The mathematical modelling of tumour angiogenesis and invasion

In order to accomplish the transition from avascular to vascular growth, solid tumours secrete a diffusible substance known as tumour angiogenesis factor (TAF) into the surrounding tissue. Endothelial cells which form the lining of neighbouring blood vessels respond to this chemotactic stimulus in a well-ordered sequence of events comprising, at minimum, of a degradation of their basement membrane, migration and proliferation. Capillary sprouts are formed which migrate towards the tumour eventually penetrating it and permitting vascular growth to take place. It is during this stage of growth that the insidious process of invasion of surrounding tissues can and does take place. A model mechanism for angiogenesis is presented which includes the diffusion of the TAF into the surrounding host tissue and the response of the endothelial cells to the chemotactic stimulus. Numerical simulations of the model are shown to compare very well with experimental observations. The subsequent vascular growth of the tumour is discussed with regard to a classical reaction-diffusion pre-pattern model.     

The impact of phenotypic switching on glioblastoma growth and invasion

The brain tumour glioblastoma is characterised by diffuse and infiltrative growth into surrounding brain tissue. At the macroscopic level, the progression speed of a glioblastoma tumour is determined by two key factors: the cell proliferation rate and the cell migration speed. At the microscopic level, however, proliferation and migration appear to be mutually exclusive phenotypes, as indicated by recent in vivo imaging data. Here, we develop a mathematical model to analyse how the phenotypic switching between proliferative and migratory states of individual cells affects the macroscopic growth of the tumour. For this, we propose an individual-based stochastic model in which glioblastoma cells are either in a proliferative state, where they are stationary and divide, or in motile state in which they are subject to random motion. From the model we derive a continuum approximation in the form of two coupled reaction-diffusion equations, which exhibit travelling wave solutions whose speed of invasion depends on the model parameters. We propose a simple analytical method to predict progression rate from the cell-specific parameters and demonstrate that optimal glioblastoma growth depends on a non-trivial trade-off between the phenotypic switching rates. By linking cellular properties to an in vivo outcome, the model should be applicable to designing relevant cell screens for glioblastoma and cytometry-based patient prognostics.     

A Coupled Mathematical Model of Cell Migration,Vessel Cooption and Tumour Microenvironment during the Initiation of Micrometastases

We propose a coupled mathematical model for the detailed quantitative analyses of initial microtumour and micrometastases formation by including cancer cell migration, host vessel cooption and changes in microenvironment. Migrating cells are included as a new phenotype to describe the migration behaviour of malignant tumour cells. Migration probability of a migrating cell is assumed to be influenced by local chemical microenvironment. Pre-existing vessel cooption and remodelling are introduced according to the local haemodynamical microenvironment, such as interstitial pressure and vessel wall permeability. After the tumour cells and tumour vessels distribution are updated, the chemical substances are coupled calculated with the haemodynamical environment. The simulation results clearly reproduce the tumour cells migrate and proliferate along the pre-existing vessels at the very early stage of growth, which are consistent with many published experimental observations. In addition, the model demonstrates the interactions of tumour cells with the pre-existing vessels, which are believed to be essential for initial adhesion, proliferation, invasion, and micrometastases establishment. Quantitative analysis of tumour expansion in longitudinal and transverse directions shows that the cooption and migration along host vessels will be inhibited once angiogenesis phase occurs. The influences of the ability of cell migration and the inclusion of vessel cooption on the formation of micrometastases are discussed.     

Covalently grafted VEGF(165) in hydrogel models upregulates the cellular pathways associated with angiogenesis

Angiogenesis is an important biological response known to be involved in many physiological and pathophysiological situations. Cellular responses involved in the formation of new blood vessels, such as increases in endothelial cell proliferation, cell migration, and the survival of apoptosis-inducing events, have been associated with vascular endothelial growth factor isoform 165 (VEGF(165)). Current research in the areas of bioengineering and biomedical science has focused on developing polyethylene glycol (PEG)-based systems capable of initiating and sustaining angiogenesis in vitro. However, a thorough understanding of how endothelial cells respond at the molecular level to VEGF(165) incorporated into these systems has not yet been established in the literature. The goal of the current study was to compare the upregulation of key intracellular proteins involved in angiogenesis in human umbilical vein endothelial cells (HUVEC) and human microvascular endothelial cells (HMEC) seeded on PEG hydrogels containing grafted VEGF(165) and adhesion peptides Arg-Gly-Asp-Ser (RGDS). Our data suggest that the covalent incorporation of VEGF(165) into PEG hydrogels encourages the upregulation of signaling proteins responsible for increases in endothelial cell proliferation, cell migration, and the survival after apoptosis-inducing events.     

