Homogenization Model for Aberrant Crypt Foci

Several explanations can be found in the literature about the origin of colorectal cancer. There is however some agreement on the fact that the carcinogenic process is a result of several genetic mutations of normal cells. The colon epithelium is characterized by millions of invaginations, very small cavities, called crypts, where most of the cellular activity occurs. It is consensual in the medical community, that a potential first manifestation of the carcinogenic process, observed in conventional colonoscopy images, is the appearance of Aberrant Crypt Foci (ACF). These are clusters of abnormal crypts, morphologically characterized by an atypical behavior of the cells that populate the crypts. In this work an homogenization model is proposed, for representing the cellular dynamics in the colon epithelium. The goal is to simulate and predict, in silico, the spread and evolution of ACF, as it can be observed in colonoscopy images. By assuming that the colon is an heterogeneous media, exhibiting a periodic distribution of crypts, we start this work by describing a periodic model, that represents the ACF cell-dynamics in a two-dimensional setting. Then, homogenization techniques are applied to this periodic model, to find a simpler model, whose solution symbolizes the averaged behavior of ACF at the tissue level. Some theoretical results concerning the existence of solution of the homogenized model are proven, applying a fixed point theorem. Numerical results showing the convergence of the periodic model to the homogenized model are presented.Comment: 26 pages, 4 figure

Análise numérica de métodos multiescala para problemas elíticos-parabólicos com aplicação na dinâmica celular durante a formação do câncer colorretal

Orientador: Giuseppe RomanazziTese (doutorado - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação CientíficaResumo: O cólon humano é propício ao desenvolvimento de câncer devido à sua renovação celular que consiste em um alto número de proliferações por dia, localizadas em pequenas cavidade chamadas de criptas. O epitélio do cólon é formado por milhões de criptas e é conhecido que mutações no processo de proliferação (dentro das criptas) podem conduzir à carcinogênese. A proliferação de células colônicas pode ser modelada usando multiescalas (FIGUEIREDO et al., 2013). Em particular, nós podemos usar uma cripta de referência como um domínio microescala, que é periodicamente distribuído em um domínio macroescala, onde este é associado a uma porção do epitélio do cólon. O modelo final resulta em um sistema de EDPs acoplado formado por uma equação elítica e uma parabólica na qual as variáveis são a densidade de células proliferativas e a pressão celular exercida. Apresentamos o processo de homogenização desse sistema de equações supondo a existência de uma expansão assintótica da solução e das demais funções que compõem o problema, veja (D.; P., 1999). Aplicamos um método de resolução multiescala baseado em elementos finitos (HMM-FEM) para aproximar a solução homogenizada encontrado em alguns trabalhos como (ABDULLE, 2009; ABDULLE, 2011; ABDULLE; HUBER, 2016). No cenário onde o problema é acoplado e não linear, a implementação de métodos se torna mais robusta e custosa computacionamente, portanto optamos por resolver primeiro o problema elítico e depois o parabólico como uma forma de amenizar essa complexidade. Como poderemos ver mais em frente, essa estratégia não afeta as ordens de convergência dos métodos. Em uma única escala, estudamos estabilidade e convergência de um esquema supraconvergente baseado em diferenças finitas centradas para malhas não uniformes que é equivalente à um esquema baseado em elementos finitos. Em um cenário mais simplificado, estudamos convergência e estabilidade do método apresentado. Já para um caso mais geral provamos, para s = 1, 2, ordem O(h^s) de convergência para a solução e gradiente se a solução exata está em H^(1)(omega). Para o problema homogenizado, apresentamos uma estratégia supraconvergente que permite aproximar a solução do problema homogenizado acoplado, onde numericamente obtemos uma ordem de convergência O(H^2+h^2). Por fim, buscamos apresentar o esboço de um esquema para resolver problemas multiescala usando dos bons resultados de convergência discutidos acima. Esse modelo é baseado em resolover um problema microescala que posteriormente será usado para construir uma solução macroescala para o sistema homogenizado. Os primeiros indícios de convergência surgem dos resultados numéricos obtidos.Abstract: The human colon is prone to develop a cancer due to its cell renovation that consists in a large number of cell divisions per day located in small cavities of the colon epithelium, called crypts. The colon epithelium is filled by millions of crypts, and it is known that mutations in the cell proliferation process (inside the crypts) can lead to the carcinogenesis. Colonic cell proliferation can be modeled by using multiscales (FIGUEIREDO et al., 2013). In particular, we can use a reference crypt, as a microscale domain, that is periodically distributed in a macroscale domain that is a portion of the colon epithelium. The final model results in a coupled PDE system formed by an elliptic and parabolic equations whose unknowns are the proliferative cell density and the exerted cell pressure. We present a homogenization for the final PDE model where it is supposed to exist a asymptotic expansion for the exact solution of the problem , see (D.; P., 1999). We apply a multiscale method based on finite elements (HMM-FEM) to approximate the homogenized solution as in (ABDULLE, 2009; ABDULLE, 2011; ABDULLE; HUBER, 2016). The coupling and the non-linearity of the system implies a more complex implementation and increase the computational effort, thus we first solve the elliptic problem and then the parabolic one to make it easier. As we can see later, that strategy does not affect the convergence rates. Furthermore, in a single scale, we study a supraconvergent method based on centered finite difference to nonuniform mesh which is equivalent to a fully discrete linear finite element method. Firstly we study convergence and stability of a simpler model and then we prove for s = 1, 2 order O(h^s) convergence of solution and gradient if the exact solution is in H^1(omega). Numerical results illustrate the methods above. For the multiscale problem, we present a supraconvergent scheme which provides approximations to the coupled system with O(H^2+h^2) of convergence rate. This is done by solving the homogenized problem with the supraconvergent method discussed before. Our last contribution is a multiscale model in development which can be useful to solve multiscale problems with the good convergence rates discussed above. That model is based on solving a microscale problem that will be used to construct a macroscale solution for the homogenized system. Numerical results for this model suggest a supraconvergence.DoutoradoMatemática AplicadaDoutor em Matemática Aplicada001CAPE

