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At least 145 records · Page 8Linked to original sources
PubMed1996

On the dangers of adjusting the parameters values of mechanism-based mathematical models.

Mechanism-based mathematical models describe systems in terms of identifiable physical processes, and the parameters are assumed to have fundamental physical significance. Ideally, the parameter values are measured independent of the system being modeled, but these values are often adjusted to give the best fit of model predictions to experimental data. A systematic investigation of the effects of such parameter adjustment was conducted by developing a model system comprising a known reaction mechanism and known rate constants. Simulations of experiments were run, and then attempts were made to model the system under a variety of problematic, but realistic, conditions. (1) When one rate constant was seriously in error, adjustment of a different rate constant gave the greatest improvement in the model fit. (2) When a contaminant was present in the experiment, the effects could be hidden by the adjustment of the rate constants. (3) When an incorrect reaction mechanism was assumed, the error could be hidden by parameter adjustment if the concentrations of only one of the reacting species were considered or if an unweighted fit was used for the optimization. (4) Parameter values adjusted for one set of experimental conditions gave a poorer fit than did the unadjusted parameter values when attempting to model a new set of experimental condition (addition of an inhibitor). These results show the potential dangers of adjusting parameter values and the importance of measuring as many variables as possible in a complex system.

Animals↗
PubMed1997

Mathematical modeling of biofilms.

A set of mathematical equations constitutes a mathematical model if it aims to represent a real system and is based on some theory of that system's operation. On this definition, mathematical models, some very simple, are everywhere in science. A complex system like a biofilm requires modeling by numerical methods and, because of inevitable uncertainties in its theoretical basis, may not be able to make precise predictions. Nevertheless, such models almost always give new insight into the mechanisms involved, and stimulate further investigation. The way in which diffusion coefficients are measured for use in a model, particularly whether they include effects of reversible reaction, is a key element in the modeling. Reasons are given for separating diffusion from reversible reaction effects and dealing with them in a separate subroutine of the model.

Bacterial Adhesion↗
PubMed1999

[Short-term dynamics of intertidal microphytobenthos biomass. Mathematical model].

We formulate a deterministic mathematical model to describe the dynamics of the microphytobenthos of intertidal mudflats. It is 'minimal' because it only takes into account the essential processes governing the functioning of the system: the autotrophic production, the active upward and downward migrations of epipelic microalgae, the saturation of the mud surface by a biofilm of diatoms and the global net loss rates of biomass. According to the photic environment of the benthic diatoms inhabiting intertidal mudflats, and to their migration rhythm, the model is composed of two sub-systems of ordinary differential equations; they describe the simultaneous evolution of the biomass 'S' concentrated in the mud surface biofilm--the photic layer--and of the biomass 'F' diluted in the topmost centimetre of the mud--the aphotic layer. Qualitatively, the model solutions agree fairly well with the in situ observed dynamics of the S + F biomass. The study of the mathematical properties of the model, under some simplifying assumptions, shows the convergence of solutions to a stable cyclic equilibrium, whatever the frequencies of the physical synchronizers of the production. The sensitivity analysis reveals the necessity of a better knowledge of the processes of biomass losses, which so far are uncertain, and may further vary in space and time.

Biomass↗
PubMed1995

From passive diffusion to active cellular migration in mathematical models of tumour invasion.

Mathematical models of tumour invasion appear as interesting tools for connecting the information extracted from medical imaging techniques and the large amount of data collected at the cellular and molecular levels. Most of the recent studies have used stochastic models of cell translocation for the comparison of computer simulations with histological solid tumour sections in order to discriminate and characterise expansive growth and active cell movements during host tissue invasion. This paper describes how a deterministic approach based on reaction-diffusion models and their generalisation in the mechano-chemical framework developed in the study of biological morphogenesis can be an alternative for analysing tumour morphological patterns. We support these considerations by reviewing two studies. In the first example, successful comparison of simulated brain tumour growth with a time sequence of computerised tomography (CT) scans leads to a quantification of the clinical parameters describing the invasion process and the therapy. The second example considers minimal hypotheses relating cell motility and cell traction forces. Using this model, we can simulate the bifurcation from an homogeneous distribution of cells at the tumour surface toward a nonhomogeneous density pattern which could characterise a pre-invasive stage at the tumour-host tissue interface.

