-dimensionalsolids. Attention is restricted to tissues in which one of the principal stress components is large and positive in comparison with the other negligible components. Results indicate the following: (1) If a deformed string isconstrained to lie on a surface and is free of tangential pressure, the tension is carried by rays which are geodesics of the surface. If a string or membrane isfree to deform in space without normal pressure, the tension rays are straight lines. If a membrane deforms without tangential surface loads, the tension rays are always geodesics on the deformed surface. If a solid deforms without body forces, the tension rays are straight lines. (2) The stress in a string is a constant if the string is free of tangential pressure and has constant cross-sectional area. The stress in flat tension fields free of tangential surface loads decays inversely with distance along a tension ray from the edge of regression. The stress in a spherically symmetric tension field free of body forces decays inversely with the square of the distance from the center of the sphere. (3) Stress singularities can occur in soft tissues, such as at the corners of a closed rectangular hole in a flat membrane strip. (4) The tension rays in the torsion of soft annular membranes are more steeply inclined from the radial direction than the tension rays for hard metals equally displaced.

Notes: Conclusion For all these reasons, the rediscovery of the equivalence theorem, first stated by David Ricardo in 1817, must be rejected as an adequate basis for policy, just as Ricardo had denied its applicability to the real world. Correspondingly, the concern with the adverse consequences of unfunded social insurance wealth for the supply of national saving, capital intensity, and living standards remains well founded. p ]If, as a practical matter, public pension and social security programs will never be funded actuarially in the United States and most other postindustrial countries, then government-supervised substitution of private for public retirement plans is the only way to achieve at least partial funding. If such substitution follows the British model of allowing employers to contract out of the earnings-related part of the state scheme if equivalent pensions are provided by the company plan, payroll taxes and social security wealth decline so as to reduce their adverse impact on capital formation.

Notes: Summary: Social Policy in the Italian Economy Favourable social and economic conditions constitute the essential framework for a stable development of savings. Saving in the form of insurance becomes advantageous for the individual, and private insurance can thus extend its activity, when social attitudes and the economic situation favour the propensity to save. If conditions change, the State can take over the coverage of risks through social insurance. By means of this institutions, an anti-cyclical policy can be pursued: the amount of social security contributions, for instance, can be increased during the expansion of the cycle and the amounts thus accumulated can be used to grant benefits during the recession period, when contributions can be fixed at a lower percentage of wages. Another type of policy can be pursued by government authorities: that of adjusting social security contributions to industrial profits, thereby directing the subsequent effects on economic growth. Inflation cab cause instability in decisional policies taken by private insurance companies. A solution to the unbalanced increase of costs can be found in index-linking. Life policies of this kind, for instance, cab be closely related to investments in houses, to be bought by the insured themselves in price-linked instalments. After a reference to present developments regarding risk instability and to possibilities of new forms of insurance, this paper considers the competition resulting from the opening of the EEC insurance markets as an opportunity for the Italian market to strengthen its structures.

Notes: Abstract A number of apparently different lines of inquiry into fundamental biological processes point to the central role played by the notion of observation in the theory of biological systems. Not only do we use the results of our own observations to obtain the system descriptions which are the starting-points for an understanding of biological processes, but it is a basic postulate of physics that the interactions between biological systems themselves can be regarded as observations. On this basis, it is clear that we cannot properly understand biological interactions unless the observables we employ for system description are the same as those involved in the interactions we are describing. To do this requires a general theory of observables and system description, establishing the relationship between different modes of description. A sketch of such a theory is developed in the present paper, using only two postulates: (a) that all interactions are determined by the values of observables of a system evaluated on specific states, and (b) that real-valued observables suffice. As an application, a specific test is proposed whereby it can be determined whether the observables employed to describe interacting systems are sufficient to specify the observables involved in the interaction itself.

Notes: Abstract Differential equations are derived whose solution gives the cross-sectional shape of a flexible tube as a function of the transmural pressure. These equations are solved digitally to produce a series of closed curves, each curve representing the shape of a cross section for a particular set of conditions. These are then applied to the case of systemic arteries, pulmonary arteries, and large veins. The results predict that systemic arteries must always be circular, even when the internal and external pressures are equal. In veins, a small positive internal pressure causes them to become circular, regardless of their initial state, with negligible stretching. Further increases in internal pressure cause the area of the cross section to increase due only to stretching, the shape remaining essentially circular. With pulmonary arteries, known to be noncircular, changes in the cross-sectional area result from a combination of stretching and changes of shape.