Paracrine and autocrine regulation of gene expression by Wnt-inhibitor Dickkopf in wild-type and mutant hepatocytes

BACKGROUND: Cells are able to communicate and coordinate their function within tissues via secreted factors. Aberrant secretion by cancer cells can modulate this intercellular communication, in particular in highly organised tissues such as the liver. Hepatocytes, the major cell type of the liver, secrete Dickkopf (Dkk), which inhibits Wnt/β-catenin signalling in an autocrine and paracrine manner. Consequently, Dkk modulates the expression of Wnt/β-catenin target genes. We present a mathematical model that describes the autocrine and paracrine regulation of hepatic gene expression by Dkk under wild-type conditions as well as in the presence of mutant cells. RESULTS: Our spatial model describes the competition of Dkk and Wnt at receptor level, intra-cellular Wnt/β-catenin signalling, and the regulation of target gene expression for 21 individual hepatocytes. Autocrine and paracrine regulation is mediated through a feedback mechanism via Dkk and Dkk diffusion along the porto-central axis. Along this axis an APC concentration gradient is modelled as experimentally detected in liver. Simulations of mutant cells demonstrate that already a single mutant cell increases overall Dkk concentration. The influence of the mutant cell on gene expression of surrounding wild-type hepatocytes is limited in magnitude and restricted to hepatocytes in close proximity. To explore the underlying molecular mechanisms, we perform a comprehensive analysis of the model parameters such as diffusion coefficient, mutation strength and feedback strength. CONCLUSIONS: Our simulations show that Dkk concentration is elevated in the presence of a mutant cell. However, the impact of these elevated Dkk levels on wild-type hepatocytes is confined in space and magnitude. The combination of inter- and intracellular processes, such as Dkk feedback, diffusion and Wnt/β-catenin signal transduction, allow wild-type hepatocytes to largely maintain their gene expression

Creation of an In-Vitro Generated Colonic Stem-Cell Niche Using Gradient-Generating Microdevices

The limitations of existing cell culture and animal studies have provided an impetus for the development of alternative cell based in vitro models that better mimic the complex structures and functions of living organs. This thesis lays the groundwork for the development of an in vitro model of the colonic epithelium by focusing on the development of microdevices to recreating the colonic stem-cell niche. New advances enable long-term organotypic culture of colonic epithelial stem cells that develop into structures known as colonoids. Colonoids represent a primary tissue source acting as a potential starting material for development of an in vitro model of the colon. However for that to be possible, there needs to an improved crypt isolation and 3-D colonoid protocols. In the first chapter, an incubation buffer and time are outlined, along with the finding that 50% Matrigel resulted in the highest colonoid formation efficiency. In the second chapter, threshold concentrations of the key Wnt-signaling factors are discovered. While critically important to homeostatic renewal, the threshold concentrations of factors such as Wnt-3a and R-spondin1 that promote stem cell renewal are unknown. A simple, linear gradient-generating device was used to screen a wide range of Wnt-3a and R-spondin1 concentrations for their impact on a large number of colonoids. A Wnt-3a concentration of 60 ng/mL and R-spondin1 concentration of 88 ng/mL were identified as the critical concentrations required for stem-cell renewal and colonoid expansion. The lower factor concentrations yielded the added benefit of a more morphologically appropriate colonoid possessing columnar cells surrounding a central lumen with active crypt-like bud formation. In the final chapter, a gradient-generating device was used to introduce variable concentrations of the two key Wnt-signaling proteins along the length of a single colonoid. After 5 days in culture under a combination of Wnt-3a and R-spondin gradients, novel image analysis techniques leveraged the intrinsic fluorescence of the mouse model to quantify the levels of stem cell polarity across a colonoid. The microenvironment able to create a stem cell niche within a colonoid by applying external growth factors in a graded fashion across the colonoid.Doctor of Philosoph