Brain↗
PubMed1977

Fetal O2 changes in response to hypoxic stress: a mathematical model.

We developed a mathematical model to compute the time course of PO2 changes in fetal blood vessels during hypoxia. The model represents the circulation and major organs as a system of paths and nodes. We calculated outflow O2 for a path from its inflow O2 content and its distribution of transit times. The O2 content at a given node equals the flow-weighted sum of O2 delivered by different paths. Placental O2 transfer and organ O2 consumption are related to their arterial PO2 levels. We simulated the effects of uterine contractions with Gaussian-shaped decreases in placental O2 transfer. Increasing the intensity and increasing the duration of hypoxic episodes have comparable effects. Liver O2 consumption decreases more than that of other organs during hypoxic episodes. At the peak of a contraction, fetal systemic PO2 values decrease only about one-fourth as much as those in end-capillary placental blood. This indicates that despite rapid circulation times, fetal O2 reserves protect it against severe, short term hypoxia.

Brain↗
PubMed1996

The role of N-methyl-D-aspartate (NMDA) receptors in wind-up: a mathematical model.

We present a mathematical model for the phenomenon of wind-up (Mendell, 1966, Exper. Neur. 16,316-22) which occurs in many neurons. We concentrate on its occurrence in the substantia gelatinosa of the dorsal horns of the spinal cord, where it is connected with certain pathological and nonpathological pain states. The model is a development of the model by Britton & Skevington (1989, J. Theor. Biol. 137, 91-105) for Melzack & Wall's gate control theory of pain (1965, Science, New York, 150, 971-9; 1982, The Challenge of Pain, Penguin: Harmondsworth), modified to take account of more recent information. Its variables are the electric potentials of various cells in the midbrain and the spinal cord. Britton & Skevington's original model simulated many of the phenomena observed in acute pain in humans, but not the wind-up mechanism. This is not surprising, since this model did not include the N-methyl-D-aspartate (NMDA) receptors that are now recognized as being crucial to the phenomenon. Here we rectify this omission, and obtain good agreement between the model and experimental data on wind-up. The positive feedback that NMDA receptors exhibit is shown to be the essential feature in producing wind-up. As an independent test of the model we simulate a completely different experimental set-up, and obtain good qualitative agreement with data there. Finally, we present a prediction of the model that has yet to be tested experimentally.

Animals↗
PubMed1988

Biological exposure index of styrene suggested by a physiologico-mathematical model.

We used a physiologico-mathematical model to study the biological exposure index of styrene correlated to the Threshold Limit Value (TLV) suggested by the ACGIH for 1986-87. This model allows the solvent concentrations in blood, alveolar air, fat tissue, and in other biological media to be estimated and simultaneously the kinetics of its metabolites to be followed when a specific exposure is settled. The comparison between the results obtained from the mathematical model and the numerous research projects documented in the literature suggests a reciprocal validation. Moreover, some biological parameters (particularly the alveolar ventilation) can explain the variability of results obtained from studies concerning the solvent pollution of the factories, which used biological monitoring. The ranges of styrene concentrations in blood and alveolar air and the urinary concentrations of its metabolites (mandelic and phenylglioxylic acids) are discussed in connection with the exposure at 215 mg/m3. Important differences correlated to the definition of set-levels of TLV and Biological Exposure Index (BEI) have been found: particularly the TLVs lead to different solvent uptakes according to some biological parameters; the BEI can better explain the individual solvent uptake and body burden.

Environmental Exposure↗
PubMedDate not supplied

[A mathematical model of postradiation autoimmunity].

A mathematical model of cellular autoimmune process in exposed mammals was developed. In terms of this model a study was made of the dependence of the autoimmunity kinetics on radiation dose and radiosensitivity of autologous tissues. The model simulates the experimentally observed dynamics of autoimmune diseases.

Animals↗
PubMed1982

A mathematical model of iron metabolism.

