A mathematical model of tumour angiogenesis incorporating cellular traction and viscoelastic effects

Formation of vascular-like structures using a chemotaxis-driven multiphase model

We propose a continuum model for pattern formation, based on the multiphase model framework, to explore in vitro cell patterning within an extracellular matrix (ECM). We demonstrate that, within this framework, chemotaxis-driven cell migration can lead to the formation of cell clusters and vascular-like structures in 1D and 2D respectively. The influence on pattern formation of additional mechanisms commonly included in multiphase tissue models, including cell-matrix traction, contact inhibition, and cell-cell aggregation, are also investigated. Using sensitivity analysis, the relative impact of each model parameter on the simulation outcomes is assessed to identify the key parameters involved. Chemoattractant-matrix binding is further included, motivated by previous experimental studies, and found to reduce the spatial scale of patterning to within a biologically plausible range for capillary structures. Key findings from the in-depth parameter analysis of the 1D models, both with and without chemoattractant-matrix binding, are demonstrated to translate well to the 2D model, obtaining vascular-like cell patterning for multiple parameter regimes. Overall, we demonstrate a biologically-motivated multiphase model capable of generating long-term pattern formation on a biologically plausible spatial scale both in 1D and 2D, with applications for modelling in vitro vascular network formation.

Modeling the extracellular matrix in cell migration and morphogenesis: a guide for the curious biologist

The extracellular matrix (ECM) is a highly complex structure through which biochemical and mechanical signals are transmitted. In processes of cell migration, the ECM also acts as a scaffold, providing structural support to cells as well as points of potential attachment. Although the ECM is a well-studied structure, its role in many biological processes remains difficult to investigate comprehensively due to its complexity and structural variation within an organism. In tandem with experiments, mathematical models are helpful in refining and testing hypotheses, generating predictions, and exploring conditions outside the scope of experiments. Such models can be combined and calibrated with in vivo and in vitro data to identify critical cell-ECM interactions that drive developmental and homeostatic processes, or the progression of diseases. In this review, we focus on mathematical and computational models of the ECM in processes such as cell migration including cancer metastasis, and in tissue structure and morphogenesis. By highlighting the predictive power of these models, we aim to help bridge the gap between experimental and computational approaches to studying the ECM and to provide guidance on selecting an appropriate model framework to complement corresponding experimental studies.

Computational modeling of angiogenesis: The importance of cell rearrangements during vascular growth

Angiogenesis is the process wherein endothelial cells (ECs) form sprouts that elongate from the pre-existing vasculature to create new vascular networks. In addition to its essential role in normal development, angiogenesis plays a vital role in pathologies such as cancer, diabetes and atherosclerosis. Mathematical and computational modeling has contributed to unraveling its complexity. Many existing theoretical models of angiogenic sprouting are based on the "snail-trail" hypothesis. This framework assumes that leading ECs positioned at sprout tips migrate toward low-oxygen regions while other ECs in the sprout passively follow the leaders' trails and proliferate to maintain sprout integrity. However, experimental results indicate that, contrary to the snail-trail assumption, ECs exchange positions within developing vessels, and the elongation of sprouts is primarily driven by directed migration of ECs. The functional role of cell rearrangements remains unclear. This review of the theoretical modeling of angiogenesis is the first to focus on the phenomenon of cell mixing during early sprouting. We start by describing the biological processes that occur during early angiogenesis, such as phenotype specification, cell rearrangements and cell interactions with the microenvironment. Next, we provide an overview of various theoretical approaches that have been employed to model angiogenesis, with particular emphasis on recent in silico models that account for the phenomenon of cell mixing. Finally, we discuss when cell mixing should be incorporated into theoretical models and what essential modeling components such models should include in order to investigate its functional role. This article is categorized under: Cardiovascular Diseases > Computational Models Cancer > Computational Models.

