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  • Articles  (19,596)
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  • International Union of Crystallography (IUCr)
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  • Articles  (19,596)
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  • 101
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    Electronic Resource
    Theory of initial current density in membrane voltage-clamp experiments (1972)
    Springer
    Bulletin of mathematical biology 34 (1972), S. 45-52 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract Herein we show that the voltage-clamp current density at zero time calculated from electrodiffusion equations is linear in the clamping voltage for a simple membrane (no charge structure) and for a membrae with fixed charges. Such membranes are nonexcitable. Excitable membranes can be represented by a homogeneous membrane with dipole layers at the surface. In this case the initial current density will be linear in the clamping voltage if a critical field for a dipole layer reorientation is not passed through in changing from holding to clamping potential. Otherwise, deviation from nonlinearity may occur. This is in agreement with experimental data for the squid giant axon.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02477023
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  • 102
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    A note on sporadic nonperiodic oscillations in the activity of the endocrine system and on the endocrine—Nervous system feedback (1972)
    Springer
    Bulletin of mathematical biology 34 (1972), S. 65-69 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract By generalizing a previous paper on periodicities in the endocrine system (Bull. Math. Biophysics,30, 735–749, 1968), it is shown that nonperiodic, sporadic oscillations in the system are also possible. A procedure of describing the feedback mechanism between the endocrine system and the central nervous system is suggested. It is shown that the combined system: endocrine—CNS, also may show sporadic fluctuations.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02477025
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  • 103
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    Wall shear in pulsatile flow (1972)
    Springer
    Bulletin of mathematical biology 34 (1972), S. 79-86 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract This paper deals with a mathematical attempt to determine the wall shear during normal flow of blood in the ascending and the descending thoracic aorta. A simple model is used, but the results obtained are in agreement with published experimental results for the descending thoracic aorta. It is suggested that the degree of fluctuation in the pressure gradient at a given station is the major factor in determining the level of wall shear at that point.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02477027
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  • 104
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    A neural circuit which exhibits some features of epileptic attacks (1972)
    Springer
    Bulletin of mathematical biology 34 (1972), S. 71-78 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract A neural model based on a generalization of a model proposed in 1938 in the first edition of the author'sMathematical Biophysics (Chicago: Univ. of Chicago Press) is described. It possesses the property that due to some endogenous or exogenous stimulus, which may be of random nature, a pathway may suddenly begin to fire spontaneously. This spontaneous firing may either gradually spread over other pathways and eventually cease, or it may remain localized within one or a few pathways and then cease. Which of the two types of events occurs depends on the values of a number of parameters. The case of spreading reminds one of Jacksonian epilepsy.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02477026
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  • 105
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    Diffusion exchange between a membrane-bounded sphere and its surrounding (1972)
    Springer
    Bulletin of mathematical biology 34 (1972), S. 93-102 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract The diffusion equation is solved for a membrane-bounded sphere situated in an infinite medium with different diffusion properties. The formal solution is obtained through Laplace transformation in the time variable. It is not possible to find a closed form solution in terms of analytical functions, and therefore a numerical inversion technique is applied to obtain the final solution. The application on a biological problem is discussed.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02477029
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  • 106
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    Determination of reaction rate constants in the mammillary system (1972)
    Springer
    Bulletin of mathematical biology 34 (1972), S. 87-92 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract It is shown that the individual rate constants can be determined for the composite chemical system: $$A + B_i \rightleftarrows C_i ; i = 1...N$$ with only measurements of the unbound species,A(t), required. The dissociation rate constants can be determined by direct analysis of a single steady state tracer study. The association constants then follow from the analysis of stable equilibrium determinations reported earlier (Hart, 1965). An approximate solution when tracer methods are in-applicable is also given.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02477028
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  • 107
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    Some biochemical threshold mechanisms (1972)
    Springer
    Bulletin of mathematical biology 34 (1972), S. 103-112 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract Certain arrangements of enzymatic (bimolecular) subsystems lead to characteristic threshold-type response. Two simple cases of such systems are studied here in terms of steady state behavior and explicit relationships between system and curve parameters. It is found that the curvature of the threshold curve is directly related to the equivalent Michaelis constant and, in the case of saturated threshold curve, the slope of the curve at the idealized threshold is limited by the ratio of saturation to threshold. This slope may be appreciably increased up to a stepwise response at the threshold if a multisubstrate complex of the enzyme is the only species which affects the enzyme mediated transport.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02477030
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  • 108
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    Theory of neural mediation of intraocular dynamics (1972)
    Springer
    Bulletin of mathematical biology 34 (1972), S. 113-148 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract Many experimental studies have indicated that the intraocular pressure is subject to mediation by adrenergic mechanisms affecting both the rate of formation of the aqueous humor and the resistance of the pathway through which the aqueous humor flows out of the eye. Thus, for example, the role of adrenergic drugs in glaucoma therapy is well known. How the mediation is accomplished has not been clarified in detail. Several possible mechanisms have been suggested, and all may indeed be involved. The present study is concerned with the basis and mathematical formulation of one of them and the consequences with respect to aqueous dynamics. The analysis leads to expressions for the aqueous outflow resistance and the formation rate, as well as other quantities of interest. The theoretical behavior is shown to compare favorably with the results of infusion studies and various other experiments, and to provide a unified picture of much of the pressure-flow behavior of both the living and the dead eye.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02477031
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  • 109
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    Eighth International Congress of Electroencephalography and Clinical Neurophysiology (1972)
    Springer
    Bulletin of mathematical biology 34 (1972), S. 149-150 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02477032
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  • 110
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    A fit to nerve membrane rectification curves with a double-dipole-layer membrane model (1972)
    Springer
    Bulletin of mathematical biology 34 (1972), S. 151-172 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract Following Wei's suggestion that nerve stimulation and conduction properties are due to dipole layers at the two membrane surfaces (Wei, 1969), we have done steady-state electro-diffusion calculations in the constant field approximation for a simple double-dipole-layer model. We are thereby able to quantitatively fit the recent potassium iso-osmotic rectification curves of Gilbert and Ehrenstein for the squid giant axon membrane. For the squid axon membrane in a natural ion environment, only the outside dipole layer is present in the fit to the data.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02476514
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  • 111
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    Geometrical parameters of the individual myocardial fibers (1972)
    Springer
    Bulletin of mathematical biology 34 (1972), S. 205-211 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract In this paper the left ventricle of the heart was considered as a shell of varying thickness. The first and second fundamental forms of the middle surface, of the shell, as well as Euler's theorem were used for deriving expressions giving the length and curvature of the individual myocardial fibers. Recent anatomical studies have shown that the myocardial fibers in the middle of the left ventricular wall follow a course nearly parallel to the horizontal plane (Streeteret al., 1969). In previous papers (Voukydis 1969, 1970) a mathematical description of the curvature and length of the individual myocardial fibers was presented. Unfortunately, both the curvature and the length formulas contained the cotangent of the fiber helix angle, which approaches infinity as the fiber assumes a course parallel to the horizontal plane. Consequently, these two formulas cannot be used for fibers nearly parallel to the horizontal plane. The present paper will give an alternative way for calculating the length and curvature of the individual myocardial fibers, based on the fundamental forms of surface and on Euler's theorem.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02476516
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  • 112
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    Some remarks on the central nervous system (1972)
    Springer
    Bulletin of mathematical biology 34 (1972), S. 231-242 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract It is pointed out that the successes obtained in the mathematical biology of the central nervous system are based mostly on a number of more or less complicated neuronic circuit models, each inventedad hoc for the purpose of explaining a given phenomenon. The individual models remain disconnected from each other, however, and the unity of the CNS is not apparent. (Rashevsky,Mathematical Biophysics, 3rd Edition, Vol. II, 1960. New York, Dover Publications, Inc.) Some “field theories” of the CNS, as for example that of Griffith (Bull. Math. Biophysics,25, 111–120, 1963;27, 187–195, 1965), give more expression to this unity but lose in the explanation of specific phenomena. The present paper starts with the picture thatevery neuron in the brain isdirectly or indirectly affected to some extent byevery other neuron. This leads to a system of equations with a very large number of variables. Such a system can be replaced in the limiting case by an integral equation of the first kind. At least two specific results can be obtained with this approach and suggestions for further improvement are made.
