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PubMed1979

A new mathematical approach for solving carrier-facilitated steady-state diffusion problems.

A new mathematical treatment is presented which simplifies the solution of carrier-diffusion problems. The method is generally applicable and is illustrated and tested for a specific, commonly occurring situation: facilitated diffusion of a single substrate through flat layers. Results predicted for total substrate flux are in excellent agreement with control computer calculations. The method also can be used to obtain concentration profiles for each species; here the results are good only if conditions at the boundaries are predicted correctly.

Biological Transport↗

PubMed1981

Comparison of mathematical models for the maternal age dependence of Down's syndrome rates.

The maternal age dependence of Down's syndrome rates was analyzed by two mathematical models, a discontinuous (DS) slope model which fits different exponential equations to different parts of the 20-49 age interval and a CPE model which fits a function that is the sum of a constant and exponential term over this whole 20-49 range. The CPE model had been considered but rejected by Penrose, who preferred models postulating changes with age assuming either a power function X10, where X is age or a Poisson model in which accumulation of 17 events was the assumed threshold for the occurrence of Down's syndrome. However, subsequent analyses indicated that the two models preferred by Penrose did not fit recent data sets as well as the DS or CPE model. Here we report analyses of broadened power and Poisson models in which n (the postulated number of independent events) can vary. Five data sets are analyzed. For the power models the range of the optimal n is 11 to 13; for the Poisson it is 17 to 25. The DS, Poisson, and power models each give the best fit to one data set; the CPE, to two sets. No particular model is clearly preferable. It appears unlikely that, with a data set from any single available source, a specific etiologic hypothesis for the maternal age dependence of Down's syndrome can be clearly inferred by the use of these or similar regression models.

Down Syndrome↗

PubMed1986

A mathematical model of the primary visual cortex and hypercolumn.

A mathematical model of the primary visual cortex is presented. Basically, the model comprises two features. Firstly, in analogy with the principle of the computerized tomography (CT), it assumes that simple cells in each hypercolumn are not merely detecting line segments in images as features, but rather that they are as a whole representing the local image with a certain representation. Secondly, it assumes that each hypercolumn is performing spatial frequency analyses of local images using that representation, and that the resultant spectra are represented by complex cells. The model is analyzed using numerical simulations and its advantages are discussed from the viewpoint of visual information processing. It is shown that 1) the proposed processing is tolerant to shifts in position of input images, and that 2) spatial frequency filtering operations can be easily performed in the model.

Depth Perception↗

PubMed1982

A mathematical model of the Pacinian corpuscle.

This article describes a mathematical model of the Pacinian corpuscle based on the analysis of the available experimental data and on previous theoretical research. The model includes the main anatomofunctional constituents of the corpuscle: the capsula and the mechano-to-neural transduction; its structure accounts for the formation of the receptor potential and of the spikes on the nerve terminal. Comparison of the theoretical predictions with the experimental results, in response to different types of stimuli provides a substantial validation of the model and an explanation for the basic aspects of the transduction, in particular for: a) the receptor potential time course for isolated stimuli; b) the frequency response, in terms of receptor potential; c) the frequency threshold curve for the spikes; d) the firing rate, I.S.I. and P.S.T. histograms and the synchronization coefficient, in response to sustained sinusoidal inputs. Possible lines for future experimental research are suggested from the model predictions.

Electrophysiology↗

PubMed1983

Selective stabilization of muscle innervation during development: a mathematical model.

