Velocity-curvature relationship of colliding spherical calcium waves in rat cardiac myocytes.

Colliding spherical calcium waves in enzymatically isolated rat cardiac myocytes develop new wavefronts propagating perpendicular to the original direction. When investigated by confocal laser scanning microscopy (CLSM), using the fluorescent Ca2+ indicator fluo-3 AM, "cusp"-like structures become visible that are favorably approximated by double parabolae. The time-dependent position of the vertices is used to determine propagation velocity and negative curvature of the wavefront in the region of collision. It is evident that negatively curved waves propagate faster than positively curved, single waves. Considering two perfectly equal expanding circular waves, we demonstrated that the collision of calcium waves is due to an autocatalytic process (calcium-induced calcium release), and not to a simple phenomenon of interference. Following the spatiotemporal organization in simpler chemical systems maintained under conditions far from the thermodynamic equilibrium (Belousov-Zhabotinskii reaction), the dependence of the normal velocity on the curvature of the spreading wavefront is given by a linear relation. The so-called velocity-curvature relationship makes clear that the velocity is enhanced by curvature toward the direction of forward propagation and decreased by curvature away from the direction of forward propagation (with an influence of the diffusion coefficient). Experimentally obtained velocity data of both negatively and positively curved calcium waves were approximated by orthogonal weighted regression. The negative slope of the straight line resulted in an effective diffusion coefficient of 1.2 x 10(-4) mm2/s. From the so-called critical radius, which must be exceeded to initiate a traveling calcium wave, a critical volume (with enhanced [Ca2+]i) of approximately 12 microm3 was calculated. This is almost identical to the volume that is occupied by a single calcium spark.

[Biomathematical aspects of numerical evaluation and mathematical modelling of non-monotonous exponential measuring processes]. / Biomathematische Aspekte der numerischen Auswertung und mathematischen Modellierung nichtmonoton exponentieller Messwertverläufe.

Measurements of endocrinological and pharmacological processes often yield courses of time series with exponentially saturated increasing first part followed by an exponentially decreasing part. Such measured courses may be mathematically modelled by the so-called BATEMAN function type, an expression consisting of 2 e-function terms. In this paper, the method of locally adjusted functional approximation for model-free quantitative evaluation of measured time series is sketched. By means of 2 real examples of measured data, it will be demonstrated how the results of the model-free evaluation may serve for internal regression to estimate starting parameter values for an iterative fitting of a BATEMAN function to measured data courses. Furthermore, it is shown that the model-free approach of data evaluation may give substantial hints for the mathematical model building process and for model verification.

[Estimation of median regression from measured values]. / Schätzung der Medianregression aus Messwerten.

The median regression function is defined and demonstrated by examples. A lemma with sufficient conditions for continuity and differentiation of the median regression function is proved. Its estimation from a random sample is deduced firstly based on the empirical distribution function and secondly based on a kernel estimation with Gaussian kernels. Both estimations are demonstrated by GALTONS historical example. A comparison between the empirical median regression function and the empirical regression of the first kind is made by an example. A hint for curve fitting with the median estimation equation is also given by an example.

Local approximation and its applications in statistics.

For the discrete and for the continuous case, the problem of evaluating the derivatives of a function f(x) in a given interval of x is solved by local approximation method. Examples of application of the resulting numerical procedures are quoted relating the estimation of smooth function and its derivative for measured values (of a growth process), internal regression, trend elimination of time series, Bernstein polynomial, and kernel estimation of a density function.

[Periodic local approximation for the model free description of measurement courses of biorhythmic processes--demonstration example for the selection of smoothness parameters]. / Periodische lokale Approximation zur modellfreien Beschreibung von Messwertverläufen biorhythmischer Vorgänge--Demonstrationsbeispiel für die Wahl des Glättungsparameterwertes.

Measured time series with known basic periodicity may be modelfreely evaluated by procedures of local adjusted approximation. Examples of motility time series recorded in chronobiological experiments in the seasonal course of a year demonstrate aspects of choosing values of the smoothing parameter and the resulting effects for the quantitative description of the measured variations as well as for their interpretation.

Involvement of the central nervous system in malignant lymphomas of mice.

19% of the male and 49% of the female AKR mice with spontaneously developed lymphomas show an involvement of the central nervous system (CNS) in the form of a tumor cell infiltration of the leptomeninx. A significantly higher rate of lymphomas was achieved by syngeneic transplantation of lymphoma cells (intraperitoneal inoculation) in the CNS with 87% in both sexes. Morphologically, there were no differences between the leptomeningeal infiltrations in spontaneously developed and transplanted lymphomas. Spontaneous and induced lymphomas of the AKR mouse may be suitable as a simple model for investigations on CNS lymphomas in experimental oncology.

