participants.htm

Titles and abstracts

Testing of the differences in materials microstructure

V. Benes, L. Klebanov, R. Lechnerova, M. Slamova, P. Slama

Faculty of Mathematics and Physics, Charles University of Prague, Czech Republic

In materials microstructures including particles the aim is to provide a statistical test of whether two microstructures are different or not. The information included can be related to individual particle parameters describing the size and shape. Since the assumption of independence of observations within a window is violated, we develop a test based on N-distances in probability theory. The situation with a few distant windows can be converted to a univariate degree-of-fit test. An application to metallographic samples of aluminium alloys follows

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A set-valued continuous time stochastic processes: some statistical aspects

Giacomo Aletti, Enea G. Bongiorno, Vincenzo Capasso

Department of Mathematics, Università degli Studi di Milano

In literature, birth-and-growth processes are the composition of a marked point process together with a real process that represents a local isotropic dilatation. They describe those evolution phenomena that are typically associated to crystals whose border is supposed regular enough.
The aim of this work is to redefine the framework to avoid regularity hypothesis for the border. In order to do this, a geometrical point of view is used to present a particular family of set-valued continuous time stochastic processes: the growth process is described by a bounded closed set-valued process {G_t, t>0}, at the same time the nucleation is described by a non-decreasing closed set-valued process {H_t, t>0}. The process is defined as a suitable combination of those processes. In particular, the discrete time process is derived by a maximality property of Minkowski sum, whilst the continuous one is defined thanks to a construction that is analogous to the definition of the Riemann integral.
The proposed setting allows us to infer the nucleation and the growth processes. A decomposition theorem is established to characterize the nucleation and the growth. As a logical consequence, different consistent set-valued estimators are studied for growth process. Moreover, the nucleation process is studied via the Choquet capacity, and some consistent estimators are derived. In order to perform image analysis and to test the obtained results in R^2 (a simple case), some codes are implemented. They are tested on benchmarks.

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Computer aided diagnosis systems in medical imaging

Paola Campadelli, Elena Casiraghi, Stella Pratissoli

Department of Computer Sciences, Università degli Studi di Milano

Digital images (CT, MRI, RX) are nowadays standard instruments for investigation of internal organ pathologies; this is mainly performed by checking the presence of either shape or texture abnormalities.
With the advent of digital image processing techniques, that provide several mathematical operators to statistically analyze and describe different textures and shapes, the development of computerized medical image analysis methods has gained a great importance since they could aid diagnosis either by performing time consuming tasks or by automatically selecting possible abnormalities to be presented for the last decision to diagnosticians.
In this talk we will describe two computer aided diagnosis systems we have developed.
The first one automatically detects nodules in postero-anterior chest radiographs, by searching for small objects with a brighter texture and a round shape; the second one automatically segments abdominal organs from CT volumes, by exploiting both anatomical knowledge about organ shapes and their relative position, and the gray levels characteristics of each organ.

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Form, performance, fitness: an integrated view on the biological reality

M. Daniela Candia Carnevali

Dipartimento di Biologia, Università degli Studi di Milano, Via Celoria 26, 20133, Milano, Italy

Functional morphology (Yonge, 1925) is an interdisciplinary approach addressed to study the relationships between form and function in organisms senn in aan daptive and evolutionary context. It can typically combine both traditional biological methods, namely morphology, physiology and biomechanics, to methods of mathematics, computational sciences, physics and engineering. The problem form/function permeates the whole biological reality : “Structure without function is a corpse, function without structure is a ghost (Vogel and Wainwright, 1969)”. It represents a fundamental keystone at all the hierarchic levels (from molecules to cells and tissues, to organs and apparatuses, and to whole organisms) by closely combining functional explanations to evolutionary explanations. Functional morphology focuses on the link between biological form and performance and is addressed to the study of the organisms as integrated systems statically and dynamically obeying to processes and principles of physics, in the light of the potential and the limits imposed by onthogenetic and phylogenetic constraints. In this holistic approach which explores fundamental problems of organisms not only related to constructional morphology and biomechanics, but also to developmental (growth and morphogenesis) and environmental biology, the complex reality of biological phenomena can be described and interpreted by resorting to mathematical modelling and computer simulations, which make it possible to incorporate experimental observations into consistent theory to be used in prediction, optimization and even manipulation of biological processes. This contribution gives a brief review of significant examples and models which underline the potential applications of this approach for re-exploring traditional aspects of animal biology from new powerful perspectives.