A mathematical model of iron metabolism is presented. It comprises the following iron pools within the body: transferrin-bound iron in the plasma, iron in circulating red cells and their bone marrow precursors, iron in mucosal, parenchymal and reticuloendothelial cells. The control exerted by a hormone, called erythropoietin, on bone marrow utilization of iron for hemoglobin synthesis is taken into account. The model so obtained consists of a system of functional differential equations of retarded type. Most model parameters can be estimated from radiotracer experiments, others can be measured and numerical values can be assigned to the remaining ones making few reasonable assumptions according to the available physiological knowledge. Iron metabolism behavior under different therapeutical treatments was stimulated. Model predictions were compared to experimental data collected in clinical routine.

Animals↗
PubMed1985

Phase resetting and annihilation in a mathematical model of sinus node.

A mathematical model of primary sinoatrial pacemaker activity was developed using modifications of Hodgkin-Huxley type equations of voltage- and time-dependent membrane currents. The computer simulation of action potential activity incorporates the results of several existing cardiac models and recent biological data in an attempt to generate a model that more closely approximates the phase-resetting behavior of sinoatrial pacemakers observed biologically in response to subthreshold or electrotonic stimuli. The model was also used to study annihilation, i.e., the cessation of rhythmic activity induced by a critically timed subthreshold stimulus. Perturbation analysis performed by scanning the pacemaker cycle with 50-ms subthreshold current pulses yielded biphasic phase-response relationships that closely resembled the biological data. The pacemaker current, although relatively unimportant in determining the degree of phase 4 depolarization in the simulated sinus nodal pacemaker, was nevertheless prominent in determining phase-resetting behavior. The characteristics of annihilation were studied in normal, "depressed," and hyperpolarized states. In all cases, successful annihilation depended in large part on the dynamic interaction between the slow inward and delayed rectifier outward currents. The annihilation point was found to be an unstable singularity point in which a critical combination of stimulus intensity and timing could cause the pacemaker to approach but never achieve a state of complete quiescence. The model provides the basis for further investigation of complex arrhythmias that may arise as a consequence of multiple-pacemaker interaction within the heart.

Mathematics↗
PubMed1990

A single-tube mathematical model of reactive hyperaemia.

A mathematical model of reactive hyperaemia is developed using quasi-steady flow in a single tube to represent blood flow in the vascular bed. The role of the myogenic response during reactive hyperaemia is examined by suggesting a linear relationship between tube cross-sectional area S and pressure p, in which S decreases as p increases, thereby modelling the response of the smooth muscle in the blood vessel walls to increases in p which the myogenic mechanism proposes. However, this simple relationship, together with the equations of continuity and Poiseuille flow, lead to an unstable equation for p which is inconsistent with the known boundary conditions. It is necessary to make S a function of p and delta p/delta t in order to achieve a stable response which implies that the myogenic response must be rate sensitive to pressure changes. The resulting equations are then solved for p, S, and flow Q by numerical integration and give results for Q which are in broad agreement with experiment. The model also suggests that the changing pressure gradient governs the flow in reactive hyperaemia rather than changes in the resistance of the blood vessels.

Humans↗
PubMed1990

Population dynamics of Trichostrongylus colubriformis in sheep: mathematical model of worm fecundity.

A mathematical model was constructed to predict the egg production of Trichostrongylus colubriformis worms as a function of worm age and host experience of infection. The model set egg production at zero until the worm was 14 days old, when a linear increase to maximum egg production levels occurred over 7 days. It was assumed that egg production remained at maximum levels until a threshold total worm burden was exceeded, when an exponential decline in egg production occurred. The rate of decline was assumed independent of worm age or worm burden. The estimated parameters (maximum egg production, threshold, lag and rate of decline) were optimized by fitting values predicted from the model to faecal egg counts observed in continuously infected sheep, giving R2 = 0.80. The model was validated against faecal egg counts obtained in two other continuous infection experiments, one performed at the same laboratory and the other in Britain.

Animals↗
PubMedDate not supplied

[Mathematical model of bilirubin transport].

