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PubMed2000

Mathematics and the brain: uncharted territory?

The prevalence of disorders of arithmetic skills in children of circa 6 % calls for intensive consideration of the subject by health care providers and researchers. The necessity of interdisciplinary cooperation is evident. The current classifications by the ICD-10 (Specific disorder of arithmetic skills) and DSM-IV (Mathematics disorder) represent different viewpoints. A developmental dyscalculia is exclusively diagnosed according to clinical presentation. Arithmetic and its disorders are brain functions, determined and influenced by the (cerebral) human development. With modem neuroimaging techniques, a connectionistic model of distinct arithmetic functions has been specified. The results of single case descriptions of arithmetic disorders correspond well in that context with the concept of parallel cortico-subcortical and subcortico-cortical neuronal circuits. In comparison with dyslexia, there is intense need for research in the field of arithmetic disorders. Only through analysis and understanding of the complex cerebral connections in arithmetic can improved diagnostic as well as multidimensionally therapeutic approaches to arithmetic disorders be achieved, possibly opening new ways of academic and social integration of the affected individuals.

Brain↗
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↗
PubMed1985

A comparison of sheep and human fetal oxygen delivery systems with use of a mathematical model.

Human fetal cardiac output measured with ultrasound is only about 60% of that found in the sheep. We modified a previously developed mathematical model of the fetal circulation and oxygen delivery in sheep for the human in order to study several differences. The model predicts that a human fetus can maintain its oxygen delivery with a relatively low cardiac output because of its relatively high fetal hemoglobin concentration, as compared with that of the sheep fetus. Thus an inverse relationship between fetal hemoglobin concentration and fetal cardiac output is suggested. This relationship may be mediated by the influence of red blood cell concentration on blood viscosity. Furthermore, it indicates that fetal anemia should be detectable by ultrasound measurements of increased cardiac output and/or umbilical blood flow. Dynamic responses of the model suggest that the mechanism of late and variable decelerations in the fetal heart rate pattern is mediated via a fall in arterial oxygen tension.

Animals↗
PubMed1990

Binding of substrates to cytochrome P450 enzymes: mathematical artifacts in the application of difference spectrophotometry.

Investigation of the difference absorption spectrum has been a common method for studying many aspects of hemoproteins such as cytochrome P450. The difference spectrophotometric technique, as applied in studies on binding of substrates to cytochrome P450, has resulted in certain mathematical artifacts which have been mistaken for real changes in binding equilibria. The purpose of this paper is to point out these artifacts.

Binding Sites↗
PubMed1986

Oscillatory response of photosynthesis in leaves to environmental perturbations: a mathematical model.

Oscillations in the yield of chlorophyll fluorescence, in oxygen evolution, and in CO2 uptake observed with leaves upon perturbation of steady-state conditions are suggested to be due to the interdependence of turnover of adenylates and Calvin cycle intermediates. This suggestion is quantified in a mathematical model; the behavior of the model system in the neighborhood of the singular point of the system is analyzed. The linearized system is solved analytically, a condition for the occurrence of oscillations is given, and explicit expressions for the oscillation period and the damping constant are derived. The model is shown to be capable of exhibiting oscillations with the period observed with algae or leaves, whereas calculated values of the damping constant are higher than those measured for leaves or algae.

Adenosine Triphosphate↗
PubMed1985

A mathematical model to study short-term regulation of mitochondrial energy transduction.

