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A theory of drug tolerance and dependence II: the mathematical model.

The preceding paper presented a model of drug tolerance and dependence. The model assumes the development of tolerance to a repeatedly administered drug to be the result of a regulated adaptive process. The oral detection and analysis of exogenous substances is proposed to be the primary stimulus for the mechanism of drug tolerance. Anticipation and environmental cues are in the model considered secondary stimuli, becoming primary in dependence and addiction or when the drug administration bypasses the natural-oral-route, as is the case when drugs are administered intravenously. The model considers adaptation to the effect of a drug and adaptation to the interval between drug taking autonomous tolerance processes. Simulations with the mathematical model demonstrate the model's behaviour to be consistent with important characteristics of the development of tolerance to repeatedly administered drugs: the gradual decrease in drug effect when tolerance develops, the high sensitivity to small changes in drug dose, the rebound phenomenon and the large reactions following withdrawal in dependence. The present paper discusses the mathematical model in terms of its design. The model is a nonlinear, learning feedback system, fully satisfying control theoretical principles. It accepts any form of the stimulus-the drug intake-and describes how the physiological processes involved affect the distribution of the drug through the body and the stability of the regulation loop. The mathematical model verifies the proposed theory and provides a basis for the implementation of mathematical models of specific physiological processes.

Adaptation, Physiological↗

Mathematical modelling in nuclear medicine.

Modern imaging techniques can provide sequences of images giving signals proportional to the concentrations of tracers (by emission tomography), of X-ray-absorbing contrast materials (fast CT or perhaps NMR contrast), or of native chemical substances (NMR) in tissue regions at identifiable locations in 3D space. Methods for the analysis of the concentration-time curves with mathematical models describing the physiological processes and the appropriate anatomy are now available to give a quantitative portrayal of both structure and function: such is the approach to metabolic or functional imaging. One formulates a model first by defining what it should represent: this is the hypothesis. When translated into a self-consistent set of differential equations, the model becomes a mathematical model, a quantitative version of the hypothesis. This is what one would like to test against data. However, the next step is to reduce the mathematical model to a computable form; anatomically and physiologically realistic models account of the spatial gradients in concentrations within blood-tissue exchange units, while compartmental models simplify the equations by using the average concentrations. The former are known as distributed models and the latter as lumped compartmental or mixing chamber models. Since both are derived from the same ideas, the parameters are usually the same; their differences are in their ability to represent the hypothesis correctly, quantitatively, and sometimes in their computability. In this essay we review the philosophical and practical aspects of such modelling analysis for translating image sequences into physiological terms.

Computer Simulation↗

[A mathematical model of the biomechanics of respiration during artificial ventilation of the lungs].

The authors review a mathematic model of the respiratory biomechanics during artificial lung ventilation, presenting a subsystem of the mathematic model of respiration designed at the All-Union Research Surgery Center of the USSR AMS and used in the medical information diagnostic system elaborated for the Department of Resuscitation and Intensive Care, provide differential equations of the mathematic model, the results of the numerical solution of the model and an example of using the model for solving a private problem of selecting an adequate mode of artificial lung ventilation.

Biomechanical Phenomena↗

Mathematical models of cumulative effect and optimization of fractionation regimes.

Different (most known) mathematical models are shortly described and basic assumptions the individual models are based on are discussed and critically examined. Advantages and shortages of individual models are mentioned. A semiphenomenological model is then used to demonstrate some possibilities how to make use of the mathematical models in attempts of optimizing the fractionation approaches in individual cases.

Dose-Response Relationship, Radiation↗

On the use of mathematical models of malaria transmission.

