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PubMed1997

Gene therapy of T helper cells in HIV infection: mathematical model of the criteria for clinical effect.

This paper presents a mathematical analysis of the criteria for gene therapy of T helper cells to have a clinical effect on HIV infection. The analysis indicates that for such a therapy to be successful, it must protect the transduced cells against HIV-induced death. The transduced cells will not survive as a population if the gene therapy only blocks the spread of virus from transduced cells that become infected. The analysis also suggests that the degree of protection against disease-related cell death provided by the gene therapy is more important than the fraction cells that is initially transduced. If only a small fraction of the cells can be transduced, transduction of T helper cells and transduction of haematopoietic progenitor cells will result in the same steady-state level of transduced T helper cells. For gene therapy to be efficient against HIV infection, our analysis suggests that a 100% protection against viral escape must be obtained. The study also suggests that a gene therapy against HIV infection should be designed to give the transduced cells a partial but not necessarily total protection against HIV-induced cell death, and to avoid the production of viral mutants insensitive to the gene therapy.

Cell Death↗
PubMed1996

A mathematical model for fibro-proliferative wound healing disorders.

The normal process of dermal wound healing fails in some cases, due to fibro-proliferative disorders such as keloid and hypertrophic scars. These types of abnormal healing may be regarded as pathologically excessive responses to wounding in terms of fibroblastic cell profiles and their inflammatory growth-factor mediators. Biologically, these conditions are poorly understood and current medical treatments are thus unreliable. In this paper, the authors apply an existing deterministic mathematical model for fibroplasia and wound contraction in adult mammalian dermis (Olsen et al., J. theor. Biol. 177, 113-128, 1995) to investigate key clinical problems concerning these healing disorders. A caricature model is proposed which retains the fundamental cellular and chemical components of the full model, in order to analyse the spatiotemporal dynamics of the initiation, progression, cessation and regression of fibro-contractive diseases in relation to normal healing. This model accounts for fibroblastic cell migration, proliferation and death and growth-factor diffusion, production by cells and tissue removal/decay. Explicit results are obtained in terms of the model processes and parameters. The rate of cellular production of the chemical is shown to be critical to the development of a stable pathological state. Further, cessation and/or regression of the disease depend on appropriate spatiotemporally varying forms for this production rate, which can be understood in terms of the bistability of the normal dermal and pathological steady states-a central property of the model, which is evident from stability and bifurcation analyses. The work predicts novel, biologically realistic and testable pathogenic and control mechanisms, the understanding of which will lead toward more effective strategies for clinical therapy of fibro-proliferative disorders.

Adult↗
PubMed1997

Mathematical modeling of adhesion of bacteria to host cell lines.

A mathematical model which describes adhesion of bacteria to host cell lines is presented. The model is flexible enough to account for the following situations: extracellular bacteria are either in exponential or in stationary phase. Adhesion is described as a reversible binding process in which the bacteria attach to or detach from specific receptors uniformly distributed on the cell surface. In turn, attached bacteria can either replicate or, conversely, they are restrained to remain in stationary phase. In the first case, however, we must consider the problem of whether the decrease of unoccupied receptors as adhesion progresses imposes a limit to the replicating capacity of the attached bacteria. The effect exerted by the multiplicity of infection (MOI), i.e. the ratio of the number of bacteria to the number of host cells, on the process of adhesion is also contemplated by the model. This has revealed that experiments performed at the same values of MOI can show completely different levels of adhered bacteria, depending on the number of host cells in the assays. This finding demonstrates that the report of the MOI values is insufficient to characterize comparative studies of bacterial adhesion since it could lead to a misunderstanding of the corresponding data. Simplified models based on the steady-state approximation and in equilibrium analysis by means of a Lagmuir absorption isotherm for the attached bacteria are also discussed. This allows us to define the adhesion coefficient ( beta) in a given bacterium-cell system so that, with the exception of those systems where these coefficients cannot be defined, larger values of beta are related to a greater adhesion capacity. An overview of the procedures to perform quantitative adhesion data analysis is outlined. Finally, theoretical predictions are compared with experimental results from the literature.

