A Lagrangian particle method for reaction–diffusion systems on deforming surfaces

Reaction–diffusion processes on complex deforming surfaces are fundamental to a number of biological processes ranging from embryonic development to cancer tumor growth and angiogenesis. The simulation of these processes using continuum reaction–diffusion models requires computational methods capable of accurately tracking the geometric deformations and discretizing on them the governing equations. We employ a Lagrangian level-set formulation to capture the deformation of the geometry and use an embedding formulation and an adaptive particle method to discretize both the level-set equations and the corresponding reaction–diffusion. We validate the proposed method and discuss its advantages and drawbacks through simulations of reaction–diffusion equations on complex and deforming geometries.     

Three-dimensional multispecies nonlinear tumor growth—II: Tumor invasion and angiogenesis

We extend the diffuse interface model developed in Wise et al. (2008) to study nonlinear tumor growth in 3-D. Extensions include the tracking of multiple viable cell species populations through a continuum diffuse-interface method, onset and aging of discrete tumor vessels through angiogenesis, and incorporation of individual cell movement using a hybrid continuum-discrete approach. We investigate disease progression as a function of cellular-scale parameters such as proliferation and oxygen/nutrient uptake rates. We find that heterogeneity in the physiologically complex tumor microenvironment, caused by non-uniform distribution of oxygen, cell nutrients, and metabolites, as well as phenotypic changes affecting cellular-scale parameters, can be quantitatively linked to the tumor macro-scale as a mechanism that promotes morphological instability. This instability leads to invasion through tumor infiltration of surrounding healthy tissue. Models that employ a biologically founded, multiscale approach, as illustrated in this work, could help to quantitatively link the critical effect of heterogeneity in the tumor microenvironment with clinically observed tumor growth and invasion. Using patient tumor-specific parameter values, this may provide a predictive tool to characterize the complex in vivo tumor physiological characteristics and clinical response, and thus lead to improved treatment modalities and prognosis.     

Mathematical modelling of cancer cell invasion of tissue: local and non-local models and the effect of adhesion

The ability to invade tissue is one of the hallmarks of cancer. Cancer cells achieve this through the secretion of matrix degrading enzymes, cell proliferation, loss of cell–cell adhesion, enhanced cell–matrix adhesion and active migration. Invasion of tissue by the cancer cells is one of the key components in the metastatic cascade, whereby cancer cells spread to distant parts of the host and initiate the growth of secondary tumours (metastases). A better understanding of the complex processes involved in cancer invasion may ultimately lead to treatments being developed which can localise cancer and prevent metastasis. In this paper we formulate a novel continuum model of cancer cell invasion of tissue which explicitly incorporates the important biological processes of cell–cell and cell–matrix adhesion. This is achieved using non-local (integral) terms in a system of partial differential equations where the cells use a so-called “sensing radius” R to detect their environment. We show that in the limit as R→0 the non-local model converges to a related system of reaction–diffusion–taxis equations. A numerical exploration of this model using computational simulations shows that it can form the basis for future models incorporating more details of the invasion process.     

A cell-based model exhibiting branching and anastomosis during tumor-induced angiogenesis

This work describes the first cell-based model of tumor-induced angiogenesis. At the extracellular level, the model describes diffusion, uptake, and decay of tumor-secreted pro-angiogenic factor. At the cellular level, the model uses the cellular Potts model based on system-energy reduction to describe endothelial cell migration, growth, division, cellular adhesion, and the evolving structure of the stroma. Numerical simulations show: 1), different tumor-secreted pro-angiogenic factor gradient profiles dramatically affect capillary sprout morphology; 2), average sprout extension speeds depend on the proximity of the proliferating region to the sprout tip, and the coordination of cellular functions; and 3), inhomogeneities in the extravascular tissue lead to sprout branching and anastomosis, phenomena that emerge without any prescribed rules. This model provides a quantitative framework to test hypotheses on the biochemical and biomechanical mechanisms that control tumor-induced angiogenesis.     