The key conclusions of several mathematical models of malaria are reviewed with emphasis on their relevance for control. The Ross-Macdonald model of malaria transmission has had major influence on malaria control. One of its main conclusions is that endemicity of malaria is most sensitive to changes in mosquito imago survival rate. Thus malaria can be controlled more efficiently with imagicides than with larvicides. An extension of this model shows that the amount of variability in transmission parameters strongly affects the outcome of control measures and that predictions of the outcome can be misleading. Models that describe the immune response and simulate vaccination programs suggest that one of the most important determinants of the outcome of a vaccine campaign is the duration of vaccine efficacy. Apparently malaria can be controlled only if the duration of efficacy is in the order of a human life-span. The models further predict that asexual stage vaccines are more efficient than transmission-blocking vaccines. Directions for further applications of mathematical models are discussed.

Animals↗

[A mathematical model for predicting the efficacy of glucocorticoid therapy of glomerulonephritis].

A mathematical model is proposed for prediction of the efficacy of glucocorticoid therapy developed on the basis of the Bayes theorem and successive Wald's analysis. The model uses the retrospective values of the results of radioligand determination of the number of glucocorticoid receptor of lymphocytes and static renal scintigraphy. By means of the blind method the informative value of the mathematical model to predict the inefficacy of glucocorticoid therapy in nephrotic glomerulonephritis was 96%. A nomogram was developed.

Algorithms↗

[Mathematical modeling of the interaction of local anesthetics with the surface of nerve fiber biomembranes].

Theoretical analysis and mathematical modelling of conductor anesthesia has been performed. It has been established that mathematical models explicity taking into account the form and the size of molecules (through molar volumes) and the energy of intermolecular interaction with biomembrane surface of a nerve fiber (through normal boiling temperatures) are the most close to electrophysiology data obtained by measuring of minimal blocking concentrations of anesthetics in inter- or intracell solutions, causing complete isolation of a pain spike in the fiber. Computation based on an improved additive systematics produced physical-chemical descriptors for construction of mathematical models. The determined parameters conform to experimental data in crucial features molar volumes and normal boiling temperatures for analyzed compounds. Predictions possibilities and restrictions of suggested approach for search for new effective anesthetics and structures with higher indices of biological activity has been analyzed.

Anesthetics, Local↗

Mathematical modeling of biofiltration in activated pine-bark charge of a biofilter.

AIM, SCOPE AND BACKGROUND: Human economic activities cause emissions of various pollutants of an organic nature: butanol, butyl acetate, methanol, formaldehyde, phenol, benzene, toluene, xylene, etc. These compounds are emitted to atmosphere by various enterprises of food, chemistry, wood processing industries, from transportation means, agricultural enterprises, etc. Therefore, when purifying air from these pollutants, it is necessary to apply efficient and inexpensive air purification methods. In this dimension, the biological air purification is chosen from all possible air cleaning methods. An experimental biofilter with the activated charge of pine bark was developed at the Department of Environment Protection of the Vilnius Gediminas Technical University. In the course of the experimental investigation, it was determined that this air purification method is efficient. Filter efficiency, when purifying air of volatile organic compounds (butanol, butyl acetate and xylene), to a great extent, depending on the nature and concentrations (up to 100 mg/m3) of pollutants injected, might go up to 70-98%. The mathematical model of the biofilter was developed based on the research results and fully taking into consideration physical, chemical, and biological processes going on during its operations. MAIN FEATURES: The aim of this article is to determine biodegradation constant alpha, absorption capacity beta, and half empiric expressions of filter efficiency. Knowing this, it is possible to find out the dependence of the filter efficiency on the operational parameters of the filter (i.e. on the concentrations and the height of biocharge of the initial pollutants (butanol, butyl acetate, xylene) fed through it). CONCLUSIONS: With the help of mathematical modeling, the biodegradation constants and absorption capability of volatile organic compounds (butanol, butyl acetate, and xylene) fed into the biofilter charged with the activated pine bark and used for the cleaning of volatile organic compounds, as well as the efficiency of the biofilter in half empiric expression, have been established. It has been discovered that the constant of pollutant biodegradation alpha is a value inverse to the time during which the amount of pollutants in the filter becomes n times higher. It is rather complicated to carry out theoretical calculation of the biodegradation constant at a molecular level, therefore this constant has been established based on the results obtained in the course of research. The equations describing pollutant dynamics in the filter charge and the air cleaning processes going on in it have been derived from diffusion equations in a mobile medium. The modeling helped to find out the absorption capacity beta of the examined pollutants, which by its numeric value is equal to the volume unit of the absorbed gas amount. The latter factor, the same as the biodegradation constant, was determined basing on the experimental results. Mathematical modeling brought a range of formulas expressing dependences of each pollutant's efficiency on its initial concentrations and filter charge height. RECOMMENDATIONS/OUTLOOK: Based on the experimental data, a mathematical model has been developed which will allow the measuring of the filter efficiency not only with regard to the absorption and biodegradation of the pollutants under examination, but also with regard to other pollutants and their compounds, etc., having an impact on the filter performance. The results of the mathematical modeling have revealed that the modeling of processes going on in the filter is much simpler than isthe performance of long and costly experiments. The developed mathematical model makes it possible to measure the filter efficiency at the present moment.

