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Mathematical model for the exchange of gases in the lungs with special reference to carbon monoxide.

A mathematical model has been formulated for the simultaneous exchange of gases O2, CO2, CO and N2 in the lungs. The model takes into account the physiological parameters, such as ventilation rate, diffusing capacity of the lungs, cardiac output, total volume of blood in the body and the interaction of gases in the blood. The nonlinear functions for representing O2, CO2 and CO dissociation curves have been used. The results predicted from the model are in good agreement with those based on the ventilation/perfusion relationships. The COHb build-up in the blood, computed from the model as a function of exposure time, is in good agreement with the experimental values. The consideration of capillary blood pO2 as a constant value, instead of an independent variable, is shown to introduce a maximum error of 0.25 per cent in the blood COHb. The model is applied to analyse the COHb levels at high altitude.

Carbon Monoxide↗

Mathematical modelling and simulation of aqueous two-phase continuous protein extraction.

A mathematical model has been developed to describe the continuous, steady-state operation of an aqueous two-phase system for protein extraction. The model is based on steady-state mass balances of the main components and phase equilibrium data. Experimental data on the separation of thaumatin from contaminant proteins of an homogenate of E. coli in a PEG4000/Phosphate system was used. The data shows the effect of the presence and absence of NaCl which was used to carry out the extraction of thaumatin into the PEG phase and back into the PO4(-3) phase. Simulation results showing the sensitivity to key process parameters, and the effect of process variables on performance are presented and discussed. The model can be used to predict performance and thus 'robustness' of process conditions as well as predict protein recovery yield and purity. This model can also be used to implement a suitable control strategy to maintain process stability.

Bacterial Proteins↗

A mathematical model of "plants-microorganisms" interaction on complete mineral medium and under nitrogen limitation.

A mathematical model concerning the interaction of plants and rhizospheric microorganisms on complete mineral medium and under nitrogen limitation has been constructed. The model takes into account the closeness of plants and microorganisms in terms of the matter released by the plant and consumed by the microorganisms. The effect of rhizospheric microorganisms on plant growth with normal carbon dioxide and complete mineral medium has been demonstrated. Plants interacting with microorganisms have a greater biomass than plants growing without microorganisms. Wheat growth stimulation by metabolites of rhizospheric microorganisms under laboratory conditions on artificial soil has been experimentally demonstrated (Pechurkin, 1997). Under nitrogen limitation, the biomass of plants, with or without microorganisms, is identical, and is substantially reduced as compared with the medium with standard nitrogen.

Biomass↗

[Comparison between two diagnostic methods of computer's mathematic model and clinical diagnosis on TCM syndromes of rheumatoid arthritis].

Based on the test result of 14 trace elements in hair of 163 cases of rheumatoid arthritis, we used one of the methods of the computer's mathematic model, DYNAMIC to assign the patients to five groups among multidimensional space. Then, another method, M-DEC was used to diagnose it back on one plane. The result was compared with that by the usual clinical diagnosis, also five groups were obtained, the group of Deficiency of both Liver and Kidney, syndrome of intermingled Cold with Heat, Deficiency of Qi and Yin, syndrome of Dampness-Heat stagnating in collaterals and syndrome of Phlegm-Dampness stagnating in collaterals. And Kappa was 0.77, greater than 0.6. It displayed the consistency in observation of these two methods, which was shown reliable. After that, we have detected the sensitivity of the syndrome diagnosis by computer's mathematic model method, the result was 96.67% and the specificity of that was 95.15%. So we consider this method could give us an objective judgment on TCM syndromes of rheumatoid arthritis.

Adult↗

A mathematical model of erythropoiesis in mice and rats. Part 3: Suppressed erythropoiesis.

