Search

International audience; AbstractBackgroundWe tested the premise that optimum-contribution selection with pedigree relationships to control inbreeding (POCS) realises at least as much true genetic gain as optimum-contribution selection with genomic relationships (GOCS) at the same rate of true inbreeding.MethodsWe used stochastic simulation to estimate rates of true genetic gain realised by POCS and GOCS at a 0.01 rate of true inbreeding in three breeding schemes with best linear unbiased predictions of breeding values based on pedigree (PBLUP) and genomic (GBLUP) information. The three breeding schemes differed in number of matings and litter size. Selection was for a single trait with a heritability of 0.2. The trait was controlled by 7702 biallelic quantitative-trait loci (QTL) that were distributed across a 30-M genome. The genome contained 54,218 biallelic markers that were used in GOCS and GBLUP. A total of 6012 identity-by-descent loci were placed across the genome in base populations. Unique alleles at these loci were used to calculate rates of true inbreeding. Breeding schemes were run for 10 discrete generations. Selection candidates were genotyped and phenotyped before selection.ResultsPOCS realised more true genetic gain than GOCS at a 0.01 rate of true inbreeding in all combinations of breeding scheme and prediction method. POCS realised 14 to 33% more true genetic gain than GOCS with PBLUP in the three breeding schemes. It realised 1.5 to 5.7% more true genetic gain than GOCS with GBLUP.ConclusionsPOCS realised more true genetic gain than GOCS because it managed expected genetic drift without restricting selection at QTL. By contrast, GOCS penalised changes in allele frequencies at markers that were generated by genetic drift and selection. Because these marker alleles were in linkage disequilibrium with QTL alleles, GOCS restricted changes in allele frequencies at QTL. This provides little incentive to use GOCS and highlights that we have more to learn before we can control inbreeding using genomic relationships in selective-breeding schemes. Until we can do so, POCS remains a worthy method of optimum-contribution selection because it realises more true genetic gain than GOCS at the same rate of true inbreeding.

In der vorliegenden Arbeit betrachten wir Prozessgleichungen der Form $$(X(t))_{t\in[0,1]} \d \Big(\sum_{j\geq 1}T_j X_j(\tau_j(t)) + C(t)\Big)_{t\in [0,1]}$$ und untersuchen diese in Hinblick auf Lösungen mit Pfaden in $D$, dem Raum der càdlàg-Funktionen über dem Einheitsintervall $[0,1]$. Unsere Motivation solche Gleichungen zu studieren ist die zum Quicksort-Algorithmus assoziierte Prozessgleichung $$(Q(t))_{t\in[0,1]}\d \Big(U Q_1\big(1\wedge \tfrac{t}{U}\big) + (1-U) Q_2\big(0 \vee \tfrac{t-U}{1-U}\big) + C(U,t)\Big)_{t\in [0,1]}.$$ Zu einer gegebenen Folge $(C, (T_1,\tau_1), (T_2,\tau_2),\ldots)$ charakterisieren wir die Menge aller $D$-wertigen Lösungen der allgemeinen Gleichung und zeigen, dass diese als Faltung einer Lösung der inhomogenen Prozessgleichung und einer allgemeinen Lösung der homogenen Prozessgleichung dargestellt werden können. \section*{Abstract} In this thesis we consider process equations of the form $$(X(t))_{t\in[0,1]} \d \Big(\sum_{j\geq 1}T_j X_j(\tau_j(t)) + C(t)\Big)_{t\in [0,1]}$$ and investigate them in regard to solutions with paths in $D$, the space of càdlàg functions on the unit interval $[0,1]$. Our motivation to study such kind of equations is the process equation associatied with the Quicksort algorithm, i.e. $$(Q(t))_{t\in[0,1]}\d \Big(U Q_1\big(1\wedge \tfrac{t}{U}\big) + (1-U) Q_2\big(0 \vee \tfrac{t-U}{1-U}\big) + C(U,t)\Big)_{t\in [0,1]}.$$ Given a sequence $(C, (T_1,\tau_1), (T_2,\tau_2),\ldots)$ we characterize all $D$-valued solutions of the generalized equation and show that every solution can be represented as the convolution of a solution to the inhomogeneous equation and a general solution of the homogeneous equation.