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    URL: http://dx.doi.org/10.1007/BF02476519
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  • 113
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    Fiber stress profiles in the left ventricle of the heart during diastole: Effects of distension and hypertrophy (1972)
    Springer
    Bulletin of mathematical biology 34 (1972), S. 379-392 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract Pressure-volume and volume-dimensions relationships, obtained from excised dog left ventricles were used for calculating the stresses acting along the longitudinal axis of the individual myocardial fibers. The calculations were based on a set of empirical and theoretical equations. The pressure-volume relationship as well as the volume-dimensions relationships for the excised left ventricle were expressed in the form of empirical equations; the fiber orientation was written as a function of the fiber location within the left ventricular wall; finally, the fiber stress was determined by means of theoretically derived formulas. Simultaneous solutions for the fibers of a meridian cut through the left ventricular myocardial shell were obtained by means of a digital computer and presented in the form of diagrams. The results showed that at low degrees of distension of the left ventricle there are two zones of higher stresses at the equatorial area, one near the epicardium and one near the endocardium. As the distension proceeds under the effect of progressively increasing intraventricular pressure, these two zones become less well defined, whereas a new zone of higher stresses appears near the apex. At high degrees of distension, the ventricle assumes a more spherical shape and the equatorial zones of higher stresses are replaced by zones of lower stresses. Increase in the myocardial mass results in appearance of the equatorial lower stress zones at lower degrees of distension.
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    URL: http://dx.doi.org/10.1007/BF02476449
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  • 114
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    A note on open linear systems (1974)
    Springer
    Bulletin of mathematical biology 36 (1974), S. 97-99 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract A theorem is given which states a necessary and sufficient condition for the specific activity to be uniform throughout an open compartmental system in the steady state.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02461195
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  • 115
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    Introduction to Mathematics for Life Scientists (1974)
    Springer
    Bulletin of mathematical biology 36 (1974), S. 101-102 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
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    URL: http://dx.doi.org/10.1007/BF02461196
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  • 116
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    A dynamic model of an environmental system withn interacting components andp degrees of freedom (1974)
    Springer
    Bulletin of mathematical biology 36 (1974), S. 91-96 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract The nonlinear differential equations of growth proposed by Volterra (1931) are used as the basis for a dynamic model of ann-element system withp specified, terminal conditions andp missing initial conditions. The resulting two-point boundary-value problem can be solved, ifp is not large, by the shooting method used previously by Huddleston (1967). A numerical example withn=4 andp=2 is solved on a digital computer, and some results are presented.
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    URL: http://dx.doi.org/10.1007/BF02461194
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  • 117
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    Use of a heat transfer analogy for a mathematical model of respiratory tract deposition (1974)
    Springer
    Bulletin of mathematical biology 36 (1974), S. 105-116 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract Mathematical models predicting the aerosol deposition in the respiratory tract are reviewed. Data in the literature indicated not only that the air flow in the trachea and major bronchi may not be laminar, but also that the entrance effect of the tube or airway has not been considered. A new approach to a mathematical model of respiratory tract deposition, based on the analogy of the heat and mass transfer, is discussed.
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    URL: http://dx.doi.org/10.1007/BF02458596
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  • 118
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    Isotherms forr species adsorbed at linear polymers with nearest neighbor cooperative interaction (1974)
    Springer
    Bulletin of mathematical biology 36 (1974), S. 117-126 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract The equation of the cooperative specific isotherm is derived for the general case ofr species interacting cooperatively at nearest neighboring sites. Explicit expressions for the special casesr=2–5 are given.
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    URL: http://dx.doi.org/10.1007/BF02458597
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  • 119
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    Pseudo-sound generation at atherosclerotic constrictions in arteries (1974)
    Springer
    Bulletin of mathematical biology 36 (1974), S. 143-155 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract Sounds and murmurs have long been employed to qualitatively diagnose cardiovascular disease. However, quantitative diagnosis has been hindered by the lack of understanding of the sound generation and transmission mechanisms. Clinical phonoangiographic studies have shown that simple assumptions about low frequency sound transmission through tissue surrounding an artery are inadequate for obtaining meaningful quantitative diagnosis. Therefore, a theory is developed which relates internal turbulent flow in constricted peripheral arteries to the sound observed at the surface of the skin by means of assumptions of similarity and local axial homogeneity of the internal turbulence. It is found that the spectrum of pressure at the wall of the artery is related to the spectrum of the pressure at the surface of the skin by a filtering factor approximately proportional to ω-2. This arises not because of frequency dependent volumetric absorption in the surrounding medium, as with ultrasound, but because of the manner in which stochastic signals add when observed.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02458599
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  • 120
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    Measuring a mitotic oscillator: The ARC discontinuity (1974)
    Springer
    Bulletin of mathematical biology 36 (1974), S. 171-182 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract Mitosis occurs synchronously in up to 108 nuclei in the syncytial plasmodium ofPhysarum polycephalum. Any two phases of the mitotic cycle may be mixed by fusing plasmodial pairs. A topological property of the synchronized phase of the fused pair as a function of parental phases, the arc discontinuity, characterizes the underlying oscillator, and indicates mitosis is controlled by a moderate relaxation oscillator which rotates more rapidly near its singularity than its limit cycle. A model oscillator is briefly described.
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    URL: http://dx.doi.org/10.1007/BF02458601
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  • 121
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    Genetic nets and dissipative structures: An algebraic approach (1974)
    Springer
    Bulletin of mathematical biology 36 (1974), S. 183-196 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract Genetic nets represent an attempt to model genome structure. Depending on the interaction dynamics assumed, they can constitute highly non-linear chemical systems having multiple steady states; hence their relevance to the theory of dissipative structures. Their typical size and possible complexity makes it difficult to study them by means of customary analytical techniques based on differential equations. We have therefore considered an algebraic approach derived from regarding the nets as finite-state automata. This view has revealed a surprisingly rich algebraic structure which can be used to investigate problems concerned with the relation between biological structure and function. This algebraic structure is described with particular reference to the genetic nets of Tsanev and Sendov.
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    URL: http://dx.doi.org/10.1007/BF02458602
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  • 122
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    Finite fluctuations and multiple steady states far from equilibrium (1974)
    Springer
    Bulletin of mathematical biology 36 (1974), S. 205-213 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract The role of finite fluctuations in transitions between nonequilibrium steady states in nonlinear systems is investigated. Attention is focused on a model biochemical system for which the usual deterministic chemical kinetics predicts a far-from-equilibrium region of multiple steady states. A stochastic approach to chemical kinetics is adopted to study explicitly the effect of fluctuations around the coexisting stable states on a predicted hysteresis in the transition between those states. A numerical solution of the stochastic master equation for the system yields results which differ qualitatively from predictions of the purely macroscopic theory. Possible implications of these results are considered, and several important aspects of the computational scheme are discussed in some detail.
