The development and maintenance of tissues and organs depend on the careful regulation and coordinated motion of large numbers of cells. There is substantial evidence that many complex tissue functions, such as cell division, collective cell migration and gene expression, are directly regulated by mechanical forces. However, relatively little is known about how mechanical stress is distributed within a tissue and how this may guide biochemical signalling. Working in the framework of a popular vertex-based model, we derive expressions for stress tensors at the cell and tissue level to build analytic relationships between cell shape and mechanical stress. The discrete vertex model is upscaled, providing exact expressions for the bulk and shear moduli of disordered cellular networks, which bridges the gap to traditional continuum-level descriptions of tissues. Combining this theoretical work with new experimental techniques for whole-tissue stretching of Xenopus laevis tissue, we separate the roles of mechanical stress and cell shape in orienting and cueing epithelial mitosis. We find that the orientation of division is best predicted by the shape of tricellular junctions, while there appears to be a more direct role for mechanical stress as a mitotic cue.

Existem milhares de espécies de serpentes no mundo, muitas com padrões distintos e intricados. Esta diversidade se torna um problema para usuários que precisam criar texturas de pele de serpente para aplicar em modelos 3D, pois a dificuldade em criar estes padrões complexos é considerável. Nós primeiramente propomos uma categorização de padrões de pele de serpentes levando em conta suas características visuais. Então apresentamos um modelo procedural capaz de sintetizar uma vasta gama de textura de padrões de pele de serpentes. O modelo usa processamento de imagem simples (tal como sintetizar bolinhas e listras) bem como autômatos celulares e geradores de ruído para criar texturas realistas para usar em renderizadores modernos. Nossos resultados mostram boa similaridade visual com pele de serpentes reais. As texturas resultantes podem ser usadas não apenas em computação gráfica, mas também em educação sobre serpentes e suas características visuais. Nós também realizamos testes com usuários para avaliar a usabilidade de nossa ferramenta. O escore da Escala de Usabilidade do Sistema foi de 85:8, sugerindo uma ferramenta de texturização altamente efetiva. / There are thousands of snake species in the world, many with intricate and distinct skin patterns. This diversity becomes a problem for users who need to create snake skin textures to apply on 3D models, as the difficulty for creating such complex patterns is considerable. We first propose a categorization of snake skin patterns considering their visual characteristics. We then present a procedural model capable of synthesizing a wide range of texture skin patterns from snakes. The model uses simple image processing (such as synthesizing spots and stripes) as well as cellular automata and noise generators to create realistic textures for use in a modern renderer. Our results show good visual similarity with real skin found in snakes. The resulting textures can be used not only for computer graphics texturing, but also in education about snakes and their visual characteristics. We have also performed a user study to assess the usability of our tool. The score from the System Usability Scale was 85:8, suggesting a highly effective texturing tool.

abstract: A sequence of models is developed to describe urban population growth in the context of the embedded physical, social and economic environments and an urban disease are developed. This set of models is focused on urban growth and the relationship between the desire to move and the utility derived from city life. This utility is measured in terms of the economic opportunities in the city, the level of human constructed amenity, and the level of amenity caused by the natural environment. The set of urban disease models is focused on examining prospects of eliminating a disease for which a vaccine does not exist. It is inspired by an outbreak of the vector-borne disease dengue fever in Peru, during 2000-2001. / Dissertation/Thesis / Ph.D. Applied Mathematics for the Life and Social Sciences 2012

Motivated by applications in tissue engineering, this thesis is concerned with the flow through and around a free-moving porous tissue construct (TC) within a high-aspect-ratio vessel (HARV) bioreactor. We formalise and extend various results for flow within a Hele-Shaw cell containing a porous obstacle. We also consider the impact of the flow on related nutrient transport problems. The HARV bioreactor is a cylinder with circular cross-section which rotates about its axis at a constant rate, and is filled with a nutrient-rich culture medium. The porous TC is modelled as a rigid porous cylinder with circular cross-section and is fully saturated with the fluid. We formulate the flow problem for a porous TC (governed by Darcy's equations) within a HARV bioreactor (governed by the Navier-Stokes equations). We couple the two regions via appropriate interfacial conditions which are derived by consideration of the intricate boundary-layer structure close to the TC surface. By exploiting various small parameters, we simplify the system of equations by performing an asymptotic analysis, and investigate the resulting system for the flow due to a prescribed TC motion. The motion of the TC is determined by analysis of the force and torque acting upon it, and the resulting equations of motion (which are coupled to the flow) are investigated. The short-time TC behaviour is periodic, but we are able to study the long-time drift from this periodic solution by considering the effect of inertia using a multiple-scale analysis. We find that, contrary to received wisdom, inertia affects TC drift on a similar timescale to tissue growth. Finally, we consider the advection of nutrient through the bioreactor and TC, and investigate the problem of nutrient advection-diffusion for a simplified model involving nutrient uptake.