Die Arbeit zeigt einen Ansatz zur Vorhersage von möglichen Wechselkursen. Dabei werden das Keynesianische Modell der Zinsparität und sogenannte Stochastische Prozesse verwendet. Bei der Erstellung des Modelles wird dabei untersucht ob der Devisenmarkt normalverteilt ist, wie oftmals angenommen oder hier andere passendere Verteilungen zur Verfügung stehen. Weiters folgt eine Analyse der Auswirkungen von Leitzinsen auf die Realwirtschaft. Es zeigt sich, dass die Zentralbanken hier bestimmten Regeln folgen. Grundlage des Prognosemodells ist die Geometrische Brownsche Bewegung, welche in weiterer Folge erweitert wird um am Ende auf Lévy-Prozesse und Fat-Tailed-Verteilungen zu stoßen. This master thesis offers an approach to predict potential currency exchange rates using the Keynesian interest rate parity model and so called stochastic processes. During the creation of the model it is examined whether the foreign exchange market is Gaussian distributed, as often assumed or other more suitable distributions are available here. An analysis of the effects of cash rates and their impact on the real economy follows. It turns out that central banks follow certain rules here. The basis of the forecast model is the Brownian Geometric Movement, which is subsequently extended to finally encounter Lévy processes and fat-tailed distributions. vorgelegt von Johann Frank Webersberger Wien, FH Campus Wien, Masterarb., 2018

Motivation Stochastic molecular processes are a leading cause of cell-to-cell variability. Their dynamics are often described by continuous-time discrete-state Markov chains and simulated using stochastic simulation algorithms. As these stochastic simulations are computationally demanding, ordinary differential equation models for the dynamics of the statistical moments have been developed. The number of state variables of these approximating models, however, grows at least quadratically with the number of biochemical species. This limits their application to small- and medium-sized processes. Results In this article, we present a scalable moment-closure approximation (sMA) for the simulation of statistical moments of large-scale stochastic processes. The sMA exploits the structure of the biochemical reaction network to reduce the covariance matrix. We prove that sMA yields approximating models whose number of state variables depends predominantly on local properties, i.e. the average node degree of the reaction network, instead of the overall network size. The resulting complexity reduction is assessed by studying a range of medium- and large-scale biochemical reaction networks. To evaluate the approximation accuracy and the improvement in computational efficiency, we study models for JAK2/STAT5 signalling and NFκB signalling. Our method is applicable to generic biochemical reaction networks and we provide an implementation, including an SBML interface, which renders the sMA easily accessible. Availability and implementation The sMA is implemented in the open-source MATLAB toolbox CERENA and is available from https://github.com/CERENADevelopers/CERENA. Supplementary information Supplementary data are available at Bioinformatics online.

One of the tasks of the livestock disease compensation funds of the federal states in Germany is the financial compensation of livestock holders for livestock losses and costs incurred for disease control measures due to certain diseases. Usually, one half of these services are financed through financial reserves built up with the contributions paid by the owners of the respective animal species. The other half is covered by the federal state itself. But there is hardly any reference to how to calculate aforementioned financial reserves. Basically, following an approach presented recently regarding estimations concerning the compensation fund of the federal state of Saxony-Anhalt, in a stochastic modeling of the required reserves concerning the fund of Lower Saxony the anticipated costs within the spatial restriction zones allocated to outbreaks were incorporated for the first time. The overall costs (including the federal state's stakes), the share of the comnensation fund (required reserves) and the the partial costs for a total of 25 categories and subcategories and subcategories of livestock species making up the latter were estimated. It became evident that overall costs/the share of the fund were particularly determined among the diseases by foot-and-mouth disease and among the cost factors by the costs incurred for the compensation of livestock value within the areas surrounding the outbreaks in which all susceptible animals are killed (culling zone). The 80th, 90 and 95th percentile of the established probability distribution of the overall costs referred to a financial volume of about 312, 409 and 540 million euro, while the respective percentiles of the probability distribution of the required reserves of the compensation fund amounted to 175, 225 and 296 million euro.

