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.     

Efficient solution of ordinary differential equations modeling electrical activity in cardiac cells

The contraction of the heart is preceded and caused by a cellular electro-chemical reaction, causing an electrical field to be generated. Performing realistic computer simulations of this process involves solving a set of partial differential equations, as well as a large number of ordinary differential equations (ODEs) characterizing the reactive behavior of the cardiac tissue. Experiments have shown that the solution of the ODEs contribute significantly to the total work of a simulation, and there is thus a strong need to utilize efficient solution methods for this part of the problem. This paper presents how an efficient implicit Runge-Kutta method may be adapted to solve a complicated cardiac cell model consisting of 31 ODEs, and how this solver may be coupled to a set of PDE solvers to provide complete simulations of the electrical activity.     

Dissipative resonance. An analytical solution with fixed boundaries

A simple one-dimensional model is presented in which the dissipative resonance phenomenon takes place. An analytical solution for steady-state oscillations occurring in the system due to the external periodic force was obtained. An analytical solution describing the dynamics of the particles-acceptors of the external force is also presented. Some common properties of such systems are considered. The method can be used to explore a wide class of models in which the dissipative resonance occurs and which can be used to describe the mechanisms of action of weak electromagnetic fields to biological and physiochemical objects.     

An analytical solution for the radial and tangential displacements on a thin hemispherical layer of articular cartilage

A simplified analytical solution has been obtained for the radial and tangential displacements on the surface of a thin, hemispherical layer of porous-elastic articular cartilage firmly bonded to a rigid foundation. A static pressure distributed according to a paraboloid of revolution is applied simulating cartilage compression by a porous indenter. The solution method is in the form of an asymptotic series and uses Laplace transforms. The analytical predictions are in qualitative agreement with the behaviour of biphasic articular cartilage reported in the literature. A direct comparison with numerical simulations using commercially available Finite Element Modelling (FEM) software was also carried out for conditions relevant to natural hip joints and the results show a good quantitative agreement overall.     

Modelling autonomous oscillations in the human pupil light reflex using non-linear delay-differential equations

Neurophysiological and anatomical observations are used to derive a non-linear delay-differential equation for the pupil light reflex with negative feedback. As the gain or the time delay in the reflex is increased, a supercritical Hopf bifurcation occurs from a stable fixed point to a stable limit cycle oscillation in pupil area. A Hopf bifurcation analysis is used to determine the conditions for instability and the period and amplitude of these oscillations. The more complex waveforms typical of the occurrence of higher order bifurcations were not seen in numerical simulations of the model. This model provides a general framework to study the different types of dynamical behaviors which can be produced by the pupil light reflex, e.g. edge-light pupil cycling.     

An analytical solution for percutaneous drug absorption: application and removal of the vehicle

The methods of Laplace transform were used to solve a mathematical model developed for percutaneous drug absorption. This model includes application and removal of the vehicle from the skin. A system of two linear partial differential equations was solved for the application period. The concentration of the medicinal agent in the skin at the end of the application period was used as the initial condition to determine the distribution of the drug in the skin following instantaneous removal of the vehicle. The influences of the diffusion and partition coefficients, clearance factor and vehicle layer thickness on the amount of drug in the vehicle and the skin were discussed.     

Correction: Utility of a single adjusting compartment: a novel methodology for whole body physiologically-based pharmacokinetic modelling

After our work was published, we found that some of the terms in the equations were incorrect and that there were some typographical errors in the abbreviations.     

High-pressure hydrocephalus: a novel analytical modeling approach

Hydrocephalus is an abnormal accumulation of cerebrospinal fluid (CSF) within ventricles and subarachnoid space (SAS) as a result of disturbances in secretion or absorption procedures. It is believed that arachnoid villi cells, which are microscopic projections of pia-arachnoid mater that extend into venous channels in sagittal sinus, are the main sites for CSF absorption, but it is tempting to speculate that a significant portion of CSF is removed from the SAS by nasal lymphatic vessels around olfactory nerve. Thus, in this paper, we propose an analytical model of CSF-lymphatic-blood circulation, in which these two output pathways for CSF absorption have been considered. Mathematical relations governing the pressures in different interacting compartments of the brain are considered. In addition, for increasing the similarity of our model to the physiological conditions, the bulk flow mechanism, which is supposed to occur during CSF absorption, has been considered in our model. We used our model to simulate hydrocephalus. The results indicate that the lymphatic disorders have more considerable effect in decreasing CSF absorption, compared to the disturbances in arachnoid villi cells. Based on our modeling, we believe that disorders in lymphatic pathway may be a cause of high-pressure hydrocephalus. Surely experimental studies are required to validate our hypothesis.     

An analytical model for a short-term advantage for sex

An analytical treatment is given for a model of Maynard Smith in which a short-term advantage for sex and recombination is provided by the mechanism of sib-competition. Suppose the next generation is formed by the winners of a large number of contests. Suppose a number of parents each contribute M offspring to a given contest, but the offspring of an asexual parent are identical whereas those of a sexual parent are distributed with some variance. If M is large there is a high probability that a sexual offspring will have a high enough fitness to win the contest. Calculations show that values of M around 3 and 4 are generally enough for sexual behaviour to overcome its two-fold disadvantage.     

Minimum domains for spatial patterns in a class of reaction diffusion equations

We study a general class of scalar reaction/interacting population diffusion equations in two space dimensions: convective terms, due to wind, are included. We consider boundary conditions which include a measure of the hostility to the species in the exterior of the domain. The main point of the paper is to obtain estimates for the minimum domain size which can sustain spatially heterogeneous structures and indicate the type of patterns which could appear.     

