Angiogenesis is the formation of new blood vessels from the existing vessels. During tumour angiogenesis, tumour cells secret a number of chemical substrates called tumour angiogenic factors (TAFs). These factors diffuse through the extracellular matrix (ECM) and degrade the basement membrane of nearby vasculature. The TAFs also disrupt the corresponding endothelial cell receptors and form finger like capillary sprouts. These factors also create a chemical gradient (chemotaxis) between the tumour and the surrounding blood vessels. Due to the chemotactic force, the capillary sprouts migrate towards the tumour. On the other hand, a haptotactic force generated due to fibronectin which is secreted by the endothelial cell, also acts on these sprouts. These sprouts grow through the proliferation of recruited endothelial cells from the parent vessels. Tumour angiogenesis is not fully understood yet. In this paper, we use 2-D cellular automata (CA) model to study the behavior of tumour angiogenesis using both Moore and von-Neumann neighborhood. The CA model also mimics capillary sprout branching and the fusion of two adjacent sprout tips (anastomoses). In this simulation, a couple of important points are noted: a) no two capillary sprouts are generated from adjacent locations; b) as the sprouts approach closer to the tumour, its branching tendency increases; c) chemotaxis is the most effective driving force for angiogenesis.
Resection of primary tumors is often followed by accelerated growth of metastases. Here we propose that this effect may be due to the fact that resection of primary tumor results in a decrease in the total systemic amount of angiogenesis stimulators, such as VEGF and bFGF. This in turn causes decrease in the systemic level of angiogenesis inhibitors, such as PF-4 and TSP-1, which at least temporarily relieves inhibition of secondary tumors, allowing them to grow. This construct is predicated on the notion that systemic level of angiogenesis inhibitors is regulated by the systemic level of angi…
A recent conceptual model of tumor-driven angiogenesis including branching, elongation, and anastomosis of blood vessels captures some of the intrinsic multiscale structures of this complex system, yet allowing to extract a deterministic integro-partial differential description of the vessel tip density [Phys. Rev. E 90, 062716 (2014)]. Here we solve the stochastic model, show that ensemble averages over many realizations correspond to the deterministic equations, and fit the anastomosis rate coefficient so that the total number of vessel tips evolves similarly in the deterministic and ensembl…
When modeling of tumor-driven angiogenesis, a major source of analytical and computational complexity is the strong coupling between the kinetic parameters of the relevant stochastic branching-and-growth of the capillary network, and the family of interacting underlying fields. To reduce this complexity, we take advantage of the system intrinsic multiscale structure: we describe the stochastic dynamics of the cells at the vessel tip at their natural mesoscale, whereas we describe the deterministic dynamics of the underlying fields at a larger macroscale. Here, we set up a conceptual stochastic…
Employing a novel two-dimensional computational model we have simulated the feedback between angiogenesis and tumor growth dynamics. Analyzing vessel formation and elongation towards the concentration gradient of the tumor-derived angiogenetic basic fibroblast growth factor, bFGF, we assumed that prior to the blood vessels reaching the tumor surface, the resulting pattern of tumor growth is symmetric, circular with a common center point. However, after the vessels reach the tumor surface, we assumed that the growth rate of that particular cancer region is accelerated compared to the tumor surf…
We study a simplified stochastic model for the vascularization of a growing tumor, incorporating the formation of new blood vessels at the tumor periphery as well as their regression in the tumor center. The resulting morphology of the tumor vasculature differs drastically from the original one. We demonstrate that the probabilistic vessel collapse has to be correlated with the blood shear force in order to yield percolating network structures. The resulting tumor vasculature displays fractal properties. Fractal dimension, microvascular density (MVD), blood flow and shear force has been comput…
Autopsy studies of adults dying of non-cancer causes have shown that virtually all of us possess occult, cancerous lesions. This suggests that, for most individuals, cancer will become dormant and not progress, while only in some will it become symptomatic disease. Meanwhile, it was recently shown in animal models that a tumor can produce both stimulators and inhibitors of its own blood supply. To explain the autopsy findings in light of the preclinical research data, we propose a mathematical model of cancer development at the organism scale describing a growing population of metastases, whic…
We propose a model for glioma patterns in a microlocal tumor environment under the influence of acidity, angiogenesis, and tissue anisotropy. The bottom-up model deduction eventually leads to a system of reaction-diffusion-taxis equations for glioma and endothelial cell population densities, of which the former infers flux limitation both in the self-diffusion and taxis terms. The model extends a recently introduced [34] description of glioma pseudopalisade formation, with the aim of studying the effect of hypoxia-induced tumor vascularization on the establishment and maintenance of these hist…
Due to over-metabolism, the tumour cells become hypoxic. To overcome this situation tumour cells secret several chemical substrates to attract nearby blood vessels towards it (angiogenesis). Transition from avascular to vascular tumour is possible with the initiation of angiogenesis. Angiogenesis also plays a crucial role to spread the cancer cells and its colonization at the distant locations of the body (metastasis). In this paper, we briefly review the processes and factors which directly affect tumour angiogenesis or may get affected by it. A model based on cellular automata is developed t…
