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PubMed1984

A mathematical model of exoprotein production in bacteria.

We present a simple mathematical model for the synthesis of extracellular proteins by a class of bacteria which secrete significant quantities of this exoprotein in late-exponential and stationary phases. This model is the simplest generalization of Michaelis-Menten kinetics (the Monod model) and agrees well with laboratory experiments in batch culture. The model may serve as a simple prototype for the analysis of certain virulent bacterial infections in vivo, particularly that of Pseudomonas aeruginosa in burn wounds.

Bacteria↗

PubMed1993

A mathematical model of the human ventilatory response to isocapnic hypoxia.

A mathematical model of the ventilatory response to a period of sustained isocapnic hypoxia in humans has been developed. After a step into hypoxia, there is an initial rapid increase in ventilation (on-transient) followed by a slow decline. At the relief of hypoxia, there is a rapid decrease in ventilation (off-transient); the magnitude of this off-transient is smaller than that of the on-transient. Previously, the asymmetry between the on- and off-transients has been dealt with by modeling the steps into and out of hypoxia separately. The current objective was to model the whole of the response by allowing the peripheral sensitivity to hypoxia to decline during the sustained exposure to hypoxia. The model was fitted to breath-by-breath data from 20-min periods of hypoxia (end-tidal oxygen 50 Torr) at two different levels of end-tidal carbon dioxide tension from five subjects. The model was able to describe the features of the ventilatory changes well, including the slow decline and the asymmetry.

Carbon Dioxide↗

PubMedDate not supplied

[A mathematical model of the physiological estrous cycle in outbred rats].

The mathematical model of the physiological estrous cycle of the laboratory rat was developed. The data on physiological concentration of the hormones, participating in the estrous cycle regulation and admittances on the growth and selection of the ovarian follicles made the base of the model. The developed system of differentiated equations allowed to reproduce quiet precisely the duration of the estrous cycle phases, to follow up the behaviour of the apparent cohort of the growing ovarian follicles during several cycles. The model is a base for the studying of the effect of the different factors on the estrous cycle.

Animals↗

PubMed1979

Modification of a mathematical model for survival curves in photobiology.

A modification of the Haynes' mathematical model (1966) can be suggested as a result of bacterial UV-survival experiments carried out. A new equation is proposed: lnS = Fi(x), + Fr(x), wherein x is the radiation dose, Fi(x) = -- kox describes how radiation damage is produced on the cell and Fr(x) = (krec + kc)x + Re(x) describes how repair systems act. ko is the relative inactivation efficiency of radiation on each cell. krec is the recombination recovery system efficiency. kc gives the additional repair increasing per cell when excision and recombination systems act simultaneously. Re(x) = a(1-e-bs) describes solely the excision repair process. The asymptote of non-exponential curves intercepts the survival fraction axis on the ordinate so (extrapolated point). In the modified model a=lnSo instead of a=So (Haynes' model).

Bacteria↗

PubMed2001

A new mathematical model for relative quantification in real-time RT-PCR.

Use of the real-time polymerase chain reaction (PCR) to amplify cDNA products reverse transcribed from mRNA is on the way to becoming a routine tool in molecular biology to study low abundance gene expression. Real-time PCR is easy to perform, provides the necessary accuracy and produces reliable as well as rapid quantification results. But accurate quantification of nucleic acids requires a reproducible methodology and an adequate mathematical model for data analysis. This study enters into the particular topics of the relative quantification in real-time RT-PCR of a target gene transcript in comparison to a reference gene transcript. Therefore, a new mathematical model is presented. The relative expression ratio is calculated only from the real-time PCR efficiencies and the crossing point deviation of an unknown sample versus a control. This model needs no calibration curve. Control levels were included in the model to standardise each reaction run with respect to RNA integrity, sample loading and inter-PCR variations. High accuracy and reproducibility (<2.5% variation) were reached in LightCycler PCR using the established mathematical model.