A novel nonlocal partial differential equation model of endothelial progenitor cell cluster formation during the early stages of vasculogenesis

The formation of new vascular networks is essential for tissue development and regeneration, in addition to playing a key role in pathological settings such as ischemia and tumour development. Experimental findings in the past two decades have led to the identification of a new mechanism of neovascularisation, known as cluster-based vasculogenesis, during which endothelial progenitor cells (EPCs) mobilised from the bone marrow are capable of bridging distant vascular beds in a variety of hypoxic settings in vivo. This process is characterised by the formation of EPC clusters during its early stages and, while much progress has been made in identifying various mechanisms underlying cluster formation, we are still far from a comprehensive description of such spatio-temporal dynamics. In order to achieve this, we propose a novel mathematical model of the early stages of cluster-based vasculogenesis, comprising of a system of nonlocal partial differential equations including key mechanisms such as endogenous chemotaxis, matrix degradation, cell proliferation and cell-to-cell adhesion. We conduct a linear stability analysis on the system and solve the equations numerically. We then conduct a parametric analysis of the numerical solutions of the one-dimensional problem to investigate the role of underlying dynamics on the speed of cluster formation and the size of clusters, measured via appropriate metrics for the cluster width and compactness. We verify the key results of the parametric analysis with simulations of the two-dimensional problem. Our results, which qualitatively compare with data from in vitro experiments, elucidate the complementary role played by endogenous chemotaxis and matrix degradation in the formation of clusters, suggesting chemotaxis is responsible for the cluster topology while matrix degradation is responsible for the speed of cluster formation. Our results also indicate that the nonlocal cell-to-cell adhesion term in our model, even though it initially causes cells to aggregate, is not sufficient to ensure clusters are stable over long time periods. Consequently, new modelling strategies for cell-to-cell adhesion are required to stabilise in silico clusters. We end the paper with a thorough discussion of promising, fruitful future modelling and experimental research perspectives.

From tumour perfusion to drug delivery and clinical translation of in silico cancer models

In silico cancer models have demonstrated great potential as a tool to improve drug design, optimise the delivery of drugs to target sites in the host tissue and, hence, improve therapeutic efficacy and patient outcome. However, there are significant barriers to the successful translation of in silico technology from bench to bedside. More precisely, the specification of unknown model parameters, the necessity for models to adequately reflect in vivo conditions, and the limited amount of pertinent validation data to evaluate models' accuracy and assess their reliability, pose major obstacles in the path towards their clinical translation. This review aims to capture the state-of-the-art in in silico cancer modelling of vascularised solid tumour growth, and identify the important advances and barriers to success of these models in clinical oncology. Particular emphasis has been put on continuum-based models of cancer since they - amongst the class of mechanistic spatio-temporal modelling approaches - are well-established in simulating transport phenomena and the biomechanics of tissues, and have demonstrated potential for clinical translation. Three important avenues in in silico modelling are considered in this contribution: first, since systemic therapy is a major cancer treatment approach, we start with an overview of the tumour perfusion and angiogenesis in silico models. Next, we present the state-of-the-art in silico work encompassing the delivery of chemotherapeutic agents to cancer nanomedicines through the bloodstream, and then review continuum-based modelling approaches that demonstrate great promise for successful clinical translation. We conclude with a discussion of what we view to be the key challenges and opportunities for in silico modelling in personalised and precision medicine.

Three-dimensional Image-based Mechanical Modeling for Predicting the Response of Breast Cancer to Neoadjuvant Therapy

The use of quantitative medical imaging data to initialize and constrain mechanistic mathematical models of tumor growth has demonstrated a compelling strategy for predicting therapeutic response. More specifically, we have demonstrated a data-driven framework for prediction of residual tumor burden following neoadjuvant therapy in breast cancer that uses a biophysical mathematical model combining reaction-diffusion growth/therapy dynamics and biomechanical effects driven by early time point imaging data. Whereas early work had been based on a limited dimensionality reduction (two-dimensional planar modeling analysis) to simplify the numerical implementation, in this work, we extend our framework to a fully volumetric, three-dimensional biophysical mathematical modeling approach in which parameter estimates are generated by an inverse problem based on the adjoint state method for numerical efficiency. In an in silico performance study, we show accurate parameter estimation with error less than 3% as compared to ground truth. We apply the approach to patient data from a patient with pathological complete response and a patient with residual tumor burden and demonstrate technical feasibility and predictive potential with direct comparisons between imaging data observation and model predictions of tumor cellularity and volume. Comparisons to our previous two-dimensional modeling framework reflect enhanced model prediction of residual tumor burden through the inclusion of additional imaging slices of patient-specific data.