Notes: Abstract Stochastic models of human reproduction are beginning to play significant roles in the evaluation of family planning programs. A class of stochastic processes called absorbing, agedependent, semi-Markov processes frequently arises in the construction of such models. The paper begins with a discussion of some technicalities regarding absorbing, age-dependent, semi-Markov processes. Then, an algorithm due to Littman, which makes possible the computerization of this class of stochastic processes, is presented. Briefly, Littman’s algorithm provides an efficient method for numerically solving systems of renewal type integral equations, provided the system does not contain a large number of equations. After setting down a concrete model for a large clinical trial of intrauterine devices conducted in Taiwan, the paper concludes with a discussion of a method for validating the model based on the data collected in the clinical trial.

Notes: Abstract Some theoretical requirements for the valid use of hemolytic plaque inhibition as a method for studying cellular selection and recognition in an immune response are reviewed. Aside from providing a rational basis for the use of this technique, the theory resolves a growing body of apparently conflicting results on the affinity regulation of IgM secreting cells and can also be used to deduce structural (in particular, morphological) features of the IgM molecule. The diverse predictions of the theory are examined in light of experimental results and find support in a wide data base. Additional specific experiments are proposed which can be used in conjunction with the theory to help clarify the relation between cellular proliferation and maturation.

Notes: Abstract A theory of the cell volume is presented with emphasis on the swelling effect of high concentrations of KCl and other chloride salts. In this theory a particular cell volume represents a state of balance between the tendency of the cell water to build deeper layers of polarized water and the restraining forces exerted by the salt linkages and H-bonds. Taking into account also the different structure-breaking effects of different salts, theoretical curves can be constructed which describe the complex multiple peak-plateau of swelling curve observed in frog muscle in response to increasing concentrations of different chloride salts.

Notes: Abstract Non-steady-state equations for kidney models are stated. General conservation relations for these equations are derived. Transient equations for the central core model of the renal medulla are developed. Solution of the equations by Laplace transform methods for time invariant volume flows is discussed. The general theory of solving models with time dependent flows by finite difference methods is developed.

Notes: Abstract The behavior of large systems of randomly-interacting variables is examined using an intentionally simplified model. The stable positive solutions are found to exhibit to a significant degree some well-known properties of ecological systems. This resemblance (including for example the predominance of “predator-prey” interactions) is all the more striking in view of the lack of biological “data” at the input end. The findings suggest it advisable to distinguish two kinds of properties in ecosystems. One kind would depend on specifically biological mechanisms; the other would characterize a wide class of persistent systems, and arise from the need for a dynamic balance between positive and negative feedback.

Notes: Abstract For the tumor model of Skipper and Zubrod, which has been analyzed previously for the theoretical FLM function and the effect of chemotherapy against tumors of known or assumed kinetic characteristics, the theoretical continuous labeling (CL) function is derived by considering an equivalent tumor (in terms of unlabeled cell populations) in which the density function of phase duration of cells inS-phasef 2(a 2)=δ(a 2−∞) and the loss functionL 2(t)=0. This mathematical concept of blocking is applied to the analysis of synchronization in tumor growth and blocking effects in cancer chemotherapy. These effects of chemical agents on the cell cycle progression are being incorporated into a previously written computer simulation program for cancer chemotherapy. Whereas, a program is written and used to simulate the CL functions for L1210 leukemia, and primary and metastatic Lewis lung carcinoma.

Notes: Abstract Global stability is established in a class of prey-predator models. This includes a prey-predator model in which the predator has Type 2 functional response and no intraspecific interactions. Two simple examples demonstrate that Kolmogoroff’s theorem does not apply to some members of this class of models.

Notes: Abstract The skeletal muscle is regarded as a periodically deformed plate. An equation of energy for the biological tissue in the system is used in the Lagrange variables. With the heat exchange through the above tissues to the exterior media and also with account of some relations between the physiological factors the approximate analytical solution for this equation is obtained and compared with the physiological data.

Notes: Abstract One of Bobisud's (1976) models for the evolution, of cannibalism is discussed. His analysis is criticised for not being based on the principle of individual selection. Assuming the operation of that principle, we show by simulating his model that cannibals may establish themselves in a noncannibal population. This will happen both in cases where Bobisud concluded cannibalism to be optimal and in cases where he concluded cannibalism not to be optimal.

Notes: Abstract The D'Arcy Thompson concept of biological transformations is developed in a form analogous to such physical concepts as the Law of Corresponding States in thermodynamics, and the Principles of Similitude found in engineering. We find that such concepts depend on a distinction between fundamental and derived quantities, in which the values assigned to the fundamental quantities set the natural scales for the derived ones. Among other things, we see that critical phenomena, such as phase transitions, arise as an immediate consequence of this distinction. In a biological context, we explore the implications of Thompson's hypothesis that closely related organisms are phenotypically similar, assuming that the organisms we see are the result of selection processes operating on phenotypes.