Nonlinear simulation of tumor necrosis,neo-vascularization and tissue invasion via an adaptive finite-element/level-set method

We present a multi-scale computer simulator of cancer progression at the tumoral level, from avascular stage growth, through the transition from avascular to vascular growth (neo-vascularization), and into the later stages of growth and invasion of normal tissue. We use continuum scale reaction-diffusion equations for the growth component of the model, and a combined continuum-discrete model for the angiogenesis component. We use the level set method for describing complex topological changes observed during growth such as tumor splitting and reconnection, and capture of healthy tissue inside the tumor. We use an adaptive, unstructured finite element mesh that allows for finely resolving important regions of the computational domain such as the necrotic rim, the tumor interface and around the capillary sprouts. We present full nonlinear, two-dimensional simulations, showing the potential of our virtual cancer simulator. We use microphysical parameters characterizing malignant glioma cells, obtained from recent in vitro experiments from our lab and from clinical data, and provide insight into the mechanisms leading to infiltration of the brain by the cancer cells. The results indicate that diffusional instability of tumor mass growth and the complex interplay with the developing neo-vasculature may be powerful mechanisms for tissue invasion.     

Invasion and adaptive evolution for individual-based spatially structured populations

The interplay between space and evolution is an important issue in population dynamics, that is particularly crucial in the emergence of polymorphism and spatial patterns. Recently, biological studies suggest that invasion and evolution are closely related. Here, we model the interplay between space and evolution starting with an individual-based approach and show the important role of parameter scalings on clustering and invasion. We consider a stochastic discrete model with birth, death, competition, mutation and spatial diffusion, where all the parameters may depend both on the position and on the phenotypic trait of individuals. The spatial motion is driven by a reflected diffusion in a bounded domain. The interaction is modelled as a trait competition between individuals within a given spatial interaction range. First, we give an algorithmic construction of the process. Next, we obtain large population approximations, as weak solutions of nonlinear reaction–diffusion equations. As the spatial interaction range is fixed, the nonlinearity is nonlocal. Then, we make the interaction range decrease to zero and prove the convergence to spatially localized nonlinear reaction–diffusion equations. Finally, a discussion of three concrete examples is proposed, based on simulations of the microscopic individual-based model. These examples illustrate the strong effects of the spatial interaction range on the emergence of spatial and phenotypic diversity (clustering and polymorphism) and on the interplay between invasion and evolution. The simulations focus on the qualitative differences between local and nonlocal interactions.      

Formulation and numerical simulations of a continuum model of avascular tumor growth

In this paper we present a continuum mathematical model for a multicellular spheroid that mimics the micro-environment within avascular tumor growth. The model consists of a coupled system of non-linear convection-diffusion-reaction equations. This system is solved using a previously developed conservative Galerkin characteristics method. In the model considered, there are three cell types: the proliferative cells, the quiescent non-dividing cells which stay in the G₀ phase of the cell cycle and the necrotic cells. The model includes viable cell diffusion, diffusion of cellular material and the removal of necrotic cells. We assume that the nutrients diffuse passively and are consumed by the proliferative and quiescent tumor cells depending on the availability of resources (oxygen, glucose, etc.). The numerical simulations are performed using different sets of parameters, including biologically realistic ones, to explore the effects of each of these model parameters on reaching the steady state. The present results, taken together with those reported earlier, indicate that the removal of necrotic cells and the diffusion of cellular material have significant effects on the steady state, reflecting growth saturation, the number of viable cells, and the spheroid size.     

Epidermal Growth Factor Receptor Inhibition Reduces Angiogenesis via Hypoxia-Inducible Factor-1α and Notch1 in Head Neck Squamous Cell Carcinoma

Angiogenesis, a marker of cancer development, affects response to radiotherapy sensibility. This preclinical study aims to understand the receptor tyrosine kinase-mediated angiogenesis in head neck squamous cell carcinoma (HNSCC). The receptor tyrosine kinase activity in a transgenic mouse model of HNSCC was assessed. The anti-tumorigenetic and anti-angiogenetic effects of cetuximab-induced epidermal growth factor receptor (EGFR) inhibition were investigated in xenograft and transgenic mouse models of HNSCC. The signaling transduction of Notch1 and hypoxia-inducible factor-1α (HIF-1α) was also analyzed. EGFR was overexpressed and activated in the Tgfbr1/Pten deletion (2cKO) mouse model of HNSCC. Cetuximab significantly delayed tumor onset by reducing tumor angiogenesis. This drug exerted similar effects on heterotopic xenograft tumors. In the human HNSCC tissue array, increased EGFR expression correlated with increased HIF-1α and micro vessel density. Cetuximab inhibited tumor-induced angiogenesis in vitro and in vivo by significantly downregulating HIF-1α and Notch1. EGFR is involved in the tumor angiogenesis of HNSCC via the HIF-1α and Notch1 pathways. Therefore, targeting EGFR by suppressing hypoxia- and Notch-induced angiogenesis may benefit HNSCC therapy.     