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    URL: http://dx.doi.org/10.1007/BF02458604
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  • 123
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    The global asymptotic stability of prey-predator systems with second-order dissipation (1974)
    Springer
    Bulletin of mathematical biology 36 (1974), S. 215-217 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract Models of biological development, evolution and control should take into account that very small numbers of cells or chemicals or individuals eventually grow into stable, large populations. The simplified two-component model used in these studies includes the following: (1) first-order decay; (2) first-order autocatalysis; (3) negative feedback; (4) positive feedback; (5) second-order decay; (6) second-order autocatalysis. A positive definite Lyapunov function is constructed and shown to have a negative definite total derivative. The stationary statex〉0,y〉0, therefore possesses global asymptotic stability. This means that sustained oscillations cannot occur. Another stationary state,x=y=0, is shown to be unstable. This means that infinitesimally small perturbations ofx=y=0 will result in evolution of the variables to the stable stationary state. This result contrasts with that obtained with the Lotka-Volterra model in that small perturbations ofx=y=0 for that model result in sustained, oscillating excursions; the smaller the initial perturbations, the larger these excursions will be. A simulation illustrates that stable populations of 1020 x's andy's can arise from a singlex andy.x grows more or less continuously, buty remains extremely small for 80 per cent of the time interval required for the variables to approach their stable populations.
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  • 124
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    Semiconductor theory of ion transport in thin lipid membranes: II. Surface recombination (1974)
    Springer
    Bulletin of mathematical biology 36 (1974), S. 247-264 
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    ISSN: 1522-9602
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    Topics: Biology , Mathematics
    Notes: Abstract Following the theory of surface recombination in semiconductors, we have derived an expression for the rate of ion recombination at the membrane surface. The surface recombination rate is used in the boundary conditions of current flows at the interfaces. Expressions for the ion fluxes are then derived as functions of environmental variables and membrane parameters. Our analysis strongly suggests that the ion flow through a thin lipid membrane consists of two major components: the surface barrier jumping current and the surface recombination current that are controlled decisively by surface barrier height, surface trap density and surface recombination rates. These general formulations are useful not only for the calculation but also for the understanding of ion transport in thin lipid membranes under a variety of experimental conditions. The implications of this theory to biological membranes and its possible extensions are discussed.
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    URL: http://dx.doi.org/10.1007/BF02461327
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  • 125
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    Transport of O2 along a model pathway through the respiratory region of the lung (1974)
    Springer
    Bulletin of mathematical biology 36 (1974), S. 275-303 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract A pathway through the system of branching in the respiratory region of the lung is modelled by a circular cylinder, closed at one end, with partitions which define the component respiratory units. In this model the transport of O2 during inspiration, generated by diffusion is compared with that produced by diffusion together with convection and the importance of convection in the respiratory region in promoting oxygen uptake at the alveolar wall is discussed. For this discussion it is only necessary to consider inspiration. The equations are solved numerically for flow rates of 10, 85 and 200 liters/min. O2 uptake at the wall and curves of constant O2 concentration are shown to illustrate the influence of convection. It is found that after a 2 sec inspiration from an O2 tension of 98 mm Hg and a lung volume of 2300 ml, convection is about 12 per cent as important as diffusion at a flow rate of 85 liters/min, whereas at 10 liters/min convection is only about 0.4 per cent as important as diffusion.
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    URL: http://dx.doi.org/10.1007/BF02461329
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  • 126
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    Theoretical relationship between the optimal models of the vascular tree (1974)
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    Bulletin of mathematical biology 36 (1974), S. 311-323 
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    ISSN: 1522-9602
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    Topics: Biology , Mathematics
    Notes: Abstract Two models of optimal branching structure of the vascular tree are compared. Murray’s minimum work model derived from minimum energy loss due to flow and volume in the duct system is proved to be included as a mathematical group in the authors’ model defined by the minimum volume under determinant pressure, flow and position at the terminals. The problem about heterotypical trees which are identical at the terminal conditions but different in the topological order of branch combinations are discussed, applying the results of analyses on the equivalent duct of uniform terminal pressure trees. It is proved that the minimum work tree has the least energy loss compared with its heterotypical minimum volume trees and is a better model of branching structure of the vascular tree.
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  • 127
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    A morphogenetic system matched to a group (1973)
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    Bulletin of mathematical biology 35 (1973), S. 615-625 
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    ISSN: 1522-9602
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    Topics: Biology , Mathematics
    Notes: Abstract A period of morphogenesis is operationally defined as a system. This period precedes cell differentiation and is axially expressed. Six interacting elements of structure are defined utilizing Spemann and Mangold's heteroplastic transplantation experiments. The system generates thirty-six interactions, given a rule of composition and specific premises. The element set and operation is matched to the symmetric group S3.
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  • 128
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    Asymptotic behavior in a deterministic epidemic model (1973)
    Springer
    Bulletin of mathematical biology 35 (1973), S. 607-614 
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    ISSN: 1522-9602
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    Notes: Abstract The effects of a periodic contact rate and of carriers are considered for a generalization of Bailey's simple epidemic model. In this model it is assumed that individuals become susceptible again as soon as they recover from the infection so that a fixed population can be divided into a class of infectives and a class of susceptibles which vary with time. If the contact rate is periodic, then the number of infectives as time approaches infinity either tends to zero or is asymptotically periodic depending on whether the total population size is less than or greater than a threshold value. The behavior for large time of the number of infectives is determined for three modifications of the model which involve carriers.
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    Electron-phonon (trapped photon) coupling and infrared coaxial transmission line theory of energy transport in mitochondria and nerve (1973)
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    Bulletin of mathematical biology 35 (1973), S. 627-644 
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    Notes: Abstract This paper develops quantitatively for the kinetics of mitochondrial oxidation the implications of the hypothesis that phosphorylation is accomplished by phonons in the mitochondrial solid. The concept is used that the phonon acts like a trapped photon to produce a reverse photocurrent, which inhibits oxidation, in the absence of phonon dissipation due to phosphorylation or uncoupling. The same conceft in reverse is shown to explain qualitatively the generation or membrane potentials in nerve axon. Infrared optical phonons generated in the above processes are expected to have limited mibility, but it is shown that the structure and geometry of the lipid bilayer membrane are well adapted to the rapid and efficient dissemination of this energy throughout the mitochondrion or nerve axoni n the form of infrared electromagnetic waves by a coaxial transmission line mechanism. The mitochondrion may act like an infrared coaxial resonant cavity.
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    Mathematical analysis of the asymptotic behavior of the Leslie population matrix model (1973)
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    Bulletin of mathematical biology 35 (1973), S. 645-661 
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    Notes: Abstract LetL be a Leslie population matrix. Leslie (1945) and others have shown that the matrixL has a leading positive eigenvalueλ 0 and that in general: (1) $$\mathop {\lim }\limits_{t \to \infty } \frac{{L^t X}}{{\lambda _0^t }} = \gamma X_{\lambda _0 } $$ whereX λ 0 is an eigenvector corresponding toλ 0,X is any initial population vector, and γ is a scalar quantity detormined byX. In this article we generalize (1) exhaustively by removing the mild restrictions on the fertility rates which most writers impose. The result is an oscillatory limit of a kind first noted by Bernardelli (1941) and Lewis (1942) and described by Bernardelli as “population waves”. We calculate in terms ofλ 0 and the entries of the matrixL the values of this oscillatory limit as well as its time-independent average over one period. This calculation includes as its leading special case the result of (1), confirming incidentally that γ is nonzero. To stabilize a population, the matrixL must be adjusted so thatλ 0=1. The limits calculated for the oscillatory and non-oscillatory cases then have maximum significance since they represent the limiting population vectors. We discuss a simple scheme for accomplishing stanbilization which yields as a byproduct an easily accessible scalar measure ofL's tendency to promote population growth. The reciprocal of this measure is the familiar net reproduction rate.
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    The Danziger-Elmergreen theory of periodic catatonic schizophrenia (1973)
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    Bulletin of mathematical biology 35 (1973), S. 689-707 
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    Notes: Abstract Properties of the solutions of the mathematical model introduced by Danziger and Elmergreen for the study of periodic catatonic schizophrenia are studied, and it is shown that the properties established suggest that the model can be used to study other forms of catatonic schizophrenia besides periodic catatonic schizophrenia.
