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PubMed1989

A mathematical model of erythropoiesis in mice and rats. Part 2: Stimulated 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 is regulated depending on the oxygen supply of the tissue. Quantitative dose-response relationships for in vivo erythropoiesis are suggested. Here, we demonstrate that this model adequately reproduces data obtained in situations of stimulated erythropoiesis in mice and rats. In detail, this implies a quantitative description of the following processes: (1) Changes in tissue oxygen tension (Pto2) following removal of red cells (bleeding, haemolytic anaemia) or increase in plasma volume (dilution anaemia) or decrease in atmospheric oxygen pressure (hypoxia). (2) Pto2 dependent erythropoietin (EPO) production. (3) Dose-response of EPO on erythropoietic amplification (up to two to four additional mitoses). (4) The changes of the marrow transit time. Model simulations are compared with experimental data for changes of erythropoiesis during hypoxia, EPO-injection, and different forms of anaemia. A satisfactory agreement suggests that the model adequately describes and correlates different direct and indirect ways to stimulate 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 stimulation under various conditions. Furthermore, the model may allow to optimize the scheme of EPO-administration and to find the maximum increase of erythropoiesis for a given amount of erythropoietin.

Anemia↗
PubMed1990

Mathematical Model of Plasmid Transfer between Strains of Streptomycetes in Soil Microcosms.

A mathematical model was developed and used to simulate the long-term dynamics of growth and plasmid transfer in nutrient-limited soil microcosms of Streptomyces lividans TK24 carrying chromosomal resistance to streptomycin, S. lividans 1326; and S. violaceolatus ISP5438. Donor, recipient, and transconjugant survival was modelled by an extension to the Verhulst logistic equation which takes account of nutrient limitation, and plasmid transfer was modelled by a mass action model. Rate parameters were derived from experimental data on the early stages of the development of sterile systems. The model predicted donor, recipient, and transconjugant populations in 2.4-h (0.1-day) steps and was tested against the long-term behavior of the experimental sterile systems and independent experimental data on nonsterile systems. Bacteria were periodically enumerated onto selective media over a 20-day period. The effects of long-term nutrient-moisture depletion were correctly predicted.

Journal Article↗
PubMed2005

Resonance in a mathematical model of baroreflex control: arterial blood pressure waves accompanying postural stress.

A mathematical model of the arterial baroreflex was developed and used to assess the stability of the reflex and its potential role in producing the low-frequency arterial blood pressure oscillations called Mayer waves that are commonly seen in humans and animals in response to decreased central blood volume. The model consists of an arrangement of discrete-time filters derived from published physiological studies, which is reduced to a numerical expression for the baroreflex open-loop frequency response. Model stability was assessed for two states: normal and decreased central blood volume. The state of decreased central blood volume was simulated by decreasing baroreflex parasympathetic heart rate gain and by increasing baroreflex sympathetic vaso/venomotor gains as occurs with the unloading of cardiopulmonary baroreceptors. For the normal state, the feedback system was stable by the Nyquist criterion (gain margin = 0.6), but in the hypovolemic state, the gain margin was small (0.07), and the closed-loop frequency response exhibited a sharp peak (gain of 11) at 0.07 Hz, the same frequency as that observed for arterial pressure fluctuations in a group of healthy standing subjects. These findings support the theory that stresses affecting central blood volume, including upright posture, can reduce the stability of the normally stable arterial baroreflex feedback, leading to resonance and low-frequency blood pressure waves.

Adult↗
PubMed1997

A simple mathematical model of the interaction between intracranial pressure and cerebral hemodynamics.