A mathematical model is presented which includes the following elementary process of mitochondrial energy transduction: hydrogen supply, proton translocation by the respiratory chain, proton-driven ATP synthesis by the F0F1-ATPase, passive back-flow of protons (leak) and carrier-mediated exchange of adenine nucleotides and phosphate. For these processes empirical rate laws are used. The model is applied to calculate time-dependent states of energy transduction in isolated rat liver mitochondria. From the general agreement of the computational results with experimental data (Ogawa, S. and Lee, T.M. (1984) J. Biol. Chem. 259, 10004-10011) the following conclusions can be drawn. (1) The length of the time interval during which mitochondria are able to maintain a relatively high and constant delta pH in the absence of oxygen (anaerobiosis) is limited by the availability of intramitochondrial ATP. (2) The overshoot kinetics of delta pH which appear when reoxigenating mitochondria after a preceeding anaerobiosis might be due to a lag phase kinetics of the F0F1-ATPase. (3) In phosphorylating mitochondria the homeostasis of delta pH is brought about by a high sensitivity of the respiration rate and the rate of the F0F1-ATPase as to changes of delta pH. (4) Analysis of the mean transient times shows that the rate of ATP synthesis in State 3 is controlled to almost the same extent by the hydrogen supply, the respiratory chain, the adenine nucleotide translocator and the proton leak.

Adenine Nucleotides↗
PubMed1979

Mathematical modelling of ATP, K+ and Na+ interactions with (Na+ + K+)-ATPase occurring under equilibrium conditions.

The controlling effect of ATP, K+ and Na+ on the rate of (Na+ + K+)-ATPase inactivation by 7-chloro-4-nitrobenzo-2-oxa-1,3-diazole (NBD-C1) is used for the mathematical modelling of the interaction of the effectors with the enzyme under equilibrium conditions. 1. Of a series of conceivable interaction models, designed without conceptual restrictions to describe the effector control of inactivation kinetics, only one fits the experimental data described in a preceding paper. 2. The model is characterized by the coexistence of two binding sites for ATP and the coexistence of two separate binding sites for K+ and Na+ on the enzyme-ATP complex. On the basis of this model, the effector parameters fitting the experimental data most closely are estimated by means of nonlinear least-squares fits. 3. The apparent dissociation constants for ATP fo the enzyme-ATP complex or of the enzyme-(ATP)2 complex are computed to lie near 0.0024 mM and 0.34 mM, respectively, irrespective of whether K+ and Na+ were absent or K+ and K+ plus Na+, respectively, were present in the experiments. 4. The origin of the high and the low affinity site for binding of ATP to the (Na+ + K+)-ATPase molecule is traced back to the coexistence of two catalytic centres which, although primarily equivalent as to the reactivity of their thiol groups with NBD-C1, are induced into anticooperative communication by ATP binding and thus show an induced geometric asymmetry. 5. On the basis of the interaction model outlined under item 2 the apparent dissociation constant for K+ or Na+ in the (K+ + Na+)-liganded enzyme-ATP complex are computed to be 1.7 mM and 3.5 mM, respectively. 6. The conclusions concerning the coexistence of two primarily equivalent but anticooperatively interacting catalytic centres and the coexistence of two separate ionophoric centres for Na+ and K+ correspond to the appropriate basic postulates of the flip-flop concept of (Na+ + K+)-ATPase mechanism.

4-Chloro-7-nitrobenzofurazan↗
PubMed1984

A mathematical model of the kinetics of 5-fluorouracil and its catabolites in freshly isolated rat hepatocytes.

A mathematical model for the kinetics of 5-fluorouracil (FUra) catabolism in liver cells is proposed. It is based on published data for the metabolism of FUra by isolated rat hepatocytes. The model relies on biochemical knowledge of the catabolic pathway. The key-steps are: the cellular uptake and the conversion of the unchanged drug to dihydrofluorouracil (FUH2) and subsequently to alpha-fluoro-beta-alanine (FBAL); the cellular fluxes of the 2 catabolites, FUH2 and FBAL. Water is partitioned between the extracellular and intracellular spaces. The first step is described by Michaelis-Menten kinetics and the other processes by first-order kinetics. Satisfactory fitting of the model validates these simplifications and provides values for the parameters describing the process. The model indicates that the kinetics of FUra disappearance are non linear, the Vmax of the first step being between 3.1 and 5.0 microM/min and the Km between 12 and 37 microM; the rate limiting step is the degradation of FUH2 (the major intracellular catabolite) with a rate constant of 0.1 to 0.02 min-1; the FUH2 transmembrane exchange is active; the exchange of the final catabolite FBAL is by diffusion.

