Lattice-gas cellular automata for the analysis of cancer invasion

Cancer cells display characteristic traits acquired in a step-wise manner during carcinogenesis. Some of these traits are autonomous growth, induction of angiogenesis, invasion and metastasis. In this thesis, the focus is on one of the latest stages of tumor progression, tumor invasion. Tumor invasion emerges from the combined effect of tumor cell-cell and cell-microenvironment interactions, which can be studied with the help of mathematical analysis. Cellular automata (CA) can be viewed as simple models of self-organizing complex systems in which collective behavior can emerge out of an ensemble of many interacting "simple" components. In particular, we focus on an important class of CA, the so-called lattice-gas cellular automata (LGCA). In contrast to traditional CA, LGCA provide a straightforward and intuitive implementation of particle transport and interactions. Additionally, the structure of LGCA facilitates the mathematical analysis of their behavior. Here, the principal tools of mathematical analysis of LGCA are the mean-field approximation and the corresponding Lattice Boltzmann equation. The main objective of this thesis is to investigate important aspects of tumor invasion, under the microscope of mathematical modeling and analysis: Impact of the tumor environment: We introduce a LGCA as a microscopic model of tumor cell migration together with a mathematical description of different tumor environments. We study the impact of the various tumor environments (such as extracellular matrix) on tumor cell migration by estimating the tumor cell dispersion speed for a given environment. Effect of tumor cell proliferation and migration: We study the effect of tumor cell proliferation and migration on the tumor’s invasive behavior by developing a simplified LGCA model of tumor growth. In particular, we derive the corresponding macroscopic dynamics and we calculate the tumor’s invasion speed in terms of tumor cell proliferation and migration rates. Moreover, we calculate the width of the invasive zone, where the majority of mitotic activity is concentrated, and it is found to be proportional to the invasion speed. Mechanisms of tumor invasion emergence: We investigate the mechanisms for the emergence of tumor invasion in the course of cancer progression. We conclude that the response of a microscopic intracellular mechanism (migration/proliferation dichotomy) to oxygen shortage, i.e. hypoxia, maybe responsible for the transition from a benign (proliferative) to a malignant (invasive) tumor. Computing in vivo tumor invasion: Finally, we propose an evolutionary algorithm that estimates the parameters of a tumor growth LGCA model based on time-series of patient medical data (in particular Magnetic Resonance and Diffusion Tensor Imaging data). These parameters may allow to reproduce clinically relevant tumor growth scenarios for a specific patient, providing a prediction of the tumor growth at a later time stage.Krebszellen zeigen charakteristische Merkmale, die sie in einem schrittweisen Vorgang während der Karzinogenese erworben haben. Einige dieser Merkmale sind autonomes Wachstum, die Induktion von Angiogenese, Invasion und Metastasis. Der Schwerpunkt dieser Arbeit liegt auf der Tumorinvasion, einer der letzten Phasen der Tumorprogression. Die Tumorinvasion ensteht aus der kombinierten Wirkung von den Wechselwirkungen Tumorzelle-Zelle und Zelle-Mikroumgebung, die mit die Hilfe von mathematischer Analyse untersucht werden können. Zelluläre Automaten (CA) können als einfache Modelle von selbst-organisierenden komplexen Systemen betrachtet werden, in denen kollektives Verhalten aus einer Kombination von vielen interagierenden "einfachen" Komponenten entstehen kann. Insbesondere konzentrieren wir uns auf eine wichtige CA-Klasse, die sogenannten Zelluläre Gitter-Gas Automaten (LGCA). Im Gegensatz zu traditionellen CA bieten LGCA eine einfache und intuitive Umsetzung der Teilchen und Wechselwirkungen. Zusätzlich erleichtert die Struktur der LGCA die mathematische Analyse ihres Verhaltens. Die wichtigsten Werkzeuge der mathematischen Analyse der LGCA sind hier die Mean-field Approximation und die entsprechende Lattice - Boltzmann - Gleichung. Das wichtigste Ziel dieser Arbeit ist es, wichtige Aspekte der Tumorinvasion unter dem Mikroskop der mathematischen Modellierung und Analyse zu erforschen: Auswirkungen der Tumorumgebung: Wir stellen einen LGCA als mikroskopisches Modell der Tumorzellen-Migration in Verbindung mit einer mathematischen Beschreibung der verschiedenen Tumorumgebungen vor. Wir untersuchen die Auswirkungen der verschiedenen Tumorumgebungen (z. B. extrazellulären Matrix) auf die Migration von Tumorzellen dürch Schätzung der Tumorzellen-Dispersionsgeschwindigkeit in einem gegebenen Umfeld. Wirkung von Tumor-Zellenproliferation und Migration: Wir untersuchen die Wirkung von Tumorzellenproliferation und Migration auf das invasive Verhalten der Tumorzellen durch die Entwicklung eines vereinfachten LGCA Tumorwachstumsmodells. Wir leiten die entsprechende makroskopische Dynamik und berechnen die Tumorinvasionsgeschwindigkeit im Hinblick auf die Tumorzellenproliferation- und Migrationswerte. Darüber hinaus berechnen wir die Breite der invasiven Zone, wo die Mehrheit der mitotischer Aktivität konzentriert ist, und es wird festgestellt, dass diese proportional zu den Invasionsgeschwindigkeit ist. Mechanismen der Tumorinvasion Entstehung: Wir untersuchen Mechanismen, die für die Entstehung von Tumorinvasion im Verlauf des Krebs zuständig sind. Wir kommen zu dem Schluss, dass die Reaktion eines mikroskopischen intrazellulären Mechanismus (Migration/Proliferation Dichotomie) zu Sauerstoffmangel, d.h. Hypoxie, möglicheweise für den Übergang von einem gutartigen (proliferative) zu einer bösartigen (invasive) Tumor verantwortlich ist. Berechnung der in-vivo Tumorinvasion: Schließlich schlagen wir einen evolutionären Algorithmus vor, der die Parameter eines LGCA Modells von Tumorwachstum auf der Grundlage von medizinischen Daten des Patienten für mehrere Zeitpunkte (insbesondere die Magnet-Resonanz-und Diffusion Tensor Imaging Daten) ermöglicht. Diese Parameter erlauben Szenarien für einen klinisch relevanten Tumorwachstum für einen bestimmten Patienten zu reproduzieren, die eine Vorhersage des Tumorwachstums zu einem späteren Zeitpunkt möglich machen