A simple mathematical model of intracranial pressure (ICP) dynamics oriented to clinical practice is presented. It includes the hemodynamics of the arterial-arteriolar cerebrovascular bed, cerebrospinal fluid (CSF) production and reabsorption processes, the nonlinear pressure-volume relationship of the craniospinal compartment, and a Starling resistor mechanism for the cerebral veins. Moreover, arterioles are controlled by cerebral autoregulation mechanisms, which are simulated by means of a time constant and a sigmoidal static characteristic. The model is used to simulate interactions between ICP, cerebral blood volume, and autoregulation. Three different related phenomena are analyzed: the generation of plateau waves, the effect of acute arterial hypotension on ICP, and the role of cerebral hemodynamics during pressure-volume index (PVI) tests. Simulation results suggest the following: 1) ICP dynamics may become unstable in patients with elevated CSF outflow resistance and decreased intracranial compliance, provided cerebral autoregulation is efficient. Instability manifests itself with the occurrence of self-sustained plateau waves. 2) Moderate acute arterial hypotension may have completely different effects on ICP, depending on the value of model parameters. If physiological compensatory mechanisms (CSF circulation and intracranial storage capacity) are efficient, acute hypotension has only negligible effects on ICP and cerebral blood flow (CBF). If these compensatory mechanisms are poor, even modest hypotension may induce a large transient increase in ICP and a significant transient reduction in CBF, with risks of secondary brain damage. 3) The ICP response to a bolus injection (PVI test) is sharply affected, via cerebral blood volume changes, by cerebral hemodynamics and autoregulation. We suggest that PVI tests may be used to extract information not only on intracranial compliance and CSF circulation, but also on the status of mechanisms controlling CBF.

Biomechanical Phenomena↗
PubMed1984

A mathematical model of the effects of acetylcholine pulses on sinoatrial pacemaker activity.

A mathematical model of dynamic vagus-sinus interactions was devised based on Hodgkin and Huxley-type equations of time- and voltage-dependent membrane currents. Brief vagal pulses were modeled with a concentration-dependent, acetylcholine-activated, potassium current. Single acetylcholine ("vagal") pulses scanning the sinus cycle induced changes in pacemaker rhythm that depended on pulse magnitude, duration, and time of occurrence during the cycle. Phase-response curves summarizing these effects are strikingly similar to experimental results. Notably, appropriately timed acetylcholine pulses could produce an acceleratory response. With repetitive acetylcholine input, the model produced various patterns of synchronization of the sinus pacemaker. There was stable entrainment at harmonic (i.e., 1:1, 2:1, etc.) relations, as well as more complex arrhythmic patterns that depended on the relationship between the acetylcholine cycle length and the sinus pacemaker period. In some cases, shortening of the acetylcholine input cycle length led to "paradoxical" acceleration of the sinus pacemaker. Simulations suggest that many clinically observed sinus rhythm disturbances can be explained by dynamic vagus-sinus interactions.

Acetylcholine↗
PubMed1985

A mathematical model for the mechanics of saccular aneurysms.

We constructed and discussed a mathematical model of intracranial saccular aneurysms based on the static mechanics of hollow vessels and were able to focus on three variables that are fundamental to the process of enlargement and rupture of these lesions. They are blood pressure (P), wall strength (sigma), and total wall substance (VT), which, if assigned values of 150 mm Hg, 10 MPa, and 1.0 mm3, lead to model-predicted values of 8 mm for the diameter and 40 micron for the wall thickness for the critical geometry of aneurysmal rupture. These are quantitatively similar to published measurements. The model is based on the assumption of a uniform thin spherical shell for the saccular aneurysm. The interrelationship of the variables, expressed in the equation for critical size at rupture (dc) (i.e., dc = [4 sigma VT/(pi P)]1/3), draws attention to the need for quantitative studies on aneurysmal geometry and on the stereology of the structural fraction of the aneurysmal wall. We concluded that tissue recruitment from around the initial site or hypertrophy of the wall tissue is commonly involved in the aneurysmal process. We identify the paradox of elastic stiffness and stability, which are characteristic of autopsy specimens in the laboratory, in contrast to plastic behavior and irreversible strain, which are essential to the natural process of enlargement of saccular aneurysms.

Biomechanical Phenomena↗
PubMed1999

Mathematical modeling of solute kinetics and body fluid changes during profiled hemodialysis.

A mathematical model of solute kinetics oriented to improve hemodialysis treatment is presented. It includes a two-compartment description of the main solutes (K+, Na+, Cl-, urea, HCO3-, H+, CO2), acid-base equilibrium through two buffer systems (bicarbonate and non-carbonic buffers) and a three-compartment model of body fluids (plasma, interstitial and intracellular). The main model parameters can be individually assigned a priori, on the basis of body weight and plasma concentration values measured before beginning the session. Model predictions are compared with clinical data obtained during 11 different hemodialysis sessions performed on six patients with profiled sodium concentration in the dialysate and profiled ultrafiltration rate. In all cases, the agreement between the time pattern of model solute concentrations in plasma and clinical data turns out fairly good as to urea, sodium, chloride and potassium kinetics. Finally, the time patterns of plasma bicarbonate concentration and pH can be reproduced fairly well with the model, provided CO2 concentration remains constant. Only in two sessions, blood volume was directly measured in the patient, and in both cases the agreement with model predictions was good. In conclusion, the model allows a priori computation of the amount of sodium removed during hemodialysis, and may enable the prediction of plasma volume changes and plasma osmolarity changes induced by a given sodium concentration profile in the dialysate and by a given ultrafiltration profile. Hence, it can be used to improve the dialysis session taking the characteristics of individual patients into account, in order to minimize intradialytic imbalances (such as hypotension or disequilibrium syndrome).