Animals↗
PubMed1993

A mathematical model of the kinetics and tissue distribution of 2-fluoro-beta-alanine, the major catabolite of 5-fluorouracil.

2-Fluoro-beta-alanine (FBAL) is the major metabolite of 5-fluorouracil (FUra), one of the most widely used anticancer drugs. It has been suggested previously that FBAL and/or its metabolites may have a role in the hepatotoxicity, neurotoxicity and cardiotoxicity resulting from FUra chemotherapy. Studies in patients and experimental animals have demonstrated that FBAL has a prolonged elimination compared with the parent drug, FUra. In the present manuscript, a mathematical model is developed for the kinetics and tissue distribution of FBAL. This model is based on recently published data from a study of the pharmacokinetics and disposition of FBAL in rats (Zhang et al., Drug Metab Dispos 20: 113-119, 1992). Satisfactory agreement was achieved between predicted and measured values, permitting an accurate evaluation of the kinetic and distribution parameters for FBAL. This model indicates that: (1) FBAL accumulates in several tissues including brain, heart, spleen, and enterohepatic system; and (2) enterohepatic circulation of FBAL and its bile acid conjugates has an important role in FBAL kinetics and distribution as demonstrated by a model in which enterohepatic circulation parameters were deleted.

Animals↗
PubMed1989

Static and dynamic computerised radioactive tracer studies, vital dye staining and theoretical mathematical calculations to ascertain the mode of survival of single cephalad channel venous island flaps.

Experiments were performed to ascertain the perfusion and drainage of single cephalad channel saphenous venous flaps in dogs. In the first experiment a standard saphenous flap and a single cephalad channel saphenous venous island flap were dissected on contralateral thighs of dogs. A radioactive tracer, 99mTc, was injected into a foreleg vein and its appearance in both flaps was monitored with the use of a static and dynamic scanner. After this experiment, Evans blue, a vital dye, was injected into the foreleg vein of the dog. Five minutes later both flaps were excised and the dye in the flaps was assayed quantitatively. In the third experiment radioactive tracer was injected into the single cephalad channel venous flap subcutaneously and its decay in the flap as well as its uptake in the rest of the body were measured dynamically. The experiments show that the single cephalad channel venous island flap is perfused by and drains through its single cephalad vein. A mathematical model to justify the above conclusion was worked out independently, based on standard values involved in mammalian circulatory mechanisms as in dog and man.

Animals↗
PubMed1983

Mathematical studies of complement activation.

Kinetic studies of complement activation were followed by hemolytic assay. Mathematical analysis shows that the curve is composed of two exponents: the first one, which occurs during a short span of time, represents the classical pathway, the second the alternative pathway. We were therefore able to foretell the respective participation of each activator used: inulin, zymosan, and aggregated immunoglobulins.

Complement Activation↗
PubMed1984

A mathematical insulin-secretion model and its validation in isolated rat pancreatic islets perifusion.

A mathematical insulin-secretion model was constructed by analyzing the glucose-induced insulin secretion using a control theory. To validate the model, experimental data obtained in the isolated rat pancreatic islet perifusion system were compared with the simulation results. The simulation study of the model was found to fit quite well to insulin responses under various experimental conditions. A glucose-induced insulin secretion from pancreatic islets' beta cell was found to be expressed as a function of proportional plus derivative response to glucose input with the first order time lag. And the negative rate of change in glucose concentration was found to suppress insulin secretion to the extent of the negative stimuli, regardless of glucose concentration. This model would be useful as a research tool for analysis of physiological significance of insulin secretory dynamics and clinical investigation of insulin secretion.

Animals↗
PubMed1984

Mathematical modelling of the growth processes in the developing chick wing bud.