A mathematical model of erythropoietic cell production and its regulation process has been proposed in a preceding paper. It is primarily based on the assumption that the number of cell divisions taking place in the CFU-E and erythropoietic precursor stages can be regulated depending on the oxygen supply to the tissue. Here we provide evidence that this model adequately describes situations of suppressed erythropoiesis. In detail this implies a quantitative description of the following processes: (1) changes in tissue oxygen tension (Pto2) due to increase in red cell numbers (red cell transfusion, posthypoxia), decrease in plasma volume (dehydration) or increase in atmospheric oxygen pressure (hyperoxia), (2) Pto2 dependent reduction of erythropoietin (EPO) production, (3) dose-response of reduced EPO-levels on erythropoietic amplification (omission of three to five mitoses). Model simulations are compared to experimental data obtained from red cell transfusion, posthypoxia, hyperoxia and dehydration. A satisfactory agreement suggests that the model adequately describes and correlates different ways to suppress erythropoiesis. It quantifies the role and relative contribution of the haematocrit, haemoglobin concentration, atmospheric oxygen pressure, tissue oxygen pressure and plasma volume as triggers in erythropoietic suppression under various conditions. In conjunction with the preceding two papers it could be shown that one unique set of model parameters is sufficient to describe erythropoiesis in steady state, stimulation and suppression. Limitations of the model are discussed and experiments for a more detailed investigation of the feedback mechanisms are proposed.

Animals↗

Mathematical modeling and numerical solutions for functionally dependent bone remodeling.

The phenomenon of bone remodeling is a complex biological process which is dependent on genetic, hormonal, metabolic, and age-related factors as well as functional requirements. The possibility of successfully developing a mathematical model to describe and predict the adaptive response of bone to load will be significantly increased after identification of the nature of the transducer(s) which senses functional requirements and provides signals for the cellular processes responsible for bone synthesis and bone removal. In spite of the present limitations in knowledge about the functional dependence of bone remodeling, a phenomenological model has been developed that assumes that the output signal from the (as yet unspecified) transducer is a remodeling potential that can be modulated by genetic, hormonal, and metabolic factors. An attempt has been made to cast the mathematical model in such a form that the constants and variables appearing in the equations are not mere abstractions, but can be related to biological parameters. In order to use the adaptive hypothesis with specific structural model examples, a numerical procedure has been developed to determine the strain distribution, predict the remodeling (assuming that the remodeling rate is related to the strain history), and update the model by changing the geometry and material properties in response to the remodeling. This numerical procedure is repeatedly iterated to determine the structural architecture at subsequent times. The numerical approach allows use of the remodeling concepts with models of irregular geometry, inhomogeneous material distribution, and anisotropic material properties.

Adaptation, Physiological↗

Mechanisms of atrial fibrillation termination by pure sodium channel blockade in an ionically-realistic mathematical model.

The mechanisms by which Na+-channel blocking antiarrhythmic drugs terminate atrial fibrillation (AF) remain unclear. Classical "leading-circle" theory suggests that Na+-channel blockade should, if anything, promote re-entry. We used an ionically-based mathematical model of vagotonic AF to evaluate the effects of applying pure Na+-current (I(Na)) inhibition during sustained arrhythmia. Under control conditions, AF was maintained by 1 or 2 dominant spiral waves, with fibrillatory propagation at critical levels of action potential duration (APD) dispersion. I(Na) inhibition terminated AF increasingly with increasing block, terminating all AF at 65% block. During 1:1 conduction, I(Na) inhibition reduced APD (by 13% at 4 Hz and 60% block), conduction velocity (by 37%), and re-entry wavelength (by 24%). During AF, I(Na) inhibition increased the size of primary rotors and reduced re-entry rate (eg, dominant frequency decreased by 33% at 60% I(Na) inhibition) while decreasing generation of secondary wavelets by wavebreak. Three mechanisms contributed to I(Na) block-induced AF termination in the model: (1) enlargement of the center of rotation beyond the capacity of the computational substrate; (2) decreased anchoring to functional obstacles, increasing meander and extinction at boundaries; and (3) reduction in the number of secondary wavelets that could provide new primary rotors. Optical mapping in isolated sheep hearts confirmed that tetrodotoxin dose-dependently terminates AF while producing effects qualitatively like those of I(Na) inhibition in the mathematical model. We conclude that pure INa inhibition terminates AF, producing activation changes consistent with previous clinical and experimental observations. These results provide insights into previously enigmatic mechanisms of class I antiarrhythmic drug-induced AF termination. The full text of this article is available online at http://circres.ahajournals.org

Algorithms↗

Mathematical modeling of the forces between an intraocular lens and the capsule.