Animals↗
PubMed1996

Making sense of the combined effect of interleukin-2 and interleukin-4 on lymphocytes using a mathematical model.

The cytokines are the information superhighway of the immune system. They are an important component of the integrated behavior of the system. In order to be able to have a good understanding of the immune system, we must be able to model the effect of cytokines and their combined effect. This work is a step in that direction. We study the combined effect of two cytokines: interleukin-2 (IL-2) and interleukin-4 (IL-4) on some cells of the immune system. Interleukin-2 and interleukin-4 are important growth and differentiation factors for B and T cells. Interleukin-4 antagonizes the effect of interleukin-2 on B cells and some T cells while it synergizes with interleukin-2 on other T cells. We build a mathematical model of the interaction of both cytokines on T and B cells as a building block toward a model of the Th1/Th2 cross-regulation. The response of a given cell to the combination of interleukin-2 and interleukin-4 is shown to involve competing dynamical effects which can lead to either antagonistic or synergistic combined effect.

Animals↗
PubMed1995

Mathematical models for tumour angiogenesis: numerical simulations and nonlinear wave solutions.

To ensure its sustained growth, a tumour may secrete chemical compounds which cause neighbouring capillaries to form sprouts which then migrate towards it, furnishing the tumour with an increased supply of nutrients. In this paper a mathematical model is presented which describes the migration of capillary sprouts in response to a chemoattractant field set up by a tumour-released angiogenic factor, sometimes termed a tumour angiogenesis factor (TAF). The resulting model admits travelling wave solutions which correspond either to successful neovascularization of the tumour or failure of the tumour to secure a vascular network, and which exhibit many of the characteristic features of angiogenesis. For example, the increasing speed of the vascular front, and the evolution of an increasingly developed vascular network behind the leading capillary tip front (the brush-border effect) are both discernible from the numerical simulations. Through the development and analysis of a simplified caricature model, valuable insight is gained into how the balance between chemotaxis, tip proliferation and tip death affects the tumour's ability to induce a vascular response from neighbouring blood vessels. In particular, it is possible to define the success of angiogenesis in terms of known parameters, thereby providing a potential framework for assessing the viability of tumour neovascularization in terms of measurable quantities.

Animals↗
PubMed1992

A mathematical model of the P-glycoprotein pump as a mediator of multidrug resistance.

Cells displaying the classic multidrug resistant (MDR) phenotype possess a transmembrane protein (p170 or P-glycoprotein) which can actively extrude cytotoxic agents from the cytoplasm. A mathematical model of this drug efflux pump has been developed. Outward transport is modeled as a facilitated diffusion process. Since energy-dependent efflux of cytotoxic agents requires that ATP also bind to p170, the model includes a dynamic calculation for efflux rate which considers Michaelis-Menten kinetics for both the substrate agent and ATP. The final system consists of one partial differential equation (PDE) for the facilitated diffusion of substrate agents out of the cell, a 2 x 2 ordinary differential equation (ODE) system for the dynamic calculation of the ATP-ADP pool, and a dynamic algebraic calculation of the efflux rate given substrate levels at the interior cell membrane interface and ATP levels in the cell. A stability analysis of the ATP-ADP pool distribution and a simplistic closed form solution of the linearized PDE are included. Numerical simulations are also provided.

ATP Binding Cassette Transporter, Subfamily B, Mem↗
PubMed1995

Mathematical analysis of a proposed mechanism for oscillatory insulin secretion in perifused HIT-15 cells.

Oscillatory secretion of insulin has been observed in many different experimental preparations ranging from pancreatic islets to the whole pancreas. Here we examine the mathematical features underlying a possible model for oscillatory secretion from the perifused, insulin-secreting cell line, HIT-15. The model includes the kinetics of uptake of glucose by GLUT transporters, the rate of glucose metabolism within the cell, and the effect of glucose on the rate of insulin secretion. Putative feedback by insulin on the rate of glucose transport into the cells is treated phenomenologically and leads to insulin oscillations similar to those observed experimentally in HIT cells. The resulting set of ordinary differential equations is simplified by time-scale analysis to a two-variable set of ordinary differential equations. Because of this simplification we can explore, in great detail, the characteristics of the oscillations and their sensitivity to parameter variation using phase plane analysis.

