A new ghost cell/level set method for moving boundary problems:application to tumor growth

In this paper, we present a ghost cell/level set method for the evolution of interfaces whose normal velocity depend upon the solutions of linear and nonlinear quasi-steady reaction-diffusion equations with curvature-dependent boundary conditions. Our technique includes a ghost cell method that accurately discretizes normal derivative jump boundary conditions without smearing jumps in the tangential derivative; a new iterative method for solving linear and nonlinear quasi-steady reaction-diffusion equations; an adaptive discretization to compute the curvature and normal vectors; and a new discrete approximation to the Heaviside function. We present numerical examples that demonstrate better than 1.5-order convergence for problems where traditional ghost cell methods either fail to converge or attain at best sub-linear accuracy. We apply our techniques to a model of tumor growth in complex, heterogeneous tissues that consists of a nonlinear nutrient equation and a pressure equation with geometry-dependent jump boundary conditions. We simulate the growth of glioblastoma (an aggressive brain tumor) into a large, 1 cm square of brain tissue that includes heterogeneous nutrient delivery and varied biomechanical characteristics (white matter, gray matter, cerebrospinal fluid, and bone), and we observe growth morphologies that are highly dependent upon the variations of the tissue characteristics—an effect observed in real tumor growth

Mathematical modelling of cancer cell invasion of tissue

Cancer cell invasion of tissue is a complex biological process during which cell migration through the extracellular matrix, facilitated by the secretion of degradative enzymes, is a central process. Cells can deform their cytoplasm to produce pseudopodia, anchor these pseudopodia to neighbouring spatial locations in the tissue and detach earlier bonds, to enable them to move and therefore migrate in a specified direction. Genetic mutations, chemoattractant gradients or a lack of nutrients in their current location can stimulate cell motility and cause them to migrate. When cancer cells migrate they degrade the surrounding extracellular matrix, thereby invading new territory. In this paper we propose a hybrid discrete-continuum two-scale model to study the early growth of solid tumours and their ability to degrade and migrate into the surrounding extracellular matrix. The cancer cells are modelled as discrete individual entities which interact with each other via a potential function, while the spatio-temporal dynamics of the other variables in the model (extracellular matrix, matrix degrading enzymes and degraded stroma) are governed by partial differential equations.</p

Quantifying the role of angiogenesis in malignant progression of gliomas:In Silico modeling integrates imaging and histology

Gliomas are uniformly fatal forms of primary brain neoplasms that vary from low-grade to high-grade (glioblastoma). While low-grade gliomas are weakly angiogenic, glioblastomas are among the most angiogenic of tumors. Thus, interactions between glioma cells and their tissue microenvironment may play an important role in aggressive tumor formation and progression. To quantitatively explore how tumor cells interact with their tissue microenvironment, we incorporated the interactions of normoxic glioma cells, hypoxic glioma cells, vascular endothelial cells, diffusible angiogenic factors, and necrosis formation into a first-generation, biologically-based mathematical model for glioma growth and invasion. Model simulations quantitatively described the spectrum of in vivo dynamics of gliomas visualized with medical imaging. Further, we investigated how proliferation and dispersal of glioma cells combine to induce increasing degrees of cellularity, mitoses, hypoxia-induced neo-angiogenesis and necrosis, features that characterize increasing degrees of “malignancy”, and we found that changes in the net rates of proliferation ({lower case rho}) and invasion (D) are not always necessary for malignant progression. Thus, although other factors, including the accumulation of genetic mutations, can change cellular phenotype (e.g. proliferation and invasion rates), this study suggests that these are not required for malignant progression. Simulated results are placed in the context of the current clinical World Health Organization grading scheme for studying specific patient examples. This study suggests that through the application of the proposed model for tumor-microenvironment interactions, predictable patterns of dynamic changes in glioma histology distinct from changes in cellular phenotype (e.g. proliferation and invasion rates) may be identified, thus providing a powerful clinical tool