Tumour angiogenesis as a chemo-mechanical surface instability

The hypoxic conditions within avascular solid tumours may trigger the secretion of chemical factors, which diffuse to the nearby vasculature and promote the formation of new vessels eventually joining the tumour. Mathematical models of this process, known as tumour angiogenesis, have mainly investigated the formation of the new capillary networks using reaction-diffusion equations. Since angiogenesis involves the growth dynamics of the endothelial cells sprouting, we propose in this work an alternative mechanistic approach, developing a surface growth model for studying capillary formation and network dynamics. The model takes into account the proliferation of endothelial cells on the pre-existing capillary surface, coupled with the bulk diffusion of the vascular endothelial growth factor (VEGF). The thermo-dynamical consistency is imposed by means of interfacial and bulk balance laws. Finite element simulations show that both the morphology and the dynamics of the sprouting vessels are controlled by the bulk diffusion of VEGF and the chemo-mechanical and geometric properties at the capillary interface. Similarly to dendritic growth processes, we suggest that the emergence of tree-like vessel structures during tumour angiogenesis may result from the free boundary instability driven by competition between chemical and mechanical phenomena occurring at different length-scales.

Structural biology response of a collagen hydrogel synthetic extracellular matrix with embedded human fibroblast: computational and experimental analysis

Adherent cells exert contractile forces which play an important role in the spatial organization of the extracellular matrix (ECM). Due to these forces, the substrate experiments a volume reduction leading to a characteristic shape. ECM contraction is a key process in many biological processes such as embryogenesis, morphogenesis and wound healing. However, little is known about the specific parameters that control this process. With this aim, we present a 3D computational model able to predict the contraction process of a hydrogel matrix due to cell-substrate mechanical interaction. It considers cell-generated forces, substrate deformation, ECM density, cellular migration and proliferation. The model also predicts the cellular spatial distribution and concentration needed to reproduce the contraction process and confirms the minimum value of cellular concentration necessary to initiate the process observed experimentally. The obtained continuum formulation has been implemented in a finite element framework. In parallel, in vitro experiments have been performed to obtain the main model parameters and to validate it. The results demonstrate that cellular forces, migration and proliferation are acting simultaneously to display the ECM contraction.

Mesoscopic and continuum modelling of angiogenesis

Angiogenesis is the formation of new blood vessels from pre-existing ones in response to chemical signals secreted by, for example, a wound or a tumour. In this paper, we propose a mesoscopic lattice-based model of angiogenesis, in which processes that include proliferation and cell movement are considered as stochastic events. By studying the dependence of the model on the lattice spacing and the number of cells involved, we are able to derive the deterministic continuum limit of our equations and compare it to similar existing models of angiogenesis. We further identify conditions under which the use of continuum models is justified, and others for which stochastic or discrete effects dominate. We also compare different stochastic models for the movement of endothelial tip cells which have the same macroscopic, deterministic behaviour, but lead to markedly different behaviour in terms of production of new vessel cells.

A hybrid model of the role of VEGF binding in endothelial cell migration and capillary formation

Vascular endothelial growth factor (VEGF) is the most studied family of soluble, secreted mediators of endothelial cell migration, survival, and proliferation. VEGF exerts its function by binding to specific tyrosine kinase receptors on the cell surface and transducing the effect through downstream signaling. In order to study the influence of VEGF binding on endothelial cell motion, we develop a hybrid model of VEGF-induced angiogenesis, based on the theory of reinforced random walks. The model includes the chemotactic response of endothelial cells to angiogenic factors bound to cell-surface receptors, rather than approximating this as a function of extracellular chemical concentrations. This allows us to capture biologically observed phenomena such as activation and polarization of endothelial cells in response to VEGF gradients across their lengths, as opposed to extracellular gradients throughout the tissue. We also propose a novel and more biologically reasonable functional form for the chemotactic sensitivity of endothelial cells, which is also governed by activated cell-surface receptors. This model is able to predict the threshold level of VEGF required to activate a cell to move in a directed fashion as well as an optimal VEGF concentration for motion. Model validation is achieved by comparison of simulation results directly with experimental data.