The aim of this paper was to study the performance of a novel fish breeding scheme, which is a combination of walk-back and optimum contribution selection using stochastic simulation. In this walk-back selection scheme, batches of different sizes (50, 100, 1000, 5000 and 10 000) with the phenotypically superior fish from one tank with mixed families were genotyped to set up the pedigree. BLUP estimated breeding values were calculated. The optimum contribution selection method was used with the rate of inbreeding (ΔF) constrained to 0.005 or 0.01 per generation. If the constraint on ΔF could not be held, a second batch of fish was genotyped etc. Compared with the genotyping of all selection candidates (1000, 5000 or 10 000), the use of batches saves genotyping costs. The results show that two batches of 50 fish were often necessary. With a batch size of 100, genetic level was 76–92% of the genetic level achieved for schemes with all fish being genotyped and thus candidates for the optimum contribution selection step. More parents were selected for schemes with larger batches, resulting in a higher genetic gain, especially when all selection candidates were genotyped. There was little extra genetic gain in genotyping of 1000 fish instead of 100 for the larger schemes of 5000 and 10 000 candidates. The accuracy of breeding values was similar for all batch sizes (~0.30), but higher (~0.5) when all candidates were included. Since only the phenotypically most superior fish were genotyped, BLUP-EBV were biased. Compared with genotyping of all selection candidates, the use of batches saves genotyping costs, while simultaneously maintaining high genetic gains.

A general model for the development of cell-populations of bone marrow, based on present knowledge of haematocytopoiesis, is derived. In evaluating the various parameters of the model (transmission probabilities etc.) two different ways wer~ involved: a) Observation of labeled cells during sub-sequent development, b) determination of the function of distribution at various time intervals by cell counting. Although the final solution of the problem is not yet possible for various reasons, however, some detailed aspects have been worked out ~uccessfully. Two attempts, i.e. the mathematical treat-ment of FlRESCU and TAUTU (Bukarest) and that of FUCHS (Berlin). will be discussed with regards to the underlying basic arguments, the practical applicability, and the limitations.

Lung cancer staging according to the TNM system is based on morphological assessment of the primary cancer, lymph nodes and metastases. All aspects of this important oncological classification are measurable with MRI. Pulmonary nodules can be detected at the clinically relevant size of 4-5 mm in diameter. The extent of mediastinal, hilar and supraclavicular lymph node affection can be assessed at the same time. The predominant metastatic spread to the adrenal glands and spine can be detected in coronal orientation during dedicated MRI of the lungs. Search focused whole body MRI completes the staging. Various additional MR imaging techniques provide further functional and clinically relevant information during a single examination. In the oncological context the most important techniques are imaging of perfusion and tumor motion. Functional MRI of the lungs complements the pure staging and improves surgical approaches and radiotherapy planning.

Modern electronic tacheometers offer the possibility to capture kinematic processes in real time. In case when the kinematic process is observed with only one measurement system, we have no possibility to perform redundant observations that would enable the accuracy estimation of observations and computed values. The Kalman filter represents a method of advanced geodetic analysis and as such adjusts the redundant data in an optimum way. Incorporating a time component directly into a processing of terrestrial kinematic observations demands good knowledge about the procedure of processing kinematic terrestrial observations and the electronic tacheometer capabilities. For this purpose the developed model of Kalman filter for processing kinematic terrestrial observations - discrete Wiener process acceleration model - was tested on reference trajectory in the Geodetic Laboratory of the Technical University Munich.

When a patient's statement of reduced visual acuity (VA) does not correspond to the ophthalmologic findings, the validity of the statement should be checked before intensive diagnostics are initiated, because the statement may be untrue or may overstate a factual reduction in VA. A primary question is whether the patient's statement differs from his or her true VA, and if so, by how much. Especially in examinations for medical opinion, individuals may be well informed on how to check the validity of VA statements. Thus, modifications of standard test strategies are required. We examined a new test that was designed to allow probabilistic evaluation.The optotype board has the form of a poster measuring 150 cm in height and 125 cm in width. It contains 16 lines corresponding to the logarithmic VA levels from 0.12 up to 4.0 at a distance of 5 m. Each line consists of nine Landolt-Cs with four alternative orientations. VA was determined by a 1/1 criterion for each single measurement at each of the nine optotype columns. In cases of low vision (VA0.25), the test distance was reduced accordingly. The test was given to 100 patients (VA 0.012-2.75), 20 healthy subjects (VA1.0) who were requested to simulate a VA of 0.1, and 13 individuals who were examined for medical opinion, nine of whom were suspected (by results of other testing) of stating nonorganic visual deterioration. The mean and the individual distribution of the nine single values were analyzed and compared with the distribution within a typical psychometric function as determined by a Monte Carlo simulation on 100,000 single measurements.None of the 100 normally cooperative patients yielded more than seven idential single values, and only one yielded seven. The single values scattered by one to three lines. Of the 20 pseudomalingerers (examined at a distance of 2.5 m), in seven subjects (35%) we found an atypical pattern of single values (seven or nine identical values). VA results of those 10 subjects who could read the calibration column on the board (reference distance 5 m) scattered around 0.06. Results of those 10 subjects who were examined after the calibration column had been replaced with lower values (reduced by a factor of 4) scattered around 0.13. Of the nine individuals examined for medical opinion who were suspect of stating too low a VA, six (67%) showed an atypical distribution of single values.This convenient test is a practical tool both for regular determination of visual acuity and for detection of false statements by means of an atypical psychometric function. It also allows for application of further validity checks. Intentionally false statements can be influenced by misleading calibration values.

