Analysis of a mathematical model for the growth of tumors

 In this paper we study a recently proposed model for the growth of a nonnecrotic, vascularized tumor. The model is in the form of a free-boundary problem whereby the tumor grows (or shrinks) due to cell proliferation or death according to the level of a diffusing nutrient concentration. The tumor is assumed to be spherically symmetric, and its boundary is an unknown function r=s(t). We concentrate on the case where at the boundary of the tumor the birth rate of cells exceeds their death rate, a necessary condition for the existence of a unique stationary solution with radius r=R ₀ (which depends on the various parameters of the problem). Denoting by c the quotient of the diffusion time scale to the tumor doubling time scale, so that c is small, we rigorously prove that (i) lim inf _t→∞ s(t)>0, i.e. once engendered, tumors persist in time. Indeed, we further show that (ii) If c is sufficiently small then s(t)→R ₀ exponentially fast as t→∞, i.e. the steady state solution is globally asymptotically stable. Further, (iii) If c is not “sufficiently small” but is smaller than some constant γ determined explicitly by the parameters of the problem, then lim sup _t→∞ s(t)<∞; if however c is “somewhat” larger than γ then generally s(t) does not remain bounded and, in fact, s(t)→∞ exponentially fast as t→∞. Received: 25 February 1998 / Revised version: 30 April 1998     

A spatially discrete model for aggregating populations

 In this paper we introduce a spatially discrete model for aggregating populations described by a system of ODEs. We study the long time behavior of the solutions and we show that the model contains mechanisms by which individuals in the population aggregate at particular points in space. Received: 29 June 1996 / Revised version: 5 August 1997     

A mathematical model of the stress induced during avascular tumour growth

In this paper a mathematical model is developed to describe the effect of nonuniform growth on the mechanical stress experienced by cells within an avascular tumour. The constitutive law combines the stress-strain relation of linear elasticity with a growth term that is derived by analogy with thermal expansion. To accommodate the continuous nature of the growth process, the law relates the rate of change of the stress tensor to the rate of change of the strain (rather than relating the stress to the strain directly). By studying three model problems which differ in detail, certain characteristic features are identified. First, cells near the tumour boundary, where nutrient levels and cell proliferation rates are high, are under compression. By contrast, cells towards the centre of the tumour, where nutrient levels are low and cell death dominant, are under tension. The implications of these results and possible model developments are also discussed. Received: 15 November 1999 / Published online: 5 May 2000     

A new explicit stability criterion for human periodic breathing<--RID="*"--><--ID="*" This research was supported by grants from DRED (96-920).-->

The aim of this paper is to carry out a stability analysis for periodic breathing in humans that incorporates the dynamic characteristics of ventilation control. A simple CO₂ model that takes into account the main elements of the respiratory system, i.e. the lungs and the ventilatory controller with its dynamic properties, is presented. This model results in a three-dimensional non-linear delay differential system for which there exists a unique equilibrium point. Our stability analysis of this equilibrium point leads to the definition of a new explicit stability criterion and to the demonstration of the existence of a Hopf bifurcation. Numerical simulations illustrate the influence of physiological parameters on the stability of ventilation, and particularly the major role of the dynamic characteristics of the respiratory controller. Received: 2 February 1999 / Revised version: 18 June 1999 / Published online: 23 October 2000     

A weakly nonlinear analysis of a model of avascular solid tumour growth

 In this paper we study a mathematical model that describes the growth of an avascular solid tumour. Our analysis concentrates on the stability of steady, radially-symmetric model solutions with respect to perturbations taken from the class of spherical harmonics. Using weakly nonlinear analysis, previous results are extended to show how the amplitudes of the asymmetric modes interact. Attention focuses on a special case for which the model equations simplify. Analysis of the simplified model equations leads to the identification of a two-parameter family of asymmetric steady solutions, the dimensions of whose stable and unstable manifolds depend on the system parameters. The asymmetric steady solutions limit the basin of attraction of the radially-symmetric steady state when it is linearly stable. On the basis of these numerical and analytical results we postulate the existence of fully nonlinear steady solutions which are stable with respect to time-dependent perturbations. Received: 25 October 1998 / Revised version: 20 June 1998     