Computational models for mechanics of morphogenesis

In the developing embryo, tissues differentiate, deform, and move in an orchestrated manner to generate various biological shapes driven by the complex interplay between genetic, epigenetic, and environmental factors. Mechanics plays a key role in regulating and controlling morphogenesis, and quantitative models help us understand how various mechanical forces combine to shape the embryo. Models allow for the quantitative, unbiased testing of physical mechanisms, and when used appropriately, can motivate new experimentaldirections. This knowledge benefits biomedical researchers who aim to prevent and treat congenital malformations, as well as engineers working to create replacement tissues in the laboratory. In this review, we first give an overview of fundamental mechanical theories for morphogenesis, and then focus on models for specific processes, including pattern formation, gastrulation, neurulation, organogenesis, and wound healing. The role of mechanical feedback in development is also discussed. Finally, some perspectives aregiven on the emerging challenges in morphomechanics and mechanobiology.

Physical determinants of vascular network remodeling during tumor growth

The process in which a growing tumor transforms a hierarchically organized arterio-venous blood vessel network into a tumor specific vasculature is analyzed with a theoretical model. The physical determinants of this remodeling involve the morphological and hydrodynamic properties of the initial network, generation of new vessels (sprouting angiogenesis), vessel dilation (circumferential growth), vessel regression, tumor cell proliferation and death, and the interdependence of these processes via spatio-temporal changes of blood flow parameters, oxygen/nutrient supply and growth factor concentration fields. The emerging tumor vasculature is non-hierarchical, compartmentalized into well-characterized zones, displays a complex geometry with necrotic zones and "hot spots" of increased vascular density and blood flow of varying size, and transports drug injections efficiently. Implications for current theoretical views on tumor-induced angiogenesis are discussed.

On the modelling of biological patterns with mechanochemical models: Insights from analysis and computation

The diversity of biological form is generated by a relatively small number of underlying mechanisms. Consequently, mathematical and computational modelling can, and does, provide insight into how cellular level interactions ultimately give rise to higher level structure. Given cells respond to mechanical stimuli, it is therefore important to consider the effects of these responses within biological self-organisation models. Here, we consider the self-organisation properties of a mechanochemical model previously developed by three of the authors in Acta Biomater. 4, 613-621 (2008), which is capable of reproducing the behaviour of a population of cells cultured on an elastic substrate in response to a variety of stimuli. In particular, we examine the conditions under which stable spatial patterns can emerge with this model, focusing on the influence of mechanical stimuli and the interplay of non-local phenomena. To this end, we have performed a linear stability analysis and numerical simulations based on a mixed finite element formulation, which have allowed us to study the dynamical behaviour of the system in terms of the qualitative shape of the dispersion relation. We show that the consideration of mechanotaxis, namely changes in migration speeds and directions in response to mechanical stimuli alters the conditions for pattern formation in a singular manner. Furthermore without non-local effects, responses to mechanical stimuli are observed to result in dispersion relations with positive growth rates at arbitrarily large wavenumbers, in turn yielding heterogeneity at the cellular level in model predictions. This highlights the sensitivity and necessity of non-local effects in mechanically influenced biological pattern formation models and the ultimate failure of the continuum approximation in their absence.

Nonlinear modelling of cancer: bridging the gap between cells and tumours

Despite major scientific, medical and technological advances over the last few decades, a cure for cancer remains elusive. The disease initiation is complex, and including initiation and avascular growth, onset of hypoxia and acidosis due to accumulation of cells beyond normal physiological conditions, inducement of angiogenesis from the surrounding vasculature, tumour vascularization and further growth, and invasion of surrounding tissue and metastasis. Although the focus historically has been to study these events through experimental and clinical observations, mathematical modelling and simulation that enable analysis at multiple time and spatial scales have also complemented these efforts. Here, we provide an overview of this multiscale modelling focusing on the growth phase of tumours and bypassing the initial stage of tumourigenesis. While we briefly review discrete modelling, our focus is on the continuum approach. We limit the scope further by considering models of tumour progression that do not distinguish tumour cells by their age. We also do not consider immune system interactions nor do we describe models of therapy. We do discuss hybrid-modelling frameworks, where the tumour tissue is modelled using both discrete (cell-scale) and continuum (tumour-scale) elements, thus connecting the micrometre to the centimetre tumour scale. We review recent examples that incorporate experimental data into model parameters. We show that recent mathematical modelling predicts that transport limitations of cell nutrients, oxygen and growth factors may result in cell death that leads to morphological instability, providing a mechanism for invasion via tumour fingering and fragmentation. These conditions induce selection pressure for cell survivability, and may lead to additional genetic mutations. Mathematical modelling further shows that parameters that control the tumour mass shape also control its ability to invade. Thus, tumour morphology may serve as a predictor of invasiveness and treatment prognosis.