A Tensor-Based Formulation of Hetero-functional Graph Theory

Recently, hetero-functional graph theory (HFGT) has developed as a means to mathematically model the structure of large flexible engineering systems. In that regard, it intellectually resembles a fusion of network science and model-based systems engineering. With respect to the former, it relies on multiple graphs as data structures so as to support matrix-based quantitative analysis. In the meantime, HFGT explicitly embodies the heterogeneity of conceptual and ontological constructs found in model-based systems engineering including system form, system function, and system concept. At their foundation, these disparate conceptual constructs suggest multi-dimensional rather than two-dimensional relationships. This paper provides the first tensor-based treatment of some of the most important parts of hetero-functional graph theory. In particular, it addresses the "system concept", the hetero-functional adjacency matrix, and the hetero-functional incidence tensor. The tensor-based formulation described in this work makes a stronger tie between HFGT and its ontological foundations in MBSE. Finally, the tensor-based formulation facilitates an understanding of the relationships between HFGT and multi-layer networks

A diffusion tensor imaging study in HIV patients with and without apathy

Thesis (MScMedSc (Biomedical Sciences. Medical Physiology))--University of Stellenbosch, 2010.ENGLISH ABSTRACT: HIV/AIDS is a global epidemic that accounts for a large percentage of the mortality in South Africa every year. Since the implementation of anti-retroviral treatment, HIV positive individuals have been living longer, and the cognitive impairment associated with the disease is becoming increasingly apparent. During the initial systemic infection of HIV, the virus migrates through the blood-brain barrier and inflicts axonal injury by causing upregulation of cytokines and neurotoxic proteins. HIV-associated dementia is a neuropsychological classification of cognitive impairment in HIV and a variety of symptoms have been classified as a part of the dementia complex. One of these is apathy, which is thought to be a precursor for dementia in HIV patients. Three groups of individuals have been recruited and scanned using magnetic resonance imaging (MRI) to examine changes in the brain. These are an HIV non-apathetic cohort, an HIV apathetic cohort and a healthy control cohort. Diffusion tensor imaging (DTI) is an MRI technique used to quantitatively assess white matter (WM) integrity using metrics such as fractional anisotropy (FA). Voxel-based analysis, tract-based spatial statistics (TBSS) and tractography are three established DTI analysis methods that have been applied in numerous studies. However, there are certain methodological strengths and limitations associated with each technique and therefore all three of these techniques were used to compare WM differences across groups. The frontal-subcortical pathways are known to be abnormal in apathy, and this has been demonstrated in a number of imaging studies. Most of these studies have examined apathy in the context of neurodegenerative disorders such as Alzheimer’s disease and Parkinson’s. However, to our knowledge this is the first DTI study in HIV apathetic patients. With the tractography method, the anterior thalamic radiation and the corpus callosum were reconstructed for each individual to determine whether there were any global changes in these tracts. No significant changes were found. However, a variety of regions in the WM were significantly abnormal in the HIV cohorts when comparing the data at a voxel-based level and using TBSS. This included areas such as the genu and splenium of the corpus callosum, the internal capsule and corona radiata. Changes in frontal WM for the HIV apathy group are an indication of dysfunction in the frontal-striatal circuits, and previous literature has implicated these circuits in the neuropathology of apathy in a variety of central nervous system (CNS) disorders.AFRIKAANSE OPSOMMING: MIV/VIGS is `n wêreldwye epidemie wat verantwoordelik is vir `n hoë sterftesyfer in Suid- Afrika elke jaar. Sedert die inleiding van anti-retrovirale behandeling, het die MIV-positiewe populasie se lewensduur verleng. Tesame met langer lewensduur, het die kognitiewe verswakking wat geassosieer word met die siekte ook meer prominent na vore gekom. Gedurende die beginstadium van sistemiese infeksie in MIV is daar `n migrasie van die virus