Cell kinetic studies of endothelial cells in the adenocarcinoma EO 771 and the effect of cyclophosphamide

Cell kinetic studies of endothelial cells in the adenocarcinoma EO 771 growing in C57bl/6j mice and after transplantation into Balb/c-nu/nu mice, as well as of the effect of cyclophosphamide treatment have been carried out. The³H-thymidine labelling index of endothelial cells decreases from about 8% 3–6 days after tumour inoculation to about 3% at 18 days. This decrease parallels that of the labelling index of tumour cells, i.e. there is a positive correlation between the labelling index of endothelial cells and that of tumour cells. The labelling index of endothelial cells in the tumour periphery is two to three times as high as that in the tumour centre reflecting corresponding differences in the rate of proliferation. There is no difference in the proliferation of endothelial cells whether the tumour grows in C57bl/6j or in Balb/c-nu/nu mice. After treatment with cyclophosphamide the labelling index of endothelial cells decreases within 2 days to 1–2% and remains that low despite regrowth of the tumour with increased tumour cell proliferation, indicating that tumour relapse does not depend on tumour angiogenesis.     

Modelling the internalization of labelled cells in tumour spheroids

In this paper a mathematical model is developed to describe the migration of labelled particles within a multicell spheroid. In the model, spatial variations in cell proliferation and death create an internal velocity field which leads to redistribution of the labelled and unlabelled cells. By applying a range of numerical and analytical techniques to the model equations, it is possible to show that, whilst the speed with which the labelled cells migrate through the tumour is independent of the type of cells that are labelled, their limiting distribution depends crucially on whether inert polystyrene microspheres or live tumour cells are labelled. These predictions are shown to be in good qualitative agreement with independent experimental results.     

The effect of fibrin on endothelial cell migration in vitro

Endothelial cells are known to migrate and come into contact with fibrin during numerous physiological processes, such as in wound healing and in tumor growth. The present study was initiated to investigate the effect of fibrin on endothelial cell migration in vitro. Endothelial cell migration was assayed by wounding confluent monolayers of bovine aortic endothelial cells with a razor blade and counting the number of cells crossing the wound per unit time. Wound-induced proliferation of endothelial cells was inhibited by mitomycin C-treatment without affecting endothelial cell migration, indicating that in this assay migration could be measured independent of proliferation. Migration of endothelial cells in vitro was inhibited by fibrin in a concentration dependent manner. Endothelial cell migration under fibrin was further reduced by plasminogen depletion of the serum, and fibrin still inhibited the migration of mitomycin C-treated endothelial cells. Kadish et al. (Tissue and Cell, 11, 99, 1979) previously reported that fibrin did not affect EC migration in vitro. The inability to inhibit EC migration with fibrin appears to be due to their assay system which employed agarose, since pre-treating the wounded monolayer with agarose eliminated the inhibition of EC migration by fibrin. The present results indicate that EC migration in vitro can be used as a model system for studying the interaction of fibrin with EC.     

Cell kinetic studies of endothelial cells in the adenocarcinoma EO 771 and the effect of cyclophosphamide.

Cell kinetic studies of endothelial cells in the adenocarcinoma EO 771 growing in C57bl/6j mice and after transplantation into Balb/c-nu/nu mice, as well as of the effect of cyclophosphamide treatment have been carried out. The 3H-thymidine labelling index of endothelial cells decreases from about 8% 3-6 days after tumour inoculation to about 3% at 18 days. This decrease parallels that of the labelling index of tumour cells, i.e. there is a positive correlation between the labelling index of endothelial cells and that of tumour cells. The labelling index of endothelial cells in the tumour periphery is two to three times as high as that in the tumour centre reflecting corresponding differences in the rate of proliferation. There is no difference in the proliferation of endothelial cells whether the tumour grows in C57bl/6j or in Balb/c-nu/nu mice. After treatment with cyclophosphamide the labelling index of endothelial cells decreases within 2 days to 1-2% and remains that low despite regrowth of the tumour with increased tumour cell proliferation, indicating that tumour relapse does not depend on tumour angiogenesis.     