AIM To estimate and evaluate the risks for the offspring due to the administration of radiopharmaceuticals to women during the first pregnancy weeks after conception (weeks p.c.).The in-utero exposition of the embryo due to diagnostic nuclear medicine procedures, for which diagnostic reference levels (DRL) are specified, as well as due to radio iodine therapy (RIT) was determined. To this end, it is assumed that the activity of the diagnostic radiopharmaceuticals administered to the mother corresponds with the DRL and amounts to 600 MBq or 4 GBq 131I for RIT of benign or malignant thyroid disease, respectively. Based on these data, the radiation risk for the offspring was assessed and compared with the spontaneous risks (R0).The dose for the offspring does not exceed 7.8 mSv for the diagnostic procedures considered, resulting in an excess risk for the offspring of less than 0.12% (R0 approximately 25%) to die from cancer during life, of less than 0.07% (R0 approximately 0.2%) to develop cancer up to the age of 15 years, and of less than 0.16% (R0 approximately 2%) for hereditary effects. RIT during the first 8 weeks p.c. results in doses for the offspring of about 100-460 mSv, resulting in an excess risk for malformations of the child of 3.4%-22% (R0 approximately 6%).The risk of stochastic radiation effects for the offspring due to a diagnostic nuclear medicine procedure of the mother during the first 8 weeks p.c. is--compared with the spontaneous risks--very small; deterministic effects are unlikely. In contrast, deterministic effects for the offspring may occur following RIT. In order to decide on a possibly indicated abortion after RIT, an individual risk assessment is mandatory.

Aerosol bolus dispersion, which has excited enormous interest in lung medicine due to its possible use as an efficient toolfor the non-invasive clinical diagnosis of lung function, was simulated by a Monte Carlo model based on the concept of effective diffusivities and a stochastic lung geometry. The mathematical approach enabled the computation of essential characteristics of the exhaled bolus (half width, standard deviation, skewness, and mode shift) as well as the estimation of their dependence upon the volumetric lung depth (VLD) of the inhaled bolus. Results of the dispersion model generally show a very good correspondence with preliminary published experimental data. Half width and standard deviation of the exhaled bolus increase with VLD according to specific functions, whereas skewness and mode shift are subject to a decrease. While no correlation between bolus dispersion and flow rate could be worked out with the model, dispersion linearly increased with total lung capacity (TLC).

The present paper provides a detailed description of an extension of the Monte Carlo computer model IDEAL, which enables the generation of spatial particle deposition patterns in the human lung. Randomly determined particle trajectories are transformed to a Cartesian coordinate system within which the orientation of the trachea corresponds to the z-direction. At the same time, the determination of particle deposition sites is carried out by application of a grid composed of specific volume elements (voxel). The calculation of particle deposition are based on the physical mechanisms of Brownian diffusion, sedimentation, and inertial impaction. After their storage in a matrix, the deposition data are subject to graphical processing. This allows the presentation of spatial particle distribution patterns (3d-density plots) and of the lung surface itself as well as the generation of two-dimensional distributions by sectioning the three-dimensional structures at pre-defined positions. The results of different model simulations (variation of particle size and breathing conditions) are discussed in detail.