The biochemical model presented concerns a critical step of the development of skeletal muscle innervation. After invasion of the muscle by exploratory motor axons, several nerve terminals converge from different motoneurons onto each muscle fibre at a single endplate. During the following weeks the redundant innervation disappears: a single nerve ending per muscle fibre becomes stabilized. The model is based on the assumption that the numbers of motoneurons and of muscle fibres remain constant during this evolution and that the selective stabilization of the adult connectivity results from the competition of the active nerve terminals for a postsynaptic retrograde factor mu. At the peak of the multiple innervation, the synthesis of mu by the muscle fiber stops, possibly as a consequence of muscle electrical and/or mechanical activity. The stock of mu becomes limited; a retrograde trans-synaptic diffusion of mu from the muscle to the nerve endings takes place. Within each nerve ending, mu enters into a chemical autocatalytic reaction which results in the production of a presynaptic stabilization factor s. The nerve impulses reaching the nerve terminal initiate this reaction. Any given nerve terminal become stabilized when the concentration of s reaches a threshold value. The mathematical analysis of the model shows that there exists a unique solution which is physically acceptable. Its application and computer simulation predict that only one nerve terminal becomes stabilized per muscle fibre. The model accounts for the experimental observations that the reduction in size of the motor units is not necessarily accompanied by a reduction in the variability of their size. The model also accounts for the acceleration or delay in regression which follows modifications of the chronic activity of the nerve endings and for the variability of the pattern of innervation observed in isogenic organisms. Plausible biochemical hypotheses concerning the factors engaged in the "selective stabilization" of the nerve-endings are discussed.

Animals↗

PubMed1985

Mathematical analysis of the stimulus for the lateral line organ.

Behavioral studies have shown that a blind fish is capable of detecting and recognizing stationary objects in its surroundings. It is proposed that the displacement of water caused by the fish as it moves is the basis for this detection capability. Alterations in the displacement of water around the fish, caused by the obstacle, act as stimuli for the lateral line organ. The question of how these stimuli acting on the skin of the fish, image the environment and what information is thus made available to the fish is the concern of this paper. The stimuli for the lateral line organ are derived mathematically. Two cases are treated: that of a fish gliding past an obstacle and that of one approaching an obstacle.

Animals↗

PubMed1987

Mathematical analysis of the changes in the parameters of the action potentials, membrane and ionic currents of frog muscle fibre during the recovery cycle.

The method of mathematical modelling was used to study the excitability changes of the membrane of a frog skeletal muscle fibre and the parameters of the action potentials, membrane and ionic currents during the first 30 ms of the recovery cycle. The threshold current for a fibre at rest was found to be 0.32 microA and the durations of the absolute and relative refractory periods were respectively 4 ms and 5.2 ms. With increasing interpulse interval, the subnormality of the membrane excitability is followed by supernormality. Under the same condition the supernormality in the velocity recovery cycle is not obtained. In the recovery cycle, the shape (polarity, sequence and number of phases) of the action potentials, of the membrane and ionic currents and their conductances, are unchanged. Only the time and amplitude parameters of the quantities listed above are known to vary. With increasing the interpulse interval, the amplitudes of the quantities increase and their durations are shortened attaining the values of the corresponding quantities of the initial action potential. The membrane properties are recovered 30 ms after application of the initial pulse, but the supernormality of the excitability is still preserved.

Action Potentials↗

PubMed1980

[Mathematical description of the reaction of color sensitive neural networks to intensity and color steps (author's transl)].

The reaction of color sensitive neural networks to intensity and color steps on logarithmic transformation of the input signals is calculated mathematically. The networks consist of opponent-color cells respectively with (duple system 1) or without a surround (duple system 2) or of double opponent-color cells (quadruple system). The output signals are independent of the intensity level. Both duple systems are able to code the color of homogeneous areas on a dichromatic level. The hue corresponds to the sign, the saturation to the absolute value of the output signal. The coding of saturation becomes incorrect at intensity borders only with duple system 1 (due to a Mach band response) at color borders however with duple system 1 and 2 (due to low-pass properties). The quadruple system (like duple system 2) is insensitive to intensity differences. It only responds to color differences, which are transferred according to a band-pass filter. The system therefore is able to function as a detector of color borders. The results are used in a new model for the processing of color and color borders. A linear transformation has been found to be less suited for color coding.

Color↗

PubMed1987

A linear mathematical model for computerized analyses of mood curves. An empirical investigation on mood courses in depressive and schizophrenic inpatients.

The repetitious measurement of actual mood alterations becomes more and more common in research oriented psychiatric departments. In this article a new method for the mathematical description and statistical evaluation of time series data is presented. This computerized approach is less restrictive than other procedures and seems to be very appropriate to the complex structure of time series. First results on mood courses of depressive and schizophrenic patients are presented to demonstrate the usefulness and validity of this new method.