deur die bloed-breinskans. MIV kan indirek verantwoordelik wees vir aksonale beskadiging deur verhoging van neurotoksiese proteine en sitokinien te induseer. MIV-geassosieerde demensie is `n neurosielkundige klassifikasie van kognitiewe verswakking in MIV en verskeie simptome is al geïdentifiseer as deel van die demensie kompleks. Een van die simptome is apatie en daar word gespekuleer dat dit `n voorloper is vir demensie in MIV pasiënte. Drie groepe individue was gewerf vir die studie en geskandeer deur magnetiese resonansie beeldvorming (MRB) om sodoende veranderinge in die brein te ondersoek. Die groepe was onderskeidelik `n HIV nie-apatiese kohort, `n HIV apatiese kohort en `n gesonde kontrole kohort. Diffusie tensor beelding (DTB) is `n MRB tegniek wat toegepas word om witstof integriteit te meet deur gebruik te maak van maatstawwe soos fraksionele anisotropie (FA). “Voxel-based analysis”, “tract-based spatial statistics (TBSS)” en “tractography” is drie gevestigde DTB analitiese metodes wat al in talle studies toegepas was. Daar is egter sekere metodologiese voordele en beperkings verbonde aan elke tegniek en daarom is al drie tegnieke gebruik om witstof verskille tussen groepe te vergelyk. Die frontale-subkortikale roetes in die brein is bekend vir abnormaliteite in apatie en dit was ook al gedemonstreer in verskeie studies. Die meeste van die studies het apatie ondersoek in die konteks van neurodegeneratiewe siektes soos Alzheimer se siekte en Parkinson se siekte. Maar sover ons weet is hierdie die eerste DTB studie in MIV pasiënte met apatie. Met die “tractography” metode was die anterior thalamic radiation en corpus callosum herbou vir elke individu. Dit was om te bepaal of daar enige globale veranderinge is in hierdie gebiede, maar geen beduidende veranderinge is gevind nie.`n Verskeidenheid van gebiede in die witstof was beduidend abnormaal in die MIV kohorte wanneer die data vergelyk was met “TBSS” en “voxel-based analysis.” Dit het gebiede ingesluit soos die genu en splenium van die corpus callosum, die internal capsule en die corona radiata. Veranderinge in die frontale witstof vir die MIVapatie groep is `n aanduiding van disfunksie in die frontale-striatale bane. Vorige literatuur impliseer dat hierdie bane betrokke is in die neuro-patologie van apatie in verskeie sentrale senuweestelsel (SS) steurings

Représentations continue et discrète de la connectivité structurelle des fibres en U du sillon central

International audienceU-shape fibers are superficial white matter fibers connecting adjacent gyri. In this paper, we present a method to characterize the connectivity of U-shape fibers coursing around the central sulcus. Pre-and post-central gyral crests are semi-automatically drawn and used to build a connectivity space that is identical between subjects. This group space provides a dense representation of the short-range connectivity between the two gyri, as well as a structured representation after clustering

An orientation corrected shaking method for the microstructure generation of short fiber-reinforced composites with almost planar fiber orientation

We present an algorithm for generating short fiber-reinforced microstructures with almost planar fiber orientation. The orientation corrected shaking (OCS) method achieves a high accuracy regarding the volume fraction, fiber length distribution and fiber orientation state. Additionally, the algorithm is capable of generating microstructures for industrial materials, e.g., for a PA66GF35 material with a volume fraction of 19.3% and an aspect ratio of 33. For typical manufacturing processes, short fiber-reinforced composites show a mainly planar fiber arrangement. Therefore, we extend the two-step shaking algorithm of Li et al. [J. Ind. Text. 51(1), pp. 506–530, 2022] for a user-selected rectangular size of the unit cell and periodic boundary conditions. Additionally, the hidden closure structure of the algorithm is uncovered and a precise realization of the fiber orientation state achieved. We examine the representative volume element size for the OCS method, observing representative errors below 2% even for unit cells with edge lengths smaller than the mean fiber length. Additionally, the influence of different closure approximations on the stiffness is investigated. When applied to an industrial PA66GF35 material with sandwich structure, the OCS method demonstrates differences below 2% and 9% for the computed directional Young’s moduli $E_1$ and $E_2$ compared to experimental data