Air Pollutants↗

Quantitative evaluation of hemodialysis therapy using a simple mathematical model and a programmable pocket calculator.

1. A single pool mathematical model has been clinically tested and found to give values similar to those previously reported for volumes of distribution of creatinine and urea. 2. Calculated generation rates for creatinine and urea approximated values obtained by independent measurement of removal rate. 3. Preliminary observations suggest that the model may be empirically useful in predicting interdialytic serum creatinine urea concentrations. 4. The potential clinical usefulness of the mathematical model has been enhanced by development of a solution suitable for a programmable pocket calculator.

Adult↗

Concerted regulation of all hyphal tips generates fungal fruit body structures: experiments with computer visualizations produced by a new mathematical model of hyphal growth.

Filamentous hyphal growth is inherently suited to kinetic analysis, and in many respects the fungal mycelium can be viewed as a very mechanical biological system, which lends itself to mathematical modelling. The mathematics of hyphal tip extension growth are well-established. However, even though a hyphal growth equation can be written with confidence, and we have a good understanding of the effects of tropisms on growth, it is not easy to form a mental picture of the behaviour of large populations of hyphal tips. What is required, and what we believe we have produced, is a mathematical model that is sufficiently sophisticated to produce a realistic visualization of fungal hyphal growth. This provides us with a cyberfungus that can be used for experimentation on the theoretical rules that might govern hyphal patterning, hyphal interactions, and tissue formation and organ development by actually visualizing the virtual hyphal growth patterns that result from different regulatory scenarios. From a series of model experiments the most significant observation is that complex fungal fruit body shapes can be simulated by applying the same regulatory functions to all of the growth points active in a structure at any specific time. No global control of fruit body geometry is necessary. No localized regulation is necessary. The shape of the fruit body emerges from the concerted response of the entire population of hyphal tips, in the same way, to the same signals.

Computer Graphics↗

A mathematical model representing the extraneuronal O-methylating system of the perfused rat heart.

1. A mathematical model was developed to mimic the function of the extraneuronal O-methylating system of the rat heart. Its essential features are: a saturable uptake process (uptake 2), a saturable, intracompartmental enzyme (COMT), the ability of the catecholamine to penetrate the membrane of the model compartment by a diffusional flux obeying first-order kinetics, and the ability of the metabolite to leave the compartment by an efflux obeying first-order kinetics. 2. Of the six kinetic constants of the model compartment five are known from experiments with hearts perfused with 3H-isoprenaline (Kmuptake, Vmaxuptake, Vmaxenzyme, k for amine, k for metabolite); only one constant is unknown (Kmenzyme) for the intact heart cells. 3. Results calculated with the help of the mathematical model were compared with results obtained from rat hearts perfused with 3H-isoprenaline. Although full congruency of results cannot be expected, there was satisfactory agreement between the two sets of results. Apparently, the mathematical model is able to simulate the function of the O-methylating system of the rat heart. 4. Comparison of the two sets of results leads to a definition of the function of the O-methylating system of the perfused rat heart. if all cells of the rat heart participate in the O-methylating system, the Km of the COMT of intact heart cells must be very low (i.e., somewhere between 2 and 5 microM isoprenaline). However, if the O-methylating system comprises only a small fraction of all cells, the COMT of the intact heart cells may well have a correspondingly higher Km.