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    On the stochastic theory of compartments: Solution forn-compartment systems with irreversible, time-dependent transition probabilities (1974)
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    Bulletin of mathematical biology 36 (1974), S. 489-504 
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    Notes: Abstract The bivariate distribution of a two-compartment stochastic system with irreversible, time-dependent transition probabilities is obtained for any point in time. The mean and variance of the number of particles in any compartment and the covariance between the number of particles in each of the two compartments are exhibited and compared to existing results. The two-compartment system is then generalized to ann-compartment catenary and to ann-compartment mammillary system. The multivariate distributions of these two systems are obtained under two sets of initial conditions: (1) the initial distribution is known; and (2) the number of particles in each compartment of the system at timet=0 is determined. The moments of these distributions are also produced and compared with existing results.
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    Dustbathing as a regulatory mechanism (1974)
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    Bulletin of mathematical biology 36 (1974), S. 545-553 
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    Notes: Abstract Three mathematical models were developed to describe how some species of birds, such as bobwhite quail (Colinus virginianus) regulate the amount of oil on their plumage. The models assume that dustbathing plays a significant role in this regulatory process. They differed primarily in their assumptions about the relationship between oiling and dustbathing behavior. Several experiments were run to check on the implications of the models.
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    Observations on the hydrodynamics and swimming motions of mammalian spermatozoa (1974)
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    Bulletin of mathematical biology 36 (1974), S. 555-565 
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    Notes: Abstract An asymptotic solution of the Stokes Flow equations for a self-propelling filament is presented. An explicit expression for the propulsive velocity is obtained for the case of an infinite filament undergoing small amplitude sinusoidal motions. The asymptotic solution is then used to obtain drag coefficients to be used in a simpler approximate analysis which can be applied to experimentally observed motions.
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    Stochastic theory of compartments: One and two compartment systems (1974)
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    Bulletin of mathematical biology 36 (1974), S. 577-587 
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    Notes: Abstract Previous work on compartmental systems is generalized (i) to allow the particles present at time zero to have a different lifetime distribution than those which arrive after time zero, and (ii) to allow a particle which enters the system at timet to have a lifetime distribution which is a function oft but is otherwise quite general. The one and two compartment models are analyzed under the above conditions and compared to previous results of Thakuret al. (1974), Purdue (1974) and Cardenas and Matis (1974). Finally, some results for the two compartment, reversible system are given. The analysis used is a blend of direct random variable and queueing theoretic techniques.
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    Bounds on the total particle number for species governed by reaction-diffusion equations in the infinite spatial domain (1974)
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    Bulletin of mathematical biology 36 (1974), S. 589-593 
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    Notes: Abstract Differential inequality methods are developed for establishing upper and lower bounds on the total particle numberN(t)=∫θ(x,t) d3 x associated with solutions to nonlinear reaction-diffusion equations of the form ∂θ/∂t=D∇2θ+fθ-gθ n+1 , whereD(〉0),n(〉0),f andg are constant parameters. If finite in a neighborhood oft=0,N(t) is bounded below for allt≥0 by a certain derived function oft for equations withg≥0. An upper bound onN(t) is obtained for equations withn=1,f〈0 andg〈0. These results provide general preservation and extinction criteria for the total particle number.
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    A neuron field theory: Mathematical approaches to the problem of large numbers of interacting nerve cells (1973)
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    Bulletin of mathematical biology 35 (1973), S. 345-357 
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    Notes: Abstract A field theoretical approach to the problem of continuously distributed and simultaneously active nerve cells is presented, starting with a differential-integral field equation of the form $$\frac{\partial }{{\partial t}}\psi (r,t) = H\psi (r,t) + F(r,t)$$ which relates the field ψ to its inhibitory and excitatory sourcesF by means of the field operatorH. General solutions are represented with the aid of Green's functions. The Green's function, giving the response of the system to a very short point stimulation, is calculated in the absence of interaction between neurons and for a special case of non-local interaction. Possible applications of the theory are demonstrated for receptive fields and neuronal mechanisms in the vertebrate retina.
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    Some proposals in cardiac muscle mechanics and energetics (1973)
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    Bulletin of mathematical biology 35 (1973), S. 375-399 
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    Notes: Abstract Based on A. V. Hill's three-component model, mechanical properties of the contractile element (CE), such as velocity and tension, were determined as muscle shortening and loads in quick-release or afterloaded isotonic contraction. The method is applicable for studying cardiac mechanics, to obtain force-velocity data of the same CE length at varous afterloads. Analysis of the energetics of cardiac muscle was based on simulation studies of cardiac mechanics (Wong 1971, 1972). By proper derivation, the conventional contractile element work (CEW) was found to be a minor energy determinant. The tension-time integral and tension-independent heat (Ricchiuti and Gibbs, 1965) represent energy utilization for activation and maintenance of tension, the primary energy determinant.
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    The effect of the torso boundary in electrocardiography (1973)
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    Bulletin of mathematical biology 35 (1973), S. 401-405 
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    Notes: Abstract This paper outlines and contrasts three different (external) “infinite medium” potential fields which may be obtained (in principle) from known geometry, conductivities, and bounded medium torso potentials. Their electrocardiographic implications are considered.
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    An attitude to nucleic acid models (1973)
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    Bulletin of mathematical biology 35 (1973), S. 407-409 
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    Nucleic acid structure (1973)
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    Bulletin of mathematical biology 35 (1973), S. 411-413 
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    Passive transfer of low-molecular nonelectrolytes across deformable semipermeable membranes—I: Equations of convective-diffusion transfer of nonelectrolytes across deformable membranes of large curvature (1974)
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    Bulletin of mathematical biology 36 (1974), S. 365-377 
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    Notes: Abstract The passive transfer of a low-molecular nonelectrolytes mixture across a semipermeable deformable membrane of large curvature is considered. The equations obtained here make it possible to describe the concentration of species redistribution and the change of the membrane shape.
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    A new model for interacting populations—II: Principle of competitive exclusion (1974)
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    Bulletin of mathematical biology 36 (1974), S. 355-364 
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    Notes: Abstract The principle of competitive exclusion is investigated within the framework of the solvable model proposed earlier for two-species systems. The results elucidate the recent controversy over the interpretation of experimental data onDrosophila equilibrium. It is shown that the necessary and sufficient conditions for stable coexistence of competing species is that the product of intraspecific rate constants be greater than the product of interspecific rate constants. Inequalities between rate constants for the occurrence of stable equilibriumbelow andabove the line joining single species equilibria are derived. The availability of larger domain of coexistence suggests that the model presented here has the potential to accommodate a wider class of phenomena than the Gause—Volterra model according to which coexistence is possible only above the line of single species equilibrium.
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    Théorie chromosimique de l'inbreeding: Modèle probabiliste (1974)
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    Bulletin of mathematical biology 36 (1974), S. 417-434 
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    Notes: Abstract A molecular theory of inbreeding involving interactions between zones of gametic recognition (ZRG) localized on the chromosomal sets of male and female gametes is developed. This theory accounts for inbreeding effects and heterosis and raises the problem of control of embryonic development and the biological validity of the inbreeding coefficient. The mathematical analysis of this interaction system with sib matings permits the estimation of viability depression in inbred organisms. The biological validity of the model is discussed.
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    On the probability of reaching a threshold in a stochastic mammillary system (1974)
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    Bulletin of mathematical biology 36 (1974), S. 445-454 
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    Notes: Abstract The multivariate distribution over time of a particular stochastic mammillary compartmental model is obtained for any point in time. The maximum expectation of the peripheral compartments is then derived and used to determine lower bounds on the probability that the maximum of the peripheral compartments reaches any arbitrary threshold level. A bound on the probability is illustrated by an example and some of its implications are explored.