Animals↗
PubMed1991

Contributions to the mathematical theory of epidemics--I. 1927.

(1) A mathematical investigation has been made of the progress of an epidemic in a homogeneous population. It has been assumed that complete immunity is conferred by a single attack, and that an individual is not infective at the moment at which he receives infection. With these reservations the problem has been investigated in its most general aspects, and the following conclusions have been arrived at. (2) In general a threshold density of population is found to exist, which depends upon the infectivity, recovery and death rates peculiar to the epidemic. No epidemic can occur if the population density is below this threshold value. (3) Small increases of the infectivity rate may lead to large epidemics; also, if the population density slightly exceeds its threshold value the effect of an epidemic will be to reduce the density as far below the threshold value as initially it was above it. (4) An epidemic, in general, comes to an end, before the susceptible population has been exhausted. (5) Similar results are indicated for the case in which transmission is through an intermediate host.

Epidemiology↗
PubMed1991

Contributions to the mathematical theory of epidemics--II. The problem of endemicity.1932.

(1) A mathematical investigation has been made of the prevalence of a disease in a population from which certain individuals are being removed as the result of the disease, whilst fresh individuals are being introduced as the result of birth or immigration. Allowance is made for the effects of the immunity produced as the result of an attack of the disease, but the effect of deaths from other causes is not taken into account, and the action of the disease is supposed to be independent of the age of the individual. (2) As a special case of the above, results have been obtained for a closed population in which no deaths occur and to which no fresh individuals are added, but in which the individuals after being infected acquire immunity, and then may be again infected. A threshold density of population exists analogous to that described in the previous paper, which is such that no disease can exist in a population, the density of which is below the threshold. (3) In other special cases investigated when either immigration or birth is operative in the supply of fresh individuals, as well as in the general case, only one steady state of disease is possible. To reach this state the population must be of a certain density which will be determined by the functions characterizing the infectivity, morbidity, etc., of the disease. (4) Increase of the immigration rate or of the birth-rate results in an increase in the rate of infection of the healthy individuals and also in the percentage rate of infection, the percentage of sick, and in the percentage of mortality from the disease. This result is, of course, a necessary consequence of our assumption that the disease is the only cause of death. (5) More particular results have been obtained by substituting constants in the place of the undetermined functions assumed in the general theory. Further, under these conditions the nature of the steady states has been more fully investigated and it has been shown that in all cases, except one, the steady states are stable ones. In the exception, a disturbance would result in purely periodic oscillations about the steady state.

Animals↗
PubMed1991

Contributions to the mathematical theory of epidemics--III. Further studies of the problem of endemicity. 1933.

(1) The mathematical investigation of the progress of an infectious disease in a community of susceptible individuals has been extended to include the case where members of the community are removed as the result of some general cause of death acting according to constant non-specific death rates, as well as by death from the disease itself. Under the more general conditions here dealt with the main conclusions arrived at in the previous paper remain qualitatively unaltered. The limitations which remain are that the susceptibility and the infective power of the individual are supposed to be independent of his age, and further that specific individual immunity does not exist in the sense that the part of the population which escapes infection is assumed to be just as susceptible as the whole population would have been if it had not been infected. (2) In the general case a unique steady state is found to exist provided that certain relatively simple conditions are satisfied. In the special cases considered a unique steady state in general exists when these conditions continue to be satisfied; but in particular instances, when these conditions are not satisfied, unique steady states will exist provided that certain other requirements are fulfilled. (3) Increase of birth rates, in general, increases both the absolute and the relative prevalence of the disease in its steady state. The effect of increase in the non-specific death rates is less simple, but has been worked out at some length. Decrease in the infectivity of the disease or in the susceptibility of the uninfected results in an increase in the whole population density as well as in an increase in the number of infected. The effect upon the relative incidence of the disease cannot be simply expressed, but it has been worked out in detail in the text. In the absence of immigration, and with the birth rates and also the non-specific death rates equal for virgins and recovered, variation in infectivity or susceptibility will not alter the relative incidence of the disease. The total population, however, will increase with decrease of either of these two factors, whilst the number of diseased will also increase proportionately. (4) Two types of threshold values have been encountered. In the first type the quantity in question must initially exceed the threshold value if the event or process is to occur in the population. Two examples of this type have been found, namely, in Cases (4) and (2'd).(ABSTRACT TRUNCATED AT 400 WORDS)

