Good manners in good modeling: mathematical models and computer simulations of physiological systems.
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Results for “mathematical modelling”
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Development of phage populations in a bacterial culture: a mathematical model.
A mathematical model for the interaction kinetic in phage-bacterium culture is proposed. An appropriate analytical relationship between phage and bacterial concentrations has been derived. Characteristic kinetic constants have been obtained by comparing the experimental growth curves with computer-simulation analysis. The model allows also to evaluate the phage-yield as a function of the initial concentrations.
Bacterial Physiological Phenomena↗
[Double-frequency oscillation in the glycolytic system. Mathematical model].
A mathematical model describing the generation mechanism of double-frequency oscillations in the glycolytic system is proposed. Interaction of two connected glycolytic oscillation generators are put in the basis of this mechanism. It is assumed in the model that the first oscillation generator is formed due to product activation of phosphofructokinase (PFK) with adenosine diphosphate (ADP), while the second one is based on substrate inhibition of glyceraldehydephosphatededhydrogenase (GAPDH) with glyceraldehydephosphate (GAP).
Adenosine Diphosphate↗
[Substrate inhibition as a cause of oscillations in an open irreversible enzymic reaction S1 + S2 in the presence of E(R,T) leads to S1' + S2'. A mathematical model].
A mathematical model of an open irreversible reaction S1 + S2 (formula: see text) catalysed by an olygomeric enzyme E(R, T) has been analysed. It is assumed that the enzyme undergoes the concerted conformational transitions R in equilibrium T in conformity with the theory of Monod, Wyman and Changeux, and one of the substrates (S2) produces inhibition of the enzyme, thus shifting the equilibrium between the two enzyme forms in the direction of T formation. A simple graphical explanation is given to the hysteresis of the input characteristic approximately v ([S1]) (approximately v is the reaction rate at d[S2]/dt=O) which gives rise to self--oscillations. The hysteresis occurs both in the case of allosteric and isosteric substrate inhibition.
Allosteric Regulation↗
Carbon monoxide exchanges between the human fetus and mother: a mathematical model.
A mathematical model was developed to calculate maternal and fetal carboxyhemoglobin concentrations, [HbCO], as functions of time during and after exposure of the mother to various inspired CO concentrations. Effects of variation in alveolar ventilation rates, pulmonary and placental fiffusing capacities, cardiac output, endogenous carbon monoxide production and other factors were studied. Following a change in the inspired CO concentration, fetal HbCO lags behind maternal HbCO by several hours. During CO uptake, fetal HbCO eventually overtakes maternal, and approaches an equilibrium value as much as 10% higher than the mother's. During CO washout the fetal levels again lag behind the mothers. Results indicate that treatment of pregnant women who have elevated HbCO levels with 100% oxygen reduces the time necessary to reduce the maternal HbCO level as expected, but that the rate of fetal CO elimination is not increased as much as that of the mother. Changes in maternal and fetal HbCO were also calculated for a representative exposure to changing inspired CO levels produced by fluctuating levels of air pollution. Finally, the effects of carboxyhemoglobin on fetal oxygenation were studied, including the effects of high altitude and exercise.
Carbon Monoxide↗
[Nerve impulse conduction along myelinated fibers while internodal vary (mathematical model)].
The mathematical model of a myelinated fibre (Hille, 1971) was used to study the dependence of the velocity of nerve impulse propagation (theta) and of some parameters of the action potential on the properties of internodes. Calculations have shown that with increasing of the length (L) of internodes over the range of 0.75-3 mm, theta rises and then declines; in the fibre with the external diameter, D = 14 mu the maximum of theta falls on L = 1.5 mm. With decreasing of d/D (d = internal diameter of the fibre) at expense of D (simulation of the myelin sheath thickening) theta grows up monotonically, while the safety factor N (defined as the ratio of the potential (V) in the 6th node to V in the 8th node at a moment when V in the 8th node reaches its maximum) rises steeply only up to d/D approximately 0.75; with further increasing of D, N increases insignificantly. The raising of the longitudinal resistance (ri + r0) leads to the gradual decrease of theta; at ri + r0 = 70 mohm/cm the nerve impulse propagation ceased. The estimation of the longitudinal resistance of the intercellular clefts suggests that in the nerve trunks with a compact packing of nerve fibres the flow of the local currents through the axoplasm of the neighbouring fibres is a prerequisite for impulse conduction. The possibility of electrical (electronical) interaction between the membranes of nodes and internodes has been studied. Calculations have shown that if the generation of a membrane potential in the internode were absent, the resting potential of the node would by 10 mv lower than the potential created by the nodal "generator".