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    Dynamics of populations ofBiomphalaria glabrata and the von forerster equation (1974)
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    Bulletin of mathematical biology 36 (1974), S. 29-37 
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    Notes: Abstract Changes in time of populations ofBiomphalaria glabrata due to changes in the rate of infection bySchistosoma mansomi are investigated. This is done by applying Von Foerster equations with boundary conditions derived from experiment. The resulting equation is solved in some simplified cases and applications of the formalism to ecological problems is suggested.
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    Electrically induced phase transitions via the dipole model in excitable membranes (1974)
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    Bulletin of mathematical biology 36 (1974), S. 17-28 
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    Notes: Abstract Calculations based upon the membrane dipole model have been carried out and indicate the possibility of cooperative behavior during the membrane excitation process. An explicit expression is obtained for the critical electric field,E, which must be present to initiate the cooperative structural transition assumed to occur during membrane excitation. An hypothesis concerning the occurrence of two distinct phase transitions in the membrane resulting in the rapid influx of sodium ions and sodium ion inactivation, respectively, is presented.
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    A note on the shape of the erythrocyte (1974)
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    Bulletin of mathematical biology 36 (1974), S. 55-58 
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    Notes: Abstract Variational calculus is used to derive an equation for the shape of the cross section of a human red blood cell, the objective being the maximization of the surface area/volume ratio. Comparison to previous work is presented.
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    An intergral method for the analysis of flow in arterial stenoses (1974)
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    Bulletin of mathematical biology 36 (1974), S. 39-53 
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    Notes: Abstract An approximate solution is presented to the problem of incompressible flow through an axisymmetric constriction. The geometry is intended to simulate an arterial stenosis, and the solution is applicable to both mild and severe stenoses for Reynolds numbers below transition. Theoretical results obtained for specific geometries are given for the velocity distribution, pressure drop, wall shearing stress, and separation phenomena. These results reveal the significant alterations in flow caused by a stenosis. Experiments using model stenoses are described and compared with the theoretical results. Theoretical predictions of pressure drop and separation characteristics are in reasonably good agreement with the experimental observations.
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    Matematicheskie problemy v biologii (1974)
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    Bulletin of mathematical biology 36 (1974), S. 615-616 
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    The University of Texas Health Science Center at Houston, Division of Continuing Education and The Region V of IEEE announces Eleventh Symposium on Biomathematics and Computer Science in the Life Sciences (1974)
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    Bulletin of mathematical biology 36 (1974), S. 617-617 
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    The simulation of multi-neurone networks: Modelling of the lateral inhibition of the eye and the generation of respiratory rhythm (1974)
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    Bulletin of mathematical biology 36 (1974), S. 77-89 
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    Notes: Abstract We describe a model of a single neurone, and a method of interconnecting such neurone models suitable for multi-neurone network simulations on a digital computer. Two simulations are described; the lateral inhibition in the eye of the Limulus and the generation of respiratory rhythm in the brain stem of the cat.
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    Pulse velocities in cylindrical, tapered and curved anisotropic elastic arteries (1974)
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    Bulletin of mathematical biology 36 (1974), S. 103-103 
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    Interaction of peristaltic motion with Poiseuille flow (1974)
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    Bulletin of mathematical biology 36 (1974), S. 127-141 
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    Notes: Abstract The effect of Poiseuille flow on peristaltic transport has been investigated in a two-dimensional mathematical model of peristalsis for the case when the wall of the channel executes a sinusoidal motion of small amplitude. Closed-form solutions have been obtained for limiting values of Reynolds number and the Poiseuille flow parameter, while the method of Frobenius series solution has been used for the general case. It is found that the mean flow reversal is strongly dependent on the Poiseuille flow. The position of flow reversal may change drastically from the center of the channel to the boundaries. Numberical results are reported for various values of the physical parameters of interest.
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    Enzyme kinetics in cells (1974)
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    Bulletin of mathematical biology 36 (1974), S. 157-169 
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    Notes: Abstract Classical enzymology ignores the role which cellular membranes may play in regulating reaction kineticsin vivo. The correct description of cellular metabolic processes must derive from a mass balance equation for each reacting species. An elementary mathematical model, called the “continuous flow stirred tank reactor” in the chemical engineering literature, has been applied to Michaelis-Menten kinetics, substrate inhibition kinetics, and kinetics involving hydrogen ion as a byproduct. A number of remarkable phenomena, including multiple stationary states, threshold effects, temporal patterns, homeostatic regulation, amplification, and irreversible differentiation can result. Predictions of the model are in qualitative accord with experimental and theoretical studies of insolubilized enzymes, which are conventionally modeled by a more difficult mathematical formalism.
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    Stability and enzyme separation: Integral representation of the solutions (1974)
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    Bulletin of mathematical biology 36 (1974), S. 197-203 
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    Notes: Abstract Two enzymes, cross-coupled through their respective products, are membrane-bound and thereby separated from each other. The cross-coupling is of opposite character and diffusion limited, i.e. an activation-inhibition couple with a diffusion-induced time-lag. In this paper an integral representation for the solutions of the dynamic equations is developed using the Green's function technique, and an application to morphological changes in the mitochrondrion is discussed.
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    Stochastic kinetic models for small irreversible systems (1974)
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    Bulletin of mathematical biology 36 (1974), S. 219-227 
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    Notes: Abstract The probability distribution of the process level is derived for a general class of irreversible reaction systems. A method of approximating the mean and variance of the process level is developed and shown to be more accurate for small irreversible systems than other approximations used in the stochastic approach to kinetics.
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    Semiconductor theory of ion transport in thin lipid membranes: I. Potential and field distributions (1974)
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    Bulletin of mathematical biology 36 (1974), S. 229-246 
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    Notes: Abstract In the theory as presented in this paper and the following one, we shall attempt to apply the semiconductor principles and methods to the study of ion transport in thin lipid membranes. Detailed formulations are given on the potential energy barriers at the interfaces, voltage drops in the polar and non-polar regions, and potential and field distributions in the diffuse double layer and within a charged membrane. These results will be used mainly as the boundary conditions for the solution of ion flow as to be given in the following paper. The analysis clearly indicates that the ion transport is interface-limited and is profoundly influenced by the presence of surface charges. An explanation of Na+ extrusion in nerve membrane is given based on the field distribution analysis. The theory also suggests that the “membrane potential” depends mainly on surface charges but not necessarily on ion permeation through the membrane.
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    Flagellar elongation as a moving boundary problem (1974)
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    Bulletin of mathematical biology 36 (1974), S. 265-273 
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    Notes: Abstract Movement of a proposed controlling factor along the growing flagellum is considered as a special case of one-dimensional diffusion with a moving boundary. Flagellar elongation, which involves polymerization of the building units at the distal tip, is viewed as taking place in a series of steps. The concentration of the controlling factor at the tip is computed as a function of the distance of the tip from the base, and the time between polymerization reactions. It is proposed that the concentration of this factor at the tip is responsible for regulating the flagellar growth rate and final length.
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    Stochastic theory of compartments (1974)
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    Bulletin of mathematical biology 36 (1974), S. 305-309 
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    Notes: Abstract The stochastic model of a compartment developed by Thakur, Rescigno and Schafer is discussed without using generating functions. The behavior of the mean and variance of the number of particles present as a function of time is also discussed. We also allow both the input and output to be time dependent.
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    A new model for interacting populations—I: Two-species systems (1974)
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    Bulletin of mathematical biology 36 (1974), S. 347-353 
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    Notes: Abstract The exact solvable model presented here describes two interacting species whose populations oscillate periodically in time in the absence of self-interaction and attain saturation with fluctuations if self-interactions are included. While sharing broad features of the classical Volterra system, the model proposed here has the advantage of possessing exact solutions. The model thus provides a testing ground for general principles employed in the study of dynamics of populations.
