Bifurcation analysis of a free boundary model of vascular tumor growth with a necrotic core and chemotaxis

A considerable number of research works has been devoted to the study of tumor models. Several biophysical factors, such as cell proliferation, apoptosis, chemotaxis, angiogenesis and necrosis, have been discovered to have an impact on the complicated biological system of tumors. An indicator of the aggressiveness of tumor development is the instability of the shape of the tumor boundary. Complex patterns of tumor morphology have been explored by Lu, Min-Jhe et al. [Nonlinear simulation of vascular tumor growth with chemotaxis and the control of necrosis, Journal of Computational Physics 459 (2022): 111153]. In this paper, we continue to carry out a bifurcation analysis on such a vascular tumor model with a controlled necrotic core and chemotaxis. This bifurcation analysis, to the parameter of cell proliferation, is built on the explicit formulas of radially symmetric steady-state solutions. By perturbing the tumor free boundary and establishing rigorous estimates of the free boundary system, %applying the Hanzawa transformation, we prove the existence of the bifurcation branches with Crandall-Rabinowitz theorem. The parameter of chemotaxis is found to influence the monotonicity of the bifurcation point as the mode $l$ increases both theoretically and numerically.Comment: 26 pages, 4 figure

Biomodeling of pancreatic tumor mass.

Pancreatic adenocarcinoma is the fourth leading cause of cancer death in the United States. It is identified by its rapid, invasive progression with a profound resistance to treatments such as chemotherapy. Unfortunately, there is a lack of information on how to effectively inhibit and control the rapid growth of pancreatic tumors, as well as limited information for diagnostics. With current methods, pancreatic cancer will continue to prevail as a leading cause of cancer death. We propose to study the complexity of pancreatic tumors with a systematic and analytical approach. Cancer is an abnormal growth of tissue caused by uncontrolled cell division. Observing the growth of these cells would prove to have a good basis to monitor the growth of a tumor. Here we create a 3-D simulation of tumor growth through mathematical modeling, using data from pancreatic cells grown in vitro. Using 3-D models will help to understand pancreatic tumors at cellular and molecular levels. The project aims to observe realistic growth of the tumor, accomplished from growing tumor cells on a monolayer in order to find parameters for our 3D mathematical model. This method will prove more beneficial than testing only on a monolayer cell line. Although cell death and the toxicity of drug dosage can be tested using a cell monolayer alone, it does not meet the demands of testing drug delivery in a realistic tumor environment that the mathematical model would provide. The monolayer lacks the dimensions that the drug would have to travel if it were delivered to a real in vivo tumor. A possible continuation of this project in the future could be to utilize the mathematical based approach to predict optimal therapy for the pancreatic tumor in order to develop models that can better test patient care for tumors. Computer modeling, another stepping stone through mathematical modeling, will possibly lead to testing the toxic effects of drugs on a 3-D model through computer modeling will aid in understanding the delivery of drugs throughout the tumor in vivo

Computational studies of vascularized tumors

Cancer is a hard problem touching numerous branches of life science. One reason for the complexity of cancer is that tumors act across many different time and length scales ranging from the subcellular to the macroscopic level. Modern sciences still lack an integral understanding of cancer, however in recent years, increasing computational power enabled computational models to accompany and support conventional medical and biological methods bridging the scales from micro to macro. Here I report a multiscale computational model simulating the progression of solid tumors comprising the vasculature mimicked by artificial arterio-venous blood vessel networks. I present a numerical optimization procedure to determine radii of blood vessels in an artificial microcirculation based on physiological stimuli independently of Murray’s law. Comprising the blood vessels, the reported model enables the inspection of blood vessel remodeling dynamics (angiogenesis, vaso-dilation, vessel regression and collapse) during tumor growth. We successfully applied the method to simulated tumor blood vessel networks guided by optical mammography data. In subsequent model development, I included cellular details into the method enabling a computational study of the tumor microenvironment at cellular resolution. I found that small vascularized tumors at the angiogenic switch exhibit a large ecological niche diversity resulting in high evolutionary pressure favoring the colonal selecion hypothesis.Krebs ist ein schwieriges Thema und tritt in zahlreichen Gebieten auf. Ein Grund für die Komplexität des Tumorwachstums sind die unterschiedlichen Zeit- und Längenskalen. In der aktuellen Forschung fehlt immernoch ein ganzheitliches Verständnis von Krebs, obwohl die computergestützten Methoden in den vergangenen Jahren die konventionellen Methoden der Medizin und der Biologie erweitern und unterstützen. Damit wird die Kluft zwischen subzellulären und makroskopischen Prozessen bereits verringert. In der vorliegenden Arbeit dokumentiere ich ein computergestütztes Verfahren, welches das Tumorwachstum auf mehreren Skalen simuliert. Insbesondere wird das Blutgefäßsystem durch künstliche Gefäße nachgeahmt. Es wurde ein numerisches Optimierungsverfahren zur Bestimmung der Gefäßradien eines künstlichen Blutkreislaufes entwickelt, welches auf physiologischen Reizen basiert und unabhängig von Murray‘s Gesetz ist. Da das beschriebene Verfahren zur Simulation von Tumoren Blutgefäße beinhaltet, kann die Umbildung des Gefäßbaumes während des Tumorwachstums untersucht werden. Das Modell wurde erfolgreich mit krankhaften Gefäßsystemen verglichen. In der darauffolgenden Weiterentwicklung des Modells berücksichtigte ich zelluläre Feinheiten, die es mir erlaubten das Mikromilieu in zellulärer Auflösung zu untersuchen. Meine Resultate zeigen, dass bereits kleine Tumore eine hohe ökologische Vielfalt besitzen, was den Selektionsdruck erhöht und damit die Klon-Selektionstheorie begünstigt

Polynomial-Time Amoeba Neighborhood Membership and Faster Localized Solving

We derive efficient algorithms for coarse approximation of algebraic hypersurfaces, useful for estimating the distance between an input polynomial zero set and a given query point. Our methods work best on sparse polynomials of high degree (in any number of variables) but are nevertheless completely general. The underlying ideas, which we take the time to describe in an elementary way, come from tropical geometry. We thus reduce a hard algebraic problem to high-precision linear optimization, proving new upper and lower complexity estimates along the way.Comment: 15 pages, 9 figures. Submitted to a conference proceeding