Computer Simulation↗

PubMed1975

Methods of mathematical differentiation in tonography.

Our bulbar-compressure isotonographical method was applied to 28 normal and 40 glaucomatous eyes. This method was carried out twice on all patients using two pressure gradients: Pt=Po+10 and Pt=Po+30. Also, the outflow facility (C) was determined by Grant's tonography. Thus, 3 C-values were obtained with 3 varying pressure differences (Pt-Po=deltaP). With this differential tonography the mathematical determination and the graphic representation of a real C-value (Cdiff) by deltaP=O are possible. With normal eyes the average C-values were indirectly proportional to the pressure differences. This relationship was found to be directly proportional on glaucomatous eyes. The tonographical coefficients Po/Co-4, P3/CL3-7, Po-10/Co-4, and Po/Cintegral were calculated on 104 normal and 312 glaucomatous eyes. The results show small differences between the average values of the various coefficients for normal and glaucomatous eyes. The percentage of the pathological values on glaucomatous eyes varies from 57.4% to 67.3%. The diagnostic efficiency of the tonography is increased by the determination of Cdiff (percentage of positive values=70%) and the calculation of Po/Cdiff value (95%).

Glaucoma↗

PubMed1984

Mathematical kinematics in engineering of endoprostheses. Evaluation of the results of gait analysis.

Certain optical methods (photogrammetry) achieve great precision in analysis of body motion. However, this study clearly shows that the tracing of body surface movement is unsuitable for analysis of actual joint motion, at least in the knee joint, because of the soft tissue shifting which occurs. It is also shown that comparatively simple mathematical-kinematic techniques can be used to check the degree of error of methods of measurement in orthopedics. It is to be assumed that no currently available knee joint prostheses function in a kinematically perfect manner. If the movement of knee joint prostheses is to be optimized and implantation errors affecting kinematics avoided, it seems indispensable, at the current level of technology, to employ X-ray stereophotogrammetry to describe natural knee joint motion.

Biomechanical Phenomena↗

PubMed1985

On periodic responses of a mathematical neuron model.

The periodic responses of a mathematical neuron model, when periodically varying input stimuli are applied to the model, are investigated. An explicit representation of periodic responses is obtained. It is shown that a periodic response as a 0-1 string is a "uniform string". That is, the 1's of the 0-1 string are distributed uniformly in the string.

Animals↗

PubMed1976

A mathematical approach to the analysis of diversity in antibody gene families.

In this article, we develop a mathematical approach for the analysis of diversity in antibody gene families. This approach is arrived at by examing two general questions about protein populations: (1) What is a relative measure of the diversity exhibited by one protein family when compared with a second? (2) What is the probability that two protein populations were derived from a single common population? These quantitative approaches permit a variety of precise evolutionary, genetic, and developmental questions to be asked of antibody gene families. Using this methodology, we demonstrate that the diversity in mouse K-immunoglobulin chains is considerably greater than in their human K counterparts. We also show that the variable (Vl) regions of light chains associated with IgG and IgA immunoglobulins in the mouse appear to have been derived from a common population of Vl genes. This approach also can be used to analyse sequence data from other informational multigene families.

Amino Acid Sequence↗

PubMed1992

Evaluation of two unstructured mathematical models for the penicillin G fed-batch fermentation.

The mathematical model for the penicillin G fed-batch fermentation proposed by Heijnen et al. (1979) is compared with the model of Bajpai & Reuss (1980). Although the general structure of these models is similar, the difference in metabolic assumptions and specific growth and production kinetics results in a completely different behaviour towards product optimization. A detailed analysis of both models reveals some physical and biochemical shortcomings. It is shown that it is impossible to make a reliable estimation of the model parameters, only using experimental data of simple constant glucose feed rate fermentations with low initial substrate amount. However, it is demonstrated that some model parameters might be key factors in concluding whether or not altering the substrate feeding strategy has an important influence on the final amount of product. It is illustrated that feeding strategy optimization studies can be a tool in designing experiments for parameter estimation purposes.