Modeling the Influence of the E-Cadherin-β-Catenin Pathway in Cancer Cell Invasion:a multiscale approach

In this article, we show, using a mathematical multiscale model, how cell adhesion may be regulated by interactions between E-cadherin and β-catenin and how the control of cell adhesion may be related to cell migration, to the epithelial-mesenchymal transition and to invasion in populations of eukaryotic cells. E-cadherin mediates cell-cell adhesion and plays a critical role in the formation and maintenance of junctional contacts between cells. Loss of E-cadherin-mediated adhesion is a key feature of the epithelial-mesenchymal transition. β-catenin is an intracellular protein associated with the actin cytoskeleton of a cell. E-cadherins bind to β-catenin to form a complex which can interact both with neighboring cells to form bonds, and with the cytoskeleton of the cell. When cells detach from one another, β-catenin is released into the cytoplasm, targeted for degradation, and downregulated. In this process there are multiple protein-complexes involved which interact with β-catenin and E-cadherin. Within a mathematical individual-based multiscale model, we are able to explain experimentally observed patterns solely by a variation of cell-cell adhesive interactions. Implications for cell migration and cancer invasion are also discussed

Modelling the Impact of Pericyte Migration and Coverage of Vessels on the Efficacy of Vascular Disrupting Agents

Over the past decade or so, there have been a large number of modelling approaches aimed at elucidating the most important mechanisms affecting the formation of new capillaries from parent blood vessels — a process known as angiogenesis. Most studies have focussed upon the way in which capillary sprouts are initiated and migrate in response to diffusible chemical stimuli supplied by hypoxic stromal cells and leukocytes in the contexts of solid tumour growth and wound healing. However, relatively few studies have examined the important role played by blood perfusion during angiogenesis and fewer still have explored the ways in which a dynamically evolving vascular bed architecture can affect the distribution of flow within it. From the perspective of solid tumour growth and, perhaps more importantly, its treatment (e.g. chemotherapy), it would clearly be of some benefit to understand this coupling between vascular structure and perfusion more fully. This paper focuses on the implications of such a coupling upon chemotherapeutic, anti-angiogenic, and anti-vascular treatments. In an extension to previous work by the authors, the issue of pericyte recruitment during vessel maturation is considered in order to study the effects of different anti-vascular and anti-angiogenic therapies from a more rigorous modelling standpoint. Pericytes are a prime target for new vascular disrupting agents (VDAs) currently in clinical trials. However, different compounds attack different components of the vascular network and the implications of targeting only certain elements of the capillary bed are not immediately clear. In light of these uncertainties, the effects of anti-angiogenic and anti-vascular drugs are re-examined by using an extended model that includes an interdependency between vessel remodelling potential and local pericyte density. Two- and three-dimensional simulation results are presented and suggest that it may be possible to identify a VDA-specific “plasticity window” (a time period corresponding to low pericyte density), within which a given VDA would be most effective

Tumor Morphology and Phenotypic Evolution Driven by Selective Pressure from the Microenvironment

SummaryEmergence of invasive behavior in cancer is life-threatening, yet ill-defined due to its multifactorial nature. We present a multiscale mathematical model of cancer invasion, which considers cellular and microenvironmental factors simultaneously and interactively. Unexpectedly, the model simulations predict that harsh tumor microenvironment conditions (e.g., hypoxia, heterogenous extracellular matrix) exert a dramatic selective force on the tumor, which grows as an invasive mass with fingering margins, dominated by a few clones with aggressive traits. In contrast, mild microenvironment conditions (e.g., normoxia, homogeneous matrix) allow clones with similar aggressive traits to coexist with less aggressive phenotypes in a heterogeneous tumor mass with smooth, noninvasive margins. Thus, the genetic make-up of a cancer cell may realize its invasive potential through a clonal evolution process driven by definable microenvironmental selective forces. Our mathematical model provides a theoretical/experimental framework to quantitatively characterize this selective pressure for invasion and test ways to eliminate it