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    The influence of noninstantaneous mixing on the measurement of fluxes of solute and water across a biological membrane (1970)
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    Bulletin of mathematical biology 32 (1970), S. 1-24 
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    Notes: Abstract The usual method of tracer analysis for calculating the flow across a biological membrance is based on the assumption that the compartments on either side are well-stirred. Thus, the validity of the rate of flow determination is questionable for cases where the distribution of tracer is not homogeneous. In this study, a mathematical model is developed for the purpose of estimating the effect of slow mixing on the calculation of the flow rate. The model is applied to the measurement of the rate of flow of aqueous humor through the living eye by use of a fluorescent dye as a tracer. A transit time of several minutes for the passage of fluorescein through the posterior chamber and an extended period of nonuniform distribution of fluorescein in the anterior chamber was observed. The effect of slow mixing on the calculated flow rate is compared to rates derived from equations based on the assumption of rapid mixing. Aqueous flow rates determined by the two methods were found to agree to within ≈20%.
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    The dispersion of indicator in the cardio-pulmonary system (1970)
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    Bulletin of mathematical biology 32 (1970), S. 25-43 
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    Notes: Abstract This paper presents a three-parameter model of the mechanism of dispersion of an indicator in the cardio-pulmonary system, based on the postulates that this dispersion can be described by the one-dimensional diffusion equation and that dispersion continues past the sampling site. The model is tested using indicator dilution curves obtained from dogs, and the coefficient of diffusion is thus measured. It is found that this coefficient increases in magnitude non-linearly with increasing blood speed.
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    The effect of distension of the left ventricle of the heart on the length of the individual myocardial fibers (1970)
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    Bulletin of mathematical biology 32 (1970), S. 45-58 
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    Notes: Abstract Based on the ellipsoid model of the left ventricle and the helicoidal course of the left ventricular myocardial fibers, a theory has been developed for calculating the length of the individual myocardial fibers. Numerical solutions of the final equation show that when the left ventricle is distended, the increase in length of the myocardial fibers is not uniform throughout the thickness of the myocardial wall. It was shown that with increasing dimensions of the left ventricle, the distension of the myocardial fibers becomes smaller as one advances from the endocardium to the middle layer of fibers, whereas it increases as one advances from the middle layer to the epicardial layer. The mechanism by which this effect is brought about as well as its physiological implications are discussed.
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    The gas washout determination under a symmetry assumption (1970)
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    Bulletin of mathematical biology 32 (1970), S. 59-63 
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    Notes: Abstract In an open circuit gas washout determination the output of test substance is shown to be of the form ∑ i=0 ∞ A i λ i k on thekth expiration whereA i 〉0,i=1,2,... and 1 〉 λ1 ≥ λ2...≥ 0 provided the transition from inspiration to expiration has certain symmetry properties with the transition from expiration to inspiration. In general, no direct physical interpretation such as volume for theA t ’s or fraction of retained gas on expiration for the λ i is justified.
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    Mathematical biology of automobile driving IV (1970)
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    Bulletin of mathematical biology 32 (1970), S. 71-78 
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    Notes: Abstract Previous studies by this author of the mathematical biology of automobile driving have emphasized only the biological aspects, except for such mechanical factors as the size of the car. Otherwise, the ideal case of an inertialess car was considered. In this paper the first step is made toward introducing the effects of the mass of the car and the side-slip of the tires when the direction of driving is even slightly altered and combining these with the previously studied biological aspects. Some tentative comparisons with available data are made.
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    Relaxation methods for the determination of drug receptor affinities (1970)
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    Bulletin of mathematical biology 32 (1970), S. 65-69 
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    Notes: Abstract According to the occupation theory of drug receptor interaction, the response is a functionf of the number of receptors occupied by drug molecules. Considerable controversy exists regarding assumptions about this function. Without knowledge of the nature of the function, it is not possible to determine directly the rate constants, and hence the affinity constant, in the reaction between the receptor and an agonist drug. Instead, indirect determinations involving the use of antagonists have been employed, limiting the determination of affinity to those agents for which specific antagonists exist. The present paper discusses a method for the direct determination of affinity of an agonist drug. It is a “relaxation method,” i.e., the equilibrium is perturbed and the kinetics of the restoration process are studied. Assuming only thatf is non-decreasing and approximately linear over a limited domain of concentrations, it is shown that the change in response obeys first order kinetics, permitting a determination of the rate constants from the time course of the restoration process.
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    A remark on the course of development of organismic sets (1970)
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    Bulletin of mathematical biology 32 (1970), S. 79-81 
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    Notes: Abstract The basic postulate la, which governs the development of organismic sets, introduced previously (Bulletin of Mathematical Biophysics,31, 159–198, 1969), is generalized so as to contain also the rates of changes of the number and variety of differentQ-relations which determine an organismic set. It is thus brought closer to the Lagrangian principle in physics. It is pointed out that the postulate also provides a criterion of stability of an organismic set.
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    In-vivo estimations of the nonlinear pressure-volume relationship of the aorta and instantaneous left ventricular volume (1970)
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    Bulletin of mathematical biology 32 (1970), S. 155-172 
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    Notes: Abstract A method is presented in this paper for the in-vivo estimation of the nonlinear pressure-volume relationship of the human aorta. The method is based on nonlinear elastic reservoir theory and utilizes clinical data that can be obtained with a high degree of accuracy, namely stroke volume, end diastolic ventricular volume and aortic pressure trace data. The computational procedure is described and then carried out for six cardiac patients. A method for the estimation of instantaneous left ventricular volume during the ejection period based on the considered nonlinear elastic reservoir theory is also presented. The method is applied for the six cardiac patients cited and the results compared with those obtained for the same subjects by a method of estimation based on linear elastic reservoir theory described in a previous paper by the author (1969).
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    The application of the minimal energy hypothesis to a casson fluid (1970)
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    Bulletin of mathematical biology 32 (1970), S. 249-262 
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    Notes: Abstract Deakin (1967b) suggested that flow of blood might obey a law of minimal energy dissipation. The present paper presents a simpler derivation of Deakin’s equations pointing out several previously unrecognized features. It is shown that these equations are unlikely to be applicable. In particular, the solution obtained by Deakin and Jones (1968) does not yield a true minimum for energy dissipation. The solution for which energy dissipation is actually minimized is shown to possess features which render it unlikely to apply to a real flow.
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    Some remarks on the Kedem-Katchalsky equations for non-electrolytes (1970)
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    Bulletin of mathematical biology 32 (1970), S. 237-247 
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    Notes: Abstract The Kedem-Katchalsky equation for the flow of a non-electrolyte through a homogeneous membrane is shown to be a first order expansion of an exact integral of the Spiegler-Bearman-Kirkwood frictional equations under the assumption that the partial frictional coefficients, ζ ij , are concentration independent. The equations are solved in terms of volume flow; there are no water-to-volume flow correction terms for the permeability, ω, or the reflection coefficient, σ. The precision of the expansion depends upon the magnitude of the water flow. The frictional coefficientsf sm andf sw are given as functions of the experimentally determined parameters ω and σ; the frictions, are shown to be independent ofL p .
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    On the average cellular volume in synchronized cell populations (1970)
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    Bulletin of mathematical biology 32 (1970), S. 459-473 
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    Notes: Abstract As was done by Sinclair and Ross (1969(, we consider a cellular population that consists initially (at time zero) ofN 0 newborn cells, all with the same volumev o. It is assumed that the occurrence of cell division is determined only by a cell’s age, and not by its volume. The frequency function of interdivision times, τ, is denoted byf(τ). If cell death is negligible, the expected number of cells,N(t), will increase according to the laws of a simple age-dependent branching process. The expression forN(t) is obtained as a sum over all generations; thevth term of this sum, in turn, is a multiple convolution integral, reflecting the life history ofvth generation cells (i.e., the lengths of thev successive interdivision periods plus the age of the cell at timet). Assuming that cell volume is a given function of cell age, e.g., linear or exponential, and that cellular volume is exactly halved at each division, it is possible to calculate the volume of a cell with a given life history, and thus the average cellular volume of the whole population as a function of time. If at time zero the volumes differ from cell to cell, the final equation must be modified by averaging over initial volumes. In the case of linear volume increase with age, a very simple asymptotic expression is found for the average cellular volume ast→∞. The case of exponential volume increase with age also leads to a simple asymptotic formula, but the resulting volume distribution is unstable. The mean cellular volume at birth and the second moment of the volume distribution can be calculated in a similar manner.