An analytical solution countercurrent heat transfer between parallel vessels with a linear axial temperature gradient

Presented in this paper is a solution for countercurrent heat exchange between two parallel vessels embedded in an infinite medium with a linear temperature gradient along the axes of the vessels. The velocity profile within the vessel is assumed to be parabolic. This solution describes the temperature field within the vessels, as well as in the tissue, and establishes that the intravessel temperature is not uniform, as is generally assumed to be the case. An explicit expression for the intervessel thermal resistance based on the difference between cup-mixed mean temperatures is derived.     

An exact solution of transient equations describing slow axonal transport

An exact analytical solution of equations describing slow axonal transport of cytoskeletal elements (CEs) injected in an axon is presented. The equations modelling slow axonal transport are based on the stop-and-go hypothesis. The simplest model implementing this hypothesis postulates that CEs switch between pausing and running kinetic states, and that the probabilities of CE transition between these two states are described by first-order rate constants. It is assumed that initially CEs are injected such that they form a uniform pulse of a given width. All injected CEs are initially attributed to the pausing state. It is shown that within 30 s kinetic processes redistribute CEs between pausing and running states; after that the process occurs under quasi-equilibrium conditions. The parameter accessible to experiments is the total concentration of CEs (pausing plus running). As the initial rectangular-shaped pulse moves, it changes its shape to become a bell-shaped wave that spreads out as it propagates. The wave's amplitude is decreasing during the wave's propagation. It is also shown that the system forgets its initial condition, meaning that if one starts with pulses of different widths, after sometime they converge to the same bell-shaped wave.     

An approximate solution to the periodic bidomain equations in one dimension

An approximate, computationally tractable solution is proposed for the potentials in the bidomain model with periodic intracellular junctions (the periodic bidomain model). This new approach is based on the one-dimensional rigorous spectral method described previously by Trayanova and Pilkington (IEEE Trans. Biomed. Eng., May 1993). The total solution to the one-dimensional periodic bidomain problem is decomposed in the spectral domain into solutions to (1) the single-fiber classical bidomain problem in which the intracellular conductivity value incorporates the average contribution from cytoplasm and junction and (2) the “junctional” potential problem due to the presence of junctions at discrete locations alone. Solving for the junctional term rigorously requires most of the numerical effort in the solution for the periodic bidomain potentials. Here the junctional potential is found approximately with little numerical effort. A comparison between the rigorous and the approximate solutions serves as a justification for the proposed approximate solution procedure. The procedure outlined in this paper is applicable to higher spatial dimensions where both tissue anisotropy and junctional inhomogeneities play a role in establishing the transmembrane potential distribution.     

An environ analysis for nonlinear compartment models

Environ analysis, an input-output analysis for models of ecological systems, has been previously formulated for linear systems. This note has a twofold purpose: first, we indicate that a variation of parameters technique can be applied, at least in principle, to computeboth input and output environs; and second, we show that this technique may be used for computation of environs in nonautonomous, nonlinear compartment models. This nonlinear theory, obtained as a direct extension of dynamical system developments, allows the traditional environ partitioning of compartmental storages and flows. An example of a nonlinear nutrient-producer-consumer system whose output environs can be computed asymptotically is presented to illustrate these concepts. This research was supported by the U.S. Environmental Protection Agency under cooperative agreement R806727030.     

Errors-in-variables in joint population pharmacokinetic/pharmacodynamic modeling

Pharmacokinetic (PK) models describe the relationship between the administered dose and the concentration of drug (and/or metabolite) in the blood as a function of time. Pharmacodynamic (PD) models describe the relationship between the concentration in the blood (or the dose) and the biologic response. Population PK/PD studies aim to determine the sources of variability in the observed concentrations/responses across groups of individuals. In this article, we consider the joint modeling of PK/PD data. The natural approach is to specify a joint model in which the concentration and response data are simultaneously modeled. Unfortunately, this approach may not be optimal if, due to sparsity of concentration data, an overly simple PK model is specified. As an alternative, we propose an errors-in-variables approach in which the observed-concentration data are assumed to be measured with error without reference to a specific PK model. We give an example of an analysis of PK/PD data obtained following administration of an anticoagulant drug. The study was originally carried out in order to make dosage recommendations. The prior for the distribution of the true concentrations, which may incorporate an individual's covariate information, is derived as a predictive distribution from an earlier study. The errors-in-variables approach is compared with the joint modeling approach and more naive methods in which the observed concentrations, or the separately modeled concentrations, are substituted into the response model. Throughout, a Bayesian approach is taken with implementation via Markov chain Monte Carlo methods.     

An approximate analytical solution to the lag period of monophenolase activity of tyrosinase

Tyrosinase shows a lag period in its action on monophenols (l-tyrosine). We propose an approximate analytical solution for the lag period, which fulfils the dependences with regard to initial enzyme concentration, and initial monophenol concentration. Furthermore, from a study of the dependences of the lag period on these variables, we can determine experimentally the o-diphenol concentration in the steady state. The Michaelis constant of the o-diphenol in the presence of the monophenol can be determined from the relationship between the o-diphenol concentration in the steady state and the initial monophenol concentration, taking into consideration the experimentally calculated Michaelis constant for the monophenol substrate. Although this Michaelis constant is much lower than the Michaelis constant for diphenol in the absence of monophenol, the binding site is the same. A kinetic analysis of the action mechanism of tyrosinase explains this difference in the values of the Michaelis constants.