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PubMed1981

Finite difference solution of a two-dimensional mathematical model of the cochlea.

A current, linear, two-dimensional mathematical model of the mechanics of the cochlea is solved numerically by using a finite difference approximation of the model equations. The finite-difference method is used to discretize Laplace's equation over a rectangular region with specified boundary conditions. The resulting matrix equation for fluid pressure is solved by using a Gaussian block-elimination technique. Numerical solutions are obtained for fluid pressure and basilar membrane displacement as a function of distance from the stapes. The finite difference method is a direct, versatile, and reasonably efficient means of solving the two-dimensional cochlear model.

Basilar Membrane↗
PubMed1986

Mathematics achievement of Chinese, Japanese, and American children.

American kindergarten children lag behind Japanese children in their understanding of mathematics; by fifth grade they are surpassed by both Japanese and Chinese children. Efforts to isolate bases for these differences involved testing children on other achievement and cognitive tasks, interviewing mothers and teachers, and observing children in their classrooms. Cognitive abilities of children in the three countries are similar, but large differences exist in the children's life in school, the attitudes and beliefs of their mothers, and the involvement of both parents and children in schoolwork.

Aptitude↗
PubMed1983

Sex differences in mathematical reasoning ability: more facts.

Almost 40,000 selected seventh-grade students from the Middle Atlantic region of the United States took the College Board Scholastic Aptitude Test as part of the Johns Hopkins regional talent search in 1980, 1981, and 1982. A separate nationwide talent search was conducted in which any student under age 13 who was willing to take the test was eligible. The results obtained by both procedures establish that by age 13 a large sex difference in mathematical reasoning ability exists and that it is especially pronounced at the high end of the distribution: among students who scored greater than or equal to 700, boys outnumbered girls 13 to 1. Some hypothesized explanations of such differences were not supported by the data.

Child↗
PubMed1994

Mathematical studies of parasitic infection and immunity.

The techniques that underpin modern molecular biology have been rapidly adopted by those interested in the major parasitic infections of humans. The parasitological literature is full of reports of genes and their amino acid sequences, of molecules, of cell membrane receptors and channels, and of the fine details of the immunological responses mounted by the host to combat infection. Much less enthusiasm has been shown for the mathematical techniques that facilitate the analysis and interpretation of dynamical processes such as transmission, evolution, and the interplay between parasite population growth and immunological responses within the host. Molecular techniques provide enormous opportunities for description, but ultimately, understanding biological systems with the precision that physicists and engineers aspire to in their own fields will require quantitative description of the many rate processes that dictate both an observed pattern and the dynamics of its change.

Animals↗
PubMed1993

Mathematical model for comparison of time-killing curves.

The relevance of mathematical modeling to investigations of the bactericidal effects of antimicrobial agents has been emphasized in many studies of killing kinetics. We propose here a descriptive model of general use, with four parameters which account for the lag phase, the initial number of bacteria, and the limit of effectiveness and bactericidal rate of antimicrobial agents. The model has been applied to several kinetic datum sets with amoxicillin, cephalothin, nalidixic acid, pefloxacin, and ofloxacin against two Escherichia coli strains. It is a useful tool to compare killing curves by taking into account model parameter confidence limits. This can be illustrated by studying drug effects, strain effects, and concentration effects. For the antibiotics used here, concentration effects had an influence mainly on the length of the lag phase and the minimum number of living cells observed. It is therefore clear that differences in the killing curves with changes in one or more parameters could occur.

Amoxicillin↗
PubMed1987

New method of serotyping Escherichia coli: mathematical development.

A mathematical analysis of serological data has provided the basis for a new serotyping routine. This paper describes the process whereby the results of agglutination reactions between the standard O antigens of Escherichia coli and rabbit antisera raised against them were used to produce a much simpler, faster, and cheaper method of serotyping. This new method for serotyping E. coli requires on average only one-fifth of the serum used in the previous method and requires less than half the laboratory time. By using this method, an efficient serotyping routine can be established for any set of comparable data regardless of the bacterial group.

Agglutination Tests↗
PubMed1997

Clinical data sets of human immunodeficiency virus type 1 reverse transcriptase-resistant mutants explained by a mathematical model.