Epidemiology↗
PubMed1994

Mathematical model of cardiovascular mechanics for diagnostic analysis and treatment of heart failure: Part 1. Model description and theoretical analysis.

The planning of drug therapy for heart failure should involve both the diagnostic analysis of the patient's defective state and a prediction of the drug effects on the identified state. We have devised a mathematical model of cardiovascular system mechanics, on which both quantitative diagnosis and evaluation of drug effects can be made. The model was composed of systemic and pulmonary circulatory networks including the dynamics of the left and right ventricles. The model of the ventricles can represent both systolic and diastolic problems in heart failure through the parameters of ventricular contractility and diastolic stiffness. Each vascular network was composed of arterial and venous resistances and total vascular capacitance. Patient's ventricular and vascular parameters were estimated simultaneously from the clinically measurable haemodynamic variables based on the model. Despite the simplicity of the model, the results showed good agreement with clinical and experimental data. The clinically significant haemodynamic classification of heart failure by Forrester et al. (Forrester et al., 1977) was simulated well by the model. This model provides a useful basis for analysing pathophysiological states in heart failure and evaluating drug effects on the disease.

Heart↗
PubMed1994

Mathematical model of cardiovascular mechanics for diagnostic analysis and treatment of heart failure: Part 2. Analysis of vasodilator therapy and planning of optimal drug therapy.

Using a mathematical model of cardiovascular mechanics, various complicated responses to vasodilator therapy for heart failure have been well accounted for through common logic: (i) the differential effects of various vasodilators on cardiac output; (ii) the opposite response of cardiac output to sodium nitroprusside in a normal state and heart failure state; (iii) the different responses of cardiac index, arterial pressure and left ventricular end-diastolic pressure to hydralazine in different types of heart failure. The response to combined vasodilator-inotropic agent therapy was simulated well by the model. The optimal therapeutic regimen was then formulated to simultaneously control the cardiac output, systemic and pulmonary arterial and venous pressures, and the degree of coronary ischaemia by multiple drug delivery, and the problem was solved using the model. We conclude that the model provides a useful basis for obtaining a guidance for more appropriate therapeutic regimen in heart failure.

Cardiotonic Agents↗
PubMed1995

Mathematical analysis of haemodilution's direct effect on rate of brain cooling during cardiopulmonary bypass.

Cerebral hypothermia is the principal means of providing neurologic protection during cardiac surgery. Better understanding is needed of ways to improve brain cooling during bypass. The goal of this study is to find whether haemodilution has a significant direct effect on the rate of brain cooling, from changes in the blood's thermal properties. The brain is cooled during hypothermic cardiopulmonary bypass, almost exclusively, by the colder blood. We use the corresponding component of the bioheat transport model to predict the proportional direct effect of changing blood density and specific heat on the amplitude of the rate of brain cooling. We find that haemodilution can significantly change blood density and specific heat. For example, haemodilution with the fluorocarbon emulsion AF0104 from a haematocrit at 45% to a haematocrit at 22% increases blood density by 18%, and decreases specific heat by 21%. Nevertheless, the mathematical model predicts that the direct effect of haemodilution on the rate of brain cooling by the cold blood is small; +7%, +6% and -7% for normal saline, 5 g dl-1 albumin in normal saline, and AF0104 fluorocarbon emulsion, respectively. We conclude that, within the haemotocrit range used clinically during bypass, haemodilution with these substances has only a small direct effect on the rate of brain cooling.