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    Dynamic analysis, with feedback characterization, of hybrid human musculoskeletal frameworks by the linegraph-flowgraph procedure (1970)
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    Bulletin of mathematical biology 32 (1970), S. 499-520 
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    Notes: Abstract The dynamic response of human musculo-skeletal framework is treated by (i) idealization of the musculo-skeletal framework as hybrid structural networks possessing feedback characteristics and then (ii) employing linegraph-flowgraph procedures for the feedback characterization of the hybrid structural networks. Topological procedures are used in which a “tree” of a network furnishes the skeleton upon which the “linkage” (muscle representing) members provide interaction. Feedback characterization (representing the sensitivity of the skeletal members to the tensile forces) is defined, between the internal “linkage” and “tree” members, by means of the flowgraph. Mikusinski operational calculus is used to facilitate representation of inertia effects by dynamic feedback characterization, with inclusion of initial conditions.
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    Organismic supercategores: II. On multistable systems (1970)
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    Bulletin of mathematical biology 32 (1970), S. 539-561 
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    Notes: Abstract The representation of biological systems in terms of organismic supercategories, introduced in previous papers (Bull. Math. Biophysics,30, 625–636;31, 59–70) is further discussed. To state more clearly this representation some new definitions are introduced. Also, some necessary changes in axiomatics are made. The conclusion is reached that any organismic supercategory has at least one superpushout, and this expresses the fact that biological systems are multistable. This way a connection between some results of Rashevsky’s theory of organismic sets and our results becomes obvious.
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    On the mass distribution model for microbial cell populations (1970)
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    Bulletin of mathematical biology 32 (1970), S. 521-537 
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    Notes: Abstract The full implications of a statistical model for growth of a microbial cell population using cell mass as the index of physiological state have been examined by solving the partial differential integral equations resulting from the model. Calculations reveal that a lag phase is predicted during the initial stages of batch growth although no specific cellular mechanism for the phenomenon of lag had been incorporated into the model. The model predicts several situations of batch and continuous growth in which the population density and biomass concentration show opposing trends due to significant variation in the cell mass distribution with time.
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    Traitement des informations chez les etres vivants: Systemes a services par essais successifs (1970)
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    Bulletin of mathematical biology 32 (1970), S. 83-148 
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    Notes: Abstract Information treatment, especially in a natural medium, may be done by a random system: given an input, possible solutions are randomly selected and successively tested until the right one is found and then emitted. This paper is a mathematical study of this type of system and its use as a model of some natural (biological) systems. The theoretical development is illustrated by several examples of possible applications to biological cases. We study, under different assumptions, the distribution of the number of trials needed to treat the input, the distribution of the time for this treatment, and the distribution of the average amount of information treated in a trial or in a unit of time. We consider the case of error, and study some optimal conditions of operation and the occurrence of memory influencing the treatment of an input according to previously processed inputs.
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    Energy of nerve impulse (1970)
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    Bulletin of mathematical biology 32 (1970), S. 151-153 
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    Comments concerning P. C. Voukydis’ “Physiological significance of the geometrical shape of the left ventricle: Course and curvature of the individual myocardial fibers” (1970)
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    Bulletin of mathematical biology 32 (1970), S. 149-149 
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    Mathematical biology of automobile driving: V (1970)
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    Bulletin of mathematical biology 32 (1970), S. 173-178 
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    Notes: Abstract The role of some inertial properties of the car, studied previously only for the case when the stimulus for the corrective turn is the perception of the angle between the direction of the car and the direction of a straight lane (Bull. Math. Biophysics,32, 71–78, 1970), is generalized to include such stimuli as the nearness to the edge of the lane and the anticipatory effect for a corrective turn, as well as the combination of all three stimuli. Conditions for stability of driving are deduced and discussed. They now depend on both biological parameters and such parameters as the position of the center of gravity of the car, its mass, and the side slip of the tires.
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    Definition d’une distance entre deux mobiles se deplacant de facon aleatoire sur une droite (1970)
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    Bulletin of mathematical biology 32 (1970), S. 179-195 
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    Description / Table of Contents: Résumé Pour le biologiste, la notion dedistance entre deux ou plusieurs éléments d’un ensemble est très utile car elle sert à mesurer laproximité, laparenté, laressemblance qui existe entre ces éléments ou parties selon le caractère auquel on s’intéresse (position géographique, situation chronologique, aptitudes, phénotypes, composition chimique etc...).
    Notes: Abstract Adistance between two mobiles performing a random walk in one dimension is defined. At a given time this distance is directly related to theprobability of encounter for the mobiles. This definition is used when the motion of the mobiles is a Wiener-Levy process, first in the case of an unrestricted random walk, then if a reflecting or absorbing barrier is introduced.
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    Effects of anisotropy and nonhomogeneity on left ventricular stresses in the intact heart (1970)
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    Bulletin of mathematical biology 32 (1970), S. 197-213 
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    Notes: Abstract The qualitative effects of anisotropy and nonhomogeneity are considered in the evaluation of left ventricular stresses in the intact heart. Maximum stresses and their location are significantly dependent on the nonhomogeneity factors and to a lesser degree on anisotropy of the ventricular wall material. If the circumferential elastic modulus is assumed to vary in a parabolic manner through the wall thickness, maximum stresses occur within the endocardial layers, a result in qualitative agreement with experimental studies.
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    A correction for the effect of the dead space in pulmonary gas washout tests (1970)
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    Bulletin of mathematical biology 32 (1970), S. 215-218 
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    Notes: Abstract Under the assumption that there is complete mixing in the dead space a series of equations are given which give the values for a washout test of the functional component of the pulmonary bellows (without a dead space) in terms of the washout values of the bellows with a dead space.
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    Respiration of tissue as a medium of heterogeneous permeability (1970)
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    Bulletin of mathematical biology 32 (1970), S. 219-235 
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    Notes: Abstract Expressions are derived for the overall oxygen consumption and the O2 penetration depth into a tissue section in terms of the basic parameters, of the system under steady-state conditions. The approach differs from many previous analyses in so far as the oxygen molecules are regarded as reaching their sites of chemical assimilation by diffusion through extracellular fluid followed by bulk diffusion into irregular cells of significantly lower permeability. This “two-phase” model would seem to be compatible with the major experimental features of steady-state respiration, and gives a ratio of cellular to extracellular diffusion coefficients of the same low order as that found for inert gases under transient conditions. The greater oxygen penetration predicted by this model is discussed in relation to the survival of ischemic tissue and is shown to be consistent with data for myocardial infarction.
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    Analytical solutions for the specific activities of a traced substance within the component parts of a simulated kidney-ureter system (1970)
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    Bulletin of mathematical biology 32 (1970), S. 263-278 
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    Notes: Abstract Two models for a kidney-ureter system are considered: one model of one vessel in which a traced substance, undergoing exchange between the vessel and an external compartment, is emptying into the ureter; the second model of two approximately parallel, identical vessels in which a traced substance, undergoing exchange between each vessel and an external compartment, is emptying into the ureter. A single impulsive input of label into a vessel is assumed. For mathematical simplicity, the major conditions imposed on each system are: (1) rapid mixing transverse to a vessel axis and no mixing longitudinal to a vessel axis within the plasma; (2) small variation of the specific activity within the plasma in the longitudinal direction to a vessel axis; (3) constant flow rate of urine into the ureter and (4) constant exchange coefficients, tubule flow velocity and traced substance concentrations within individual compartments.