In this paper we examine whether there is room for improvement in farm program decisions through the integration of formal mathematical optimisation into the planning process. Probing the potential for improvement, we investigate the cases of four Brandenburg cash crop farms over the last six years. We find that their total gross margins could have been increased significantly through a more sophisticated program planning. However, we also find that the superiority of formalised planning approaches depends on the quality of the data. The superior formal planning approach includes, in contrast to farmers’ ad hoc planning, a systematic time series analysis of gross margins and a stochastic optimisation model. For each of the six years, the formal planning approach provides optimised alternative programs based on the information available to the farmers at the respective time of planning. In order to avoid solutions that exceed the farmers’ risk tolerance, the variance of the observed program’s total gross margin which implicitly reflects the risk attitude of the individual farmer is used as an upper bound in the optimisation. Using the yields and prices realised at the end of each planning period, the total gross margins that could have been realized through the formally optimised programs in each year are then compared to those that were actually realised.

Optimization has been recognized as a powerful tool in teaching and research for a long time. In spite of its well known problem solving capacity, some methodological obstacles have persisted over the years. The main problem is that stochastic variables and their correlations cannot be adequately accounted for within traditional optimization procedures. In this paper, we develop a methodological mix of stochastic simulation and a heuristic optimization procedure which has become known as genetic algorithms. The simulation part of the mix allows for the consideration of complex information such as stochastic processes; the genetic algorithms-part ensures that the method remains manageable in terms of required time and resources. We demonstrate the decision support potential of the approach by optimizing the production program of a Brandenburg crop farm. We account for the risky environment by using existing stochastic information: on the one hand, we model man-days which are available in critical seasons (particularly harvesting) as triangular distributions according to expert estimations. On the other hand, we use empirical time series and estimate stochastic processes for the gross margins of different activities (wheat, barley etc.). Additionally, variant calculations are made in order to take into account different risk attitudes of decision-makers. Model results in terms of optimal production programs and expected total gross margins are highly sensitive both to the risk attitudes of decision-makers and the stochastic processes which are estimated for different activities.

Bispectral analysis of the electroencephalogram (EEG) has been used to monitor depth of anaesthesia. In the majority of publications this has involved the use of the so called BIS-Index TM (Aspect Medical Systems, Inc.). The exact relationship of this index to such bispectral parameters as the bispectrum and bicoherence has not yet been reported. If the EEG is considered as a linear random process, bicoherence is trivial, i.e. it is independent of the EEG frequency. The aim of this study was to determine the proportion of EEG epochs with non-trivial bicoherence during isoflurane/N20 anaesthesia. We reanalyzed 25.5 hours of digitalised EEG signal from 9 patients undergoing gynaecological surgery. The test developed by Hinich for Gaussian distribution and linearity was then applied. The test was validated using various synthetic surrogate data: Gaussian random data, the z-component of the Lorenz attractor, the phase randomized EEG and the phase randomized z-component of the Lorenz attractor. The percentage of epochs (8.192 s, 1024 data points) with non-trivial bicoherence was: Lorenz data 95.4%, phase randomized Lorenz data 9.4%, synthetic Gaussian data 14.8%, original EEG 9.1%, phase randomized EEG 5.1%. The original EEG data were not found to contain a higher percentage of epochs with non-trivial bicoherence than the phase randomized data and the synthetic Gaussian data. We conclude that bispectral analysis does not substantially add to the information obtained with other methods of quantitative EEG analysis.

To answer the question for the most effective use of all usually available information, the MAG3 clearance was determined by different methods: firstly in an optimated whole-body technique for reference, further for comparison by means of a camera background-curve and a dedicated evaluating procedure, and in addition according to the compartmental principle (Sapirstein), by the latter method also simultaneously together with the hippurate clearance. Two simplified methods were applied as well, a non-invasive one and a single-sample method. The results revealed the relatively unpretentious camera technique to be nearly equivalent to more expensive procedures, but qualitatively superior to those methods, which are unnecessarily oversimplified.