Endothelial CD146 is required for in vitro tumor-induced angiogenesis: The role of a disulfide bond in signaling and dimerization

Tumor angiogenesis, induced by tumor-secreted pro-angiogenic factors, is an essential process for cancer development and metastasis. CD146 is identified as an endothelial cell adhesion molecule and implicated in blood vessel formation, however, its exact role in angiogenesis, particularly tumor angiogenesis, and its potential function of mediating downstream signaling are still unclear. In present study, we evidenced that silencing endogenous endothelial CD146 by RNAi significantly impaired hepatocarcinoma cell secretions-promoted tubular morphogenesis and -enhanced motility of endothelial cells. Biochemical studies revealed that CD146 was required for the activation of p38/IKK/NFκB signaling cascade and up-regulation of NFκB downstream pro-angiogenic genes, notably IL-8, ICAM-1 and MMP9, in response to tumor secretions. Interestingly, specific anti-CD146 mAb AA98, which bound a conformational epitope depending on C452–C499 disulfide bond, could abrogate NFκB activation and tumor angiogenesis, whereas another anti-CD146 mAb AA1 recognizing a linear epitope containing aa50–54 did not have such effects. Further structure–function analysis identified that C452–C499 disulfide bond within the fifth extracellular Ig domain was indispensible for CD146-mediated signaling and tube formation. Moreover, dimerization of CD146, which was enhanced by tumor secretions and suppressed by AA98 but not AA1, also relied on C452 and C499. Together, this study for the first time uncovered the pro-angiogenic role of CD146 and also pinpointed the key structural basis responsible for its signaling function and dimerization. These findings also suggested that CD146 might serve as not just a cell adhesion molecule but also a membrane signal receptor in tumor-induced angiogenesis.     

3D Multi-Cell Simulation of Tumor Growth and Angiogenesis

We present a 3D multi-cell simulation of a generic simplification of vascular tumor growth which can be easily extended and adapted to describe more specific vascular tumor types and host tissues. Initially, tumor cells proliferate as they take up the oxygen which the pre-existing vasculature supplies. The tumor grows exponentially. When the oxygen level drops below a threshold, the tumor cells become hypoxic and start secreting pro-angiogenic factors. At this stage, the tumor reaches a maximum diameter characteristic of an avascular tumor spheroid. The endothelial cells in the pre-existing vasculature respond to the pro-angiogenic factors both by chemotaxing towards higher concentrations of pro-angiogenic factors and by forming new blood vessels via angiogenesis. The tumor-induced vasculature increases the growth rate of the resulting vascularized solid tumor compared to an avascular tumor, allowing the tumor to grow beyond the spheroid in these linear-growth phases. First, in the linear-spherical phase of growth, the tumor remains spherical while its volume increases. Second, in the linear-cylindrical phase of growth the tumor elongates into a cylinder. Finally, in the linear-sheet phase of growth, tumor growth accelerates as the tumor changes from cylindrical to paddle-shaped. Substantial periods during which the tumor grows slowly or not at all separate the exponential from the linear-spherical and the linear-spherical from the linear-cylindrical growth phases. In contrast to other simulations in which avascular tumors remain spherical, our simulated avascular tumors form cylinders following the blood vessels, leading to a different distribution of hypoxic cells within the tumor. Our simulations cover time periods which are long enough to produce a range of biologically reasonable complex morphologies, allowing us to study how tumor-induced angiogenesis affects the growth rate, size and morphology of simulated tumors.     

S100A4蛋白与肿瘤血管生成的研究进展

肿瘤血管生成是指肿瘤细胞诱导的微血管生长以及肿瘤中血液循环建立的过程。重要脏器的转移是恶性肿瘤致死的主要原因,而肿瘤生长、转移和复发都依赖于肿瘤血管生成.S100A4基因是近几年发现的一种具有促肿瘤作用的基因,该基因编码一种钙离子结合调节蛋白,通过与钙离子结合在肿瘤发生和发展中起重要作用。目前研究认为该蛋白在肿瘤的侵袭和转移中有促血管生成作用.本文主要就S100A4与肿瘤血管生成的有关研究进展加以综述。     

Antiangiogenic activity of the endocannabinoid anandamide: correlation to its tumor-suppressor efficacy