Brain↗
PubMed1998

EEG coherence and reference signals: experimental results and mathematical explanations.

Coherence has become an essential tool in the description of functional relationships between EEG signals generated within various brain areas. In EEG coherence analysis, the reference signal has an important influence, as an improper reference can distort the results and make them impossible to interpret. In the study, EEG are recorded from one volunteer in 11 sessions, with electrodes selected according to the international 10-20 system against FCz. Additional electrodes are placed on the nose, chin and left and right ear lobes, and recordings are made also against FCz. This enables re-referencing of the stored EEG signals for different reference sites, averaged reference signals, common average reference, Laplacian and bipolar. Coherence values using single reference electrodes depend on the reference site to a large extent. Reliable results are obtained using averaged non-cephalic signals as reference ([A1 + A2]/2). Coherence based on FCz yields slightly lower or higher values than that based on non-cephalic reference sites. Completely different results yield common average reference recordings, Laplacian and bipolar recordings, probably owing to the cancellation effect of essential signal portions using these techniques. A mathematical model for coherence based on signal-to-noise ratios is introduced to explain the experimental findings: the model demonstrates that noisy reference signals lead to coherence increase, whereas a coherent amount in the reference signal leads to coherence decrease.

Adult↗
PubMed1987

Bone age and remodeling: a mathematical treatise.

Remodeling of an element of mineralized bone is described by specifying the probability per unit time of being replaced as a function of age of the element. The functional dependence of this probability is called the "law of remodeling." The properties of this stochastic law of remodeling are discussed in terms of random, selective, and redundant remodeling. A relationship between the law of remodeling and the mean bone tissue age is derived. It is shown how the probability density of bone age for individual elements of bone depends on the law of remodeling. The mathematical formalism is exemplified by assuming a specific parametric form of the law. A procedure for experimentally determining the law is suggested based on tracing the resorption of single fluorescent labels. Finally, the extension of the model to a real skeleton with regional differences in turnover rates is discussed.

Age Determination by Skeleton↗
PubMedDate not supplied

Mathematical modelling of stimulus-secretion coupling in the pancreatic B-cell. I. Dynamics of insulin release.

A mathematical model for cyclic AMP-Ca2+ interactions in pancreatic islets, defined in a previous study, was modified to include two new features. First, the cytosol and vacuolar Ca2+ pools were stratified in a cortical layer (representing 1 to 20% of the total cellular pool) and central core. Second, the changes in Ca2+ inflow and Ca2+ fractional outflow rate evoked by a rise in glucose concentration were discontinued during the 3rd and 4th min of stimulation. In this onion leaf sheet model, the biphasic pattern of glucose-induced insulin release could be simulated. This new model emphasizes the significance of both cytosol heterogeneity and signal discontinuity in the dynamics of insulin secretion.

Animals↗
PubMed1980

Mathematical models of antibody response.

An antigen triggers the clonal expansion of B lymphocytes accompanied by antibody production. This paper presents and compares the basic ideas of three mathematical models of B cell differentiation and proliferation.

Antibody Formation↗
PubMed1975

Cyclic kinetics and mathematical expression of the primary immune response to soluble antigen. VI. The possibility of prediction of plasma cell reaction in the spleen of mice immunized with soluble antigen.

A mathematical expression of the accumulation of the plasma cells in the spleen of CBA mice immunized intraperitoneally is presented. The dependence of the plasma cell reaction in the spleen on the kinetics of antigen concentration in the blood was confirmed. For the transition from antigen to plasma cells, index A was proposed. The mean values of index A were used for comparison of the calculated and experimental values of the plasma cell reaction and the recorded differences were not great. In a similar way, index A was used for prediction of plasma cell accumulation in the spleen of animals, immunized with a mixture of two soluble antigens -- capsular antigen of Pasteurella pestis and complete antigen of Francisella tularensis. The calculated values of plasma cell reaction corresponded to experimental values.

Animals↗