Acid-Base Equilibrium↗
PubMed2001

Development and evaluation of a mathematical model for the study of sediment-related water quality issues.

A mathematical model (Sediment-Transport-Associated Nutrient Dynamics-STAND) has been developed for the study of sediment-associated water quality issues. The model is intended to simulate changes of water composition associated with sediment behavior. It has a 3-level structure. The first level accounts for the hydraulics of open-channel flow. The second computes sediment transport potential and actual rates based on the information provided by the first level. A non-equilibrium approach is used. In the third level, changes of nutrient concentrations along a studied river are computed with the consideration of nutrient transport, adsorption/desorption, and release. In order to calibrate the model, field data were collected from the Oconee River, a major tributary of the Altamaha River in Georgia, USA. Two stations, approximately 17 km distant from each other, were established along the river for the purpose of data collection. Observations of the river's hydraulics, suspended sediment, and water quality (mainly orthophosphate, nitrate, temperature, specific conductivity, oxidation-reduction potential, dissolved oxygen, and pH) were collected at the two stations. Another data set collected along a major tributary of the Yellow River in China was also used for calibration of the model's hydraulics and sediment transport parts. Calibration and validation results are encouraging, which suggests STAND may be a useful tool for the thorough study and understanding of nutrient dynamics associated with sediment behaviour.

Computer Simulation↗
PubMed1977

[Mathematical model of transient long-term adaptation processes of the heart].

A mathematical model of the heart's long-term adaptation to changing load is described. The model is based on the long-term adaptation concept which assumes the phosphorylation potential changing in the heart muscle cells with the load to control the genetic apparatus activity and thus determine structural alterations ensuring adaptation. The computerized model reproduces the transient process in four major types of the heart's long-term adaptation, i.e. in a compensatory hypertrophy development and hypertrophy regression, as well as in the training and detraining states formation. The simulation results are in agreement with the available data and predict the phenomena that are not as yet substantiated by experiments.

Adaptation, Physiological↗
PubMed2004

A mathematical model of the dynamics of odontogenic cyst growth.

OBJECTIVE: To formulate a mathematical model of odontogenic cyst growth and establish the dynamics of cyst enlargement and role of osmotic pressure forces throughout its growth. STUDY DESIGN: The model assumed a spherical cyst with a semipermeable lining of living cells and a core consisting of degraded cellular material, including generic osmotic material, fed by the continuous death of epithelial cells in the lining. The lining cells were assumed to have both elastic and viscous properties, reflecting the action of physical stresses by the surrounding cyst capsule, composed of fibroblasts and collagen fibers. The model couples the cyst radius and osmotic pressure differences resulting in a system of 2 nonlinear ordinary differential equations. RESULTS: The model predicts that in all parameter regimens the long-time behavior of the cyst is the same and that linear radial expansion results. CONCLUSION: In the early and intermediate stages of cystic growth, osmotic pressure differences play an important role; however, in very large cysts, this role becomes negligible, and cell birth in the lining dominates growth.

Algorithms↗
PubMed1991

Mathematical model of antiviral immune response regulation. I. Conceptual description of the modelled processes.

The report covers the initial steps in construction of a mathematical model, namely a conceptual description of antiviral immune response regulation. The model describes cellular and molecular levels of the basic mechanisms: interaction of virus with a sensitive cell; action and activation of nonspecific resistance factors (phagocytosis, antiviral action of interferon, humoral inhibitors, and natural killer cells in the course of the immune response; humoral and cellular immune response induction and the specific antiviral defence mechanisms. Two helper cell subpopulations are considered explicitly: cellular immune response T helper cells, also named inflammatory cells (TH1) and humoral immune response T helper cells (TH2). Regulation of interleukin 1-, 2- and 4-induced lymphocyte progression through cell cycle phases is described.