Endocannabinoids are now emerging as suppressors of key cell-signaling pathways involved in cancer cell growth, invasion, and metastasis. We have previously observed that the metabolically stable anandamide analog, 2-methyl-2'-F-anandamide (Met-F-AEA) can inhibit the growth of thyroid cancer in vivo. Our hypothesis was that the anti-tumor effect observed could be at least in part ascribed to inhibition of neo-angiogenesis. Therefore, the aim of this study was to assess the anti-angiogenic activity of Met-F-AEA, to investigate the molecular mechanisms underlying this effect and whether Met-F-AEA could antagonize tumor-induced endothelial cell sprouting. We show that Met-F-AEA inhibited bFGF-stimulated endothelial cell proliferation, in a dose-dependent manner, and also induced apoptosis, both effects reliant on cannabinoid CB1 receptor stimulation. Analyzing the signaling pathways implicated in angiogenesis, we observed that the bFGF-induced ERK phosphorylation was antagonized by Met-F-AEA, and we found that p38 MAPK was involved in Met-F-AEA-induced apoptosis. Moreover, Met-F-AEA was able to inhibit bi-dimensional capillary-like tube formation and activity of matrix metalloprotease MMP-2, a major matrix degrading enzyme. Importantly, we demonstrated that Met-F-AEA is also functional in vivo since it inhibited angiogenesis in the chick chorioallantoic neovascularization model. Finally, Met-F-AEA inhibited tumor-induced angiogenesis in a three-dimensional model of endothelial and thyroid tumor cell (KiMol) spheroids co-cultures in different 3-D polymeric matrices that resemble tumor microenvironment and architecture. Thus, our results suggest that anandamide could be involved in the control of cancer growth targeting both tumor cell proliferation and the angiogenic stimulation of the vasculature.     

Modeling and characterization of glucose-sensitive hydrogel: Effect of Young's modulus

A multiphysics model concerning the diffusion and enzyme reaction simultaneously is developed in this paper to characterize the equilibrium behavior of the glucose-sensitive hydrogel, which is called the multi-effect-coupling glucose-stimulus (MECglu) model. The responsive behavior of the hydrogel in the chemo-electro-mechanical coupled energy domains is modeled by the nonlinear coupled partial differential equations. They include the Nernst–Planck equations for the diffusion of mobile species and the enzyme reaction catalyzed by the glucose oxidase and the catalase, the Poisson equation for electric potential, and the mechanical equilibrium equation for finite deformation of the glucose-oxidase-loaded pH-sensitive hydrogel. Numerical simulations demonstrate that the MECglu model can consist well with the published experiment for the practical physiological glucose concentration ranging from 0 to 16.5 mM (300 mg/ml). The effect of Young's modulus of the hydrogel is investigated on the distributive concentrations of reacting and diffusive species and the deformation of the glucose-sensitive hydrogels.     

Reaction–Diffusion Systems and External Morphogen Gradients: The Two-Dimensional Case,with an Application to Skeletal Pattern Formation

We investigate a reaction–diffusion system consisting of an activator and an inhibitor in a two-dimensional domain. There is a morphogen gradient in the domain. The production of the activator depends on the concentration of the morphogen. Mathematically, this leads to reaction–diffusion equations with explicitly space-dependent terms. It is well known that in the absence of an external morphogen, the system can produce either spots or stripes via the Turing bifurcation. We derive first-order expansions for the possible patterns in the presence of an external morphogen and show how both stripes and spots are affected. This work generalizes previous one-dimensional results to two dimensions. Specifically, we consider the quasi-one-dimensional case of a thin rectangular domain and the case of a square domain. We apply the results to a model of skeletal pattern formation in vertebrate limbs. In the framework of reaction–diffusion models, our results suggest a simple explanation for some recent experimental findings in the mouse limb which are much harder to explain in positional-information-type models.     

Mechanisms of normal and tumor-derived angiogenesis

Often those diseases most evasive totherapeutic intervention usurp the human body's own cellular machineryor deregulate normal physiological processes for propagation.Tumor-induced angiogenesis is a pathological condition that resultsfrom aberrant deployment of normal angiogenesis, an essential processin which the vascular tree is remodeled by the growth of newcapillaries from preexisting vessels. Normal angiogenesis ensures thatdeveloping or healing tissues receive an adequate supply of nutrients.Within the confines of a tumor, the availability of nutrients islimited by competition among actively proliferating cells, anddiffusion of metabolites is impeded by high interstitial pressure (Jain RK. Cancer Res 47: 3039-3051, 1987). As a result, tumorcells induce the formation of a new blood supply from the preexisting vasculature, and this affords tumor cells the ability to survive andpropagate in a hostile environment. Because both normal and tumor-induced neovascularization fulfill the essential role of satisfying the metabolic demands of a tissue, the mechanisms by whichcancer cells stimulate pathological neovascularization mimic thoseutilized by normal cells to foster physiological angiogenesis. Thisreview investigates mechanisms of tumor-induced angiogenesis. Thestrategies used by cancer cells to develop their own blood supply arediscussed in relation to those employed by normal cells duringphysiological angiogenesis. With an understanding of blood vesselgrowth in both normal and abnormal settings, we are better suited todesign effective therapeutics for cancer.