Vascular remodelling of an arterio-venous blood vessel network during solid tumour growth

We formulate a theoretical model to analyze the vascular remodelling process of an arterio-venous vessel network during solid tumour growth. The model incorporates a hierarchically organized initial vasculature comprising arteries, veins and capillaries, and involves sprouting angiogenesis, vessel cooption, dilation and regression as well as tumour cell proliferation and death. The emerging tumour vasculature is non-hierarchical, compartmentalized into well-characterized zones and transports efficiently an injected drug-bolus. It displays a complex geometry with necrotic zones and "hot spots" of increased vascular density and blood flow of varying size. The corresponding cluster size distribution is algebraic, reminiscent of a self-organized critical state. The intra-tumour vascular-density fluctuations correlate with pressure drops in the initial vasculature suggesting a physical mechanism underlying hot spot formation.

A hybrid model for three-dimensional simulations of sprouting angiogenesis

Recent advances in cancer research have identified critical angiogenic signaling pathways and the influence of the extracellular matrix on endothelial cell migration. These findings provide us with insight into the process of angiogenesis that can facilitate the development of effective computational models of sprouting angiogenesis. In this work, we present the first three-dimensional model of sprouting angiogenesis that considers explicitly the effect of the extracellular matrix and of the soluble as well as matrix-bound growth factors on capillary growth. The computational model relies on a hybrid particle-mesh representation of the blood vessels and it introduces an implicit representation of the vasculature that can accommodate detailed descriptions of nutrient transport. Extensive parametric studies reveal the role of the extracellular matrix structure and the distribution of the different vascular endothelial growth factors isoforms on the dynamics and the morphology of the generated vascular networks.

Modeling the VEGF–Bcl-2–CXCL8 Pathway in Intratumoral Agiogenesis

Recent experiments show that vascular endothelial growth factor (VEGF) is the crucial mediator of downstream events that ultimately lead to enhanced endothelial cell survival and increased vascular density within many tumors. The newly discovered pathway involves up-regulation of the anti-apoptotic protein Bcl-2, which in turn leads to increased production of interleukin-8 (CXCL8). The VEGF-Bcl-2-CXCL8 pathway suggests new targets for the development of anti-angiogenic strategies including short interfering RNA (siRNA) that silence the CXCL8 gene and small molecule inhibitors of Bcl-2. In this paper, we present and validate a mathematical model designed to predict the effect of the therapeutic blockage of VEGF, CXCL8, and Bcl-2 at different stages of tumor progression. In agreement with experimental observations, the model predicts that curtailing the production of CXCL8 early in development can result in a delay in tumor growth and vascular development; however, it has little effect when applied at late stages of tumor progression. Numerical simulations also show that blocking Bcl-2 up-regulation, either at early stages or after the tumor has fully developed, ensures that both microvascular and tumor cell density stabilize at low values representing growth control. These results provide insight into those aspects of the VEGF-Bcl-2-CXCL8 pathway, which independently and in combination, are crucial mediators of tumor growth and vascular development. Continued quantitative modeling in this direction may have profound implications for the development of novel therapies directed against specific proteins and chemokines to alter tumor progression.

Tumor growth instability and the onset of invasion

Motivated by experimental observations, we develop a mathematical model of chemotactically directed tumor growth. We present an analytical study of the model as well as a numerical one. The mathematical analysis shows that: (i) tumor cell proliferation by itself cannot generate the invasive branching behavior observed experimentally, (ii) heterotype chemotaxis provides an instability mechanism that leads to the onset of tumor invasion, and (iii) homotype chemotaxis does not provide such an instability mechanism but enhances the mean speed of the tumor surface. The numerical results not only support the assumptions needed to perform the mathematical analysis but they also provide evidence of (i), (ii), and (iii). Finally, both the analytical study and the numerical work agree with the experimental phenomena.