Antibody Formation↗
PubMed1990

A mathematical model to describe the risk of infection from sharing injection equipment.

A mathematical model is presented that describes the probability of a subject becoming infected as a result of sharing equipment used for percutaneous injections. This risk is a function of (a) the number of syringes shared by the subject, (b) the probability of a syringe becoming contaminated after being used once by an infective person, (c) the probability of a person becoming infected after using a contaminated syringe once, (d) the number of persons who have previously used each syringe shared by the subject, (e) the prevalence of infectivity in the group from which these previous users are drawn, (f) the number of times each syringe has been used by each previous user, and (g) the number of times the subject uses it. Simulations with the model suggest mainly that (i) when each shared syringe has been used previously by only one person, the number of persons with whom syringes are shared is more important than the number of syringes; (ii) the reverse is true when each shared syringe has previously been used by many persons; and (iii) for an equal number of injections, the "shooting gallery" type of sharing can be much more risky than other kinds of sharing, but this difference decreases as the infections becomes more prevalent.

Humans↗
PubMed1980

[Mathematical model of carbohydrate energy metabolism. Interaction between glycolysis, the Krebs cycle and the H-transporting shuttles at varying ATPase load].

A simple mathematical model for carbohydrate energy metabolism based on the stoichiometic structure of glycolysis, the Krebs cycle and oxidative phosphorylation is proposed. The only allosteric regulation involved in the model is phosphofructokinase activation by AMP. Simple as it is, the model can explain the following properties of carbohydrate metabolism: a drastic rise of the rate of glucose consumption during transition to a higher level of ATPase load; stabilization of ATP and an increase of the steady state rates of glycolysis and oxidation of cytoplasmic NADH by the H-transporting shuttles and of pyruvate in the Krebs cycle with increasing rate of the ATPase load; activation of glycolysis and a decrease of the rate of oxidative phosphorylation following an inhibition of the H-transporting shuttles. The mechanisms of the coordinated changes in the steady state rates of glycolysis, the H-transporting shuttles and the Krebs cycle at varying ATPase load in the cell are discussed.

Adenosine Triphosphatases↗
PubMed1984

A mathematical model of the hemoglobin-oxygen dissociation curve of human blood and of the oxygen partial pressure as a function of temperature.

A mathematical model is described giving the oxygen saturation fraction (s) as a function of the oxygen partial pressure (p): y - y0 = x - x0 + h X tanh [k X (x - x0)], where y = kn[s/(1-s)] and x = ln(p/kPa). The parameters are: y0 = 1.875; x0 = 1.946 + a + b; h = 3.5 + a; k = 0.5343; b = 0.055 X [T/(K - 310.15)]; a = 1.04 X (7.4 - pH) + 0.005 X Cbase/(mmol/L) + 0.07 X [[CDPG/(mmol/L)] - 5], where Cbase is the base excess of the blood and CDPG is the concentration of 2,3-diphosphoglycerate in the erythrocytes. The Hill slope, n = dy/dx, is given by n = 1 + h X k X [1 - tanh2[k X (x - x0)]]. n attains a maximum of 2.87 for x = x0, and n----1 for x----+/- infinity. The model gives a very good fit to the Severinghaus standard oxygen dissociation curve and the parameters may easily be fitted to other oxygen dissociation curves as well. Applications of the model are described including the solution of the inverse function (p as a function of s) by a Newton-Raphson iteration method. The po2-temperature coefficient is given by dlnp/dT = [A X alpha X p + CHb X n X S X (1 - s) X B]/[alpha X p + CHB X n X s X (1 - s)], where A = -dln alpha/dT approximately equal to 0.012 K-1; B = (lnp/T)s = 0.073 K-1 for y = y0; alpha = the solubility coefficient of O2 in blood = 0.0105 mmol X L-1 X kPa-1 at 37 degrees C; CHb = concentration of hemoglobin iron in the blood. Approximate equations currently in use do not take the variations of the po2-temperature coefficient with p50 and CHb into account.

Chemical Phenomena↗
PubMed1983

[Mathematical model of the kinetics of radiopharmaceutical preparations in the right compartments of the heart].