[Case studies of the use of the locally adjusted polynomial regression]. / Fallstudien zur Anwendung der lokal angepassten polynomialen Regression.

Firstly the operation mode of the procedures of local adjusted polynomial regression is explained by an artificial example of data. After that, the properties of these nonparametric estimation procedures of unknown continuous functions from measured data are demonstrated by some examples arising in experimental examinations. Especially the aspect is discussed how far the main task of separating the deterministic component from the random one in the course of measured values may be fulfilled by using different degrees of polynomials in connection with different values of the smoothing parameter.

Improvement of the method of kernel estimation by local polynomial approximation of the empirical distribution function and its application to empirical regression.

The kernel estimation of an unknown density function derived from measured values has at least one disadvantage: the second central moment of the estimated density is greater than the sample variance. This is unsatisfactory. It is possible to reduce the difference between both variances by an improvement of the method of kernel estimation. But for improved GAUSS-kernels it is not possible to reduce it to zero.

[Model-free biomathematical analysis and description of correlations]. / Modellfreie biomathematische Analyse und Beschreibung von Zusammenhängen.

Modelfree methods for nonparametric estimation of unknown continuous functions from measured values and their advantages as against fitting of analytical expressions are discussed. The main features are stated especially of the empirical regression and of the local adjusted polynomial approximation. These topics and problems of application of the procedures are illustrated by 3 examples coming from chronobiological, neuromorphometrical, and histochemical examinations.

[Behavior of HDL cholesterol and the apoproteins A1 A2 and B in the central circulation in patients with coronary disease and genetically determined hyperlipoproteinemias]. / Zum Verhalten des HDL-Cholesterols und der Apoproteine A1 A2 und B im zentralen Kreislauf bei Patienten mit koronarer Herzkrankheit und genetisch determinierten Hyperlipoproteinämien.

Seven lipid parameters, including apoproteins A1, A2 and B were determined in blood from different parts of the cardiovascular system in two groups of patients one with and the other without coronary heart disease. In addition, HDL cholesterol levels were compared in patients with coronary heart disease and primary monogenic combined hyperlipoproteinaemia. None of the seven parameters showed significant differences in samples from the pulmonary artery (mixed venous blood) and the hepatic vein of the control and patient groups. The expected significant concentration gradient between the two vascular regions was confirmed in the groups with and without coronary heart disease. HDL components showed a contrary behaviour: the hepatic vein and pulmonary artery exhibited significantly lower values of HDL cholesterol in patients with coronary heart disease than in the controls, but this was not true for HDL apoproteins A1 and A2. Patients with coronary heart disease and primary hyperlipoproteinaemia showed a significant concentration gradient between the two vascular regions for total and HDL cholesterol. These observations reveal the importance of genetic disturbances of lipid metabolism, particularly the implication of various lipid fractions in the pathogenesis of atherosclerosis.

[Procedures of local approximation for nonparametric estimation of unknown continuous functions from given measurements]. / Verfahren der lokalen Approximation zur nichtparametrischen Schätzung unbekannter stetiger Funktionen aus Messdaten.

There exists a lot of literature about the nonparametric estimation of unknown continuous functions from measured values. Many formulas of different shape were discussed. In this paper, we show that a great part of those formulas may be developed by using a uniformly mathematical principle which we name the local approximation by the least squares method. The knowledge of the principle enables one to develop new estimating formulas with special properties. The LOKONREG-, LOLINREG- and LOQUAREG-estimation are compared with each other by means of two biometric examples.

[Nonparametric estimation of 1-dimensional continuous distribution density functions using the continuous LOLINREG approximation]. / Nichtparametrische Schätzung eindimensionaler stetiger Verteilungsdichten mittels stetiger LOLINREG-Approximation.

A nonparametric method for estimation of one-dimensional continuous probability distribution functions is presented. Procedures for calculation of estimation of the unknown distribution function and the distribution density will be discussed in their application. 2 items are what type of weight function may be chosen for the proposed local-linear continuous approximation of the empirical distribution function by the least squares method (LOLINREG), and upon what value of bandwidth- or smoothing parameter one optimally should settle. The latter problem is practically very important with respect to the quality of the estimation results. Examples of simulated measurements which come from a standardized normal distribution as random numbers serve to demonstrate the mode of working, the advantages as well as the limits of the presented continuous LOLINREG-approximation.

[A new procedure for estimating the distribution of measured values--demonstrated by morphometric examples]. / Ein neues Verfahren zur Schätzung der Verteilung aus Messwerten--dargestellt an morphometrischen Beispielen.