Notes: Abstract If information is coded in the combination of activities of many neurons operating in parallel, then information present in a network can be defined by the correlation of present network activity with the activity which had been elicited by a stimulus in the past; a high correlation indicates the presence of the previously encoded stimulus. Information is distributed in the network because coding is dependent upon the activities of all cells. A model based on Hartline-Ratliff lateral inhibition with a time delay shows that lateral inhibition can distribute information across a parallel network, reduce output noise, and also briefly store information. With no changes in model parameters, and the use of a correlation measure for recognition, the model can stimulate psychophysical results in eleven variations of the metacontrast masking paradigm.

Notes: Abstract A theory is derived to calculate the regional and total deposition of aerosol particles in the nasal passages during inhalation. The particle size studied range from 0.2 to 10.0 μm diameter. The deposition is calculated in five regions; (I) the region filled with nasal hair, (II) the nasal valve, (III) the expansion region, (IV) the turbinate region and (V) the posterior bend. Equations are derived to determine the deposition caused by direct impaction on the nasal hairs and bends of the passages. The calculations show the deposition due to direct impaction does not account for the amount or location of deposited particles measured in experiments. Secondary flows have been speculated to exist in the expansion region after the nasal valve and an equation is derived to estimate the deposition caused by the secondary flows. The calculated deposition, due to direct impaction and secondary flows, shows general agreement with the experiment as to the predicted amount and location of deposited particles.

Notes: Abstract The paper reviews a series of models for circadian clocks and discusses their conclusions and predictions. Attention is focused on Pittendrigh's empirical model, two mathematical models by the author and Winfree's work.

Notes: Abstract From a model of facilitated ionic transport across axonal membranes proposed by McIlroy (1975), the equipotentials and electric field distributions in the vicinity of a potassium conducting pore and of a sodium conducting pore are computed and presented as two-dimensional mappings. The model is then extended to include the effect of impurity ions in the conducting pores viz. of potassium ions in a sodium pore and of sodium ions in a potassium pore. The ionic selectivities and permeabilities of the transported species are discussed in relation to the extended model. Bounds are deduced for the ionic selectivity coefficients for both the sodium and potassium current-carrying systems in squid giant axons and the electric-field distributions in the vicinities of the pores are computed for the extended model and compared with the impurity-free fields first calculated. Finally the permeability coefficients defined in terms of the extended model are shown to reconcile the results of attempts to measure permeability by means of radioactive tracer techniques, with the classical description of the resting nerve.

Notes: Abstract The partial differential equations describing transcapillary exchange and subsequent removal of solutis from an idealized liver sinusoid are amenable to solution by similarity analysis. The power and utility of this technique, which is not widely appreciated as a method for solving biological models, is illustrated here for a system whose Laplace transforms is difficult to invert.

Notes: Abstract The phenomenological theory for the chemotaxis and consumption of oxygen by motile aerobic bacteria is shown to yield a remarkably simple one-dimensional steady-state solution for a congregation of bacteria close to the surface of an oxygen-depleted aqueous medium.

Notes: Abstract In this paper biological compartmental models are considered which take into account the intrinsic randomness of the transport rate parameters and their possible variability in time. An identification procedure is presented for the estimation of the stochastic processes representing the transport rate parameters of a compartmental model from a noisy input-output experiment. The problem is formulated in terms of nonlinear filtering. A simple model is discussed for the case in which the transport rate parameters are independent of each other. The possibility of testing possible important features of the behavior of the transport rate parameters is also evidenced.

Notes: Abstract For any essentially nonlinear system of reaction-diffusion equations of the generic form ∂ci/∂t=Di∇2ci+Qi(c,x,t) supplemented with Robin type boundary conditions over the surface of a closed bounded three-dimensional region, it is demonstrated that all solutions for the concentration distributionn-tuple function c=(c 1(x,t),...,c n (x,t)) satisfy a differential variational condition. Approximate solutions to the reaction-diffusion intial-value boundary-value problem are obtainable by employing this variational condition in conjunction with a Galerkin-Ritz procedure. It is shown that the dynamical evolution from a prescribed initial concentrationn-tuple function to a final steady-state solution can be determined to desired accuracy by such an approximation method. The variational condition also admits a systematic Galerkin-Ritz procedure for obtaining approximate solutions to the multi-equation elliptic boundary-value problem for steady-state distributions c=−c(x). Other systems of phenomenological (non-Lagrangian) field equations can be treated by Galerkin-Ritz procedures based on analogues of the differential variational condition presented here. The method is applied to derive approximate nonconstant steady-state solutions for ann-species symbiosis model.