The paper is concerned with the description of a mathematical model of the kinetics of radiopharmaceutical drugs (RPD) in the right heart with due regard for its two-compartmental structure. A formula has been obtained to describe a right heart radiogram (RHRG) for any RPD entry into the heart (with any type of entry function). It has been shown that in case of an impulsed entry a RHRG curve is described by the following formula (formula; see text) which is a biexponential one. The above formula shows that the determination of phi 1 and phi 2 factor on a curve gives an opportunity to obtain exact information on the correlation of cardiac chamber volumes (the auricle and ventricle). The authors point a possibility of using a RHRG curve for the determination of cardiac output and rate. The formulas obtained are of particular value for analysis of the curves of radiocardiography with 133Xe.

Heart↗
PubMed1997

A mathematical model of sialylation of N-linked oligosaccharides in the trans-Golgi network.

A mathematical model is developed of the compartmentalized sialylation of N-linked oligosaccharides in order to understand and predict the outcome of sialylation reactions. A set of assumptions are presented, including Michaelis-Menten-type dependency of reaction rate on the concentration of the glycoprotein substrate. The resulting model predicts the heterogeneous outcome of a posttranslational oligosaccharide biosynthesis step, a critical aspect that is not accounted for in the modeling of the cotranslational attachment of oligosaccharides to glycosylation sites (Shelikoff et al., Biotech. Bioeng., 50, 73-90, 1996) or general models of the secretion process (Noe and Delenick, J. Cell Sci., 92, 449-459, 1989). In the steady-state for the likely case where the concentration of substrate is much less than the Km of the sialyltransferase, the model predicts that the extent of sialylation, x, will depend upon the enzyme concentration, enzyme kinetic parameters and substrate residence time in the reaction compartment. The value of x predicted by the model using available literature data is consistent with the values of x that have been recently determined for the glycoproteins CD4 (Spellman et al., Biochemistry, 30, 2395-2406, 1991) and t-PA (Spellman et al., J. Biol. Chem., 264, 14100-14111, 1989) secreted by Chinese hamster ovary cells. For the unsaturated case, the model also predicts that x is independent of the concentration of secreted glycoprotein in the Golgi. The general modeling approach outlined in this article may be applicable to other glycosylation reactions and posttranslational modifications.

Animals↗
PubMed1996

Mathematical model of beta-lactam penetration into a biofilm of Pseudomonas aeruginosa while undergoing simultaneous inactivation by released beta-lactamases.

We present a mathematical model that describes penetration of an antibacterial agent into a bacterial biofilm and, in particular, the penetration of a beta-lactam compound into a biofilm of Pseudomonas aeruginosa. As well as dealing with this penetration, and the consequent bacterial lysis, the model considered diffusion of the released beta-lactamases in the extracellular space and the consequent inactivation there of further incoming antibiotic; it also allowed for any chosen fraction of the total beta-lactamase to be permanently accessible to exogenous substrate. The modelling scheme was validated against analytical solutions under appropriately simplified conditions. Using published experimental data we show here that lysis of cells in the surface layers of a film could have an important protective effect on the viability of underlying bacteria, especially in thick biofilms.

Anti-Bacterial Agents↗
PubMed1980

Passage of molecules through the wall of the gastrointestinal tract. II. Application of low-molecular weight polyethyleneglycol and a deterministic mathematical model for determining intestinal permeability in man.

The intestinal permeability to low molecular weight polyethyleneglycol (PEG) has been evaluated by means of a simple mathematical model and computer-aided curve-fitting procedures. Macrogolum 400, a mixture of 11 PEGs with molecular weights ranging from 194 to 634 daltons, was taken together with a liquid meal and a six-hour portion of urine collected. The different PEGs were then extracted from the urine, separated from each other by gas-liquid chromatography, and the relative peak area of each individual PEG determined. The distribution of different PEGs in the urine was then compared with the original PEG-distribution in three different ways: (1) by comparing the median values of the molecular weights, (2) by comparing the mean and standard deviation after curve fitting to the normal distribution, and (3) by curve fitting to mathematical filter functions demonstrating molecular exclusion due to size. It thus appeared that molecules were excluded both in the high and in the low molecular weight range, possibly by a combined effect of the intestinal permeability barrier and an escape to other compartments than the urine. However, relatively more of the larger PEGs passed from the intestine to the urine in a patient with Crohn's disease than in an apparently healthy individual.

Adult↗