PURPOSE: To model the pressure relationship between an intraocular lens (IOL) and the capsular bag. SETTING: Department of Physics, Kings' College, London University, London, United Kingdom. METHODS: A mathematical model was made of the forces between the capsule and IOL showing that the pressure related to the local radius of curvature of the IOL at any given point. The local radius of curvature for round-edged and square-edged IOLs was measured from electron micrographs of the IOL profiles, and the corresponding pressure profiles were calculated and compared. RESULTS: The pressure between an IOL and the capsular bag was proportional to the quotient of the tension in the capsule divided by the local radius of curvature of the IOL, with a constant of proportionality that depended on the coefficient of friction between the capsule and IOL. Measuring the local radius of curvature of the 2 IOL types suggested a pressure increase of at least 69% +/- 6% at the optic edge with the square-edged IOL. CONCLUSIONS: The mathematical model predicted that IOLs with square-edged optic profiles exerts higher pressure on the posterior capsule than IOLs with round-edged optic profiles. The higher pressure may form a physical barrier to lens epithelial cell migration onto the posterior capsule.

Humans↗

Mathematic model to estimate change in burn scar length required for joint range of motion.

Burn scar contracture results from an insufficient amount of extensible tissue to permit complete range of motion. The purpose of this study was to develop a mathematic model to estimate additional tissue length required for full range of motion in the presence of a scar contracture. Seven areas with a known predilection for burn scar contracture were assessed. Twenty-five volunteers with normal range of motion had the length of their limbs measured at predetermined angles. Changes in limb length through range of motion were documented. On the basis of these changes, a mathematic model was developed to estimate the additional amount of tissue length required to complete range of motion for each area. This information may be useful to determine burn patient rehabilitation potential or need for reconstructive surgery.

Adult↗

Mathematical modelling of aqueous humour outflow from the eye through the pores in the lining endothelium of Schlemm's canal.

Aqueous humour leaves the eye mainly by way of pore openings in vacuolar swellings in the endothelial cells which line the wall of Schlemm's canal. Calculation of the resistance to fluid flow offered by these small pores requires data on their incidence and dimensions linked to a suitable mathematical model. Data are derived from scanning electron microscopy (SEM) studies conducted on rhesus monkey eyes maintained at an intraocular pressure (IOP) of either 8 or 15 mmHg and on human eyes either untreated controls or treated with pilocarpine. Mathematical models of flow through the pore openings are developed. Application of a Venturi tube model to the SEM data shows that the pore resistance of rhesus monkey eyes at 8 mmHg was seventeen times greater than that at 15 mmHg while the pore resistance in untreated human eyes was five times greater than in eyes which received pilocarpine.

Animals↗

Dissolution of nonuniformly distributed immiscible liquid: intermediate-scale experiments and mathematical modeling.