Animals↗

PubMed1984

A mathematical model for a thermal clearance probe.

We introduce a new mathematical model for a thermal clearance probe for the measurement of skin blood flow. It is concluded that the technique is more sensitive to differences in the thermal conductivity of the skin than to differences in blood flow. The depth of measurement is also considered.

Body Temperature Regulation↗

PubMed1992

Evaluation of two unstructured mathematical models for the penicillin G fed-batch fermentation.

The mathematical model for the penicillin G fed-batch fermentation proposed by Heijnen et al. (1979) is compared with the model of Bajpai & Reuss (1980). Although the general structure of these models is similar, the difference in metabolic assumptions and specific growth and production kinetics results in a completely different behaviour towards product optimization. A detailed analysis of both models reveals some physical and biochemical shortcomings. It is shown that it is impossible to make a reliable estimation of the model parameters, only using experimental data of simple constant glucose feed rate fermentations with low initial substrate amount. However, it is demonstrated that some model parameters might be key factors in concluding whether or not altering the substrate feeding strategy has an important influence on the final amount of product. It is illustrated that feeding strategy optimization studies can be a tool in designing experiments for parameter estimation purposes.

Fermentation↗

PubMed2000

A mathematical model quantifying GnRH-induced LH secretion from gonadotropes.

A mathematical model is developed to investigate the rate of release of luteinizing hormone (LH) from pituitary gonadotropes in response to short pulses of gonadotropin-releasing hormone (GnRH). The model includes binding of the hormone to its receptor, dimerization, interaction with a G protein, production of inositol 1,4, 5-trisphosphate, release of Ca(2+) from the endoplasmic reticulum, entrance of Ca(2+) into the cytosol via voltage-gated membrane channels, pumping of Ca(2+) out of the cytosol via membrane and endoplasmic reticulum pumps, and release of LH. Cytosolic Ca(2+) dynamics are simplified (i.e., oscillations are not included in the model), and it is assumed that there is only one pool of releasable LH. Despite these and other simplifications, the model explains the qualitative features of LH release in response to GnRH pulses of various durations and different concentrations in the presence and absence of external Ca(2+).

Calcium↗

PubMed1990

Mathematical model of the cell cycle regulation in budding yeasts.

A mathematical model of the cell cycle regulation in S. cerevisiae is proposed. The model is based on the assumption of the G1----S phase transition control mediated by two signals. One of them is correlated with the cellular energy level--its messenger could be cAMP; the second one depends on the change of the cellular growth rate (reaching the critical size) and remains hypothetical.

Cell Cycle↗

PubMed1997

A mathematical model of breast and ovarian cancer treated with paclitaxel.

A mathematical model that describes the effects of cell-cycle-specific drugs on cancer and normal tissue is developed. The model takes into account the proliferating cells, which are sensitive to the treatment, and the quiescent cells, which are resistant to the treatment. With the use of information from the medical literature, model parameters are estimated for breast and ovarian cancer as well as for bone marrow. Then, with the use of the model and the estimated parameters, some acceptable treatment strategies are discussed in terms of treatment period, drug-infusion time, and proliferative fraction of cancer mass. Finally, these results are compared with current clinical practices for treatment with Taxol, and possible improvements on current treatment strategies are suggested.

Antineoplastic Agents, Phytogenic↗

PubMed1980

A mathematical model relating cohort and period mortality.

This paper presents a mathematical model for changing mortality in functional form. This model may be used to obtain cohort forces of mortality and cohort survivorship functions from a period force of mortality and a period life table under conditions of gradually changing mortality if an estimate of the amount of change in mortality is available. An example is given to show how the cohort functions are derived from the period functions.

Age Factors↗

PubMed1991

On the application of mathematical models of schistosome transmission dynamics. I. Natural transmission.