     

Simulating PDGF-Driven Glioma Growth and Invasion in an Anatomically Accurate Brain Domain

Gliomas are the most common of all primary brain tumors. They are characterized by their diffuse infiltration of the brain tissue and are uniformly fatal, with glioblastoma being the most aggressive form of the disease. In recent years, the over-expression of platelet-derived growth factor (PDGF) has been shown to produce tumors in experimental rodent models that closely resemble this human disease, specifically the proneural subtype of glioblastoma. We have previously modeled this system, focusing on the key attribute of these experimental tumors—the “recruitment” of oligodendroglial progenitor cells (OPCs) to participate in tumor formation by PDGF-expressing retrovirally transduced cells—in one dimension, with spherical symmetry. However, it has been observed that these recruitable progenitor cells are not uniformly distributed throughout the brain and that tumor cells migrate at different rates depending on the material properties in different regions of the brain. Here we model the differential diffusion of PDGF-expressing and recruited cell populations via a system of partial differential equations with spatially variable diffusion coefficients and solve the equations in two spatial dimensions on a mouse brain atlas using a flux-differencing numerical approach. Simulations of our in silico model demonstrate qualitative agreement with the observed tumor distribution in the experimental animal system. Additionally, we show that while there are higher concentrations of OPCs in white matter, the level of recruitment of these plays little role in the appearance of “white matter disease,” where the tumor shows a preponderance for white matter. Instead, simulations show that this is largely driven by the ratio of the diffusion rate in white matter as compared to gray. However, this ratio has less effect on the speed of tumor growth than does the degree of OPC recruitment in the tumor. It was observed that tumor simulations with greater degrees of recruitment grow faster and develop more nodular tumors than if there is no recruitment at all, similar to our prior results from implementing our model in one dimension. Combined, these results show that recruitment remains an important consideration in understanding and slowing glioma growth.     

Some implications of Scale Relativity theory in avascular stages of growth of solid tumors in the presence of an immune system response

We present a traveling-wave analysis of a reduced mathematical model describing the growth of a solid tumor in the presence of an immune system response in the framework of Scale Relativity theory. Attention is focused upon the attack of tumor cells by tumor-infiltrating cytotoxic lymphocytes (TICLs), in a small multicellular tumor, without necrosis and at some stage prior to (tumor-induced) angiogenesis. For a particular choice of parameters, the underlying system of partial differential equations is able to simulate the well-documented phenomenon of cancer dormancy and propagation of a perturbation in the tumor cell concentration by cnoidal modes, by depicting spatially heterogeneous tumor cell distributions that are characterized by a relatively small total number of tumor cells. This behavior is consistent with several immunomorphological investigations. Moreover, the alteration of certain parameters of the model is enough to induce soliton like modes and soliton packets into the system, which in turn result in tumor invasion in the form of a standard traveling wave. In the same framework of Scale Relativity theory, a very important feature of malignant tumors also results, that even in avascular stages they might propagate and invade healthy tissues, by means of a diffusion on a Newtonian fluid.     

Exclusion processes on a growing domain

A discrete model provides a useful framework for experimentalists to understand the interactions between growing tissues and other biological mechanisms. A cellular automata (CA) model with domain growth, cell motility and cell proliferation, based on cellular exclusion processes, is developed here. Average densities can be defined from the CA model and a continuum representation can be determined. The domain growth mechanism in the CA model gives rise to a Fokker-Planck equation in the corresponding continuum model, with a diffusive and a convective term. Deterministic continuum models derived from conservation laws do not include this diffusive term. The new diffusive term arises because of the stochasticity inherited from the CA mechanism for domain growth. We extend the models to multiple species and investigate the influence of the flux terms arising from the exclusion processes. The averaged CA agent densities are well approximated by the solution of nonlinear advection-diffusion equations, provided that the relative size of the proliferation processes to the diffusion processes is sufficiently small. This dual approach provides an understanding of the microscopic and macroscopic scales in a developmental process.