A non-local model for a swarm

 This paper describes continuum models for swarming behavior based on non-local interactions. The interactions are assumed to influence the velocity of the organisms. The model consists of integro-differential advection-diffusion equations, with convolution terms that describe long range attraction and repulsion. We find that if density dependence in the repulsion term is of a higher order than in the attraction term, then the swarm profile is realistic: i.e. the swarm has a constant interior density, with sharp edges, as observed in biological examples. This is our main result. Linear stability analysis, singular perturbation theory, and numerical experiments reveal that weak, density-independent diffusion leads to disintegration of the swarm, but only on an exponentially large time scale. When density dependence is put into the diffusion term, we find that true, locally stable traveling band solutions occur. We further explore the effects of local and non-local density dependent drift and unequal ranges of attraction and repulsion. We compare our results with results of some local models, and find that such models cannot account for cohesive, finite swarms with realistic density profiles. Received: 17 September 1997 / Revised version: 17 March 1998     

A mathematical model of a crocodilian population using delay-differential equations

The crocodilia have multiple interesting characteristics that affect their population dynamics. They are among several reptile species which exhibit temperature-dependent sex determination (TSD) in which the temperature of egg incubation determines the sex of the hatchlings. Their life parameters, specifically birth and death rates, exhibit strong age-dependence. We develop delay-differential equation (DDE) models describing the evolution of a crocodilian population. In using the delay formulation, we are able to account for both the TSD and the age-dependence of the life parameters while maintaining some analytical tractability. In our single-delay model we also find an equilibrium point and prove its local asymptotic stability. We numerically solve the different models and investigate the effects of multiple delays on the age structure of the population as well as the sex ratio of the population. For all models we obtain very strong agreement with the age structure of crocodilian population data as reported in Smith and Webb (Aust. Wild. Res. 12, 541-554, 1985). We also obtain reasonable values for the sex ratio of the simulated population.     

A mathematical model to describe the dynamic response of a spruce tree to the wind

 A mathematical, computer-based, dynamic sway model of a Sitka spruce (Picea sitchensis) tree was developed and tested against measurements of the movement of a tree within a forest. The model tree was divided into segments each with a stiffness, mass and damping parameter. Equations were formulated to describe the response of every segment which together form a system of coupled differential equations. These were solved with the aid of matrices and from the resulting modes, the transfer function of the tree was found and used to calculate the movement of the tree in the wind. Comparison of the modelled movement of a tree in response to the measured wind speed above a forest canopy gave good agreement with the measured movement of the top of the tree but less satisfactory agreement close to the base. The comparison also pointed to the complexity of tree response to the wind and inadequacies in the model. In particular, the branches need to be treated as coupled cantilevers attached to the stem rather than simply as masses lumped together. Received: 18 February 1997 / Accepted: 16 December 1997     

Periodic orbits arising from Hopf bifurcations in a Volterra prey-predator model

 In this paper we derive a formula which enables the stability of periodic solutions to a Volterra integro-differential system to be determined. This system which has been studied by Cushing [1], models a predator-prey interaction with distributed delays. The results are obtained by using the algorithm developed by Kazarinoff, Wan and van den Driessche [2] based on the centre manifold formulas of Hassard and Wan [3]. We discuss an example of the formula for the case of weak kernels and show that under certain conditions stable periodic solutions arising from Hopf bifurcations at different critical values of the parameters can exist together. Received 30 December 1994; received in revised form 12 December 1995     

Mathematical modelling of the intravenous glucose tolerance test

 Several attempts at building a satisfactory model of the glucose-insulin system are recorded in the literature. The minimal model, which is the model currently mostly used in physiological research on the metabolism of glucose, was proposed in the early eighties for the interpretation of the glucose and insulin plasma concentrations following the intravenous glucose tolerance test. It is composed of two parts: the first consists of two differential equations and describes the glucose plasma concentration time-course treating insulin plasma concentration as a known forcing function; the second consists of a single equation and describes the time course of plasma insulin concentration treating glucose plasma concentration as a known forcing function. The two parts are to be separately estimated on the available data. In order to study glucose-insulin homeostasis as a single dynamical system, a unified model would be desirable. To this end, the simple coupling of the original two parts of the minimal model is not appropriate, since it can be shown that, for commonly observed combinations of parameter values, the coupled model would not admit an equilibrium and the concentration of active insulin in the “distant” compartment would be predicted to increase without bounds. For comparison, a simple delay-differential model is introduced, is demonstrated to be globally asymptotically stable around a unique equilibrium point corresponding to the pre-bolus conditions, and is shown to have positive and bounded solutions for all times. The results of fitting the delay-differential model to experimental data from ten healthy volunteers are also shown. It is concluded that a global unified model is both theoretically desirable and practically usable, and that any such model ought to undergo formal analysis to establish its appropriateness and to exclude conflicts with accepted physiological notions. Received: 22 June 1998 / Revised version: 24 February 1999     

