Asymptotic solution of the model of the erythrocyte shape as an autowave process

An asymptotic solution was plotted for a model of erythrocyte forms assuming that the biomembrane is anisotropic and of "small" thickness. This leads to small non-linearity and low diffusion, therefore the solution is unrelaxational. The model was investigated qualitatively assuming that the liquid current directed inside the spheric membrane induces its "distension", while that directed outside-its "crumpling". In the spherical system of coordinates the lines of solution level at theta = const are circumferences, while at phi-const-trochoids (Pascal coil, for example). Trochoids rotation areas show stomacyte and discocyte forms. Several hypotheses based on the analysis performed are advanced.     

Numerical study of the stability of some autowave solutions in a mathematical model of blood clotting

The stability of some types of solution for a set of equations representing a blood clotting model is examined using numerical methods of solving perturbation problems and the apparatus of adjoint equations. It is concluded that spiral waves are stable in this system, whereas traveling pulses are unstable at parameters corresponding to a chaotic regime in the punctual problem.     

Fermentation process diagnosis using a mathematical model

Summary Intriguing physiology of a solvent-producing strain ofClostridium acetobutylicum led to the synthesis of a mathematical model of the acetone-butanol fermentation process. The model presented is capable of describing the process dynamics and the culture behavior during a standard and a substandard acetone-butanol fermentation. In addition to the process kinetic parameters, the model includes the culture physiological parameters, such as the cellular membrane permeability and the number of membrane sites for active tansport of sugar. Computer process simulation studies for different culture conditions used the model, and quantitatively pointed out the importance of selected culture parameters that characterize the cell membrane behaviour and play an important role in the control of solvent synthesis by the cell. The theoretical predictions by the new model were confirmed by experimental determination of the cellular membrane permeability.     

Theoretical model of thalassemic erythrocyte shape transformation

Our earlier model of reticulocyte shape transformation [Pawlowski, P.H., Burzynska, B., Zielenkiewicz, P., 2006. Theoretical model of reticulocyte to erythrocyte shape transformation. J. Theor. Biol. 243, 24-38] was applied to explain the morphological properties of thalassemic erythrocytes. Modification of the standard set of parameters of the model, describing minimal cell volume, membrane bending rigidity, and membrane tension, allowed for simulation of development of α- and β-thalassemic cells from splenectomized and nonsplenectomized individuals. This resulted in observation of thin rim discocytes, tailed erythrocytes and oval forms, as well as in differentiation of time of the cell shape metamorphosis. A comparative analysis of the susceptibility of thalassemic and normal erythrocytes to undergo deformation as well of their stability was performed.     

Theoretical model of reticulocyte to erythrocyte shape transformation

A theoretical model describing the kinetics of reticulocyte shape transformation was developed. The model considers the evolution of a simple cellular shape under transmembrane pressure difference, and proposes a four-parameter axisymmetric approximation of the cell surface. The mathematical analysis considers plasma membrane tension in the plane of bilayer leaflets, membrane spontaneous curvature and transmembrane transport of water. Cytoskeleton dilatational and shear rigidity, and the energetic barrier preventing the decrease of cell volume below a certain minimum are also incorporated. The set of adequate physical assumptions allowed for formulation of the equation for free energy of the investigated system. Computer simulations of cell shape changes, down to the state of free energy minimum, together with estimation of the time needed for the resulting transport of water, revealed a complex, three-phase picture of temporal alterations in cellular geometry with a wide spectrum of final results, and led to propose a standard model of reticulocyte-erythrocyte transformation. According to the model, both cell volume and surface undergo changes, and the work of the pressure, initially accumulated in the cytoskeleton, is consumed for local bending of the cell membrane. Further simulations with modified initial shape or parameters of the standard model show the trajectories of system evolution and help in better understanding the conditions for the erythro-, sphero-, ovalo-, stomato-, and leptoidal metamorphosis of maturing red blood cells. The stability of the final biconcave shape was also verified. Spherogenic modifications were discussed in the context of spherocytosis. Future development of the model was proposed.     

Some mathematical aspects of the medical diagnostic process. I: a general mathematical model

Making a medical diagnosis consists of correlating knownpatterns of disease with the various classes of clinical data elicited from the history, physical examination, and batteries of tests relative to the diagnostic dynamics symbolized by atree branching into the various possible diagnostic decisions. In this paper a relational mathematical model of the reasoning aspects of the conventional medical diagnostic process is suggested as a way of extracting a general, formal concept of medical diagnosis. Computer implementation of the model is discussed briefly.     

Avian erythrocyte development: microtubules and the formation of the disk shape

The development of avian erythrocytes involves a spheroid to discoid transformation in shape. The disk shape of the young erythroid cells is dependent on the presence of microtubules in a marginal bundle in the early stages of postmitotic maturation. Disassembly of microtubules with colchicine, vincristine, sulfate or cold temperature produces the spheroidal shape. Erythrocytes which have acquired the flattened ellipsoidal shape do not alter their shape with disassembly of the microtubules. The number of microtubules decreases as cell maturation occurs. The correlation coefficient for the number of microtubular profiles in one end of erythrocytes and the concentration of ribosomes (cell age) is 0.88. Microtubules of immature erythrocytes disappear more rapidly at 0°C than do microtubules of mature cells.It is concluded that microtubules play little or no role in the maintenance of mature red cell shape; however, they play an important role in the development of the flat discoid shape of avian erythrocytes during maturation.     