Animals↗

Mathematical modelling of the composting process: a review.

In this paper mathematical models of the composting process are examined and their performance evaluated. Mathematical models of the composting process have been derived from both energy and mass balance considerations, with solutions typically derived in time, and in some cases, spatially. Both lumped and distributed parameter models have been reported, with lumped parameter models presently predominating in the literature. Biological energy production functions within the models included first-order, Monod-type or empirical expressions, and these have predicted volatile solids degradation, oxygen consumption or carbon dioxide production, with heat generation derived using heat quotient factors. Rate coefficient correction functions for temperature, moisture, oxygen and/or free air space have been incorporated in a number of the first-order and Monod-type expressions. The most successful models in predicting temperature profiles were those which incorporated either empirical kinetic expressions for volatile solids degradation or CO2 production, or which utilised a first-order model for volatile solids degradation, with empirical corrections for temperature and moisture variations. Models incorporating Monod-type kinetic expressions were less successful. No models were able to predict maximum, average and peak temperatures to within criteria of 5, 2 and 2 degrees C, respectively, or to predict the times to reach peak temperatures to within 8 h. Limitations included the modelling of forced aeration systems only and the generation of temperature validation data for relatively short time periods in relation to those used in full-scale composting practice. Moisture and solids profiles were well predicted by two models, but oxygen and carbon dioxide profiles were generally poorly modelled. Further research to obtain more extensive substrate degradation data, develop improved first-order biological heat production models, investigate mechanistically-based moisture correction factors, explore the role of moisture tension, investigate model performance over thermophilic composting time periods, provide more information on model sensitivity and incorporate natural ventilation aeration expressions into composting process models, is suggested.

Biodegradation, Environmental↗

Mathematical modelling of lipid production by oleaginous yeasts in continuous cultures.

A mathematical model was constructed to describe the influence of the carbon to nitrogen ratio (C/N-ratio) of the growth medium on lipid production by oleaginous yeasts. To test this model and to determine some relevant model parameters, the oleaginous yeast Apiotrichum curvatum ATCC 20509 was grown in continuous cultures at various C/N-ratios and dilution rates. It appeared that when nitrogen is limiting for the formation of biomass, the remaining glucose can be converted to storage carbohydrate and storage lipid. No clear dependence of carbohydrate yield on the C/N-ratio could be demonstrated, but lipid yield increased gradually with increasing C/N-ratios. The maximal dilution rate for lipid producing yeast cells appeared to be optimal at relatively low C/N-ratios. It can be concluded that the experimental results fitted well with the mathematical model. By using this model, lipid yield and lipid production rate can be calculated at any C/N-ratio of the growth medium and optimum operation conditions can be predicted for the production of microbial lipids.

Candida↗

On the solution of mathematical models of herd immunity in human helminth infections.

The general solution of the mathematical model of herd immunity to human helminth infections recently proposed by Anderson and May is obtained. The numerical solution of a more accurate biological model is indistinguishable from the corresponding exact solution of a more tractable mathematical model. Computer simulations of some particular cases of this model support the notion that both ecological and immunological factors determine the observed convex patterns of age-prevalence and age-intensity curves of human helminth infections.

Animals↗

Mathematical model of antiviral immune response. I. Data analysis, generalized picture construction and parameters evaluation for hepatitis B.