Fermentation↗

PubMed1986

Histopathological diagnosis of brain tumours with the help of a computer: mathematical fundaments and practical application.

With the help of a personal computer we have subjected 1684 tumours of the central and peripheral nervous systems to a discriminant analysis to test whether the different brain tumours can be classified. By ascribing a coefficient value which belongs to a linear function to the objective criteria for each tumour these can be ordered in groups, and a classification is then possible. This has been termed discriminant analysis. Fifty histological characteristics were the criteria taken into consideration for each tumour. By using Bayes' formula with a correction for binomial distribution we obtained a correct diagnosis in 98% of the cases studied. The differential diagnosis with its percentage of probabilities is given by the computer. The mathematical fundaments are given. The valuability and precision of this method are discussed.

Brain Neoplasms↗

PubMed1996

Effect of different mathematical methods on etoposide area under the curve estimations and pharmacodynamic response predictions.

Different methods to calculate interval area under the curve (AUC) data may produce substantial error. The purpose of this study was to compare methods of calculating etoposide AUC and determine the effect of these values on white blood cell (WBC) count nadir predictions calculated from a previously reported equation. Three AUC calculation methods were used: (1) the linear trapezoidal method, (2) a combination of the linear and logarithmic trapezoidal methods, and (3) the Lagrange method. Since none of the methods for determining the AUC could be considered the standard, the methods were evaluated by comparing differences between pairs of calculated AUC values by each method. The 95% CI for differences between all pairs of AUC values were greater than zero (no difference) indicating significance. Consistent with the smoother fitting function between data points, the Lagrange method tended to produce a larger AUC, lower clearance values, and lower WBC nadir count predictions than the other methods. The largest difference encountered was between the Lagrange and the linear-log AUC methods with a mean value of 16.9 micrograms h/ml (95% CI 9.4-24.3) This difference would account for approximately 11% of the total AUC. Using a previously published equation, where WBC nadir = -0.057 +0.048 x etoposide clearance, with clearance determined as dose/AUC, mean differences in calculated WBC nadir count values between the three AUC methods ranged from 80 to 220 cells/microliters, which would be expected to be of little clinical consequence. The precision of this equation, using data derived from linear trapezoidal AUC calculations, had a mean absolute error of 0.93 x 10(3)/microliters (95% CI 0.53-1.32). Our findings suggest that any of the three mathematical methods studied would produce similar etoposide AUC values and pharmacodynamic predictions. Further, these findings also suggest that the major limitation in predicting etoposide leukopenia lies with the imprecision of the pharmacodynamic model more so than the ability to accurately determine the AUC. However, our findings may not be applicable if other factors intervene which dramatically alter the shape of the etoposide concentration-time curve.

Etoposide↗

PubMed1976

[The astigmatism bioblique. II. Mathematical description (author's transl)].

Optical surface showing an astigmatism bioblique could only approximately be described by mathematical formulas. The best approximation is that one which shows a close relation between regular and bioblique astigmatism. It allows the calculation of the best correcting cylinder, which brings the anavoidable residual astigmatism to a minimum.

Astigmatism↗

PubMed1984

Mathematical basis for the measurement of absolute and fractional cardiac output with diffusible tracers by compartmental analysis methods.

Using compartmental analysis methods, a mathematical basis is given for the measurement of absolute and fractional cardiac output with diffusible tracers. Cardiac output is shown to be the product of the blood volume and the sum of the rate constants of tracer egress from blood, modified by a factor reflecting transcapillary diffusibility, the transfer fraction. The return of tracer to the blood and distant (intracellular) events are shown to play no role in the solution. Fractional cardiac output is the ratio of the rate constant of tracer egress from blood to an organ, divided by the sum of the egress constants from blood. Predominantly extracellular ions such as sodium or bromide are best suited for this technique, although theoretically any diffusible tracer whose compartmental model can be solved may be used. It is shown that fractional cardiac output is independent of the transfer fraction, and therefore can be measured accurately by tracers which are not freely diffusible.

Animals↗