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Shape Analysis and Molecule Matching

Ian Dryden
University of Nottingham

The statistical analysis of geometrical objects is increasingly important in a wide variety of disciplines, for example in
comparing molecules in bioinformatics. Key aspects of shapeanalysis involve dealing with geometrical invariances and correspondences between parts of objects. Shape data are inherently non-Euclidean, but with care a wide range of practical analyses can be undertaken.

We consider Bayesian methodology for comparing two or more steroid molecules, where the labelling correspondence between atoms is unknown. We initially match a pair of molecules, where one molecule is regarded as random and the other fixed. A type of mixture model is proposed for the point set co-ordinates, and the parameters of the distribution are a labelling matrix (indicating which pairs of points match) and a concentration parameter. An important property of the likelihood is that it is invariant under rotations and translations of the data. Bayesian inference for the parameters is carried out using Markov chain Monte Carlo simulation, and an approximation is considered for speeding up the simulation algorithm. Extensions to multiple molecule alignment are also introduced, and properties of the shape of the steroid molecules are explored in relation to binding activity.

Reference:

Dryden, I.L., Hirst, J.D. and Melville, J.L. (2007). Statistical analysis of unlabelled point sets: comparing molecules in chemoinformatics. Biometrics, 63, 237--251.

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Different approaches to identify skeletal muscle fibre types and to study muscle fibre type plasticity, resulting from thirty year collaboration among biologists, mathematicians and engineers will be presented.

The overview encompasses (i) fibre typing through a series of differently stained muscle sections, (ii) the distribution of fibre types within muscle fascicles, (iii) expression and co-expression of different antigens that leads to fibre type transformations and adaptation of muscles to changed workloads, (iv) counting very rare structures; (v) capillary supply of muscles and muscle fibre types.

Estimation of geometrical characteristics of biological structures using confocal microscopy, image analysis and stereology

Lucie Kubinova and Jiri Janacek
Department of Biomathematics, Institute of Physiology, v.v.i., Academy of Sciences of the Czech Republic, Prague, Czech Republic
e-mail: kubinova@biomed.cas.cz

[2] Parra-Denis E., Ducottet, Ch., Jeulin D. (2007) 3D morphological analysis of intermetallic inclusions, accepted for publication in International Journal of Microstructure and Materials Properties (IJMMP), Special issue : Stereology and Image Analysis in Materials Science.

[3] Parra-Denis E., Moulin N., Jeulin D. (2007) Three Dimensional complex shapes analysis from 3D local curvature measurements: application to intermetallic particles in Aluminium alloy 5XXX, Image Analysis and Stereology: 26, pp. 157-164.

[4] Parra-Denis E., Jeulin D. (2006) Modélisation morphologique 3D des particules intermétalliques dans les alliages d'aluminium. In : Matériaux 2006, Dijon, 13-17 Nov 2006.

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Morphological analysis of tissue microarchitecture

Gabriel Landini

School of dentistry, University of Birmingham (UK)

Image analysis applied to histological images has mainly been used in describing the size, shape and staining texture patterns in individual cells and nuclei. Beyond this first level of morphological characterisation, it is

well established that cells within tissues organise into higher structural and functional hierarchies. These (if quantified appropriately) may provide convenient morphological descriptions to be used in discrimination,

classification and diagnosis.

The methods and examples presented in this talk make use of an algorithmic segmentation of the tissue compartments based on markers (cell nuclei) to provide a theoretical partition of the tissue into exclusive virtual cells (or v-cells). Such segmentation into discrete elements facilitates the computation of new types of quantifiable spatial relationships (architectural features) such as layers, neighbourhoods, tissue borders, as well as allowing to define orientation landmarks using minimal assumptions or user interaction.

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A method of fractal coding for measure-valued images

Davide La Torre

Dept. of Economical and Statistical Sciences, Universita' degli studi di Milano, Italy

Fractal image coding generally seeks to express an image as a union of spatially contracted and greyscale modified copies of subsets of itself. Generally, images are represented as functions u(x) and the fractal coding method is conducted in the framework of L2 or L1. Here we formulate a method of fractal image coding on measure-valued images: At each point x, u(x) is a probability measure over the range of allowed greyscale values. It is natural to consider the possibility of more than one domain subblock mapping onto a given range block Rj. We consider in particular the case in which all subblocks from a common domain pool D can contribute to the image supported on each range block. Using an appropriate weighting of these subblocks, in invariant measure that reflects the local self-similarity properties of the image can be constructed.