Notes: Abstract A method, based on symmetry, is suggested for determining the information content of systems. A comparison has been made between the information for symmetry, topology, and chemical composition. The new information measure increases when the asymmetry of the molecules and the number of atoms in the latter increases. It can distinguish between different molecular conformations, and give a linear correlation with the absolute entropy for homologous series of chemical compounds.

Notes: Abstract The micromorphic theory of Eringen is applied to study the tube flow of blood. The blood is considered to be a deformable suspension, with constitutive relations of the form of those of simple microfluids. By means of energy consideration, a relation is established between the local concentration parameter and the measure of rotationality involving both macro-and micromotions. The tube flow problem is then solved with some analyses on viscosity coefficients and boundary conditions. The results obtained indicate an integrated explanation of various important physical phenomena associated with blood flow, such as the tube size dependence of the apparent viscosity and the non-uniform concentration distribution over a tube cross section.

Notes: Abstract By observing that the n-tuple of rate functionsQ(c) is orthogonal to the c-space gradients of each of the (n - 1) constants of the motion Φ v (c), a generic canonical expression for the rate functions is given in terms of the exterior product of the gradients of the (n - 1) Φ v 's. For models withQ so prescribed from the outset, an analytical general solution is obtainable directly for the system of autonomous ordinary differential equations dc/dt =Q(c). Thus, the generic canonical expression for the rate functions can be utilized to construct analytically solvable models for interacting biological species, as ilIus~rated by examples here.

Notes: Abstract A model for the extraocular plant of the human visual eye tracking mechanisms is discussed. Its sensitivity to variation of controller signal nervous activity is studied in order to determine the type of activity that yields realistic simulations characteristic of typical saccadic eye movements.

Notes: Abstract Mathematical models of predator-prey systems in which the prey species has a three-stage life cycle are studied. Certain stages of the prey life history are allowed to use younger stages as food. It is shown that sufficiently restricted cannibalism can result in an increase in the numbers of adult prey on a sustained basis when cannibalism decreases the vulnerability of a stage subject to predation or increases overall productivity.

Notes: Abstract A general analysis is presented for the thermal behavior of a biological tissue. Energy transport by the circulatory system is assumed to be represented by a modified Fick's law. General boundary conditions are assumed for the two-dimensional model and solutions are obtained for rectangular, cylindrical, and spherical geometries. The effects of blood perfusion rate, metabolic rate, arterial temperature and heat exchange with the environment are considered. Results indicate a region of almost constant temperature in the deeper layers of the tissue and reaffirm the important role which blood flow plays in maintaining homeostasis.

Notes: Abstract The oscillatory aspect in a system having two steady states is studied theoretically using a model of excitable nerve membrane. The condition for the occurrence of oscillatory instability is discussed on the basis of the kinetic picture of nerve excitation in consideration of the non-Markoffian effect caused by ion transport in the system. Small oscillations around a steady state as well as a giant fluctuation between two states are obtained. Results are compared with experiments carried out with squid giant axons perfused intracellularly.

Notes: Abstract An expression for the variance in birth volumes during balanced growth of a cell population is derived. The requirement of this expression being positive and finite allows a discussion of some of the requirements imposed on the mechanisms of growth and division.

Notes: Abstract The phenomenon of axonal transport of material has been well documented (Ochs, 1971; Lasek, 1970; and Grafstein, 1967). This report seeks to establish the role of diffusion—if any—in such a transport process. We report that diffusion cannot account for the observed build-up of material as reported in the literature.

Notes: Abstract The question of how to fit a general cubic model of a multicomponent, interactive growth system to observed data is addressed. A multidimensional-polynomial type of regression analysis is used, with a least-squares criterion. By testing the scheme on a problem with known solution, the way in which the accuracy of the results varies with the number of datum points used is investigated in an heuristic manner.

Notes: Abstract The describing function method is used as a guide to the behaviour of the solutions of the equations of Danziger and Elmergreen, proposed as a model of periodic catatonia. The method suggests that whenever the equilibrium point is unstable it is surrounded by a stable closed periodic orbit. This is confirmed in specific cases by computation.