The purpose of this work is to examine the effect of nonuniform distributions of immiscible organic liquid on dissolution behavior, with a specific focus on the condition dependency of dissolution (i.e., mass transfer) rate coefficients associated with applying mathematical models of differing complexities to measured data. Dissolution experiments were conducted using intermediate-scale flow cells packed with sand in which well-characterized zones of residual trichloroethene (TCE) and 1,2-dichloroethane (DCA) saturation were emplaced. A dual-energy gamma radiation system was used for in-situ measurement of NAPL saturation. Aqueous concentrations of TCE and DCA measured in the flow-cell effluent were significantly less than solubility, due primarily to dilution associated with the nonuniform immiscible-liquid distribution and bypass flow effects associated with physical heterogeneity. A quantitative analysis of flow and transport was conducted using a three-dimensional mathematical model wherein immiscible-liquid distribution, permeability variability, and sampling effects were explicitly considered. Independent values for the initial dissolution rate coefficients were obtained from dissolution experiments conducted using homogeneously packed columns. The independent predictions obtained from the model provided good representations of NAPL dissolution behavior and of total TCE/DCA mass removed, signifying model robustness. This indicates that for the complex three-dimensional model, explicit consideration of the larger scale factors that influenced immiscible-liquid dissolution in the flow cells allowed the use of a dissolution rate coefficient that represents only local-scale mass transfer processes. Conversely, the use of simpler models that did not explicitly consider the nonuniform immiscible-liquid distribution required the use of dissolution rate coefficients that are approximately 3 orders of magnitude smaller than the values obtained from the column experiments. The rate coefficients associated with the simpler models represent composite or lumped coefficients that incorporate the effects of the larger scale dissolution processes associated with the nonuniform immiscible-liquid distribution, which are not explicitly represented in the simpler models, as well as local-scale mass transfer. These results demonstrate that local-scale dissolution rate coefficients, such as those obtained from column experiments, can be used in models to successfully predict dissolution and transport of immiscible-liquid constituents at larger scales when the larger scale factors influencing dissolution behavior are explicitly accounted for in the model.

Ethylene Dichlorides↗

The effects of intense pulsed light (IPL) on blood vessels investigated by mathematical modeling.

BACKGROUND AND OBJECTIVES: Intense pulsed light (IPL) sources have been successfully used for coagulation of blood vessels in clinical practice. However, the broadband emission of IPL hampers the clinical evaluation of optimal light parameters. We describe a mathematical model in order to visualize the thermal effects of IPL on skin vessels, which was not available, so far. STUDY DESIGN/MATERIALS AND METHODS: One IPL spectrum was shifted towards the near infrared range (near IR shifted spectrum: NIRSS) and the other was heavily shifted toward the visible range (visible shifted spectrum: VSS). The broadband emission was separated in distinct wavelengths with the respective relative light intensity. For each wavelength, the light and heat diffusion equations were simultaneously solved with the finite element method. The thermal effects of all wavelengths at the given radiant exposure (15 or 30 J/cm2) were added and the temperature in the vessels of varying diameters (60, 150, 300, 500 microm) was calculated for the entire pulse duration of 30 milliseconds. RESULTS: VSS and NIRSS both provided homogeneous heating in the entire vessel. With the exception of the small vessels (60 microm), which showed only a moderate temperature increase, all vessels exhibited a temperature raise within the vessel sufficient for coagulation with each IPL parameter. The time interval for effective temperature raise in larger vessels (diameter >60 microm) was clearly shorter than the pulse duration. In most instances, the vessel temperature was higher for VSS when compared to NIRSS. CONCLUSIONS: We presented a mathematical model capable of calculating the photon distribution and the thermal effects of the broadband IPL emission within cutaneous blood vessels.

Blood Vessels↗

A mathematical model for cadmium in the stone loach (Noemacheilus barbatulus L.) from the River Ecclesbourne, Derbyshire.

A mathematical model which linked metabolism of stone loach with cadmium dynamics was developed using information from laboratory experiments and field studies. Three possible sources of cadmium were distinguished: water, food, and sediment. Predicted results over 1 year were compared with field observations from three sites in Derbyshire. The model adequately predicted growth of fish for all three sites. Predicted cadmium levels in loach were in good agreement with measured levels in fish from all three sites despite the fact that concentrations of cadmium in the environment were kept constant during each simulation. The weight of fish affected the relative importance of different pathways of cadmium intake under similar conditions. Uptake from water contributed substantially to body burden even though the concentration in water was lower than that in food or in sediment. However, uptake from both food and sediment could not be ignored given the measured levels of cadmium in the field. The relative importance of uptake from the three sources also differed with site. The model showed that metabolism, affected by temperature, is important to the dynamics of cadmium in the stone loach.

Animals↗

[Mathematical modeling of human cardiovascular reactions under postural test and physical loading conditions].