After an introduction, three selected contributions to the 10th Course on Radiation Protection held at the University Hospital of Basel are presented. The principles of radiation protection and new Swiss legislation are discussed as the basis for radiological protection. Ways are proposed of reducing radiation exposure while optimizing the X-ray picture with a minimum dose to patient and personnel. Radiation effects from low doses. From the beginning, life on this planet has been exposed to ionizing radiation from natural sources. For about one century additional irradiation has reached us from man-made sources as well. In Switzerland the overall annual radiation exposure from ambient and man-made sources amounts to about 4 mSv. The terrestrial and cosmic radiation and natural radionuclids in the body cause about 1.17 mSv (29%). As much as 1.6 mSv (40%) results from exposure to radon and its progenies, primarily inside homes. Medical applications contribute approximately 1 mSv (26%) to the annual radiation exposure and releases from atomic weapons, nuclear facilities and miscellaneous industrial operations yield less than 0.12 mSv (5%) to the annual dose. Observations of detrimental radiation effects from intermediate to high doses are challenged by observations of biopositive adaptive responses and hormesis following low dose exposure. The important question, whether cellular adaptive response or hormesis could cause beneficial effects to the human organism that would outweigh the detrimental effects attributed to low radiation doses, remains to be resolved. Whether radiation exerts a detrimental, inhibitory, modifying or even beneficial effect is likely to result from identical molecular lesions but to depend upon their quantity, localization and time scale of initiation, as well as the specific responsiveness of the cellular systems involved. For matters of radiation protection the bionegative radiation effects are classified as deterministic effects or stochastic effects respectively. The various histopathological reactions of tissues and organs following localized tissue irradiation, and the radiation syndromes following total body irradiation, constitute the deterministic effects. There will be a threshold below which deterministic effects do not appear and spontaneous incidences are not known. For low dose risk considerations deterministic effects are of no significance. Genetic effects and carcinogenesis are said to be stochastic effects. Characteristically the probability of stochastic effects increases with dose but the severity of the effects is independent of the dose. The shape of the dose-response relationship at intermediate to high dose levels is linear-quadratic. For exposure to low doses the response becomes linear, as is to be expected for a linear-quadratic function at low dose. No threshold is assumed for stochastic effects. The estimate of probability of fatal cancer by the ICRP is 4 x 10(-2) per Sv for the working population and 5 x 10(-2) per Sv for the total population. Their estimate of probability of serious hereditary disorders within the first two generations is 1 x 10(-2) per Sv. The highest probability coefficient is attributed to mental retardation following exposure in utero. Within the sensitive period at 8-15 weeks of gestation, a risk probability of 40 x 10(-2) per Sv is assumed but a threshold at 0.1 Sv is not excluded. Conclusions drawn from experiments, clinical observations and epidemiological studies following intermediate to high radiation exposures attribute a mutagenic and carcinogenic competence to all radiation doses. Microdosimetric considerations support this assumption. This conclusion cannot be confirmed experimentally nor by epidemiological studies of populations living under different conditions from natural sources of radiation. Nevertheless, a change in the present restrictive radiation protection policy does not yet appear appropriate.

To answer this question it is important to explain the mode of development of the different maximum life span potentials (MLP) during evolution. We know by empiric-historical investigations, that at least later in evolution appearing species of mammals possess a higher MLP. What does then mean a "regular process"? The aging process is a regular one, being a physiologic event, in the character of that what all we can determine at it, as far as it allows explanation and prediction. This did succeed in a high degree during the until now short tradition of experimental gerontology. We refer to experimental results not only from the literature but also from our own work on repair, genetic and metabolic changes and multivariate investigations into the biological age, to decide from two hypotheses of the causes of aging, how the title question can be answered: a) aging as a programmed event, brought about by aging genes, controlled by an intrinsic clock. These genes should restrict life expectancy or the replicative capacity of cells--hence the adaptive hypothesis; b) aging as a consequence of the accumulation of errors counteracted by antibiosenescent strategies (AS)--hence nonadaptive hypotheses, the most plausible one being that of the disposable soma. Later species are adapted better to changing environmental conditions by their other development, their ecological niches are larger, their environment contains specifically lesser dangers and risks, because they better cope with them--therefore it is worth while to spend more energy for the maintenance of their soma than for their reproduction: they develop better and more AS.(ABSTRACT TRUNCATED AT 250 WORDS)

Second-order stereology combines stereology with stochastic geometry and digital image analysis to obtain an insight into the spatial architecture (texture) of tissues. In twenty radical prostatectomy specimens from patients with invasive adenocarcinomas of the prostate gland, the pair correlation function of the epithelial tissue component was estimated both for the carcinomatous and the tumour-free domains. After interactive segmentation of the gray level images, the method works fully automatically. The mean pair correlation function of carcinomatous tissue shows less clustering of epithelium at short distances and less rhythmicity of epithelium at long distances as compared to tumour-free tissue. This loss of glandular integrity is higher in cribriform than in tubular carcinomas. The results suggest that second-order stereology is a powerful tool for the objective evaluation of glandular differentiation.