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PubMed1990

The true canalicular angle: a mathematical model.

A mathematical formula that allows for the computation of the true angle between the upper and lower canaliculi using dacryocystograms is described. It was used to determine the true canalicular angle in 33 patients. The mean calculated angle at the 1.0 mm distance was 57.2 degrees +/- 13.0 degrees, and at the 0.5 mm distance was 65.2 degrees +/- 16.2 degrees. The true calculated angle was highly correlated with the angle measured in the Waters view. There was no statistically significant correlation between the right and left sides in the same patient. There was no statistically significant difference between the canalicular angle in males and females, and there was no correlation between canalicular angle and patient's age. A clinical application of this model is discussed.

Adult↗
PubMed1988

Autoimmunity and its therapy: mathematical modelling.

A mathematical description of autotolerance and autoimmunity based on the previous model of immune response for normal antigen stimulation is given. In particular, the clonal deletion theory and non-specific stimulation of T-helper cells are included. Thus, an idea about the origin of autoimmune disease and the qualitative description of its course is presented. Possible therapies, such as immunosuppression and extracorporeal removal of autoantibodies, are also discussed.

Antigens↗
PubMed1997

Laminar structure of the heart: a mathematical model.

A mathematical description of cardiac anatomy is presented for use with finite element models of the electrical activation and mechanical function of the heart. The geometry of the heart is given in terms of prolate spheroidal coordinates defined at the nodes of a finite element mesh and interpolated within elements by a combination of linear Lagrange and cubic Hermite basis functions. Cardiac microstructure is assumed to have three axes of symmetry: one aligned with the muscle fiber orientation (the fiber axis); a second set orthogonal to the fiber direction and lying in the newly identified myocardial sheet plane (the sheet axis); and a third set orthogonal to the first two, in the sheet-normal direction. The geometry, fiber-axis direction, and sheet-axis direction of a dog heart are fitted with parameters defined at the nodes of the finite element mesh. The fiber and sheet orientation parameters are defined with respect to the ventricular geometry such that 1) they can be applied to any heart of known dimensions, and 2) they can be used for the same heart at various states of deformation, as is needed, for example, in continuum models of ventricular contraction.

Animals↗
PubMed1991

Age, time since menopause, and body parameters as determinants of female spinal bone mass: a mathematical model.

The study of mathematical models to describe bone mass behavior throughout life is a possibility for assessing the main factors of peak bone mass and bone loss. We developed a mathematical model to predict spinal bone mass behavior on a sample of 181 healthy Italian women whose lumbar bone mineral content was determined by Gd-153 dual photon absorptiometry. This model proved to be both efficient, showing the best fit (r = 0.7 on spinal bone mineral content) when compared to other previously suggested models, and also reliable as its fit remained the best when applied to a subsequent sample of 519 women whose lumbar spine was measured by dual X-ray photon absorptiometry. This model suggests that body height and body weight (but not age) are determinants of bone mass in premenopausal women. In postmenopausal women, an accelerated phase of bone loss starting at menopause is dependent on age and time since menopause, whereas body mass index acts as a protective factor. This model confirms the influence on spinal bone mass not only of age and time since menopause but also of body size parameters.

Absorptiometry, Photon↗
PubMed1996

A possible role of adenylate metabolism in human erythrocytes: simple mathematical model.

A simplified mathematical model of cell metabolism describing ion pump, glycolysis and adenylate metabolism was developed and investigated in order to clarify the functional role of the adenylate metabolism system in human erythrocytes. The adenylate metabolism system was shown to be able to function as a specific regulatory system stabilizing intracellular ion concentration and, hence, erythrocyte volume under changes in the permeability of cell membrane. This stabilization is provided via an increase in adenylate pool in association with ATPases rate elevation. Proper regulation of adenylate pool size might be achieved even in the case when AMP synthesis rate remains constant and only AMP degradation rate varies. The best stabilization of intracellular ion concentration in the model is attained when the rate of AMP destruction is directly proportional to ATP concentration and is inversely proportional to AMP concentration. An optimal rate of adenylate metabolism in erythrocytes ranges from several tenths of a percent to several percent of the glycolytic flux. An increase in this rate results in deterioration of cell metabolism stability. Decrease in the rate of adenylate metabolism makes the functioning of this metabolic system inefficient, because the time necessary to achieve stabilization of intracellular ion concentration becomes comparable with erythrocyte life span.

Adenine Nucleotides↗
PubMed1999

Numerical simulation of motility patterns of the small bowel. 1. formulation of a mathematical model.

A complete mathematical model of the periodic myoelectrical activity of a functional unit of the small intestine is presented. Based on real morphological and electrophysiological data, the model assumes that: the functional unit is an electromyogenic syncytium; the kinetics of L-type Ca2+, T-type Ca2+, Ca2+-activated K+, voltage dependent K+and Cl-channels determine the electrical activity of the functional unit; the enteric nervous system is satisfactorily represented by an efferent cholinergic neuron that provides an excitatory input to the functional unit through receptor-linked L-type Ca2+channels and by an afferent pathway composed of the primary and secondary sensory neurons; the dynamics of propagation of the wave of depolarization along the unmyelinated nerve axons satisfy the Hodgkin-Huxley model; the electrical activity of the neural soma reflects the interaction of N-type Ca2+channels, Ca2+-activated K+and voltage dependent Na+, K+and Cl-channels; the smooth muscle syncytium of the locus is a null-dimensional contractile system. With the proposed model the dynamics of active force generation are determined entirely by the concentration of cytosolic calcium. The model describes: the mechanical excitation of the free nerve endings of the mechanoreceptor of the receptive field of the pathway; the electrical processes of the propagation of excitation along the afferent and efferent neural circuits; the chemical mechanisms of nerve-pulse transmission at the synaptic zones; the slow wave and bursting type electrical activity; cytosolic calcium concentration; the dynamics of active force generation. Numerical simulations have shown that the model can display different electrical patterns and mechanical responses of the locus. The results show good qualitative and quantitative agreement with the results of experiments conducted on the small intestine.

Enteric Nervous System↗
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↗
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↗
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↗