Treatment of human immunodeficiency virus type 1 (HIV-1) infection during the clinical latency phase with drugs inhibiting reverse transcriptase (RT) reduces the HIV-1 RNA load and increases the CD4+ T-cell count. Typically, however, the virus evolves mutations in the RT gene that circumvent the drugs. We develop a mathematical model for this situation. The model distinguishes quiescent from activated CD4+ T cells, incorporates the fact that only activated cells can become productively infected by HIV-1, embodies empirical estimates for the drug resistance and the mutation frequency for each of the HIV-1 drug-resistant mutants, and assumes the antiviral immune response to remain constant over the course of the experiments. We analyze clinical data on the evolution of drug-resistant mutants for the RT inhibitors lamivudine and zidovudine. The results show that the evolutionary sequence of the drug-resistant mutants in both data sets is accounted for by our model, given that lamivudine is more effective than zidovudine. Thus, current empirical estimates of the mutation frequencies and the drug resistances of the mutants suffice for explaining the data. We derive a critical treatment level below which the wild-type HIV-1 RNA load can rebound before the first drug-resistant mutant appears. Our zidovudine data confirm this to be the case. Thus, we demonstrate in the model and the data that the rebound of the HIV-1 RNA load in the case of zidovudine is due to the outgrowth of wild-type virus and the first drug-resistant mutant, whereas that in the case of lamivudine can only be due to the drug-resistant mutants. The evolution of drug resistance proceeds slower in the case of zidovudine because (i) zidovudine is not as effective as lamivudine and (ii) the first zidovudine drug-resistant mutant is competing with the rebounding wild-type virus.

Drug Resistance, Microbial↗
PubMed1995

The mathematical relationship between the number of events in which people are injured and the number of people injured.

In this paper the mathematical relationship between the number of times people sustained injuries and the number of people injured is discussed. The number of injury-causing events in a population will be larger than the number of people injured as people can be injured on more than one occasion. The Poisson distribution is used as a way to study the relationship between the two different sets of numbers. A practical example is given of injuries sustained while running. From this it seems that injury proneness needs to be considered in studying those who were injured on more than one occasion while running during the past year.

Accidents↗
PubMed1973

A new mathematical model for fitting an HPL radioimmunoassay curve.

A number of mathematical models have been tested for their suitability in representing the dose response curve of a specific assay of the hormone human placental lactogen (HPL).(1)A new equation Y = B[0.14 + 1/[1 + C log [1 + exp[X - D]]] where Y is the percentage activity or counts bound to the antibody and X is the HPL concentration is proposed as representing the overall shape of the curve. This model is shown to give both an accurate representation of the curve and to allow reproducible determination of an unknown over a number of occasions. A number of other models are compared. The new model allows automatic calculation of HPL concentrations from a standard curve using a computer.

Computers↗
PubMed2006

Mathematical constraints for building learning rules in the Purkinje cell system.

We first present a method to mathematically build a learning rule for closed-loop neural networks. This rule is then applied to climbing fibers in the cerebellar cortex. Our analytical study is based on previous experimental non-analytical studies, which suggests that climbing fibers carry out an error signal to the brain. Thus, our goal is to find the class of functions for the activity propagated by climbing fibers, allowing the output of the Purkinje cell to converge towards a desired output. These functions must tend towards zero when the objective is reached. Our techniques are generalized to other network models.

Afferent Pathways↗
PubMed1976

Shaping of isodose curves in intracavitary irradiation using the afterloading method. A feasibility study involving mathematical simulation.

The possibility of adjusting the irradiation scheme to shape the isodose curve around an intracavitary source arrangement was studied mathematically. The total number of millicurie hours of irradiation for each source position was derived by a linear programming approach, taking into account the shape of the tumor in three different cases. The practical isodose curves obtainable with the source arrangement chosen were compared with the shape desired.

Humans↗
PubMed1979

A mathematical index of uniformity (IOU) for sensitivity and resolution.

The authors define a mathematically precise index of uniformity (IOU) and show that the IOU is a convenient tool to quantitate uniformity of sensitivity and resolution response for any given area of interest on the detector face. They also show that the IOU is an appropriate factor for weighing the mean values of sensitivity and resolution for nonuniformity of response; these weighted sensitivity and resolution factors are used to derive a weighted figure of merit which takes into account, in a single parameter, the nonuniformities of response inherent in the determination of sensitivity and resolution.

Mathematics↗
PubMed1985

Cytoplasmic and nuclear uptake of aldosterone in toad bladder: a mathematical modeling approach.

The mechanism of aldosterone uptake in the epithelial cells of toad bladder was studied using mathematical modeling. Two complementary approaches were used. The first involved analysis of cytosolic aldosterone binding at steady state according to models defined by the sum of independent noninteractive binding sites. The best model describing the experimental data corresponded to two specific binding sites with mean dissociation constant values of 0.20 and 60 nM for types 1 and 2, respectively. The second approach was based on the analysis of cytoplasmic and nuclear aldosterone uptake kinetics at 25 and 0 degrees C in intact bladder. Two models (A and B) were studied. They both implied the existence of two types of aldosterone binding sites as precursors of the corresponding chromatin bound complexes. In model A, nuclear translocation of the two types of receptors was assumed to obey first-order kinetics. In model B, the translocation process for type 1 sites involved a time lag leading to delayed binding to chromatin. Both models were found to fit the experimental data satisfactorily. The fit obtained for model B appeared to be better at low aldosterone concentrations.

