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PubMed2002

Cardiovascular response to dynamic aerobic exercise: a mathematical model.

An original mathematical model of the cardiovascular response to dynamic exercise is presented. It includes the pulsating heart, the pulmonary and systemic circulation, a separate description of the vascular bed in active tissues, the local metabolic vasodilation in these tissues and the mechanical effects of muscular contractions on venous return. Moreover, the model provides a description of the ventilatory response to exercise and various neural regulatory mechanisms working on cardiovascular parameters. These mechanisms embrace the so-called central command, the arterial baroreflex and the lung inflation reflex. All parameters in the model have been given in accordance with physiological data from the literature. In this work, the model has been used to simulate the steady-state value of the main cardiorespiratory quantities at different levels of aerobic exercise and the temporal pattern in the transient phase from rest to moderate exercise. Results suggest that, with suitable parameter values the model is able accurately to simulate the cardiorespiratory response in the overall range of aerobic exercise. This response is characterised by a moderate hypertension (10-30%) and by a conspicuous increase in systemic conductance (80-130%), heart rate (64-150%) and cardiac output (100-200%). The transient pattern exhibits three distinct phases (lasting approximately 5s, 15s and 2 min), that reflect the temporal heterogeneity of the mechanisms involved. The model may be useful to improve understanding of exercise physiology and as an educational tool to analyse the complexity of cardiovascular and respiratory regulation.

Baroreflex↗

PubMed1989

The analysis of extracellular calcium exchange in perfused myocardium using mathematical modeling.

1. A mathematical model of diffusional Ca exchange in a continuously perfused heart has been formulated. Based on biochemical studies, sarcolemmal Ca binding on the extracellular surface of cardiomyocytes is taken into account. The changes in sarcolemmal Ca binding may affect the kinetics of Ca washout from the myocardium. The model is consistent with the real dynamics of 45Ca washout from rabbit heart septum reported by Philipson and Langer (F Mol Cell Cardiol 11, 857 (1979)). 2. The changes in the kinetics of Ca washout calculated according to the proposed model agree with the real changes in the kinetics of 45Ca washout from rabbit heart septum at two coronary flow rate values reported by Shine et al. (Am J Physiol 221, 1408 (1971)). 3. The calculated dynamics of the decrease in sarcolemmal Ca content is close to the real dynamics of the myocardial contractility decrease demonstrated by Philipson and Langer (1979). 4. The model offers an estimation of the contribution of different myocardial compartments to the kinetic components of Ca washout curves resolved by the method of Solomon (In: Mineral Metabolism 1A, p. 119, New York, Academic Press, (1960)). According to the results of the modeling, more than 80% of the fast exchanging pool 0 is composed of sarcolemmal Ca. 85 and 95% of the slowly exchanging pools 2 and 3 are composed of intracellular Ca; pool 1 is determined by both sarcolemmal and intracellular Ca.

Calcium↗

PubMed2001

Plasmodium falciparum parasitaemia described by a new mathematical model.

A new mathematical model of Plasmodium falciparum asexual parasitaemia is formulated and fitted to 35 malaria therapy cases making a spontaneous recovery after primary inoculation. Observed and simulated case-histories are compared with respect to 9 descriptive statistics. The simulated courses of parasitaemia are more realistic than any previously published. The model uses a discrete time-step of 2 days. Its realistic behaviour was achieved by the following combination of features (i) intra-clonal antigenic variation, (ii) large variations of the variants' baseline growth rate, depending on both variant and case, (iii) innate autoregulation of the asexual parasite density, variable among cases, (iv) acquired variant-specific immunity and (v) acquired variant-transcending immunity, variable among cases. Aspects of the model's internal behaviour, concerning variant dynamics, as well as the respective contributions of the three control mechanisms (iii) - (v), are displayed. Some implications for pathogenesis and control are discussed.

Animals↗

PubMed2002

Mathematical models of vaccination.

Mathematical models of epidemics have a long history of contributing to the understanding of the impact of vaccination programmes. Simple, one-line models can predict target vaccination coverage that will eradicate an infectious agent, whilst other questions require complex simulations of stochastic processes in space and time. This review introduces some simple ordinary differential equation models of mass vaccination that can be used to address important questions about the predicted impact of vaccination programmes. We show how to calculate the threshold vaccination coverage rate that will eradicate an infection, explore the impact of vaccine-induced immunity that wanes through time, and study the competitive interactions between vaccine susceptible and vaccine resistant strains of infectious agent.

Child↗

PubMed1989

[Hemoglobin glycosylation in prolonged hyperglycemic episodes. Interpretation of results using mathematical modeling].