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The fractal configuration of normal and pathologic cell tissues

Gabriele Losa

Institute of Scientific Interdisciplinary Studies (ISSI), Locarno (CH)
e-mail: glosa@cerfim.ch

The extension of the concepts of the Fractal Geometry (Mandelbrot,1983) toward the life sciences has brought a significant progress in understanding complex morphological features and functional properties that characterise cells, tissues and organs. It has even been argued that fractal geometry might provide a coherent description of the design principle underlying living organisms (Weibel,1991). The fractal geometry may help interpreting the dynamics of morphogenetic process and the role of the architectural organization in normal and pathologic /carcinogenetic tissues. In this regard it might also provide support to the < tissue organization field theory [TOFT] >, a recent theory (Soto & Sonnenschein, 2005) that regards carcinogenesis as a developmental process in which the complex architectural organization is broken and the communication among the various layers of a tissue dishevelled. Morphological structures and functional processes of biological tissues are considered fractal elements when they fulfil a series of theoretic and methodological criteria. These includes an high level of organization, shape irregularity, statistical self-similarity, configuration and shape invariance within a limited scaling range of measurements, iterative pathway, and a peculiar non-integer fractal dimension invariant with respect to changes of magnification or resolution which can be estimated from a logarithmic equation described by a power law (Losa & Nonnenmacher, 1996). From an heuristic point of view the fractal geometry implies two main features. First, it enables to determine efficiently and accurately the optimal conditions for an objective evaluation of irregular and self-similar morphologic and structural traits, and second, it makes feasible the comparison of the sequential changes that occur in cells and tissues provided that relative morphometric data are collected under appropriate conditions as defined above. It is worth stressing the potential role of fractal geometry in estimating the spatial, geometric, dimensional and architectural arrangement of cells, sub-cellular organelles and tissues during neoplastic transformation and proliferation and, in unravelling developmental dynamics and phenotypic changes which may occur in tumours, in several physiological states and other pathologic lesions as well (Losa, 2006).

References

E. R. Weibel. Fractal geometry: a design principle for living organisms. Am J Physiol 261,361-369,1991

B. Mandelbrot. The fractal geometry of nature. Freeman, San Francisco USA, 1983

A. Soto & C. Sonnenschein. Emergentism as a default: Cancer as a problem of tissue organization. J Biosci. 30(1),1001-116, 2005

G.A. Losa & T.F. Nonnenmacher. Self-similarity and fractal irregularity in pathologic tissues. Mod Pathol 9(3),174-182,1996

G.A. Losa. Do complex cell structures share a fractal-like organization ? Microscopie 5, 53-56, 2006

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Simulating the formation of keratin networks by a piecewise-deterministic Markov process

S. Lueck, Michael Beil, Frank Fleischer, Stéphanie Portet, Wolfgang Arendt, and Volker Schmidt

Institute of Stochastics, Ulm University
Germany
e-mail: sebastian.lueck@uni-ulm.de

Keratin networks are part of the cytoskeleton in epithelial tissues. In previous studies their architecture was found to regulate the viscoelastic properties of individual cells and to influence cell motility. Therefore, dynamic modifications of network architecture were linked to cancer progression and metastasis. Understanding the macromolecular mechanisms defining the structure of keratin networks could thus identify points of intervention for reducing cancer cell migration.

We present a piecewise-deterministic Markov process that has been constructed in order to model de novo formation of a keratin network. The model combines deterministic processes, i.e. diffusion of soluble keratins in the cytoplasm, which is described by a partial differential equation, with stochastic processes governing time points, locations, and macromolecular mechanisms of the annealing process forming the keratin network.

The impact of various hypothetical network formation mechanisms on network morphology has been analyzed in a simulation study. For this purpose, simulated networks in final state have been investigated statistically by techniques from graph theory and point-process analysis. This approach allowed for identifying network formation mechanisms generating morphological key features observed in electron microscopy data of keratin networks.