abstract: Cancer is a major health problem in the world today and is expected to become an even larger one in the future. Although cancer therapy has improved for many cancers in the last several decades, there is much room for further improvement. Mathematical modeling has the advantage of being able to test many theoretical therapies without having to perform clinical trials and experiments. Mathematical oncology will continue to be an important tool in the future regarding cancer therapies and management. This dissertation is structured as a growing tumor. Chapters 2 and 3 consider spheroid models. These models are adept at describing 'early-time' tumors, before the tumor needs to co-opt blood vessels to continue sustained growth. I consider two partial differential equation (PDE) models for spheroid growth of glioblastoma. I compare these models to in vitro experimental data for glioblastoma tumor cell lines and other proposed models. Further, I investigate the conditions under which traveling wave solutions exist and confirm numerically. As a tumor grows, it can no longer be approximated by a spheroid, and it becomes necessary to use in vivo data and more sophisticated modeling to model the growth and diffusion. In Chapter 4, I explore experimental data and computational models for describing growth and diffusion of glioblastoma in murine brains. I discuss not only how the data was obtained, but how the 3D brain geometry is created from Magnetic Resonance (MR) images. A 3D finite-difference code is used to model tumor growth using a basic reaction-diffusion equation. I formulate and test hypotheses as to why there are large differences between the final tumor sizes between the mice. Once a tumor has reached a detectable size, it is diagnosed, and treatment begins. Chapter 5 considers modeling the treatment of prostate cancer. I consider a joint model with hormonal therapy as well as immunotherapy. I consider a timing study to determine whether changing the vaccine timing has any effect on the outcome of the patient. In addition, I perform basic analysis on the six-dimensional ordinary differential equation (ODE). I also consider the limiting case, and perform a full global analysis. / Dissertation/Thesis / Doctoral Dissertation Applied Mathematics 2016

Existem milhares de espécies de serpentes no mundo, muitas com padrões distintos e intricados. Esta diversidade se torna um problema para usuários que precisam criar texturas de pele de serpente para aplicar em modelos 3D, pois a dificuldade em criar estes padrões complexos é considerável. Nós primeiramente propomos uma categorização de padrões de pele de serpentes levando em conta suas características visuais. Então apresentamos um modelo procedural capaz de sintetizar uma vasta gama de textura de padrões de pele de serpentes. O modelo usa processamento de imagem simples (tal como sintetizar bolinhas e listras) bem como autômatos celulares e geradores de ruído para criar texturas realistas para usar em renderizadores modernos. Nossos resultados mostram boa similaridade visual com pele de serpentes reais. As texturas resultantes podem ser usadas não apenas em computação gráfica, mas também em educação sobre serpentes e suas características visuais. Nós também realizamos testes com usuários para avaliar a usabilidade de nossa ferramenta. O escore da Escala de Usabilidade do Sistema foi de 85:8, sugerindo uma ferramenta de texturização altamente efetiva. / There are thousands of snake species in the world, many with intricate and distinct skin patterns. This diversity becomes a problem for users who need to create snake skin textures to apply on 3D models, as the difficulty for creating such complex patterns is considerable. We first propose a categorization of snake skin patterns considering their visual characteristics. We then present a procedural model capable of synthesizing a wide range of texture skin patterns from snakes. The model uses simple image processing (such as synthesizing spots and stripes) as well as cellular automata and noise generators to create realistic textures for use in a modern renderer. Our results show good visual similarity with real skin found in snakes. The resulting textures can be used not only for computer graphics texturing, but also in education about snakes and their visual characteristics. We have also performed a user study to assess the usability of our tool. The score from the System Usability Scale was 85:8, suggesting a highly effective texturing tool.

abstract: There has been important progress in understanding ecological dynamics through the development of the theory of ecological stoichiometry. This fast growing theory provides new constraints and mechanisms that can be formulated into mathematical models. Stoichiometric models incorporate the effects of both food quantity and food quality into a single framework that produce rich dynamics. While the effects of nutrient deficiency on consumer growth are well understood, recent discoveries in ecological stoichiometry suggest that consumer dynamics are not only affected by insufficient food nutrient content (low phosphorus (P): carbon (C) ratio) but also by excess food nutrient content (high P:C). This phenomenon, known as the stoichiometric knife edge, in which animal growth is reduced not only by food with low P content but also by food with high P content, needs to be incorporated into mathematical models. Here we present Lotka-Volterra type models to investigate the growth response of Daphnia to algae of varying P:C ratios. Using a nonsmooth system of two ordinary differential equations (ODEs), we formulate the first model to incorporate the phenomenon of the stoichiometric knife edge. We then extend this stoichiometric model by mechanistically deriving and tracking free P in the environment. This resulting full knife edge model is a nonsmooth system of three ODEs. Bifurcation analysis and numerical simulations of the full model, that explicitly tracks phosphorus, leads to quantitatively different predictions than previous models that neglect to track free nutrients. The full model shows that the grazer population is sensitive to excess nutrient concentrations as a dynamical free nutrient pool induces extreme grazer population density changes. These modeling efforts provide insight on the effects of excess nutrient content on grazer dynamics and deepen our understanding of the effects of stoichiometry on the mechanisms governing population dynamics and the interactions between trophic levels. / Dissertation/Thesis / Ph.D. Applied Mathematics 2014