The paper presents a mathematical model of the human cardiovascular system consisting of 34 compartments that represent arterial and venous subsystems of the systemic and pulmonary circulation, arterio-venous capillaries, right and left ventricles and atria of the heart. The model describes pulsating blood flows, changes in the pressures and volumes during a cardiac cycle in each compartment. The model has been used to study cardiovascular reactions to exercises and to determine efficiency of different compensatory reactions to postural tests.

Blood Circulation↗

A mathematical model of the effect of aging on bone marrow cells colonizing the thymus.

The process of T cell generation in the thymus involves complex cell-cell interactions between the various types of thymic stromal cells, thymocyte progenitors, thymocytes at different stages of differentiation and external factors. We applied the tool of mathematical modelling to analyze hypotheses and direct experiments concerning mechanisms underlying the observed developmental inferiority of bone-marrow thymocyte progenitors from old mice. Previous experimental data showed that lower cell numbers were obtained from old bone marrow-derived thymocyte progenitors, compared to young bone marrow-derived progenitors, when colonizing simultaneously the same fetal thymus. In this study, simulations based on the mathematical model indicate that the developmental inferiority of old bone marrow-derived progenitors cannot be explained by a change in a single parameter, such as the observed differences in progenitor frequency, an increase in cell cycle duration, a reduction in the fraction of proliferating cells in old age, and/or an increase in the rate of cell death. We have performed experimental measurements of the fractions of cycling cells. No significant difference was found between these fractions in young and old bone marrow-derived thymocytes. The difference in developmental patterns of young and old bone marrow-derived thymocytes may be due to a combination of more than one mechanism, possibly including interactions between competing thymocytes of old and young bone marrow origin.

Aging↗

Mathematical model of Ehrlich ascites tumor growth from in vitro treated cells.

A mathematical model is proposed describing the dependence of lethality and average life-span in mice on the number of Ehrlich ascites tumor (EAT) cells inoculated both intact or treated in vitro with various agents. Particularly, the effect of ionizing radiation is discussed, and the effect of combined action of two agents is also considered.

Animals↗

Mathematical model for human myeloma relating growth kinetics and drug resistance.

We present a computer-based mathematical model that can simulate characteristic features of the clinical time course of human myeloma. It asserts that therapy resistance in myeloma cells is an inherited trait associated with the longer inter-mitotic times of some cells and that the strength of this trait affects tumour growth characteristics. These kinetic differences within the malignant cell clone may also influence therapeutic efficacy. In the model, the same total therapy, administered in different time-dose fractions, could be 'curative' or 'minimally effective' depending on kinetic properties. For example, as others have shown, in myeloma pulsed intermittent therapy is often more effective than low dose continuous therapy. According to our model this finding is compatible with a high coefficient of inheritability of resistance from one cell generation to the next. The model also suggests that if there are subclones of varying resistance, a therapy must have some effect on each of them if it is to be employed in a curative fashion. While many aspects of the model are not yet clinically testable, exploration of its concepts might increase knowledge about fundamental neoplastic mechanisms.

Antineoplastic Agents↗

Mathematical model of steroidogenesis in rat and rabbit testes.

The purpose of this study was to develop a mathematical model of testicular steroidogenesis which could not only predict steroid secretion but also could be implemented to test the validity of assumptions used in studies of testicular testosterone biosynthesis in rats and rabbits. Since the two predominant fates of testicular steroids are metabolism and secretion, we hypothesized that the data for construction of the model could be observed testicular steroid secretions and that these data could then be used to elucidate intratesticular steroid conversions. Equations based on steroid secretion by testes perfused in vitro were developed to estimate transition probabilities corresponding to steroid secretion or conversion. The model was tested by comparing the predicted steroid secretion rates with those observed for control testes perfused in vitro. In addition, the transition probabilities determined by the model were used to indicate the predominant series of reactions for converting pregnenolone to testosterone. The results presented herein confirm the capability of the model to predict steroid secretion rates and to predict preferred pathways for testosterone biosynthesis in maximally stimulated testes perfused in vitro. Moreover, the model construction required only algebra and steady-state measurements.

Animals↗