Action Potentials↗
[Effect of NAD recirculation on the mechanism of ATP stabilization in cytoplasm. Mathematical models].
A mathematical model of the glycolytic system with the cytoplasmic coenzymes NAD+ and NADH as essential variables is proposed. It has been shown that any increase in the steady-state concentration of NADH will reduce the range of activity of the "generalized" ATPase, wherein the level of ATP is stabilized. Such a reduction in the range of ATP stabilization may be caused by an increasing rate of the pyruvate loss into non-glycolytic pathways, in particular, into mitochondria. This effect may be compensated by increasing oxidation of NADH by the dehydrogenases of H+-transferring cytosol-mitochondrial shuttles (malate-aspartate or alpha-glycerophosphate). The properties of the complete model were compared with those of its simplified version, which takes account only of the phosphotransferase reactions of glycolysis. The effects of various factors, which do not alter the level of NADH in the system, may be studied within the scope of the simplified model.
Adenosine Triphosphatases↗
[Kinetic study of a heterogeneous tumor cell population using a mathematical model].
A mathematical model of a heterogenous tumor as a system of interrelating cell populations is described, including a pool of quiescent cells, cell-to-cell variability in maturation rates, and cell migration from growth area to necrotic one. Computer simulation results are given, model labeled mitoses and labeled index curves for the Lewis carcinoma are compared with experimental data.
Animals↗
Drug elimination interactions: analysis using a mathematical model.
A mathematical model was developed to analyze the elimination kinetics of drug interactions in the rat. The model is based on physiological blood flow rates and organ weights and includes Michaelis-Menten equations for enzymatic processes which are involved in the elimination of the drug; competitive inhibition interactions are computed for shared pathways. Using data from the single drugs, the model can simulate the results of experiments of the acute warfarin-BSP interactions in rats.
Animals↗
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↗
A mathematical model of a blood-gas service.
A mathematical model depicting operation of a blood-gas workstation was developed by two systems analysts working closely with two clinical pathologists. This model was used to provide estimates of average as well as maximum turnaround times under various conditions of workload, specimen types (capillary vs. syringe), methodology (use of IL 513 vs. IL 313 for capillary samples), and reporting procedures (report each sample as analyzed vs. report after analysis of all samples in batch). These estimates have been validated against actual experience in our laboratory. Such an objective mathematical model can be used to plan optimal service.
Blood Gas Analysis↗
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↗
A comparison of mathematical models for estimating right ventricular volumes in animals and man.
Volume of 19 right ventricular canine casts and 11 right ventricular human casts were obtained by water displacement and compared to three different mathematical models for estimating right ventricular volumes by biplane cineangiography. In the canine studies, significant linear correlation coefficients were obtained using the longest measured length method (r = 0.92), the triangular modification of Simpson's rule (r = 0.93), and the elliptical modification of Simpson's rule (r = 0.93). The human studies resulted in similar significant correlation coefficients of 0.96, 0.97, and 0.97, respectively. Although the highest correlation with the lowest standard error of estimate was obtained using the triangular model, all three mathematical models produced volume estimations that feel within acceptabe biological limits of accuracy.
Animals↗
[Mathematical model of adaptation of the energy metabolism of a cell. Calculation of the influence of ATP on the activity and concentration of the initiator stage enzyme].