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Statistical Shape analysis applied to problems of classification

Alessandra Micheletti

Dipartimento di Matematica
Università degli Studi di Milano

In applications, objects rarely have exactly the same shape within measurement error; hence the randomness of shapes need to be taken into account.
The solution to the problem of describing a “shape” via functions taking values in a finite dimensional space, without loosing relevant information, is needed for the mathematical and statistical analysis of the objects.
Some of the techniques which are present in literature make use of landmarks, which usually are specific points, angles, distances, … on the objects, chosen by an expert; other techniques are based on the use of proper measuring functions, usually chosen on the basis of the invariance properties that the geometrical descriptors must satisfy (e.g. invariance with respect to rotations, translations, scaling, etc.). All these techniques have at some extent some degree of subjectivity on which the statistical analysis must rely. The aim of this talk is to compare some of the existing techniques for the statistical shape analysis on some test cases coming both from biology and materials science and to propose extensions which may reduce the degree of subjectivity in the analysis, leading to the definition of automatic procedures of classification based on shape analysis.

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Stochastic networks in angiogenesis

V.Capasso, D. Morale

Dipartimento di Matematica,
Universita' degli Studi di Milano

In biology and medicine it is possible to observe a wide spectrum of formation of patterns and clustering, usually due to self-organization phenomena. Some interesting examples may be found in the process of tumour growth and in particular in angiogenesis, where at an individual level, cells interact and perform a branching process during the formation of new vessels, under the stimulus of a chemical field produced by a tumour. In this way formation of aggregating networks are shown as a consequence of collective behaviour. Aggregation patterns are usually explained in terms of forces, external and/or internal, acting upon individuals. Here we discuss the formation of a stochastic morphology due to the coupling of a continuum field with a system of N(t) stochastic differential equations, where N(t) is a counting process. As a matter of example, we present a simplified stochastic geometric model for a spatially distributed angiogenic process, strongly coupled with a set of relevant underlying fields.

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New imaging strategies in medicine

Miguel Moscoso

Modeling, Numerical Simulation and Industrial Mathematics Group
Universidad Carlos III de Madrid
Avda. de la Universidad 30, Leganés 28911, Spain

Several alternative imaging techniques have been proposed recently in medicine that may provide significant advantages compared to more traditional techniques. These imaging techniques offer new points of view and need to be investigated thoroughly in order to finally find their way into their routine use. Among them, microwave imaging is showing significant promise for early breast cancer detection. Its physical basis is the high contrast between
the dielectric properties of the healthy breast tissue and the malignant tumors at microwave frequencies. In this talk, I will present a shape-based approach for this problem that uses level-set techniques. A shape-based approach offers several advantages compared to more traditional pixel-based approaches, as for example, well defined boundaries and the incorporation of an intrinsic regularization that reduces the dimensionality of the inverse problem and thereby
stabilizes the reconstruction. I will show numerical results that demostrate the potential of these approach in various simulated realistic situations.

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Diffusion Tensor Imaging and its applications to basic neuroscience research and

neuroimaging

Giovanni B Frisoni 1, Donatella Giuliani 2, Giovanni Naldi 2,3, Michela Pievani1

1 LENITEM Laboratory of Epidemiology, Neuroimaging, & Telemedicine - IRCCS Centro S. Giovanni di Dio - FBF, Brescia, Italy

2 Dipartimento di Matematica ``F. Enriques’’, Università degli studi di Milano,Milano, Italy

3 Centro ADAMSS, Università degli studi di Milano, Milano, Italy

Diffusion of molecules in brain and other tissues is important in a wide rangeof biological processes and measurements ranging from the delivery of drugs to diffusion-weighted magnetic resonance imaging. Diffusion tensor imaging is a powerful noninvasive method to characterize neuronal tissue in the human brain
in vivo. In fact, diffusion tensor MRI may be used to characterize the orientational dependence of diffusion in a medium. In white matter the measured diffusion of water appears to be greatest along the fiber direction and more restricted in the perpendicular direction. Connectivity between different brain regions may be estimated from the long-range continuity in the diffusion tensor field, and it is believed to be correlated to the underlying white matter fiber systems. Then, water diffusion magnetic resonance imaging allows tissue structure to be probed and imaged on a microscopic scale, providing unique clues to the fine architecture of neural tissues and to changes associated with various physiological and pathological states, such as acute brain ischaemia. Moreover, because diffusion is anisotropic in brain white matter, reflecting its organization in bundles of fibres running in parallel, diffusion tensor imaging can also be used to map the orientation in space of the white matter tracks in the brain, opening a new point of view on brain connectivity and brain maturation studies. The advent of diffusion magnetic resonance imaging methods has advanced the field of neuroimaging, particularly the radiologic assessment of white matter patency, microstructure, architectural
organization, and orientation. This work provides an introduction to the key concepts that underlie diffusion tensor imaging, and reviews its potential applications in the neurosciences and associated clinical fields. We also point out some applications involved in a collaboration between the Laboratory of Epidemiology, Neuroimaging, and Telemedicine in Brescia and the Department of Mathematics of the University of Milano.