Hybrid mathematical model of glioma progression

Objectives:  Gliomas are an important form of brain cancer, with high mortality rate. Mathematical models are often used to understand and predict their behaviour. However, using current modeling techniques one must choose between simulating individual cell behaviour and modeling tumours of clinically significant size.
Materials and Methods:  We propose a hybrid compartment-continuum-discrete model to simulate glioma growth and malignant cell invasion. The discrete portion of the model is capable of capturing intercellular interactions, including cell migration, intercellular communication, spatial cell population heterogeneity, phenotype differentiation, epigenetic events, proliferation, and apoptosis. Combining this with a compartment and continuum model allows clinically significant tumour sizes to be evaluated.
Results and Conclusions:  This model is used to perform multiple simulations to determine sensitivity to changes in important model parameters, specifically, the fundamental length parameter, necrotic cell degradation rate, rate of cell migration, and rate of phenotype transformation. Using these values, the model is able to simulate tumour growth and invasion behaviour, observed clinically. This mathematical model provides a means to simulate various tumour development scenarios, which may lead to a better understanding of how altering fundamental parameters can influence neoplastic progression.     

Modelling the Role of Angiogenesis and Vasculogenesis in Solid Tumour Growth

Recent experimental evidence suggests that vasculogenesis may play an important role in tumour vascularisation. While angiogenesis involves the proliferation and migration of endothelial cells (ECs) in pre-existing vessels, vasculogenesis involves the mobilisation of bone-marrow-derived endothelial progenitor cells (EPCs) into the bloodstream. Once blood-borne, EPCs home in on the tumour site, where subsequently they may differentiate into ECs and form vascular structures. In this paper, we develop a mathematical model, formulated as a system of nonlinear ordinary differential equations (ODEs), which describes vascular tumour growth with both angiogenesis and vasculogenesis contributing to vessel formation. Submodels describing exclusively angiogenic and exclusively vasculogenic tumours are shown to exhibit similar growth dynamics. In each case, there are three possible scenarios: the tumour remains in an avascular steady state, the tumour evolves to a vascular equilibrium, or unbounded vascular growth occurs. Analysis of the full model reveals that these three behaviours persist when angiogenesis and vasculogenesis act simultaneously. However, when both vascularisation mechanisms are active, the tumour growth rate may increase, causing the tumour to evolve to a larger equilibrium size or to expand uncontrollably. Alternatively, the growth rate may be left unaffected, which occurs if either vascularisation process alone is able to keep pace with the demands of the growing tumour. To clarify further the effects of vasculogenesis, the full model is also used to compare possible treatment strategies, including chemotherapy and antiangiogenic therapies aimed at suppressing vascularisation. This investigation highlights how, dependent on model parameter values, targeting both ECs and EPCs may be necessary in order to effectively reduce tumour vasculature and inhibit tumour growth.     

PV1 down‐regulation via shRNA inhibits the growth of pancreatic adenocarcinoma xenografts

PV1 is an endothelial‐specific protein with structural roles in the formation of diaphragms in endothelial cells of normal vessels. PV1 is also highly expressed on endothelial cells of many solid tumours. On the basis of in vitro data, PV1 is thought to actively participate in angiogenesis. To test whether or not PV1 has a function in tumour angiogenesis and in tumour growth in vivo, we have treated pancreatic tumour‐bearing mice by single‐dose intratumoural delivery of lentiviruses encoding for two different shRNAs targeting murine PV1. We find that PV1 down‐regulation by shRNAs inhibits the growth of established tumours derived from two different human pancreatic adenocarcinoma cell lines (AsPC‐1 and BxPC‐3). The effect observed is because of down‐regulation of PV1 in the tumour endothelial cells of host origin, PV1 being specifically expressed in tumour vascular endothelial cells and not in cancer or other stromal cells. There are no differences in vascular density of tumours treated or not with PV1 shRNA, and gain and loss of function of PV1 in endothelial cells does not modify either their proliferation or migration, suggesting that tumour angiogenesis is not impaired. Together, our data argue that down‐regulation of PV1 in tumour endothelial cells results in the inhibition of tumour growth via a mechanism different from inhibiting angiogenesis.     