Notes: Abstract A theory of ambiguous pattern perception is formulated. This theory proposes a feature selector (field of attention) based on the time-sequential discrete property of the attention, a short-term memory for storage of the selected features, and a displayer (perception) to display the consecutively stored features. Since the selected features continuously enter, and since the features can only be stored in the short-term memory for a short period, the features which can be displayed in the displayer vary with time. When all the essential features belonging to one pattern happen to be in the displayer, the picture is perceived to be that pattern; when all the essential features belonging to another pattern happen to be in the displayer, then the picture is perceived to be the other pattern. Thus the picture appears to vary with time and alternate between two patterns. A numerical calculation is presented.

Notes: Abstract The thermodynamics of irreversible processes is derived from the principles of dynamical field theory independently of all elements of thermostatics, in particular the assumption of local equilibrium. Field thermodynamics proceeds from the premise that all driving forces experienced by the molecules in a continuum are conservative and arise from scalar potential functions. Dynamically the temperature potentialT is no different from the pressure potentialp. A field is converted to a force upon multiplication by a scale factor. A potential is converted to potential energy by the same scale factor. To scale the field −∇p to the force per mole of molecular speciesk, the partial molar, volume $$\bar V_k $$ is the scale factor. Similarly the partial molar entropy, $$\bar S_k $$ , scales the temperature field. The transition from the scale factors (which are physical parameters) to the systemic variables, for example $$\bar S_k \to s\left( {x,y,z;t} \right)$$ , is not trivial. From the dynamics and the structure of the derived potential energy function are inducted the conjugate variables such as (p, V I) and (T, s). The meta-mechanical properties of the thermal variables (T, s) are discovered via the local First Law of Thermodynamics, which relates internal energy, thermal flux, and work, and from the local Second Law, which prescribes, the possible partitions of internal energy between kinetic, potential, and thermal energies. From the form of the potential energy come Maxwell's relationships. From the energy partition comes the equation of continuity for entropy, with its important source term. In contrast to earlier theories of irreversible thermodynamics, the dissipation function does not include the stress tensor, a constitutive parameter.

Notes: Abstract The transfer of solute through a membrane separating two aqueous solutions is studied with the time-dependent diffusion equation for composite media. By introducing new independent and dependent variables it is shown that the differential equations and boundary conditions can be transformed into a dimensionless form which does not explicitly depend on the diffusivities of the media. Laplace transforms are used to derive explicit solutions for the solute concentration as a function of position and time. It is shown that at large time the concentration approaches the equilibrium distribution exponentially. Explicit results are given for the decay time as a function of the parameters of the system. In addition, an accurate and simplified expression is derived for the decay time for the case of small membrane permeability. The accuracy of the analytic solutions for the concentration profiles is tested by comparing them with numerical results obtained by solving the diffusion equations by the method of finite differences. Excellent agreement is found.

Notes: Abstract Physiological systems are often modelled by a set of compartments. Alternatively they can be described by the diffusion-convection-reaction equations governing distributed systems. The problem considered here is that of identifying a continuously changing input of some metabolite )tracee), endogenous to the system and hence inaccessible, when a nonlinear or time-varying component is also introduced into the loss parameter, as for example through feedback mechanisms. A tracer is used to determine the steady-state impulse response under time-invariant, linear conditions. A known input of tracer is also administered when the system is driven out of steady state. The integral equations developed utilize the predetermined impulse response, the measured concentrations of both tracer and tracee (output) in some region of the system to estimate the changing loss parameter and the unknown input in a continuous fashion.

Notes: Abstract The equilibrium distribution for a generalQth-order multivariate reaction system is studied. The state transition intensity matrix is developed and examples are given for small numbers of reaction components. A closed-form expression for the equilibrium distribution for systems which are symmetric with respect to the order of component reactions is presented. Numerical examples for three component systems are discussed.

Notes: Abstract The model recently proposed by Dreitlein and Smoes for oscillatory kinetic systems is studied. Diffusion of the oscillating species is taken into account, and bounds on the total number of individuals of each species are determined for both two- and three-dimensional finite regions with various boundary conditons applied. It is found that in general the effect of diffusion on the system behavior is to reduce the maximum possible radius of limit cycles. In particular, in some cases global limit cycle behavior is precluded.

Notes: Abstract The Kedem-Katchalsky equations for fluid flux across membranes may not be adequate for large solvent flows. In particular, for an example of two membranes in series, it is argued that they would predict physically unreasonable behavior. An alternate equation for solute flow is proposed for a simple sieving membrane. For the same example, this equation predicts more physically reasonable results.