A simple kinetic model was constructed to study the adaptation of cell energy metabolism to a varying loading. In this model the initiatory step of energy metabolism, in which the initial substrate S is activated at the expense of ATP molecule energy, is catalyzed by an oligomeric enzyme E dissociable at high ATP concentration to monomers E1. It is assumed that the steady state level of monomers E1 in the cell is maintained by constitutive synthesis of E1 molecules, which balances their continuous hydrolysis by proteases. The properties of the kinetic model were studied using a mathematical model which is a system of nonlinear differential equations describing the change with time of the total enzyme E concentration and the concentration of ATP. The main isoclines of this system can intersect in one, two or three points. The mathematical analysis shows that the kinetic model considered exhibits adaptive properties. A sharp increase of the ATPase activity in the model initiates a transient process which leads to a rise in the total enzyme E concentration and in the efficiency of energy metabolism. As a result, the concentration of ATP drops only slightly. The establishment of a new level of the enzyme E concentration may proceed in the oscillatory fashion.
Adaptation, Physiological↗
A mathematical model for spreading cortical depression.
A mathematical model is derived from physiological considerations for slow potential waves (called spreading depression) in cortical neuronal structures. The variables taken into account are the intra- and extracellular concentrations of Na+, Cl-, K+, and Ca++, together with excitatory and inhibitor transmitter substances. The general model includes conductance changes for these various ions, which may occur at nonsynaptic and synaptic membrane together with active transport mechanisms (pumps). A detailed consideration of only the conductance changes due to transmitter release leads to a system of nonlinear diffusion equations coupled with a system or ordinary differential equations. We obtain numerical solutions of a set of simplified model equations involving only K+ and Ca++ concentrations. The solutions agree qualitatively with experimentally obtained time-courses of these two ionic concentrations during spreading depression. The numerical solutions exhibit the observed phenomena of solitary waves and annihilation of colliding waves.
Cortical Spreading Depression↗
A mathematical model of air and water caloric nystagmus.
A mathematical model is proposed to explain the induction of nystagmic eye movements in response to thermal stimulation of the ear by air and water. Laplace-transformed equations are set up to describe heat flow in the meatus lumen to the ear-drum and heat transmission into meatus wall. Heat transport to the lateral semicircular canal, resulting in convective endolymph flow, and the induction of reflectory eye movements are included in the mathematical description. Input of the model is the time-course of temperature at the irrigating tip, output is the time-course of eye position (in correspondence to experimental nystagmogramms). The predicted nystagmus is in good agreement with experimental results, thus supporting our assumptions on the thermal effects of air and water irrigations.
Air↗
Mathematical modeling and the redesign of a teaching ambulatory clinic.
Mathematical modeling was utilized in the planning and decision-making process involved in reorganizing a teaching clinic to effect continuity of care. The model interrelated physicians, time and space, facilitating value judgments and decisions. After examining multiple model runs, the authors finally selected a per-clinic-session doctor mix of 5 interns, 2.3 residents, and 1.6 fellows. Group productivity by model simulation was 14.6 pts/hour, utilizing 10.1 rooms. Subsequently, 90 house officers were each assigned to the clinic one-half day a week on a continuing basis. Time-motion data, from 10 sessions five months after the change, showed that during one week 353 patients were seen at a rate of 13.9 patients per hour in the rooms and time available. The fact that the reorganization was successful and the outcomes remarkably similar to model predictions has engendered confidence in the role of modeling in the planning process.
Ambulatory Care↗
Systems-matching by degeneration. II. Interpretation of the generation and degeneration of retinal ganglion cells in the chicken by a mathematical model.
Quantitative data on generation and degeneration of retinal ganglion cells during development (Rager and Rager, 1978) are interpreted in terms of a mathematical model which consists of a system of differential equations. By these equations we attempt to describe the formation of retinal ganglion cells and their termination domains in the tectum. Since ganglion cells seem not to degenerate before their axons have arrived at their termination site and start branching, from the arrival time on they may become competent either to continue to mature or to die. Therefore, to find the actual number of competent cells the extension of the fiber pathway between the retina and the optic tectum had also to be measured and computed. The differential equations are united by the principle that at any given time cells in excess of the number of termination domains have to die. By this model the mathematical function was determined. Several parameter values of this function were optimized with the Gauss-Newton method by which the curve was fitted to the measured values. The high correlation obtained by this method allows to conclude that, to a first approximation, the model may be satisfactory. The evidence of competition for termination sites and of systems-matching by cell death is discussed.
Age Factors↗