Current Frontiers and Perspectives in Cell Biology

A numerous internationally renowned authors in the pages of this book present the views of the fields of cell biology and their own research results or review of current knowledge. Chapters are divided into five sections that are dedicated to cell structures and functions, genetic material, regulatory mechanisms, cellular biomedicine and new methods in cell biology. Multidisciplinary and often quite versatile approach by many authors have imposed restrictions of this classification, so it is certain that many chapters could belong to the other sections of this book. The current frontiers, on the manner in which they described in the book, can be a good inspiration to many readers for further improving, and perspectives which are highlighted can be seen in many areas of fundamental biology, biomedicine, biotechnology and other applications of knowledge of cell biology. The book will be very useful for beginners to gain insight into new area, as well as experts to find new facts and expanding horizons

Psr1p interacts with SUN/sad1p and EB1/mal3p to establish the bipolar spindle

Regular Abstracts - Sunday Poster Presentations: no. 382During mitosis, interpolar microtubules from two spindle pole bodies (SPBs) interdigitate to create an antiparallel microtubule array for accommodating numerous regulatory proteins. Among these proteins, the kinesin-5 cut7p/Eg5 is the key player responsible for sliding apart antiparallel microtubules and thus helps in establishing the bipolar spindle. At the onset of mitosis, two SPBs are adjacent to one another with most microtubules running nearly parallel toward the nuclear envelope, creating an unfavorable microtubule configuration for the kinesin-5 kinesins. Therefore, how the cell organizes the antiparallel microtubule array in the first place at mitotic onset remains enigmatic. Here, we show that a novel protein psrp1p localizes to the SPB and plays a key role in organizing the antiparallel microtubule array. The absence of psr1+ leads to a transient monopolar spindle and massive chromosome loss. Further functional characterization demonstrates that psr1p is recruited to the SPB through interaction with the conserved SUN protein sad1p and that psr1p physically interacts with the conserved microtubule plus tip protein mal3p/EB1. These results suggest a model that psr1p serves as a linking protein between sad1p/SUN and mal3p/EB1 to allow microtubule plus ends to be coupled to the SPBs for organization of an antiparallel microtubule array. Thus, we conclude that psr1p is involved in organizing the antiparallel microtubule array in the first place at mitosis onset by interaction with SUN/sad1p and EB1/mal3p, thereby establishing the bipolar spindle.postprin

Removal of antagonistic spindle forces can rescue metaphase spindle length and reduce chromosome segregation defects

Regular Abstracts - Tuesday Poster Presentations: no. 1925Metaphase describes a phase of mitosis where chromosomes are attached and oriented on the bipolar spindle for subsequent segregation at anaphase. In diverse cell types, the metaphase spindle is maintained at a relatively constant length. Metaphase spindle length is proposed to be regulated by a balance of pushing and pulling forces generated by distinct sets of spindle microtubules and their interactions with motors and microtubule-associated proteins (MAPs). Spindle length appears important for chromosome segregation fidelity, as cells with shorter or longer than normal metaphase spindles, generated through deletion or inhibition of individual mitotic motors or MAPs, showed chromosome segregation defects. To test the force balance model of spindle length control and its effect on chromosome segregation, we applied fast microfluidic temperature-control with live-cell imaging to monitor the effect of switching off different combinations of antagonistic forces in the fission yeast metaphase spindle. We show that spindle midzone proteins kinesin-5 cut7p and microtubule bundler ase1p contribute to outward pushing forces, and spindle kinetochore proteins kinesin-8 klp5/6p and dam1p contribute to inward pulling forces. Removing these proteins individually led to aberrant metaphase spindle length and chromosome segregation defects. Removing these proteins in antagonistic combination rescued the defective spindle length and, in some combinations, also partially rescued chromosome segregation defects. Our results stress the importance of proper chromosome-to-microtubule attachment over spindle length regulation for proper chromosome segregation.postprin