Using a mathematical model, the authors analyze the relationship between glycemia and glycated haemoglobin concentration (GHb). This relationship is more complex that it seems at first sight as GHb concentration in erythrocytes is the outcome of two processes: glucose binding to haemoglobin and continuous turnover of erythrocytes in blood. Old erythrocytes carry information on glycemia of longer duration than do the younger ones. The result is that hyperglycemias which occurred immediately before to GHb estimation have a greater effect on GHb concentration than those that occurred former. Due to the fact that behind a certain value of GHb different hyperglycemic periods can be hidden, the compensation of a patient with diabetes mellitus cannot be assessed only on the basis of GHb concentration. The assessment can only be made when using criteria which take into consideration glycemia, glycated plasmatic protein, and glycated haemoglobin values in a complex way.

Erythrocytes↗

PubMedDate not supplied

[Study of cyclic kinetics of immunity by mathematical modeling methods].

The mathematical model of the dynamics of humoral immune responses to soluble antigens has been developed. This is a system of nonlinear differential equations describing concentrations of immunocompetent B-lymphocytes, plasma cells, antibodies and antigen. The model reproduces cyclic kinetics of the immune reaction to slowly catabolizing antigens which is observed experimentally. Within the framework of the model a description of the mechanism of origin of oscillatory modes of the dynamics of the immune response is presented. It has been shown that the feedback in the control of the antibody synthesis by antibodies is due to neutralization of the main stimulus of the immune system, i.e., free molecules of the antigen--by circulating antibodies.

Antibodies↗

PubMed1989

The post: pre-dialysis plasma urea nitrogen ratio to estimate K.t/V and NPCR: mathematical modeling.

The mathematical basis of the relationship between K.t/V and the ratio of the postdialysis (Ct) to predialysis (Co) plasma urea nitrogen levels (Ct/Co = R) is the urea kinetic model. The R vs. K.t/V relationship is modulated by the patient's normalized protein catabolic rate (NPCR), the dialysis session length, and the predialysis plasma urea nitrogen level, due to urea generated during the dialysis session; the latter increases Ct and hence raises R. The relationship between R and K.t/V is also affected by the amount of ultrafiltrate removed during the dialysis session, because convective urea removal, which is a part of K, does not result in a lowering of the (Ct). In stable adult maintenance dialysis patients receiving 3 treatments/week, with an NPCR of less than or equal to 1.1 g/kg/day and zero residual renal function, the target K.t/V is 1.05 and the target R will be about 0.41. In patients who require different K.t/V values, corresponding values of R can be computed. Based on an empiric examination of the urea kinetic equations, several formulas are proposed for estimating K.t/V from R and vice versa, which depend only on the dialysis session length t, the amount of ultrafiltrate UF, and the postdialysis weight W or V; e.g., K.t/V = - ln (R - 0.008.t-UF/W). After K.t/V has been estimated from R, t, UF and W, one can then estimate the NPCR in the residual renal urea clearance (Kru) has also been measured. From the estimated K.t/V, the Kru, and an estimated V, the total urea clearance over time corrected for V ("KT") is computed.(ABSTRACT TRUNCATED AT 250 WORDS)

Blood Urea Nitrogen↗

PubMed2000

[HIV-1 quantitative dynamics in vivo: a review of mathematical models].

Over the last years, mathematical models have been applied in HIV infection to investigate the population dynamics of HIV-1 and cells of the immune system in infected hosts. They have contributed to a better understanding of the pathogenesis of AIDS. Among the model-based works, the quantitative studies carried out by two teams, during the years 1995, 1996 and 1997, have brought important results on HIV infection dynamics, reminding that HIV belonged to the lentivirus family, which had not been integrated in research of the previous years. In the studies on HIV dynamics, solutions of mathematical models were fitted to viral and/or immunological markers data in order to estimate the parameters of the viral and cellular production kinetics in infected patients. The present paper is a critical review of these studies. The methods and the most important results are presented. We explain why their impact was so considerable, but also show how the simplicity of modelling could result in conceptual errors. We finally discuss the contribution and the limits of mathematical models in the analysis of experimental data.

Acquired Immunodeficiency Syndrome↗

PubMed2002

The use of mathematical models in teaching wastewater treatment engineering.

Mathematical modeling of wastewater treatment processes has become increasingly popular in recent years. To prepare students for their future careers, environmental engineering education should provide students with sufficient background and experiences to understand and apply mathematical models efficiently and responsibly. Approaches for introducing mathematical modeling into courses on wastewater treatment engineering are discussed depending on the learning objectives, level of the course and the time available.