Notes: Abstract The geometrical nature of the Elementary catastrophes (Thom, 1969) is reviewed. Histories of the movement of catastrophe manifolds and bifurcation sets are presented for some of the space-equivalent unfoldings described by Wassermann (1975). These unfoldings provide descriptions of the variation with time of the stability of stationary states of associated potential energy functions. Identification of these stationary energy states with stationary states of a system therefore provides a description of its behavior with time. Qualitative descriptions of this type are particularly useful when the complexity of a system prevents a detailed quantitative description. Histories of bifurcation set movements suggest different types of system behavior at different space-like coordinates. This type of theory may be a useful model for the processes leading to differentiation of cells and to emergence of adult forms of a biological organism.

Notes: Abstract Three conjectures are given which predict the existence of unique stable limit cycle oscillations in a class of piecewise linear (PL) differential equations. The equations are appropriate to model biological or other complex systems in which there are switchlike interactions between the elements of the network. Methods are presented which can be used to develop mathematical models which are conjectured to display stable limit cycle oscillations, from qualitative experimental information about relative phases of activity in the dynamical systems. Several illustrative numerical examples are given, and one experimental example from neurobiology is discussed.

Notes: Abstract The formulation and results of the Kalman State Regulator Problem are applied to a mathematical model of the arterial system of a dog to obtain an optimal control for blood pressure. The criterion for optimality is minimum energy per cycle.

Notes: Abstract A mathematical model is developed in state variable form suitable for the study of the control of blood pressure and flow in the mammalian cardiovascular system. The applicability of the model to steady state, both mean and pulsatile, and transient phenomena is demonstrated by the agreement of the results with experimental data. This model was developed to study the neural and renal-endocrine-electrolyte control of cardiovascular functions.

Notes: Abstract As a contribution to the discussion of oscillatory models for interacting species it is shown that two-species Volterra models can never have limit cycles, and a complete enumeration is given of conditions which the parameters of these models must satisfy in order that a part of the phase space be filled with a family of closed curves; sketches of phase portraits are also given. These results complement and correct older results by Bautin and by Coppel on quadratic differential systems. The paper opens with a brief discussion of some more practical aspects of the ecological application of oscillatory models.

Notes: Abstract The purpose of this paper is to present a method for the determination of a solution for a coupled system of first order initial boundary-value problems arising from some biological systems. The physical problem is to determine the suspended and the superficial molecular concentrations of a traced substance passing through an organ containing a tangle of vessels, such as the kidney-ureter system. The approach to the problem is by successive approximation which leads to a recursion formula for the determination of the solution as well as error estimates for the approximations. The recursion formula involves only direct integration which indicates a promising possibility in obtaining numerical results by using a computer. In addition to the determination of a solution, some qualitative analysis of the solution is given. This includes the existence of a unique solution, the continuous dependency of the solution on the data, and the stability problem of a steady-state solution.

Notes: Abstract In Part I, the general solution of the heat transport equation was given for a medium of constant thermal properties, containing a general convection fieldv(r,t). Starting from this general solution, an average or “macroscopic” temperature distribution is calculated in this paper, since in most physiological problems only an average or “macroscopic” temperature is accessible to the measurement. It is shown that the averaged temperature distribution is influenced alone by the order of the local symmetries of the convection field. These local symmetries can be deduced from anatomical data on the vascular architecture. In order to perform the averaging process, a detailed discussion, and an estimate of the correlation between the microscopic temperature variation and the convection field is carried out. Hereby, great emphasis is put on the experimental consequences of these results. As an illustrative example, the calculation of the macroscopic temperature distribution in a medium traversed by parallel capillaries is outlined.

Notes: Abstract The study of current flow fields in biologicaltissue requires finding the potential field from dipole sources such that the normal gradient vanishes at the exterior surface. A convenient way to determine the dipole field is by taking the gradient of the potential field set up by a point source. However, the point source problem is ill-posed when the normal gradient is required to vanish at the outer surface. In the paper the nature of this problem is discussed and several methods for overcoming the difficulty are presented.

Notes: Abstract An analysis of the hydrodynamics of aqueous flow in the posterior chamber of the anterior segment of the eye is presented. The viscid, incompressible description of fluid dynamics in a spherical geometry is utilized to reduce the problem to a biharmonic-type equation using a “Stokes Stream Function”. Analogous to Poiseuille flow in a cylindrical pipe, velocity profiles are deduced and an Ohms Law relationship between pressure and flow is derived in terms of the geometry of the assumed model. This result is then incorporated into a synthesized electric circuit analog of flow between ciliary artery and episcleral vein. Applications to open angle and pupilary block glaucoma are discussed.