Adding self-renewal in committed erythroid progenitors improves the biological relevance of a mathematical model of erythropoiesis

We propose a new mathematical model of erythropoiesis that takes a positive feedback of erythrocytes on progenitor apoptosis into account, and incorporates a negative feedback of erythrocytes on progenitor self-renewal. The resulting model is a system of age-structured equations that reduces to a system of delay differential equations where the delays account for progenitor compartment duration and cell cycle length. We compare this model with experimental data on an induced-anemia in mice that exhibit damped oscillations of the hematocrit before it returns to equilibrium. When we assume no self-renewal of progenitors, we obtain an inaccurate fitting of the model with experimental data. Adding self-renewal in the progenitor compartment gives better approximations, with the main features of experimental data correctly fitted. Our results indicate the importance of progenitor self-renewal in the modelling of erythropoiesis. Moreover, the model makes testable predictions on the lifespan of erythrocytes confronted to a severe anemia, and on the progenitors behavior.     

Stability of periodic solutions for a model of genetic repression with delays

A technique is discussed for locating the Hopf bifurcation of an n-dimensional system of delay differential equations which arises from a model for control of protein biosynthesis. Certain parameter values are shown to allow a Hopf bifurcation to periodic orbits. At the Hopf bifurcation the periodic orbits are shown to be stable either analytically or numerically depending on the parameter values.On leave from North Carolina State University.Supported in part by N.S.F. Grant # MCS 81-02828     

Analysis of an SEIRS epidemic model with two delays

 A disease transmission model of SEIRS type with exponential demographic structure is formulated. All newborns are assumed susceptible, there is a natural death rate constant, and an excess death rate constant for infective individuals. Latent and immune periods are assumed to be constants, and the force of infection is assumed to be of the standard form, namely proportional to I(t)/N(t) where N(t) is the total (variable) population size and I(t) is the size of the infective population. The model consists of a set of integro-differential equations. Stability of the disease free proportion equilibrium, and existence, uniqueness, and stability of an endemic proportion equilibrium, are investigated. The stability results are stated in terms of a key threshold parameter. More detailed analyses are given for two cases, the SEIS model (with no immune period), and the SIRS model (with no latent period). Several threshold parameters quantify the two ways that the disease can be controlled, by forcing the number or the proportion of infectives to zero. Received 8 May 1995; received in revised form 7 November 1995     

Taurine chloramine modulates cytokine production by human peripheral blood mononuclear cells

Summary.  The effect of taurine (Tau) and taurine chloramine (Tau-Cl) on the production of TNF-α, IL-1β, and IL-6 by peripheral blood mononuclear cells of healthy volunteers was examined. Cells were stimulated with bacterial lipopolysaccharide (LPS) in the presence of either Tau or Tau-Cl. After 24 h culture the cytokine concentrations were measured in both culture supernatants (secreted) and cell lysates (cell-associated) using ELISA. In LPS-stimulated cells Tau-Cl inhibited both the secreted and cell-associated IL-1β and IL-6, while exerted dual effect on TNF-α production: raising it slightly at low and reducing at higher concentration. By contrast, Tau had no significant effect on the cytokine production. These results indicate that Tau-Cl modulates synthesis of pro-inflammatory cytokines, and therefore it may play a role in the initiation and propagation of immune response. Received November 29, 2001 Accepted January 18, 2002 Published online August 30, 2002 Acknowledgments This research was supported by grants from the State Committee for Scientific Research of Poland (No 4 P05B 01018) and the Institute of Rheumatology (No I/14). The Institute of Rheumatology is supported by a core grant from the State Committee for Scientific Research of Poland. Authors' address: Ewa Kontny, Ph.D., Department of Pathophysiology and Immunology, Institute of Rheumatology, Spartanska 1, 02-637 Warsaw, Poland, E-mail: zpatiir@warman.com.pl Abbreviations: Tau, taurine; Tau-Cl, taurine chloramine; LPS, lipopolysaccharide; TNF-α, tumor necrosis factor-α; IL-1β, interleukin 1β; IL-6, interleukin 6; PBMC, peripheral blood mononuclear cells     

L-NAME administration prevents the inhibition of nucleotide hydrolysis by rat blood serum subjected to hyperargininemia

Summary. The main objective of the present study was to evaluate the in vivo and in vitro effect of Arg on serum nucleotide hydrolysis. The action of N^ω-nitro-L-arginine methyl ester (L-NAME), an inhibitor of nitric oxide synthase, on the effects produced by Arg was also examined. Sixty-day-old rats were treated with a single or a triple (with an interval of 1 h between each injection) intraperitoneal injection of saline (group I), Arg (0.8 g/kg) (group II), L-NAME (2.0 mg/kg or 20 mg/kg) (group III) or Arg (0.8 g/kg) plus L-NAME (2.0 mg/kg or 20 mg/kg) (group IV) and were killed 1 h later. The present results show that a triple Arg administration decreased ATP, ADP and AMP hydrolysis. Simultaneous injection of L-NAME (20 mg/kg) prevented such effects. Arg in vitro did not alter nucleotide hydrolysis. It is suggested that in vivo Arg administration reduces nucleotide hydrolysis in rat serum, probably through nitric oxide or/and peroxynitrite formation. Both are first authors.     