The present approach to the mathematical modelling of infectious diseases is based upon the idea that specific immune mechanisms play a leading role in development, course, and outcome of infectious disease. The model describing the reaction of the immune system to infectious agent invasion is constructed on the bases of Burnet's clonal selection theory and the co-recognition principle. The mathematical model of antiviral immune response is formulated by a system of ten non-linear delay-differential equations. The delayed argument terms in the right-hand part are used for the description of lymphocyte division, multiplication and differentiation processes into effector cells. The analysis of clinical and experimental data allows one to construct the generalized picture of the acute form of viral hepatitis B. The concept of the generalized picture includes a quantitative description of dynamics of the principal immunological, virological and clinical characteristics of the disease. Data of immunological experiments in vitro and experiments on animals are used to obtain estimates of permissible values of model parameters. This analysis forms the bases for the solution of the parameter identification problem for the mathematical model of antiviral immune response which will be the topic of the following paper (Marchuk et al., 1991, J. theor. Biol. 15).

Hepatitis B↗

A heuristic mathematical model for the dynamics of sensory conflict and motion sickness.

The etiology of motion sickness is now usually explained in terms of a qualitatively formulated "sensory conflict" hypothesis. By consideration of the information processing task faced by the central nervous system in estimating body spatial orientation and in controlling active body movement using an "internal model" referenced control strategy, a mathematical model for sensory conflict generation is developed. The model incorporates and extends models proposed by von Holst, Held, and Reason, and is congruent with multisensory models for spatial orientation developed by Young and coworkers. The model postulates a major dynamic functional role for sensory conflict signals in movement control, as well as in sensory-motor adaptation. It accounts for the role of active movement in creating motion sickness symptoms in some experimental circumstances, and in alleviating them in others. The relationship between motion sickness produced by "sensory rearrangement" and that resulting from external motion disturbances is explicitly defined. A nonlinear conflict averaging model is proposed which describes dynamic aspects of experimentally observed subjective discomfort sensation, and suggests resulting behaviours. The model admits several possibilities for adaptive mechanisms which do not involve internal model updating. Further systematic efforts to experimentally refine and validate the model are indicated.

Conflict, Psychological↗

A mathematical model of thrombopoiesis in rats.

A mathematical model of thrombopoiesis in rats is presented. This has four compartments; stem cells, megakaryocytes, thrombocytes and thrombopoietin. A high thrombopoietin concentration influences bone marrow proliferation in three ways. Firstly the stem cells are stimulated and a slow increase in megakaryocyte number follows. Secondly there are additional endomitoses in the (early) megakaryocytes resulting in an increase in megakaryocyte volume. Thirdly the megakaryocyte maturation time is shortened. The parameters of the model are determined from experimental values for the normal, maximum and minimum proliferation rates, maturation times and destruction rates. The model is tested by comparing simulated results for acute and chronic thrombocytopenia and thrombocytosis with experimental curves from the literature. The model and data agree within the limits of experimental error. Not all of the thrombopoietic regulatory system is known yet, so some important alternative hypotheses are investigated and compared with the model. Several hypotheses have been excluded in this way.

Animals↗

Interaction between opioid and muscarinic receptors in the guinea-pig ileum preparation: a mathematical model.

Fentanyl and pethidine are opioid agonists and muscarinic antagonists in the guinea-pig ileum preparation. In this preparation an opioid agonist reduces the release of acetylcholine. Therefore an opiate may influence the potency of an anticholinergic drug. A mathematical model was developed to characterize this putative interaction between opioid and muscarinic receptors. The model is based on the assumption that the drugs interact with the receptors in a competitive manner according to the law-of-mass action. Concentration-response experiments were performed in the guinea-pig ileum preparation to test the model. The mathematical model describes the concentration-response curves very well and estimates the IC50 values for the two components with good precision. The study shows that an opioid agonist can potentiate the effect of an anticholinergic drug substantially. This is interesting with regard to the central anticholinergic syndrome. The conclusion is that the model describes the interaction adequately.

Animals↗