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Vascular networks and vascularised tumors

Luigi Preziosi
Politecnico di Torino

In the talk we will present some models describing the process of aggregation of endothelial cells to form capillary networks governed by the endogenous production of VEGF and the process of dispersion of epatocytes and network formation due to the transition to the mesenchymal state triggered by the presence of exogenous HGF.
We will then describe a model of vascular network formation around capillary networks.

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Spiral growth: observations and models.

Marco Rubbo
Universita' degli Studi di Torino

In the early years of last century X-ray diffraction by crystals confirmed they are made by an ordered disposition of atomic building units as foreseen by Haüy (1784) and Bravais (1895).
The idealized picture coming out from the diffraction spectra, neglecting second order effects resulting from many different causes, gives a description of an ideal crystal. By consequence also the crystal faces structure was idealized in order to explain how crystallization occurs by addition of building units. But the kinetic models did not explain the growth of crystals in conditions close to the thermodynamic equilibrium.
In 1950 Frank, in collaboration with mathematicians and solid state physicists, taking into account real crystals’ structures characterized by dislocations in the atomic arrangements, proposed a mathematical model able to explain almost quantitatively the measured rates of growth of crystals.
For the technological importance of controlling the microscopic topography, the macroscopic appearance and all the physical properties, the phenomenological model they proposed has been developed since then, taking into account different kinetic processes: surface and volume diffusion, elastic interaction among steps, formation of steps waves, effect of impurities in the growth medium, etc.
Some of the proposed models will be exposed shortly; attention will also be drawn to some problems not yet fully resolved such as: -the formation of waves of growth steps, -the interaction of random distributed dislocation originating interacting growth spirals, -the growth when nucleation occurs randomly on the surface among the arms of growth spirals.

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Nonparametric permutation tests in shape analysis

Luigi Salmaso
Dept. of Management and Engineering,
Universita' degli Studi di Padova, Italy
salmaso@gest.unipd.it

Traditional approaches for the statistical analysis of shape involve methods assessing the difference between configurations of landmarks optimally superimposed using a least-squares procedure or methods based on interlandmark distances. All these methods are based on strong assumptions, like equality of covariance matrices, independence, multivariate normal model for landmarks. Moreover, in almost all real applications, researchers have to cope with few individuals and many landmarks, implying over-dimensioned spaces and loss of power. For these reasons we suggest a nonparametric permutation approach to shape analysis. Focussing on the two independent sample case, through a simulation study, we evaluate the behaviour of some nonparametric permutation tests and we show that the proposed tests are very powerful, both in for balanced and unbalanced sample sizes.

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Reconstruction of Shapes with A Priori Knowledge based on M-Reps

M. Fuchs, O. Scherzer
University of Innsbruck

The reconstruction of geometry or, in particular, the shape of objects is a common issue in image analysis. Starting from a variational formulation of such a problem on a shape manifold we introduce a regularization technique incorporating statistical shape knowledge. The key idea is to consider a Riemannian metric on the shape manifold which reflects the statistics of a given training set. We investigate the properties of the regularization functional and illustrate our technique by applying it to region-based and edge-based segmentation of image data. In contrast to previous works our framework can be considered on arbitrary (finite-dimensional) shape manifolds and allows the use of Riemannian metrics for regularization of a wide class of variational problems in image processing.

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Cell dynamics modelling of morphogenesis

Vitaly Volpert

Institute of Mathematics, University of Lion

We will discuss possible mechanisms of pattern formation in biology from the point of view of individual based modelling. Each cell is represented in this case as an individual object which can interact mechanically and chemically with the surrounding cells. The cells can proliferate, differentiate or die according to some deterministic or stochastic criteria. The conditions of the emergence of patterns and their characteristics can be different in comparison with continuous models.