The many mathematical models of the transmission dynamics of schistosomes that have been published since 1965 have had little impact on field studies or on the design of schistosome control programmes. At least in part, this is due to limited interaction between theoretician and field worker, resulting in unrealistic models that are not easily applied to field data. This review aims to make explicit the assumptions and limitations of existing models and their relationships with field data. A basic model is described which considers the mean number of schistosomes per person and the prevalence of patent infections of snails. Various modifications to this model are introduced. These include: prepatent infections of snails; loss of infection of snails; the effects of snail population dynamics; the effects of miracidia and cercariae population dynamics; miracidia searching efficiency; reservoir hosts; heterogeneous patterns of transmission; seasonality; and predisposition to infection. Variation in levels of infection with age and the effects of acquired immunity to infection are also considered. Published models of schistosome transmission dynamics are reviewed within this framework. Approaches to the modelling of schistosome control measures are considered in a companion paper. It is suggested that future theoretical studies give greater attention to the details of snail population dynamics, heterogeneous patterns of transmission and the effects of acquired immunity. There is a need for field studies explicitly designed to provide estimates of transmission parameters and for studies of the epidemiological effects of acquired immunity.

Age Factors↗

PubMed1999

Mathematical models of tumour and normal tissue response.

The historical application of mathematics in the natural sciences and in radiotherapy is compared. The various forms of mathematical models and their limitations are discussed. The Linear Quadratic (LQ) model can be modified to include (i) radiobiological parameter changes that occur during fractionated radiotherapy, (ii) situations such as focal forms of radiotherapy, (iii) normal tissue responses, and (iv) to allow for the process of optimization. The inclusion of a variable cell loss factor in the LQ model repopulation term produces a more flexible clonogenic doubling time, which can simulate the phenomenon of 'accelerated repopulation'. Differential calculus can be applied to the LQ model after elimination of the fraction number integers. The optimum dose per fraction (maximum cell kill relative to a given normal tissue fractionation sensitivity) is then estimated from the clonogen doubling times and the radiosensitivity parameters (or alpha/beta ratios). Economic treatment optimization is described. Tumour volume studies during or following teletherapy are used to optimize brachytherapy. The radiation responses of both individual tumours and tumour populations (by random sampling 'Monte-Carlo' techniques from statistical ranges of radiobiological and physical parameters) can be estimated. Computerized preclinical trials can be used to guide choice of dose fractionation scheduling in clinical trials. The potential impact of gene and other biological therapies on the results of radical radiotherapy are testable. New and experimentally testable hypotheses are generated from limited clinical data by exploratory modelling exercises.

Brachytherapy↗

PubMed1986

Population dynamics in echinococcosis and cysticercosis: mathematical model of the life-cycle of Echinococcus granulosus.

A mathematical model of the life-cycle of Echinococcus granulosus in dogs and sheep in New Zealand is constructed and used to discuss previously published experimental and survey data. The model is then used to describe the dynamics of transmission of the parasite, and the means by which it may be destabilized. It is found that under the conditions that prevailed in New Zealand during the late 1950s, at the time of surveys of this parasite, the dog-sheep life-cycle was not regulated by any effective density-dependent constraint. In contrast there was evidence for an effective acquisition of immunity to reinfection by cattle. The long time to maturity of the cyst in sheep, together with the practice of feeding aged sheep to dogs, provides a time delay in the intermediate host. By comparison, the time to maturity of the adult stage in dogs is short, but it is of sufficient magnitude to be a key factor in the destabilization of the system by a regular dog-dosing programme. The model used to describe the life-cycle is a linear integrodifferential equation of the Volterra type. Such equations are intrinsically unstable in that a small perturbation in parameters can drive a previous equilibrium solution to zero. At the time of the surveys, the value of the basic reproductive rate, R0, was close to 1, and it has since been reduced below 1 by control measures.

Age Factors↗

PubMed1986

[Erythrocyte hemolysis by detergents--a mathematical model and analysis of the concentration and kinetic curves].