A new mathematical model and numerical approach are proposed for the simulation of fluid and chemical exchanges between a growing capillary network and the surrounding tissue, in the context of tumor-induced angiogenesis. Thanks to proper modeling assumptions the capillaries are reduced to their centerline: a well posed mathematical model is hence worked out, based on the coupling between a three-dimensional and a one-dimensional equation (3D-1D coupled problem). Also the application of a PDE-constrained optimization formulation is here proposed for the first time for angiogenesis simulations.…
Angiogenesis, the formation of new vessels, is one of the key mechanisms in tumor development and an appealing target for therapy. Non-invasive, high-resolution, high sensitivity, quantitative 3D imaging techniques are required to correctly depict tumor heterogeneous vasculature over time. Ultrafast Doppler was recently introduced and provides an unprecedented combination of resolution, penetration depth and sensitivity without requiring any contrast agents. The technique was further extended to 3D with Ultrafast Doppler Tomography (UFD-T). In this work, UFD-T was applied to the monitoring of …
In this work, we present a coupled 3D-1D model of solid tumor growth within a dynamically changing vascular network to facilitate realistic simulations of angiogenesis. Additionally, the model includes erosion of the extracellular matrix, interstitial flow, and coupled flow in blood vessels and tissue. We employ continuum mixture theory with stochastic Cahn--Hilliard type phase-field models of tumor growth. The interstitial flow is governed by a mesoscale version of Darcy's law. The flow in the blood vessels is controlled by Poiseuille flow, and Starling's law is applied to model the mass tran…
We study a system of particles in a two-dimensional geometry that move according to a reinforced random walk with transition probabilities dependent on the solutions of reaction-diffusion equations for the underlying fields. A birth process and a history-dependent killing process are also considered. This system models tumor-induced angiogenesis, the process of formation of blood vessels induced by a growth factor released by a tumor. Particles represent vessel tip cells, whose trajectories constitute the growing vessel network. New vessels appear and may fuse with existing ones during their e…
We present a mathematical model describing the time development of a population of tumors subject to mutual angiogenic inhibitory signaling. Based on biophysical derivations, it describes organism-scale population dynamics under the influence of three processes: birth (dissemination of secondary tumors), growth and inhibition (through angiogenesis). The resulting model is a nonlinear partial differential transport equation with nonlocal boundary condition. The nonlinearity stands in the velocity through a nonlocal quantity of the model (the total metastatic volume). The asymptotic behavior of …
We present a mathematical model, based on ordinary differential equations, for the evolution of solid tumors and their response to treatment. Specifically the effects of a cytotoxic agent and a monoclonal antibody are included as control term in the equations. The variables considered here are: the number of cancerous cells sensitive to chemotherapy, the number of cancerous cells resistant to chemotherapy, the degree of angiogenesis and the average intensity of VEGF. The rules that govern the quantities mentioned above are based on a geometrical argument: we approximate the tumor mass as a sph…
Tumor growth has long been a target of investigation within the context of mathematical and computer modelling. The objective of this study is to propose and analyze a two-dimensional probabilistic cellular automata model to describe avascular solid tumor growth, taking into account both the competition between cancer cells and normal cells for nutrients and/or space and a time-dependent proliferation of cancer cells. Gompertzian growth, characteristic of some tumors, is described and some of the features of the time-spatial pattern of solid tumors, such as compact morphology with irregular bo…
Notch signaling is an evolutionary conserved cell-cell communication pathway. Besides regulating cell-fate decisions at an individual cell level, Notch signaling coordinates the emergent spatiotemporal patterning in a tissue through ligand-receptor interactions among transmembrane molecules of neighboring cells, as seen in embryonic development, angiogenesis, or wound healing. Due to its ubiquitous nature, Notch signaling is also implicated in several aspects of cancer progression, including tumor angiogenesis, stemness of cancer cells and cellular invasion. Here, we review experimental and co…
In this work, we present a mathematical model of the initiation and progression of a low-grade urinary bladder carcinoma. We simulate the crucial processes involved in tumor growth, such as oxygen diffusion, carcinogen penetration, and angiogenesis, within the framework of the urothelial cell dynamics. The cell dynamics are modeled using the discrete technique of Cellular Automata, while the continuous processes of carcinogen penetration and oxygen diffusion are described by nonlinear diffusion-absorption equations. As the availability of oxygen is necessary for tumor progression, processes of…
In this paper we consider chemotherapy in a spatial model of tumor growth. The model, which is of reaction-diffusion type, takes into account the complex interactions between the tumor and surrounding stromal cells by including densities of endothelial cells and the extra-cellular matrix. When no treatment is applied the model reproduces the typical dynamics of early tumor growth. The initially avascular tumor reaches a diffusion limited size of the order of millimeters and initiates angiogenesis through the release of vascular endothelial growth factor (VEGF) secreted by hypoxic cells in the …
Inflammatory breast cancer (IBC), a rare form of breast cancer associated with increased angiogenesis and metastasis, is largely driven by tumor-stromal interactions with the vasculature and the extracellular matrix (ECM). However, there is currently a lack of understanding of the role these interactions play in initiation and progression of the disease. In this study, we developed the first three-dimensional, in vitro, vascularized, breast tumor platform to quantify the spatial and temporal dynamics of tumor-vasculature and tumor-ECM interactions specific to IBC. Platforms consisting of colla…