abstract: Cancer is a worldwide burden in every aspect: physically, emotionally, and financially. A need for innovation in cancer research has led to a vast interdisciplinary effort to search for the next breakthrough. Mathematical modeling allows for a unique look into the underlying cellular dynamics and allows for testing treatment strategies without the need for clinical trials. This dissertation explores several iterations of a dendritic cell (DC) therapy model and correspondingly investigates what each iteration teaches about response to treatment. In Chapter 2, motivated by the work of de Pillis et al. (2013), a mathematical model employing six ordinary differential (ODEs) and delay differential equations (DDEs) is formulated to understand the effectiveness of DC vaccines, accounting for cell trafficking with a blood and tumor compartment. A preliminary analysis is performed, with numerical simulations used to show the existence of oscillatory behavior. The model is then reduced to a system of four ODEs. Both models are validated using experimental data from melanoma-induced mice. Conditions under which the model admits rich dynamics observed in a clinical setting, such as periodic solutions and bistability, are established. Mathematical analysis proves the existence of a backward bifurcation and establishes thresholds for R0 that ensure tumor elimination or existence. A sensitivity analysis determines which parameters most significantly impact the reproduction number R0. Identifiability analysis reveals parameters of interest for estimation. Results are framed in terms of treatment implications, including effective combination and monotherapy strategies. In Chapter 3, a study of whether the observed complexity can be represented with a simplified model is conducted. The DC model of Chapter 2 is reduced to a non-dimensional system of two DDEs. Mathematical and numerical analysis explore the impact of immune response time on the stability and eradication of the tumor, including an analytical proof of conditions necessary for the existence of a Hopf bifurcation. In a limiting case, conditions for global stability of the tumor-free equilibrium are outlined. Lastly, Chapter 4 discusses future directions to explore. There still remain open questions to investigate and much work to be done, particularly involving uncertainty analysis. An outline of these steps is provided for future undertakings. / Dissertation/Thesis / Doctoral Dissertation Applied Mathematics 2020

Philosophiae Doctor - PhD / Cancer is a complex disease that involves a sequence of gene-environment interactions in a progressive process that cannot occur without dysfunction in multiple systems. From a mathematical point of view, the sequence of gene-environment interactions often leads to mathematical models which are hard to solve analytically. Therefore, this thesis focuses on the design and implementation of reliable numerical methods for nonlinear, first order delay differential equations, second order non-linear time-dependent parabolic partial (integro) differential problems and optimal control problems arising in cancer modeling. The development of cancer modeling is necessitated by the lack of reliable numerical methods, to solve the models arising in the dynamics of this dreadful disease. Our focus is on chemotherapy, biological stoichometry, double infections, micro-environment, vascular and angiogenic signalling dynamics. Therefore, because the existing standard numerical methods fail to capture the solution due to the behaviors of the underlying dynamics. Analysis of the qualitative features of the models with mathematical tools gives clear qualitative descriptions of the dynamics of models which gives a deeper insight of the problems. Hence, enabling us to derive robust numerical methods to solve such models. / 2021-04-30

abstract: Patients suffering from Retinitis Pigmentosa (RP), the most common type of inherited retinal degeneration, experience irreversible vision loss due to photoreceptor degeneration. The preservation of cone photoreceptors has been deemed medically relevant as a therapy aimed at preventing blindness in patients with RP. Cones rely on aerobic glycolysis to supply the metabolites necessary for outer segment (OS) renewal and maintenance. The rod-derived cone viability factor (RdCVF), a protein secreted by the rod photoreceptors that preserves the cones, accelerates the flow of glucose into the cone cell stimulating aerobic glycolysis. This dissertation presents and analyzes ordinary differential equation (ODE) models of cellular and molecular level photoreceptor interactions in health and disease to examine mechanisms leading to blindness in patients with RP. First, a mathematical model composed of four ODEs is formulated to investigate the progression of RP, accounting for the new understanding of RdCVF’s role in enhancing cone survival. A mathematical analysis is performed, and stability and bifurcation analyses are used to explore various pathways to blindness. Experimental data are used for parameter estimation and model validation. The numerical results are framed in terms of four stages in the progression of RP. Sensitivity analysis is used to determine mechanisms that have a significant affect on the cones at each stage of RP. Utilizing a non-dimensional form of the RP model, a numerical bifurcation analysis via MATCONT revealed the existence of stable limit cycles at two stages of RP. Next, a novel eleven dimensional ODE model of molecular and cellular level interactions is described. The subsequent analysis is used to uncover mechanisms that affect cone photoreceptor functionality and vitality. Preliminary simulations show the existence of oscillatory behavior which is anticipated when all processes are functioning properly. Additional simulations are carried out to explore the impact of a reduction in the concentration of RdCVF coupled with disruption in the metabolism associated with cone OS shedding, and confirms cone-on-rod reliance. The simulation results are compared with experimental data. Finally, four cases are considered, and a sensitivity analysis is performed to reveal mechanisms that significantly impact the cones in each case. / Dissertation/Thesis / Doctoral Dissertation Applied Mathematics 2020