A mathematical model of erythrocyte lysis by detergents is developed which takes into consideration the kinetics of detergent binding to plasma membrane. Experimentally obtained sigmoidal kinetic and concentration curves of hemolysis are well described by the model. A comparative study is carried out in terms of the model of hemolytic action for five detergents: Triton X-100, sodium dodecylsulfate, sodium deoxycholate, cetyltrimethylammonium bromide, and cetylpyridinium chloride. The amount of detergent which should be bound to an erythrocyte membrane to induce lysis was found to be roughly the same for all detergents studied. However, detergents vary in their affinity to the membrane. Cetylpyridinium displays the highest affinity (and consequently the highest hemolytic activity), whereas deoxycholate has the least one.

Detergents↗

PubMed1986

A mathematical model of the flow in the circle of Willis.

A mathematical model of the flow in the circle of Willis has been designed and the effects of (a) the large anatomical variation of the communicating arteries and (b) physiological changes of the resistances of the vertebral arteries have been studied. The influence of the posterior perforating arteries on the flow in the posterior communicating arteries has been investigated as well, with special attention being paid to the possible occurrence of a 'dead point'. In the model, the influence of diameters of the communicating arteries on the flow in the afferent vessels and the segments of the circle turns out to be considerable, especially in the range of the anatomical variation of the diameters. Within this range flow reductions due to an increased resistance of the vertebral artery will be compensated for by the system. Assuming that the values and ratios of the peripheral resistances are within the physiological range, a dead point is not to be expected in the flow in the posterior communicating arteries.

Cerebrovascular Circulation↗

PubMedDate not supplied

[Mathematical model of an immune response. II. Stochastic aspects].

In the mathematical model describing the development of infection and its suppression with antibodies worked out of the course of the delayed immune response a problem of complete destruction of antigen is considered. A method of calculating the probabilities of antigen destruction is advanced. The optimal cure tactics is discussed. It is shown that the highest probability of the destruction of antigene is achieved if the serum is injected in the moment of antibodies peak and when the cure with antibiotics is started at the antigen maximum.

Anti-Bacterial Agents↗

PubMed2004

A two-dimensional mathematical model of percutaneous drug absorption.

BACKGROUND: When a drug is applied on the skin surface, the concentration of the drug accumulated in the skin and the amount of the drug eliminated into the blood vessel depend on the value of a parameter, r. The values of r depend on the amount of diffusion and the normalized skin-capillary clearance. It is defined as the ratio of the steady-state drug concentration at the skin-capillary boundary to that at the skin-surface in one-dimensional models. The present paper studies the effect of the parameter values, when the region of contact of the skin with the drug, is a line segment on the skin surface. METHODS: Though a simple one-dimensional model is often useful to describe percutaneous drug absorption, it may be better represented by multi-dimensional models. A two-dimensional mathematical model is developed for percutaneous absorption of a drug, which may be used when the diffusion of the drug in the direction parallel to the skin surface must be examined, as well as in the direction into the skin, examined in one-dimensional models. This model consists of a linear second-order parabolic equation with appropriate initial conditions and boundary conditions. These boundary conditions are of Dirichlet type, Neumann type or Robin type. A finite-difference method which maintains second-order accuracy in space along the boundary, is developed to solve the parabolic equation. Extrapolation in time is applied to improve the accuracy in time. Solution of the parabolic equation gives the concentration of the drug in the skin at a given time. RESULTS: Simulation of the numerical methods described is carried out with various values of the parameter r. The illustrations are given in the form of figures. CONCLUSION: Based on the values of r, conclusions are drawn about (1) the flow rate of the drug, (2) the flux and the cumulative amount of drug eliminated into the receptor cell, (3) the steady-state value of the flux, (4) the time to reach the steady-state value of the flux and (5) the optimal value of r, which gives the maximum absorption of the drug. The paper gives valuable information which can be obtained by this two-dimensional model, that cannot be obtained with one-dimensional models. Thus this model improves upon the much simpler one-dimensional models. Some future directions of the work based on this model and the one-dimensional non-linear models that exist in the literature, are also discussed.

Humans↗