On the role of endothelial progenitor cells in tumor neovascularization

The exact role that bone marrow (BM)-derived endothelial progenitor cells (EPCs) play in tumor neovascularization is heavily debated. We develop a quantitative three-compartment model with predictive power regarding the dynamics of tumorigenesis. There are two distinct processes by which tumor neovasculature can be built: angiogenesis is the formation of new blood vessels from preexisting vessels; vasculogenesis is the formation of new vessels by recruiting circulating EPCs. We show that vasculogenesis-driven and angiogenesis-driven tumors grow in different ways. (i) If angiogenesis is the prevailing process, then the tumor mass (and volume) will grow as a cubic power of time, and BM-derived EPCs will stay at a constant level. (ii) If vasculogenesis is the dominant process, then the tumor mass will be characterized by a linear growth in time, and the number of circulating EPCs (after possibly increasing to a maximum) will decrease to low levels. With this information, one can identify the "signature" of each of the processes in the observations of tumor growth and the dynamics of the relevant characteristics, such as the level of BM-derived EPCs. We show how our results can help explain some apparently contradictory experimental data. We also propose ways to couple this study with directed experiments to identify the exact role of vasculogenesis in tumor progression.

A Review of Vasculogenesis Models

Mechanical and chemical models of vasculogenesis are critically reviewed with an emphasis on their ability to predict experimentally measured quantities. Final remarks suggest a possibility to merge the capabilities of different models into a unified approach.

A mathematical model of tumour angiogenesis, regulated by vascular endothelial growth factor and the angiopoietins

Angiogenesis--the growth of new blood vessels from existing ones--is a prerequisite for the growth of solid tumours beyond a diameter of approximately 2 mm. In recent years, the angiopoietins have emerged as important regulators of angiogenesis. They mediate a delicate balance between vascular quiescence, regression and new growth, but their mechanism of action is not fully understood. This work attempts to provide a mathematical description of the role of the angiopoietins in angiogenesis. The model is formulated within the framework of reinforced random walks, which allows easy transition between the continuum (macroscopic) and discrete (microscopic) forms. Model predictions are in qualitative agreement with experimental observations, and may have implications for anti-cancer therapies based on the prevention of angiogenesis.

Markovian model of growth and histologic progression in prostate cancer

Models, based on bio-physical and biological considerations, may be very helpful as support tools for traditional diagnostic methodologies and interpretation of statistical data in oncology. This is particularly true when the neoplastic progression and differentiation are rather simple and regular, such as in the case of prostatic adenocarcinomas. Using clinical data as a "statistical ensemble," we propose here a Markovian model to forecast the tumor progression. After validation with clinical data, the model is applied to the determination of the temporal evolution of the risk of metastasis.

Critical conditions for pattern formation and in vitro tubulogenesis driven by cellular traction fields

In vitro angiogenesis assays have shown that tubulogenesis of endothelial cells within biogels, like collagen or fibrin gels, only appears for a critical range of experimental parameter values. These experiments have enabled us to develop and validate a theoretical model in which mechanical interactions of endothelial cells with extracellular matrix influence both active cell migration--haptotaxis--and cellular traction forces. Depending on the number of cells, cell motility and biogel rheological properties, various 2D endothelial patterns can be generated, from non-connected stripe patterns to fully connected networks, which mimic the spatial organization of capillary structures. The model quantitatively and qualitatively reproduces the range of critical values of cell densities and fibrin concentrations for which these cell networks are experimentally observed. We illustrate how cell motility is associated to the self-enhancement of the local traction fields exerted within the biogel in order to produce a pre-patterning of this matrix and subsequent formation of tubular structures, above critical thresholds corresponding to bifurcation points of the mathematical model. The dynamics of this morphogenetic process is discussed in the light of videomicroscopy time lapse sequences of endothelial cells (EAhy926 line) in fibrin gels. Our modeling approach also explains how the progressive appearance and morphology of the cellular networks are modified by gradients of extracellular matrix thickness.

Inhibition of vascularization in tumor growth

The transition to a vascular phase is a prerequisite for fast tumor growth. During the avascular phase, the neoplasm feeds only from the (relatively few) existing nearby blood vessels. During angiogenesis, the number of capillaries surrounding and infiltrating the tumor increases dramatically. A model which includes physical and biological mechanisms of the interactions between the tumor and vascular growth describes the avascular-vascular transition. Numerical results agree with clinical observations and predict the influence of therapies aiming to inhibit the transition.