Angiogenic properties of adult human thymus fat

The endogenous proangiogenic properties of adipose tissue are well recognized. Although the adult human thymus has long been known to degenerate into fat tissue, it has never been considered as a potential source of angiogenic factors. We have investigated the expression of diverse angiogenic factors, including vascular endothelial growth factor A and B, angiopoietin 1, and tyrosine-protein kinase receptor-2 (an angiopoietin receptor), and then analyzed their physiological role on endothelial cell migration and proliferation, two relevant events in angiogenesis. The detection of the gene and protein expression of the various proteins has been performed by immunohistochemistry, Western blotting, and quantitative real-time polymerase chain reaction. We show, for the first time, that adult thymus fat produces a variety of angiogenic factors and induces the proliferation and migration of human umbilical cord endothelial cells. Based on these findings, we suggest that this fat has a potential angiogenic function that might affect thymic function and ongoing adipogenesis within the thymus.     

Proliferation kinetics of endothelial and tumour cells in three mouse mammary carcinomas

Abstract. An autoradiographic study of three corded mouse tumours is reported. The proliferation characteristics of both tumour cells and endothelial cells were studied. The doubling time of these three tumours differed by a factor of 2.6 but there was only a small difference in the intermitotic time. All three tumours showed a very high cell loss factor (˜0.80) and the differences in growth rate resulted mainly from differences in the growth fraction .
The endothelial cell proliferation rates differed markedly in the three tumours, with labelling indices ranging from 18% in the faster tumours to 4.5% in the slowest. The potential doubling times for endothelium, calculated from these values, were much slower than the tumour cell cycle time or the tumour potential doubling time, but were two to four times faster than the volume doubling time of the tumour.
It appears likely that the endothelial proliferation rate influences the growth fraction, but similar high cell loss factors can occur in tumours with a four-fold difference in endothelial cell production rates. Inadequate branching of blood vessels seems likely to be at least as important as inadequate production of endothelial cells. It is not possible to determine whether slow tumour cell production evokes a slower endothelial growth or vice versa.     

The reorganization of microfilaments, centrosomes, and microtubules during in vitro small wound reendothelialization

The repair of small endothelial wounds is an important process by which endothelial cells maintain endothelial integrity. An in vitro wound model system was used in which precise wounds were made in a confluent endothelial monolayer. The repair process was observed by time-lapse cinemicrophotography. Using fluorescence and immunofluorescence microscopy, the cellular morphological events were correlated with the localization and distribution of actin microfilament bundles and vinculin plaques, and centrosomes and their associated microtubules. Single to four-cell wounds underwent closure by cell spreading while wounds seven to nine cells in size closed by initially spreading which was then followed at approximately 1 h after wounding by cell migration. These two processes showed different cytoskeletal patterns. Cell spreading occurred independent of centrosome location. However, centrosome redistribution to the front of the cell occurred as the cells began to elongate and migrate. While the peripheral actin microfilament bundles (i.e., the dense peripheral band) remained intact during cell spreading, they broke down during migration and were associated with a reduction in peripheral vinculin plaque staining. Thus, the major events characterizing the closure of endothelial wounds were precise in nature, followed a specific sequence, and were associated with specific cytoskeletal patterns which most likely were important in maintaining directionality of migration and reducing the adhesion of the cells to their neighbors within the monolayer.     

Role of copper in tumour angiogenesis--clinical implications.

The formation of new blood vessels is the initial step in progressive tumour development and metastasis. The first stage in tumour angiogenesis is the activation of endothelial cells. Copper ions stimulate proliferation and migration of endothelial cells. It has been shown that serum copper concentration increases as the cancer disease progresses and correlates with tumour incidence and burden. Copper ions also activate several proangiogenic factors, e.g., vascular endothelial growth factor, basic fibroblast growth factor, tumour necrosis factor alpha and interleukin 1. This review concerns a brief introduction into the basics of tumour blood vessel development as well as the regulatory mechanisms of this process. The role of copper ions in tumour angiogenesis is discussed. The new antiangiogenic therapies based on a reduction of copper levels in tumour microenvironment are reviewed.     

VEGF guides angiogenic sprouting utilizing endothelial tip cell filopodia

Vascular endothelial growth factor (VEGF-A) is a major regulator of blood vessel formation and function. It controls several processes in endothelial cells, such as proliferation, survival, and migration, but it is not known how these are coordinately regulated to result in more complex morphogenetic events, such as tubular sprouting, fusion, and network formation. We show here that VEGF-A controls angiogenic sprouting in the early postnatal retina by guiding filopodial extension from specialized endothelial cells situated at the tips of the vascular sprouts. The tip cells respond to VEGF-A only by guided migration; the proliferative response to VEGF-A occurs in the sprout stalks. These two cellular responses are both mediated by agonistic activity of VEGF-A on VEGF receptor 2. Whereas tip cell migration depends on a gradient of VEGF-A, proliferation is regulated by its concentration. Thus, vessel patterning during retinal angiogenesis depends on the balance between two different qualities of the extracellular VEGF-A distribution, which regulate distinct cellular responses in defined populations of endothelial cells.     