The elliptocyte: a study of the relationship between cell shape and membrane structure using the camelid erythrocyte as a model

1. The elliptocytic shape of the camelid erythrocyte is very stable and has a high resistance to modification by drugs and treatment which alter the shape of the discocytic erythrocytes of scimitar-horned oryx and man. 2. Differences in the erythrocyte membrane proteins have been found which indicate that proteins play an important role in stabilisation of the camelid elliptocyte. 3. The organisation of the cytoskeletal network in camelid elliptocytes differs from that established for human discocytes.     

A mathematical model for the leukocyte filtration process

Leukocyte filters are applied clinically to remove leukocytes from blood. In order to optimize leukocyte filters, a mathematical model to describe the leukocyte filtration process was developed by modification of a general theoretical model for depth filtration. The model presented here can be used to predict the time-dependent leukocyte filtration as a function of cell-cell interaction in the filter, filter efficiency, filter capacity, filter dimensions, and leukocyte concentration in the suspension applied to the filter. The results of different leukocyte filtration experiments previously reported in the literature could be well described by the present model. (c) 1995 John Wiley & Sons, Inc.     

A mathematical model of selection, including the energy metabolism of organisms, as an instrument for researching the evolutionary process

A system simulating the selection conditions for studying the consequences arising from the differences in metabolism rates of the individuals (E-system) has been described. The E-system is a constructive expression of indirect individual elimination (Shmal'gauzen, 1939). The notion of energy exchange and general nonselective elimination as conjugated evolutionary factors has been proposed. The partial elimination has been connected with the known ecological generalization, the temporal decrease of negative interactions. The study demonstrated the ability to use the E-system as a heuristic tool for studying the theory of evolution.     

A mathematical model of the development of an epidemic process with aerosol mechanism of infection

A model of a parasitical system with the respiratory transfer of pathogenic organisms is suggested. The mechanism of internal regulation and the conditions of the onset of seasonal rise of sickness are investigated. The influence of asymptomatic carriers on the formation of the virulent potential of the pathogen organism is shown.     

A mathematical model of venous neointimal hyperplasia formation

Background  

In hemodialysis patients, the most common cause of vascular access failure is neointimal hyperplasia of vascular smooth muscle cells at the venous anastomosis of arteriovenous fistulas and grafts. The release of growth factors due to surgical injury, oxidative stress and turbulent flow has been suggested as a possible mechanism for neointimal hyperplasia.     

The effect of interfering substances on the disinfection process: a mathematical model

AIMS: To gain a greater understanding of the effect of interfering substances on the efficacy of disinfection. METHODS AND RESULTS: Current kinetic disinfection models were augmented by a term designed to quantify the deleterious effect of soils such as milk on the disinfection process of suspended organisms. The model was based on the assumption that inactivation by added soil occurred at a much faster rate than microbial inactivation. The new model, the fat-soil model, was also able to quantify the effect of changing the initial inoculum size (1 x 10(7)-5 x 10(7) ml(-1) of Staphylococcus aureus) on the outcome of the suspension tests. Addition of catalase to the disinfection of Escherichia coli by hydrogen peroxide, resulted in changes to the shape of the log survivor/time plots. These changes were modelled on the basis of changing biocide concentration commensurate with microbial inactivation. CONCLUSIONS: The reduction in efficacy of a disinfectant in the presence of an interfering substance can be quantified through the use of adaptations to current disinfection models. SIGNIFICANCE AND IMPACT OF THE STUDY: Understanding the effect of soil on disinfection efficacy allows us to understand the limitations of disinfectants and disinfection procedures. It also gives us a mechanism with which to investigate the soil tolerance of new biocides and formulations.     

Use of a mathematical model of erythropoiesis to study the recovery process in mice exposed to acute x-ray irradiation]

The model described in the preceding paper was used (after introducing several further presumptions of a radiobiological nature) for the interpretation of the dynamics of erythropoietic recovery, experimentally studied by the authors in mice after acute X-irradiation. The solution of equations in the model was carried out by a digital computer, finding some (not yet experimentally determined) quantitative characteristics of the system by trial and error. A satisfactory degree of agreement in the behaviour of the model with the experimentally determined course of recovery seems to indicate the validity of the basic assumptions of the model, esp. the existence of the negative feedback between erythroid populations of the bone marrow and spleen.     

A mathematical simulation model of the AIDS pandemic]

In this work the mathematical model of the spread of AIDS pandemic over 5 regions of the world are presented. With the values of all parameters corresponding to the initial dynamics of the process the prognosis obtained with the use of this model is highly unfavorable. The program of mathematical modeling, developed in this investigation, may help in the evaluation of the effectiveness of different measures aimed at the liquidation of the epidemic process and in their correction, as well as in the choice of the optimum strategy of AIDS control.