Curriculum↗

PubMed1991

Mathematical model of antiviral immune response regulation. II. Mathematical formalization of the modelled processes. Imitation of acute course of hepatitis B.

The mathematical formalization of the conceptual model for antiviral immune response regulation described in the preceding report was carried out. The mathematical model is presented as a system of 30 ordinary nonlinear differential equations with delays. The algorithm for numerical integration of the mathematical model is based on Gear's methods of variable step and variable order. Initial conditions and parameters, as well as intervals of plausible values for them, were chosen for adaptation of the model for description of acute hepatitis B.

Acute Disease↗

PubMedDate not supplied

[The efficacy of universal vaccination against the hepatitis B virus. Simulation with a mathematical model].

We present a mathematical model of the hepatitis B virus (HBV) infection in a community. The main object is to analyze the effects of two different strategies of mass vaccination: newborns or adolescents. It appears that adolescents mass vaccination produces in the short-term a bigger effect than in the newborns. Mathematically, the model is a system of non linear first order differential equations, in which each function is a related class of individuals (susceptible, infectious, carrier, immune and death by HBV) in the evolution of the HBV. The solution of system is obtained in a numerical way. It should be pointed out that the model explains neatly the herd immunity effect of the vaccine and can be used in the simulation of possible changes in the HBV infection such that generalized use of discarded needles by the drug addicted population, changes in the sexual habits, etc.

Adolescent↗

PubMedDate not supplied

[Mathematical model of autoimmunity].

A mathematical model of autoimmunity is developed. This model is a system of two nonlinear differential equations, which describe the concentration dynamics of tissue cells and agressive lymphocytes. An analysis of the solutions shows that this model reproduces general behaviour of autoimmune diseases.

Autoimmune Diseases↗

PubMed1988

Quantitative assessment of cerebral blood flow using technetium-99m-hexamethyl-propyleneamine oxime: Part I, Design of a mathematical model.

To design a mathematical model for quantifying cerebral blood flow using 99mTc-hexamethyl-propyleneamine oxime (HM-PAO), basic studies were performed in animals and human volunteers. Microautoradiography revealed that HM-PAO crossed the blood-brain barrier. Thin layer chromatographic studies demonstrated the rapid disappearance of free HM-PAO in the brain tissue. Back diffusion from brain to blood was found negligible. From these observations, the familiar microsphere model was employed in the measurements of blood flow with HM-PAO. This, however, resulted in much lower flow values than simultaneously obtained values with the labeled microspheres. This underestimation was ascribed to the high affinity of HM-PAO to blood cells and serum protein. Taking the binding of HM-PAO to blood components into consideration, the following model equation was designed for quantifying cerebral blood flow: Ce(t) = Ca(t)-kCa(t)*exp(-kt), Cb(T) = F integral of T0 Ce(t)dt, where Ce and Ca are the free HM-PAO concentration in the intravascular space and the arterial whole-blood concentration of HM-PAO, respectively, as a function of time (t), Cb is the brain activity concentration, k is the rate constant for the binding of HM-PAO to the blood components, F is the blood flow value, T is time of measurement, and * denotes the operation of convolution. In clinical studies, Ca(t) and Cb(T) are obtainable from a dynamic single photon emission computerized tomographic study of the brain and multiple arterial blood sampling, respectively. The values for F and k can be estimated using a nonlinear least squares fitting method.

Animals↗

PubMed1999

Estimation of fetal weight in twins: a new mathematical model.

OBJECTIVES: Evaluation of new mathematical formula (Femur 4) derived from a twin population to estimate fetal weight in twins using ultrasound. Comparison of Femur 4 is with conventional mathematical models. DESIGN: Retrospective analysis of ultrasonic measurements of 297 twin babies from 24 to 40 weeks of gestation who were born within 10 days of ultrasound examination. SETTING: Aberdeen Maternity Hospital. METHODS: With ultrasonic measurements obtained from twin babies, estimated fetal weight was calculated using the mathematical models of Campbell, Shepard and Hadlock. The calculations were repeated for the model of Femur 4. All models were compared against Femur 4. RESULTS: The coefficient of determination of the linear regression between the actual and predicted weight was highest for Femur 4 (0.852). Femur 4 had the highest proportion of babies with estimated weights within 10% of actual birthweight (71.4%). In babies who weighed between 2000 and 3000 g, Femur 4 had the least systematic and random error of -1.69 and 8.96, respectively. For babies below the 10th centile for weight, Femur 4 had comparable positive and negative predictive values of 76.0% and 92.3%, respectively. Femur 4 was equally poor at predicting growth discordancy with positive and negative predictive values of 70.0% and 86.5% only. CONCLUSION: Femur 4 requires measurements of femur length and abdominal circumference only, hence avoiding the need to obtain difficult head measurements which is a common problem in twins. It is a good model for estimation of fetal weight in twins. However, prediction of growth discordancy remains problematic.