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    The motion of the left ventricle—III (1970)
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    Bulletin of mathematical biology 32 (1970), S. 279-291 
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    Notes: Abstract We use the concept of a layered wall, where each separate layer is to be homogeneous, isotropic, and incompressible, to derive stress-strain relations for the middle layer muscle ring at the transverse midsection of the left ventricle; a convenient method of formulation is that based on the elastic potential function. The hoop or circumferential stress in all three layers is found using dimensional and mechanical parameters derived earlier. The various parameters are expressed as Fourier series so that their behavior over a complete ventricular cycle is known analytically. The cases of simple elongation and what we termcurvilinear simple elongation are considered for the middle layer muscle ring strain, and the resulting stress-strain relations are derived. The results are compared with an incompressible rubber-like material known as a Mooney material.
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    Transient thermal behavior in biological systems (1970)
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    Bulletin of mathematical biology 32 (1970), S. 303-314 
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    Notes: Abstract The Peaceman-Rachford finite difference method is applied to cylindrically symmetric, transient heat conduction problems in biological media. Inhomogeneous media and internal sources which vary in both space and time are permitted. Boundary conditions are satisfied without sacrificing high local resolution by means of an exponentially stretched grid. Computation time on a Philco 2000/210 computer is approximately 5 msec per grid point per time step.
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    International Symposium on Mathematical Biology (1970)
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    Bulletin of mathematical biology 32 (1970), S. 601-601 
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    Physiological significance of the geometrical shape of the left ventricle: Course and curvature of the individual myocardial fibers (1970)
    Springer
    Bulletin of mathematical biology 32 (1970), S. 599-600 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02476773
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    A concentric layer model for estimating the energy expenditure of the left ventricle (1970)
    Springer
    Bulletin of mathematical biology 32 (1970), S. 581-598 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract The energy cost of the left ventricle is quantitatively analyzed on the basis of the following assumptions: (1) The left ventricle is assumed to be an isotropic, homogeneous elastic, thick, spherical shell. (2) The ventricular wall is made up of a finite number of thin concentric shells. (3) The energetics of the left ventricle is in accordance with the second law of thermodynamics. An expression for the work done during ventricular contraction is derived according to the definition of physical work. The energy liberation during isovolumic contraction is formulated parallel to the concepts of heat production in skeletal muscle during isometric contraction. This expression gives the total work done per stroke in terms of mean systolic pressure, end diastolic volume, stroke volume and wall thickness during diastolic phase.
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    Ernesto Trucco (1970)
    Springer
    Bulletin of mathematical biology 32 (1970), S. i 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02476775
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    Chorioretinal temperature increases from solar observation (1971)
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    Bulletin of mathematical biology 33 (1971), S. 1-17 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract Chorioretinal temperature increases produced by solar observations are computed digitally. The spectral characteristics of solar radiation and the spectral transmittances of the atmosphere, atmospheric water vapor and ocular media are considered. Image spread is employed in all calculations. The calculations are executed for a variety of observation angles, and the effects of pupil diameter, fixed filters and optical instruments are predicted. Temperature increases are also computed for solar eclipse observations by the appropriate superposition of unobscured solar disk solutions.
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    The kinetics of diffusion between a spherical cell and a surrounding medium with different diffusion properties (1971)
    Springer
    Bulletin of mathematical biology 33 (1971), S. 19-26 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract The diffusion equation according to Fick’s law is solved for a spherical cell, surrounded by an infinite medium with different diffusion properties. The method of Laplace transform is used to obtain the formal solution, however, no inversion can be found for all times and an expansion suitable for small times is performed. The final expression found is expanded further to be more suited for the determination of the diffusion coefficient from an experimental curve. Application to a biological problem is dissussed.
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    A mathematical model for first degree block and the Wenckebach phenomenon (1971)
    Springer
    Bulletin of mathematical biology 33 (1971), S. 27-38 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract A simple mathematical model of first and second degree heart block is developed on the assumption that eachR wave, with its associated ventricular contraction, results in the release of a substance which raises the excitation threshold of the conducting tissue lying between theS-A andA-V nodes. The conduction pathway is represented by a chain of excitatory elements, the first member of which is acted upon by a regularly occurringP wave. The rate constant of the removal of this inhibitory substance, i.e., recovery rate, is the only constant necessary to be varied in order to pass from normal to first degree block to second degree. In this latter case one can predict the progressive increase inP-R interval until anR wave is missed, as occurs in the Wenckebach phenomenon. The model indicates that an increasedP wave frequency which would have little effect on theP-R inerval for a normal individual with rapid recovery, could produce a Wenckebach pattern in an individual with even a mild first degree block.
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    Temperature dependence of the Elovich equation derived from activated electron or ion transport across a biological interface, with an application to muscle spindle adaptation (1971)
    Springer
    Bulletin of mathematical biology 33 (1971), S. 39-47 
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    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract The temperature dependence of the Elovich equation, which is−dx/dt=m exp (nx) (wherex is concentration of substrate or ion,t is time, andm andn are coefficients dependent upon temperature), has been derived from the hypothesis of electron or ion conduction across an activation energy barrier at the surface of a biological particle or membrane, driven by a difference in redox or ion potentials. Using the additional hypothesis of reversible, temperature-dependent, inactivation of sites of reduction or complexing, the theory predicts that both coefficientsm andn have linear Arrhenius plots, in agreement with experimental data for gas adsorption on inorganic solid surfaces, and with two studies of muscle spindle adaptation.
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    Some recent results inn-person game theory (1971)
    Springer
    Mathematical programming 1 (1971), S. 58-67 
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    ISSN: 1436-4646
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract The discussion will center mainly on some work on two solution concepts: the core for gaines without side payments and the nucleolus for games with side payments (characteristic funtion games). The core has become an important equilibrium concept in mathematical economics. The nucleolus is related to the theory of bargaining sets.
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    Publications (1971)
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    Mathematical programming 1 (1971), S. 117-120 
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    ISSN: 1436-4646
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
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    URL: http://dx.doi.org/10.1007/BF01584078
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    The Mathematical Programming Society (1971)
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    Mathematical programming 1 (1971), S. 123-125 
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    ISSN: 1436-4646
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
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    URL: http://dx.doi.org/10.1007/BF01584080
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    Hybrid programs: Linear and least-distance (1971)
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    Mathematical programming 1 (1971), S. 153-167 
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    ISSN: 1436-4646
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract We consider a quadratic program equivalent to the general problem of minimizing a convex quadratic function of many variables subject to linear inequality constraints. In previous work [12], one of us presented an algorithm related to such problems, and classified them under “combinatorial equivalence”. The classification contained linear programs at one end (where the function has zero quadratic part) and least-distance programs at the other. A least-distance program is a problem of finding a point of a convex polyhedron which is at least distance from a given point; such programs have been studied by one of us [15, 17] and were shown to correspond to that case of the general problem where the function has positive definite quadratic part. We now extend the work on least-distance programs to those programs intermediate between linear and least-distance (called “essentially bisymmetric” in [12]), and show that such programs are really “hybrids”, with traits inherited from both parent programs: linear and leastdistance.
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    An “out-of-kilter” algorithm for solving minimum cost potential problems (1971)
    Springer
    Mathematical programming 1 (1971), S. 275-290 
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    ISSN: 1436-4646
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract An algorithm is given for solving the optimum potential problem, which is the dual of the classical “out-of-kilter” algorithm for flow problems. Moreover, a new proof of finiteness is provided, which holds even for non-rational data; it applies to all the algorithms of network theory which include a labeling process.
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    URL: http://dx.doi.org/10.1007/BF01584092
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    Mathematical programming 2 (1972), S. 130-132 
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    ISSN: 1436-4646
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01584539
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