Aldosterone↗
PubMed2000

A mathematical model quantifying GnRH-induced LH secretion from gonadotropes.

A mathematical model is developed to investigate the rate of release of luteinizing hormone (LH) from pituitary gonadotropes in response to short pulses of gonadotropin-releasing hormone (GnRH). The model includes binding of the hormone to its receptor, dimerization, interaction with a G protein, production of inositol 1,4, 5-trisphosphate, release of Ca(2+) from the endoplasmic reticulum, entrance of Ca(2+) into the cytosol via voltage-gated membrane channels, pumping of Ca(2+) out of the cytosol via membrane and endoplasmic reticulum pumps, and release of LH. Cytosolic Ca(2+) dynamics are simplified (i.e., oscillations are not included in the model), and it is assumed that there is only one pool of releasable LH. Despite these and other simplifications, the model explains the qualitative features of LH release in response to GnRH pulses of various durations and different concentrations in the presence and absence of external Ca(2+).

Calcium↗
PubMed2002

Mathematical analysis of dynamics of cardiac memory and accommodation: theory and experiment.

Decreasing the slope of the dynamic, but not conventional, restitution curves is antifibrillatory. Cardiac memory/accommodation underlies the difference. We measured diastolic interval (DI) and action potential duration (APD) in epicardial, endocardial, and Purkinje tissue from eight dogs. Consecutive 100-stimulus trains were given to study transitions between basic cycle lengths (BCL) ranging from 400 to 1,300 ms. (DI,APD) pairs aligned immediately on the line DI + APD = BCL (64/67) or oscillated (3/67). The shifting effect of up to 10 extrastimuli on restitution curves was also measured. These curves were fit with the equation APD = alpha + beta exp(-DI/tau), where alpha is asymptote, beta is drop, and tau is time constant. Linear regression of the parameters against the number of extrastimuli showed that premature and postmature stimuli decreased and increased alpha and beta and increased and decreased tau, respectively. Analysis of a mathematical model treating memory as an exponentially decreasing shift of restitution curves shows that oscillatory DI,APD is expected with large DeltaBCL, steep restitution slope, or increased cardiac accommodation. The model explains phase shifts and suggests a common mechanism for Purkinje and myocardial electrical alternans.

Animals↗
PubMed1977

Carbon monoxide exchanges between the human fetus and mother: a mathematical model.

A mathematical model was developed to calculate maternal and fetal carboxyhemoglobin concentrations, [HbCO], as functions of time during and after exposure of the mother to various inspired CO concentrations. Effects of variation in alveolar ventilation rates, pulmonary and placental fiffusing capacities, cardiac output, endogenous carbon monoxide production and other factors were studied. Following a change in the inspired CO concentration, fetal HbCO lags behind maternal HbCO by several hours. During CO uptake, fetal HbCO eventually overtakes maternal, and approaches an equilibrium value as much as 10% higher than the mother's. During CO washout the fetal levels again lag behind the mothers. Results indicate that treatment of pregnant women who have elevated HbCO levels with 100% oxygen reduces the time necessary to reduce the maternal HbCO level as expected, but that the rate of fetal CO elimination is not increased as much as that of the mother. Changes in maternal and fetal HbCO were also calculated for a representative exposure to changing inspired CO levels produced by fluctuating levels of air pollution. Finally, the effects of carboxyhemoglobin on fetal oxygenation were studied, including the effects of high altitude and exercise.

Carbon Monoxide↗
PubMed1985

Phase resetting and annihilation in a mathematical model of sinus node.

A mathematical model of primary sinoatrial pacemaker activity was developed using modifications of Hodgkin-Huxley type equations of voltage- and time-dependent membrane currents. The computer simulation of action potential activity incorporates the results of several existing cardiac models and recent biological data in an attempt to generate a model that more closely approximates the phase-resetting behavior of sinoatrial pacemakers observed biologically in response to subthreshold or electrotonic stimuli. The model was also used to study annihilation, i.e., the cessation of rhythmic activity induced by a critically timed subthreshold stimulus. Perturbation analysis performed by scanning the pacemaker cycle with 50-ms subthreshold current pulses yielded biphasic phase-response relationships that closely resembled the biological data. The pacemaker current, although relatively unimportant in determining the degree of phase 4 depolarization in the simulated sinus nodal pacemaker, was nevertheless prominent in determining phase-resetting behavior. The characteristics of annihilation were studied in normal, "depressed," and hyperpolarized states. In all cases, successful annihilation depended in large part on the dynamic interaction between the slow inward and delayed rectifier outward currents. The annihilation point was found to be an unstable singularity point in which a critical combination of stimulus intensity and timing could cause the pacemaker to approach but never achieve a state of complete quiescence. The model provides the basis for further investigation of complex arrhythmias that may arise as a consequence of multiple-pacemaker interaction within the heart.

Mathematics↗