The local approximation of the empirical distribution function of a one-dimensional continuous random variable leads to a continuous estimation of the distribution function. The first derivative of it gives the estimation of the density function. At a sample of growth data the advantages will be demonstrated of the continuous estimation compared with the classical approaches by histogram or frequency polygon respectively. Further advantageous aspects of the new approach will be illustrated by an example of a stomatologic-morphometrical examination and at one of caryometric research. Especially for comparison of 2 samples conclusions may be drawn by inspection and discussion of the obtained graphs of the empirical distribution functions and of the estimated density functions of the samples to be compared.

[Locally adjusted linear regression and its possibilities for application]. / Die lokal angepasste lineare Regression und ihre Anwendungsmöglichkeiten.

A new numerical method is presented for a model free representation of the mean course in a series of measured data of an unknown functional connection. The usefulness of the method is demonstrated by some examples of different time series (chemical reaction kinetics, growth, damped oscillations of a physiological system, pharmacokinetics). The advantages of the new method compared with nonlinear regression or segmented linear regression are noted. For any argument within the interval of measurements, the local adjusted linear regression value may be calculated. In this way one get a continuous curve with a continuous 1st derivative, too. This local adjustment will be obtained by using a weight function which is from Gaussianlike type in our procedure. The calculated continuous approximation curve represents the mean course in the measured values and may serve for a further quantitative evaluation of the measured functional connection, for interpolative and smoothing purposes, or even for model construction and model proving procedures. The nonparametric estimation of a continuous (1 dimensional) distribution density function from realizations of a continuous random variable is another field of application of this method.

[A method of non-parametric evaluation of one-dimensional continuous distribution density]. / Ein Verfahren zur nichtparametrischen Schätzung eindimensionaler stetiger Verteilungsdichten.

A continuous density function f(x) of a one-dimensional continuous random variable X is estimated from independent measured values xi, i = 1 (1) n, by a nonparametric procedure. If the measuring precision is sufficiently high, then the density estimation f(x) will be received from the 1st derivative of the LOLINREG-approximation of the empirical distribution function which is generated by the measured values. If the measured values are granulated by a lower measuring precision, then the density estimation f(x) may be obtained from the 1st derivative of the LOLINREG-approximation of an empirical distribution function which is calculated from the natural histogram by integration. The procedure is demonstrated by examples from biometrical research.

[Lipid chemical parameters and incidence of genetically determined hyperlipoproteinemias in patients between 20 and 45 years of age with coronary heart disease]. / Zur lipidchemischen Charakteristik und Häufigkeit genetisch determinierter Hyperlipoproteinämien bei Patienten zwischen 20 und 45 Jahren mit einer koronaren Herzkrankheit.

In 35 patients at the age between 20 and 45 years with a manifest coronary heart disease various lipid-chemical parameters were established and compared with a control group of the same age of laboratory-chemically and clinically inconspicuous test persons as well as with a group of patients with infarction older than 45 years. For HDL-cholesterol, LDL-cholesterol as well as LDL significant differences of the mean value were the result in the group of juvenile patients with a coronary heart disease and in the healthy control group as well as the group of older test persons with infarction, which confirm the accumulation of lipid-chemical risk constellations in "juvenile" patient with a coronary heart disease. Moreover, the proportion of primary hyperlipoproteinaemias is essentially greater in these patients than in older test persons with infarction. Apart from the recognition of primary hypercholesterolaemias the diagnostics of autosomally combined hyperlipoproteinaemias is of particular importance.

Empirical regression for trend elimination and smoothing of time series 1,2,3.

The mathematical background of empirical regression its meaning, and its role in a first stage of data evaluation will be sketched. In the second part, applications of empirical regression procedures for time series analysis will be presented, especially for a model-free quantitative description of the trend component in nonstationary time series and for trend elimination with regard to a further spectral, decomposition and spectral characterization of the modified time series.

[Morphometric studies of the aging of human submaxillary glands (author's transl)]. / Morphometrische Untersuchungen über das Altern menschlicher Unterkieferdrüsen.

Morphometric examinations were made on hematoxylin-stained paraffin sections of 43 human submaxillary glands, in order to determine and describe histological changes of this organ during the aging thereof. The number of fat cells per unit area increases in the first decades, whereas that of the striated ducts decreases. Obviously, striated ducts differentiate earlier than serous acini, with the former thus being less subject to involution with aging than the latter. In old age, the nuclei of the epithelial cells of serous acini and striated ducts are characterized by the formation of certain classes of size, of which the association with polyploidization is, however, questionable. In a 44-year-old male, differences in the size of nuclei between different striated ducts of one section, which were due perhaps to functional peculiarities, indicated the existence of inactive and active glandular regions. The volume density of serous acini decreases continuously in the course of life, while that of the striated ducts decreases in the first decennia only. On the other hand, there was observed a continuous increase in the volume density of fat cells in the entire course of life and an increase of the volume density of connective tissue in the second half of life. The mucous fractions were observed to increase up through the fifth decade of life, after which they were seen to decrease continuously.