This paper illustrates how a simple geometric model resembling the shape of the chick wing bud at an early growth stage can be mathematically expanded to simulate subsequent growth characteristics of the developing bud. The model was tested against several sets of experimental data and gave an acceptable representation of growth over the range considered. Representing growth patterns in this form enables the determination of differential growth characteristics in different parts of the bud and provides boundary constraints which will play an important part in the eventual evaluation of internal growth mechanisms.

Animals↗
PubMed1985

A mathematical model to quantitate histamine release from basophils challenged in vitro with insect venom.

Using a two parameter mathematical model, it was possible to describe clinically obtained data from histamine release (HR) in vitro as a response to basophils challenged by insect venom. The Gompertz function used in this model not only reflects the observed HR data but can be used to predict maximal HR values from initial lower antigen concentrations HR determinations. The model was used to calculate the concentration of venom antigens releasing 25% of total intracellular histamine (HR25). This is clinically significant in that while knowledge of the HR25, or any other reference fraction, facilitates comparisons between allergic patients, it is impossible to empirically administer venom antigen concentrations that will produce exactly HR25 in a given clinical setting.

Adult↗
PubMed1995

An exact mathematical expression for describing competitive binding of two different ligands to a protein molecule.

The dissociation constant for the binding of a spectroscopically invisible or non-radioactive ligand to its protein receptor can be determined in a competition experiment by using a structural analog that contains a reporter group. Many plotting and numerical analysis methods have been developed to calculate the binding constant of unlabeled ligand from the displacement experiments. However, a common problem with these plotting methods is that the equation transformations inevitably result in non-standard error distribution, and thus simple linear regression can not be used to extract correct values for the parameters. In the case of the numerical analysis methods, one would be faced with the possible existence of multiple solutions. In this paper, the exact mathematical expression for describing competitive binding of two different ligands to a protein molecule is presented in terms of the total concentrations of species in the system. Thus, using a commercially available non-linear regression program, all unknown parameters for describing this system can be determined by fitting the experimental data to the algebraically explicit equation without any data transformations. The distribution curves of all the species in the system can also be constructed with this equation. It is particularly useful for the cases in which the concentrations of all the species in the system are comparable to each other.

Binding, Competitive↗
PubMed1976

Electrocochleography by mathematical modelling.

The aim of this work is to critically review some of the classical electrocochleographic methods in accordance with the systems theory: (a) in describing, by means of mathematical models, the different parts of the auditory system concerned in the generation of stimulus-evoked signals; and: (b) in comparing the outputs of these models with the measurements of the cochlear potentials. It must be stated clearly that we realise that many of the conclusions drawn could be criticised, it is hoped that future research in the field with be stimulated by this work.

Adaptation, Physiological↗
PubMed1979

A non-linear mathematical model for the in vivo determination of Kupffer cells number and rate of phagocytosis of radiocolloids in rats.

In order to perform a quantification of the hepatic RES function, a mathematical model of the colloids phagocytosis was constructed and validated in normal and partially hepatectomised rats. The experimental design consisted of the collection of successive blood samples for the measurement of radiocolloids time courses after the injection of different doses of gelatin colloids (0.075--5 mg of gelatin/100 g body wt). The unknown parameters were estimated by method of maximum likelihood using a second order algorithm. A good fit between experimental and simulated data was obtained for a large range of injected doses using a single set of parameters. Comparisons of parameters values between normal and hepatectomised rats were found to be -onsistent with the hepatectomy ratio. This computerised estimation of parameters provides a determination of both the total number of Kupffer cells and the mean time of a complete phagocytosis cycle which cannot be obtained by classical approaches.

Animals↗
PubMed1982

[Mathematical model of a learning process by biofeedback].

The aim of Biofeedback is to teach a subject to control some of his biological functions. This experimental method was numerous therapeutic applications. General laws of these particular learning processes have not yet been well understood. Mathematical models can be used in order to make evident these laws can lead to an optimization of the learning processes. The stochastic model presented here underlines the shortcomings of certain experimental procedures. Although it has been performed from one of our particular experiments, it can be applied to analogous experiments.

Biofeedback, Psychology↗