Properties of a "phase transition" induced by antiangiogenetic therapeutical protocols

Inhibiting angiogenesis has been found to be an interesting therapeutical strategy against cancer. In fact, the success of tumor growth is subordinated to the corresponding increase of the vascular system feeding the neoplasm. However, optimization and design of proper antiangiogenetic therapeutical strategies is still an open problem. We apply a recently developed angiogenesis model to study how variations in the relevant parameters, e.g., induced by chemicals, may cause a "phase transition" to a region in the parameter space in which angiogenesis is not succesful. To demonstrate the reliability of our approach and its usefulness, we will study some specific drugs and use our model to investigate the influence of the main variables involved in a clinical treatment: the administration time, the duration of the drug effect, and the drug dose.

Competition effects in the dynamics of tumor cords

A general feature of cancer growth is the cellular competition for available nutrients. This is also the case for tumor cords, neoplasms forming cylindrical structures around blood vessels. Experimental data show that, in their avascular phase, cords grow up to a limit radius of about 100 microm, reaching a quasi-steady-state characterized by a necrotized area separating the tumor from the surrounding healthy tissue. Here we use a set of rules to formulate a model that describes how the dynamics of cord growth is controlled by the competition of tumor cells among themselves and with healthy cells for the acquisition of essential nutrients. The model takes into account the mechanical effects resulting from the interaction between the multiplying cancer cells and the surrounding tissue. We explore the influence of the relevant parameters on the tumor growth and on its final state. The model is also applied to investigate cord deformation in a region containing multiple nutrient sources and to predict the further complex growth of the tumor.

Diffusion with evolving sources and competing sinks: development of angiogenesis

Tumors ensure their long-time growth by emitting molecular messengers that induce cellular modifications in neighboring capillaries. These modifications are conducive to the enlargement of the vascular system feeding the tumor. This phenomenon, termed angiogenesis, is controlled by the diffusion and competitive trapping of nutrients and molecular messengers by several cell species. The number, location, and properties of these traps change continuously. The angiogenic process also implies that nutrient sources are time dependent. Starting from assumptions at the cellular level, we formulate a mathematical model that predicts the evolution of angiogenesis and the increase in the blood flow to the tumor. The model also predicts the emergence of directed growth and the possibility of therapeutical synergy. Simulations permit a careful analysis of the influence of the main parameters.

Emergence of taxis and synergy in angiogenesis

Angiogenesis, the expansion of the vascular system feeding a tumor, is crucial to both primary tumors long-time growth and for the successful implantation of metastases. We formulate a model that relates the energetic requirements of the cancer cells to the production and diffusion of an angiogenic factor and to the ensuing evolution of neighboring endothelial cells. The model yields predictions for the development of neovascularization and for the increase in the blood flow to the tumor. We show that the directed growth of the vascular net is an emergent property and that therapies targeting different stages of the angiogenic process might have a synergistic effect.

Effects of anatomical constraints on tumor growth

Competition for available nutrients and the presence of anatomical barriers are major determinants of tumor growth in vivo. We extend a model recently proposed to simulate the growth of neoplasms in real tissues to include geometrical constraints mimicking pressure effects on the tumor surface induced by the presence of rigid or semirigid structures. Different tissues have different diffusivities for nutrients and cells. Despite the simplicity of the approach, based on a few inherently local mechanisms, the numerical results agree qualitatively with clinical data (computed tomography scans of neoplasms) for the larynx and the oral cavity.

Modeling the effects of transforming growth factor-beta on extracellular matrix alignment in dermal wound repair

We present a novel mathematical model for collagen deposition and alignment during dermal wound healing, focusing on the regulatory effects of transforming growth factor-beta (TGFbeta.) Our work extends a previously developed model which considers the interactions between fibroblasts and an extracellular matrix composed of collagen and a fibrin based blood clot, by allowing fibroblasts to orient the collagen matrix, and produce and degrade the extracellular matrix, while the matrix directs the fibroblasts and control their speed. Here we extend the model by allowing a time varying concentration of TGFbeta to alter the properties of the fibroblasts. Thus we are able to simulate experiments which alter the TGFbeta profile. Within this model framework we find that most of the known effects of TGFbeta, i.e., changes in cell motility, cell proliferation and collagen production, are of minor importance to matrix alignment and cannot explain the anti-scarring properties of TGFbeta. However, we find that by changing fibroblast reorientation rates, consistent with experimental evidence, the alignment of the regenerated tissue can be significantly altered. These data provide an explanation for the experimentally observed influence of TGFbeta on scarring.