Tumour-secreted miR-9 promotes endothelial cell migration and angiogenesis by activating the JAK-STAT pathway

Angiogenesis plays a crucial role during tumorigenesis and much progress has been recently made in elucidating the role of VEGF and other growth factors in the regulation of angiogenesis. Recently, microRNAs (miRNAs) have been shown to modulate a variety of physiogical and pathological processes. We identified a set of differentially expressed miRNAs in microvascular endothelial cells co-cultured with tumour cells. Unexpectedly, most miRNAs were derived from tumour cells, packaged into microvesicles (MVs), and then directly delivered to endothelial cells. Among these miRNAs, we focused on miR-9 due to the strong morphological changes induced in cultured endothelial cells. We found that exogenous miR-9 effectively reduced SOCS5 levels, leading to activated JAK-STAT pathway. This signalling cascade promoted endothelial cell migration and tumour angiogenesis. Remarkably, administration of anti-miR-9 or JAK inhibitors suppressed MV-induced cell migration in vitro and decreased tumour burden in vivo. Collectively, these observations suggest that tumour-secreted miRNAs participate in intercellular communication and function as a novel pro-angiogenic mechanism.     

Density-dependent quiescence in glioma invasion: instability in a simple reaction-diffusion model for the migration/proliferation dichotomy

Gliomas are very aggressive brain tumours, in which tumour cells gain the ability to penetrate the surrounding normal tissue. The invasion mechanisms of this type of tumour remain to be elucidated. Our work is motivated by the migration/proliferation dichotomy (go-or-grow) hypothesis, i.e. the antagonistic migratory and proliferating cellular behaviours in a cell population, which may play a central role in these tumours. In this paper, we formulate a simple go-or-grow model to investigate the dynamics of a population of glioma cells for which the switch from a migratory to a proliferating phenotype (and vice versa) depends on the local cell density. The model consists of two reaction-diffusion equations describing cell migration, proliferation and a phenotypic switch. We use a combination of numerical and analytical techniques to characterize the development of spatio-temporal instabilities and travelling wave solutions generated by our model. We demonstrate that the density-dependent go-or-grow mechanism can produce complex dynamics similar to those associated with tumour heterogeneity and invasion.     

Structure-activity relationships of the human prothrombin kringle-2 peptide derivative NSA9: anti-proliferative activity and cellular internalization

The human prothrombin kringle-2 protein inhibits angiogenesis and LLC (Lewis lung carcinoma) growth and metastasis in mice. Additionally, the NSA9 peptide (NSAVQLVEN) derived from human prothrombin kringle-2 has been reported to inhibit the proliferation of BCE (bovine capillary endothelial) cells and CAM (chorioallantoic membrane) angiogenesis. In the present study, we examined the structure-activity relationships of the NSA9 peptide in inhibiting the proliferation of endothelial cells lines e.g. BCE and HUVE (human umbilical vein endothelial). N- or C-terminal truncated derivatives and reverse sequence analogues of NSA9 were prepared and their anti-proliferative activities were assessed using the MTT [3-(4,5-dimethylthiazol-2-yl)-2,5-diphenyl-2H-tetrazolium bromide] assay. This cell proliferation assay demonstrated that both the N-terminal region and sequence orientation of NSA9 are important for inhibiting the proliferation of endothelial cells. In particular 2 C-terminal truncation derivatives of NSA9 [NSA7 (NSAVQLV) and NSA8 (NSAVQLVE)] inhibited cellular proliferation to a greater extent than did NSA9. The heptapeptide NSA7, was found to be more potent than NSA9 in inhibiting CAM angiogenesis, and tubular formation and migration of HUVE cells. In addition NSA9, NSA8 and NSA7 peptides exhibited considerable inhibitory effects on the proliferation of tumour cells such as B16F10 (murine melanoma), LLC and L929 (murine fibroblast). Also, cellular internalization studies demonstrated that NSA7 was internalized into both endothelial and tumour cells more easily than was NSA9. In conclusion, these results suggest that NSA7, residing within the full sequence of NSA9, contains the required sequence for anti-proliferative activity and cellular internalization.