Notes: Abstract With the assumption of dipole interaction with the membrane matrix, the dipole barrier under an applied field shows a minimum in its time transient. Kinetic equations governing the migration of ions are presented. Na+ activation, Na+ inactivation and K+ delay are all part of the same mechanism in this model with no other separate assumptions needed. Voltage Clamp equation and action potential equation are presented.

Notes: Abstract A metabolic model for copper is derived and presented in the context of the International Commission on Radiological Protection (ICRP) dosimetry models for the lungs and gastrointestinal tract.

Notes: Abstract The analytical conditions by which a Volterra's general system describingn interacting species can be put in the “conservative” form have been examined. The cases forn=2, 3, 4 have been analyzed in detail and a general condition for any value ofn is deduced. The analytical and biological constraints following by this approach are compared to the conclusions drawn by Leigh on the ground of purely biological considerations.

Notes: Abstract General formulation of stochastic single- and multi-compartment reversible systems with time-dependent transitions is made. The correspondence between the stochastic mean and the deterministic value is established in case of time-dependence and it is shown how the consequence of this can be utilized to compute the distribution and the moments of each individual compartment of the system. A two-compartment reversible system previously proposed by Cardenas and Matis (1975a) is analyzed on the basis of the theory.

Notes: Abstract Optimal control theory is applied to the problem of controlling pests by biological and chemical means simultaneously. The net birth rate of the pests is controlled chemically while at the same time predators are allowed to operate. Several numerical examples are included.

Notes: Abstract Transient solutions are developed for the buildup of a concentration gradient in the single loop solute cycling model of the renal medulla. The “pump” from ascending limb to descending limb is considered in both unsaturated and completely saturated modes of operation. Both analytic solutions and semianalytic solutions obtained from inverting Laplace transforms are considered. The classic representation of concentration buildup by the multiplication process is compared with calculated profiles. The “single effect” is found to vary both in time and space.

Notes: Abstract In this study the possibility of applying the asymptotic method of Krylov-Bogoliubov-Mitropolskii to problems of population dynamics is shown. Especially a general Volterra-Gause-Witt type model for prey-predator interaction is investigated. A discussion on the results obtained is given for the general model and for a particular case as well.

Notes: Abstract The analytical exploration of hierarchical structures is a central problem in modern biology. In many ways, the most basic tenet of any theory of any biological system is that its behavior lies within the laws of thermodynamics. Thus an investigation into the hierarchical structure of field thermodynamics is in effect an investigation into the underlying structures common to all biological systems. In field thermodynamics, the dynamics is prescribed at the hierarchical level of description concerned with the motions of classes of the various molecular species flowing in the system. Of crucial importance are scale factors which convert potentials into potential energies per mole of the various molecular species. On the other hand, the overall behavior of the system is presented at a higher hierarchical level of description in terms of systemic properties such as charge density or entropy density. Three questions are investigated: (1) how is the transition from the specific to the systemic level of description effected, (2) given the scale factors at the specific level how does one find those at the systemic level, and (3) what relationship can one, find between the scale factors at the two levels of description? Answering these questions necessitates an examination of the nature of the forces at the two levels of the hierarchy of descriptions.

Notes: Abstract The identification of a linear compartment model, which may describe a chemical or biological process, is a difficult task, since the available data is generally limited. In this paper we propose a method for determining the state transition matrix by minimizing a given quadratic criterion. To solve the resulting matrix equation, an assumption has to be made which constitutes a necessary condition for the identifiability of the model. Moreover when this assumption is satisfied, it is shown that the knowledge of one line or one column of the transition matrix is sufficient to define it completely.

Notes: Abstract The two-element muscle model considered consists of a contractile element defined by a hyperbolic force-velocity relation connected in series with an “exponential spring”. Differential equations for the isometrically developed force during a tetanic contraction and the corresponding contractile element shortening velocity are derived and their stability is investigated. Analytical solutions to both equations are obtained. Two numerical examples are given, the second chosen to illustrate pressure-induced hypertrophy of a cardiac muscle.

Notes: Abstract We study a two-type, age-dependent branching process in which the branching probabilities of one of the types may vary with time. Specifically this modification of the Bellman-Harris process starts with a Type I particle which may either die or change to a Type II particle depending upon a time varying probability. A Type II particle may either die or reproduce with fixed probabilities but may not return to a particle of Type I. In this way the process models the lag phenomenon observed in microbe growth subsequent to transfer to a new culture medium while the organism is adapting to its new environment. We show that if the mean reproduction rate of Type II particles exceeds 1, then the population size grows exponentially. Further the extinction probability for this process is related to that of the Bellman-Harris process. Finally the governing equations are solved for several choices of the growth parameters and the solutions are graphically displayed showing that a wide variety of behavior can be modeled by this process.