Body Weight↗

PubMed1988

Ventricular volume regulation: a mathematical model and computer simulation.

A mathematical model of ventricular volume regulation based on fluid mechanical principles has been constructed using a systems engineering approach. The parameters used in the model are based on clinical observation, laboratory investigation, and presumptions that will be tested later. The model was constructed to be the basis of a computer simulation. Using the computer simulation, information obtained from the literature and laboratory hypotheses regarding pathophysiology, several enigmatic conditions were tested. The model predicted that over-production of cerebrospinal fluid, as in the case of choroid plexus papilloma, could by itself lead to distention of the ventricular system. In simulating pseudotumor cerebri, if cerebrospinal fluid absorption at the arachnoid villi is impaired and the brain itself is rendered incompressible by swelling, intracranial pressure rises and ventricular volume diminishes. Conversely, in normal-pressure hydrocephalus, if cerebrospinal fluid flow is restricted between the spinal and cortical subarachnoid spaces and the brain is made more compressible, the ventricular volume increases with minimal increases in intracranial pressure. This mathematical model and its associated computer simulation is useful in predicting the behavior of the volume of the cerebral ventricles to a variety of pathological phenomena.

Animals↗

PubMed1984

Mathematical models for ligand-receptor binding. Real sites, ghost sites.

In the basic life sciences the term "model" implies a physical, chemical, or molecular construct that provides a representation for the interpretation of experimental observations. To the statistician, however, a model is a mathematical expression for correlating data, which may or may not have roots in a molecular picture. With regard to ligand-receptor interactions, the mathematical model used plays a crucial role in extrapolations of binding measurements. Regardless of the statistical goodness of fit of data to an equation, the relationships of the parameters of a mathematical formalism to the molecular features of ligand-receptor complexes are generally very complex. Oversimplified interpretations of the molecular significance of the constants derived from binding measurements are unwarranted, unless one has independent information from molecular probes.

Kinetics↗

PubMed2000

Brain biomechanics: mathematical modeling of hydrocephalus.

The considerable amount of literature on mathematical models of hydrocephalus and other brain abnormalities is critically reviewed. These models have various degrees of mathematical sophistication, and have influenced not only the diagnosis of hydrocephalus, but also its treatment with CSF shunts. The mathematical models are classified into two classes, pressure-volume models, and consolidation models. Advantages and disadvantages of both types are pointed out with a view to removing the confusion frequently generated by the technical aspects of the subject. The conclusion is reached that, while none of the current models are good enough to be of immediate use to the neurosurgeon, mathematical models are likely in the future to be a powerful tool for the understanding and the treatment of hydrocephalus, as well as other conditions related to brain biomechanics. The amount of mathematics has been kept to the absolute minimum, but it is cited and appended for those who would like to dig further into this fascinating area of research.

Biomechanical Phenomena↗

PubMedDate not supplied

[Mathematical modelling in medicine and biology. Theoretical basis and fundamentals].

Mathematical modelling is currently a common tool in the study of physiological and biochemical systems. Its basis and fundaments are not, however, well known by the non-specialist. Its aims are to describe, explain and predict physiological and biochemical phenomena. Mathematical models provide a concise and objective description of complex dynamic processes by defining, through mathematical equations, the relationships between quantitative measurements; they indicate, also, ways to improve experimental designs, and allow the testing of hypotheses about physiological or biochemical phenomena. Mathematical models can be developed from simple non-compartmental representations to large scale multi-compartmental models. The basic steps in the formulation of a model include conceptualization, realization and solution of the model. Each step has to be verified and validated. In the case of compartmental models, mass-balance equations are used to represent each compartment. A brief review of the theory of system's analysis and the general aims of mathematical modelling is presented here. The modelling process is usually started with a definition of the problem and a parameter identification followed by the setting up of a clear conceptual model of the system. The model consists of the description of the principal flows of material (in and out) and of the main components which store, convert or transmit these flows. A selection of the class of mathematical representation follows, i.e. linear or non-linear, in order to formulate the equations relating the input and output flows of material for each individual component of the system.(ABSTRACT TRUNCATED AT 250 WORDS)

Models, Biological↗