The mathematical model of the bilirubin transport through liver is suggested. The model is sufficiently reflects the peculiarities of the bilirubin behavior under the different physiological conditions. The model is described by the system of difference-differential equations. The coefficients of the intensity of bilirubin transfer in normal liver are calculated.

Bilirubin↗
PubMed1984

Mathematical model of visual perception.

A mathematical model of visual perception is presented with the intention of throwing some light on the problem of perceptual invariance. Two types of differential manifolds (receptive and effector) are associated with the repertoire which is the fundamental concept in the model. The elements of the repertoire carry weights which control the input-output relation in the repertoire and which can be modified by a learning process. It is shown that, under reasonable conditions, these repertoires possess good stability properties and can adjust to the various environments to which they may be subjected. In particular cases, it is shown that the stochastic learning process can be considered as deterministic to a first approximation.

Animals↗
PubMed1990

A mathematical model for insulin kinetics. III. Sensitivity analysis of the model.

A non-linear mathematical model involving four variables and several constants incorporating beta-cell kinetics, a glucose-insulin feedback system and a gastrointestinal absorption term had been applied in earlier papers to various forms of diabetes mellitus. In this paper, we examine the response of the system to variations in the parameters and to initial conditions using sensitivity analysis. It is found that such a method leads to results that are consistent with clinical findings. Further, it is suggested that such an analysis could help in making some predictions regarding future directions in the therapy of diabetes mellitus.

Diabetes Mellitus↗
PubMedDate not supplied

Insights from mathematical modeling of renal tubular function.

Mathematical models of proximal tubule have been developed which represent the important solute species within the constraints of known cytosolic concentrations, transport fluxes, and overall epithelial permeabilities. In general, model simulations have been used to assess the quantitative feasibility of what appear to be qualitatively plausible mechanisms, or alternatively, to identify incomplete rationalization of experimental observations. The examples considered include: (1) proximal water reabsorption, for which the lateral interspace is a locus for solute-solvent coupling; (2) ammonia secretion, for which the issue is prioritizing driving forces - transport on the Na+/H+ exchanger, on the Na,K-ATPase, or ammoniagenesis; (3) formate-stimulated NaCl reabsorption, for which simple addition of a luminal membrane chloride/formate exchanger fails to represent experimental observation, and (4) balancing luminal entry and peritubular exit, in which ATP-dependent peritubular K+ channels have been implicated, but appear unable to account for the bulk of proximal tubule cell volume homeostasis.

Ammonia↗
PubMedDate not supplied

Mathematical models and their applications in medicine and health.

Mathematical models have great potentialities as regards their utility in different disciplines of medicine and health. This paper attempts to elucidate their uses in the field. A brief mention of some models has also been made. Mathematical models are useful in epidemiologic research, planning and evaluation of preventive and control programmes, clinical trials, measurement of health, cost-benefit analysis, diagnosis of patients and in maximizing effectiveness of operations aimed at attaining specified goals within existing resources.

Health Services Research↗
PubMed1998

Distributing working versions of published mathematical models for biological systems via the Internet.

Mathematical models are useful tools for investigating complex systems. By representing physiological systems as models, theories can be tested quantitatively against data from the system. Models can be used to explore new theories prior to experimentation and to design studies to optimize experimental resources. They can also be used as teaching tools to illustrate physiochemical principles. In spite of their usefulness and the time invested in developing models, published models are often underused due to the difficulty in obtaining working versions of the model. To address this problem we have designed a library for mathematical models of biological systems on the Internet. The library contains published models of biological systems in formats compatible with several modeling packages, from the fields of physiology, metabolism, endocrinology, biochemistry, and chemistry. The models can be viewed graphically, model solutions can be viewed as plots against data, and models can be downloaded to be run with software on the user's own system. The address of the library is: http://biomodel.georgetown.edu/model/ Investigators are invited to submit working versions of published models to the library. Models can be submitted electronically at the time a manuscript is accepted for publication. As journals go online, articles containing models can be linked to working versions of the models in the library. By increasing access to working versions of models, more of the investment in kinetic studies and model development can be realized.

Computer Simulation↗