The effects of the functional response on the bifurcation behavior of a mite predator–prey interaction model

 A predator–prey interaction model based on a system of differential equations with temperature-dependent parameters chosen appropriately for a mite interaction on apple trees is analyzed to determine how the type of functional response influences bifurcation and stability behavior. Instances of type I, II, III, and IV functional responses are considered, the last of which incorporates prey interference with predation. It is shown that the model systems with the type I, II, and III functional responses exhibit qualitatively similar bifurcation and stability behavior over the interval of definition of the temperature parameter. Similar behavior is found in the system with the type IV functional response at low levels of prey interference. Higher levels of interference are destabilizing, as illustrated by the prevalence of bistability and by the presence of three attractors for some values of the model parameters. All four systems are capable of modeling population oscillations and outbreaks. Received 12 March 1996; received in revised form 25 October 1996     

Effects of taurine on polymorphonuclear phagocytosis activity in burned patients

Summary.  This study determines the effects of taurine (Tau) on phagocytosis of polymorphonuclear neutrophils (PMN) isolated from normal subjects (n = 41) and severely burned patients (n = 20). Phagocytosis was measured by nitroblue of tetrazolium (NBT) reduction in samples with and without latex bead stimulation. Taurine was added at doses of 0.2, 0.4, 0.8 and 1.6 mM to stimulated samples. In control cells there were statistically significant increases in phagocytosis after addition of Tau 0.8 mM and 1.6 mM to as compared to samples without Tau addition (295 ± 23% and 330 ± 35% vs. 248 ± 18%; mean ± S.E.; p < 0.05). A statistically significant increase in phagocytosis was observed in cells from the burned population after addition of Tau 1.6 mM (288 ± 38% vs. 198 ± 13%; mean ± S.E.; p < 0.05). No changes in phagocytosis were found in cells from a subgroup of burn patients (n = 13) followed over 7, 15 and 21 days. These results indicate that taurine supplementation in vitro at doses of 0.8 to 1.6 mM improves the phagocytic capacity of neutrophils in healthy subjects and in patients with severe burn injury, mainly when neutrophil function is unaltered. Received December 17, 2001 Accepted January 17, 2002 Published online August 30, 2002 Authors' address: Dr. Mireia Farriol, Centre d'Investigacions Bioquímicas i Biología Molecular (CIBBIM), Hospital General Vall d'Hebron Passeig Vall d'Hebron 119-129, E-08035 Barcelona, Spain, Fax: 34-93-2746831, E-mail: farriol@hg.vhebron.es     

Competitive exclusion in a vector-host model for the dengue fever

 We study a system of differential equations that models the population dynamics of an SIR vector transmitted disease with two pathogen strains. This model arose from our study of the population dynamics of dengue fever. The dengue virus presents four serotypes each induces host immunity but only certain degree of cross-immunity to heterologous serotypes. Our model has been constructed to study both the epidemiological trends of the disease and conditions that permit coexistence in competing strains. Dengue is in the Americas an epidemic disease and our model reproduces this kind of dynamics. We consider two viral strains and temporary cross-immunity. Our analysis shows the existence of an unstable endemic state (‘saddle’ point) that produces a long transient behavior where both dengue serotypes cocirculate. Conditions for asymptotic stability of equilibria are discussed supported by numerical simulations. We argue that the existence of competitive exclusion in this system is product of the interplay between the host superinfection process and frequency-dependent (vector to host) contact rates. Received 4 December 1995; received in revised form 5 March 1996     

Cystine/glutamate exchange serves as the source for extracellular glutamate: modifications by repeated cocaine administration

Summary.  Repeated administration of cocaine lowers the basal extracellular levels of glutamate in the nucleus accumbens as measured by microdialysis. The studies presented reveal that this long-term neuroadaptation elicited by repeated cocaine results from a decrease in the activity of cystine/glutamate exchange. Received July 6, 2001 Accepted August 6, 2001 Published online June 17, 2002