Notes: Abstract This paper studies the effect of harvesting a fraction of a population where the population growth is modelled by a linear age-dependent model, the Von Foerster equation. Two harvesting strategies are considered: the first is where a fraction of the population greater than agec is removed, and the second is where a fraction of the population of age greater thanc but less thanc+n is removed. In the case where the death rate and fertility rate are time independent, the effect of harvesting on the stable age distribution is examined.

Notes: Abstract The general means are set forth by which, in a highly speciated Volterra eco-system, the bifurcation of a species into two in a competitive-exclusion configuration, provides a description of odd as well as even systems. The the Gibbs ensemble theory earlier developed is extended to grand ensembles in which the number of ecotypical species in genera-like clusters is allowed to be variable. The probability law for the sizes of clusters is deduced, and shown very closely to imply independent Poisson statistics for the cluster sizes, in answer to a fundamental issue of bioevolution. The role of the eco-analogs of Gibbs’ chemical potentials is brought forth, with a suggestion of how to describe cluster-to-cluster eco-genetic flows, and therewith a definite relationship between the degrees of speciation of genetically linked clusters on the one hand, and amplitudes of population fluctuations of member species within the clusters on the other hand.

Notes: Abstract Solutions of the Poisson-Boltzmann equation yield potential profiles and equilibrium distributions of ions on either side of a spherical shell membrane across which there exists a separation of ionic charges. For the special case in which the membrane is permeable to only one ion the total charge separation is analyzed in terms of the potential difference given by the Nernst equation. Potential profiles and ionic charge distributions are also given for situations involving a uniform distribution of fixed charges within the membrane.

Notes: Abstract The recently proved classification theorem of René Thom has led to the development of the new science of Catastrophe Theory. This theory has been widely held to be particularly relevant in biological applications. The present paper presents a simplified proof of Thom’s Theorem and assesses its importance in biology.

Notes: Abstract A mathematical model describes the mechanical behaviour of ventricular aneurysms assuming a spherical geometry for the left ventricle. Employing pressure-volume data obtained from normal dog hearts 1–2 hours after infarction, conditions are obtained on infarct thickness and angle of damage for ventricular rupture to occur. The results indicate that rupture is more likely to occur in the early period following infarction and that the dominant factor is the per cent thickness of the infarct.

Notes: Abstract A deterministic investigation of a linear differential equation system which describes predator vs prey behavior as a function of equilibrium densities and reproductive rates is given. A more realistic structure of this model in a stochastic framework is presented. The reproductive rates and initial population sizes are considered to be random variables and their probabilistic behavior characterized by various joint probability distributions. The deterministic behaviors of the prey and predator species as functions of time are compared with the mean behaviors in the stochastic model.

Notes: Abstract The effects of ventricular geometry, muscle mass, muscle elasticity and external pressures on the pressure-volume and muscle stiffness-stress relations have been quantitated on the basis of a theoretical model. Data taken from patients before and after interventions with nitroprusside and angiotensin were applied to the model in order to explain the possible causes for the marked shifts in the pressure-volume relations. The results indicate that (a) ventricular geometry does not markedly alter the pressure-volume and stiffness-stress relations unless there is a drastic change from a spherical shape to an ellipsoidal shape orvice versa, (b) increases in muscle mass and muscle elasticity of the order of 30% result in significant alterations in the P-V relations but are not the cause for the parallel shifts unless accompanied by substantial external pressures, (c) the parallel shifts in the pressure-volume relations can be accounted for entirely by the presence of external pressures without changes in muscle mass or muscle elasticity. Thus manipulation of right ventricular pressures or pericardial pressures by drug interventions may be useful in the treatment of left heart disease and the presence of such pressures must be considered in the analysis of ventricular function curves.

Notes: Abstract This paper summarizes in a nonmathematical way the major properties of coupled oscillators which relate to circadian rhythms. For certain values of the coupling strength it is far easier to maintain synchrony than to achieve it among the various interacting units. This property not only simulates the free run period lability but also the effects of critical pulses.

Notes: Abstract A deterministic model for a multi-agent disease epidemic with asymptomatic attacks is proposed and investigated. The limitations inherent in the assumptions of the model are discussed in connection with specific agents of disease. The mathematical treatment of the model is separated into analyses of the equilibrium situation and the transient behavior of the disease outbreak. Explicit formulas are derived for the number of susceptibles in the population as well as for the numbers of each type of infective—those with and without symptoms. These theoretical results are followed by a discussion of the practical considerations which must be taken into account to obtain useful information from the model.