Mathematical modelling of dynamic adaptive tumour-induced angiogenesis: Clinical implications and therapeutic targeting strategies

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Abstract

Angiogenesis, the growth of a network of blood vessels, is a crucial component of solid tumour growth, linking the relatively harmless avascular growth phase and the potentially fatal vascular growth phase. As a process, angiogenesis is a well-orchestrated sequence of events involving endothelial cell migration, proliferation; degradation of tissue; new capillary vessel (sprout) formation; loop formation (anastomosis) and, crucially, blood flow through the network. Once there is blood flow associated with the nascent network, the subsequent growth of the network evolves both temporally and spatially in response to the combined effects of angiogenic factors, migratory cues via the extracellular matrix and perfusion-related haemodynamic forces in a manner that may be described as both adaptive and dynamic. In this paper we present a mathematical model which simultaneously couples vessel growth with blood flow through the vessels—dynamic adaptive tumour-induced angiogenesis (DATIA). This new mathematical model presents a theoretical and computational investigation of the process and highlights a number of important new targets for therapeutic intervention. In contrast to earlier flow models, where the effects of perfusion (blood flow) were essentially evaluated a posteriori, i.e. after generating a hollow network, blood flow in the model described in this paper has a direct impact during capillary growth, with radial adaptations and network remodelling occurring as immediate consequences of primary anastomoses. Capillary network architectures resulting from the dynamically adaptive model are found to differ radically from those obtained using earlier models.

The DATIA model is used to examine the effects of changing various physical and biological model parameters on the developing vascular architecture and the delivery of chemotherapeutic drugs to the tumour. Subsequent simulations of chemotherapeutic treatments under different parameter regimes lead to the identification of a number of new therapeutic targets for tumour management.

Introduction

Angiogenesis is the process by which new blood vessels develop from an existing vasculature, through endothelial cell sprouting, proliferation and fusion (Risau, 1997). Adult endothelial cells (ECs) are normally quiescent and, apart from certain developmental processes (e.g. embryogenesis) and wound healing, angiogenesis is generally a pathological process implicated in arthritis (Walsh, 1999), some eye diseases, and solid tumour development, invasion and metastasis (Folkman, 1995). Tumour-induced angiogenesis is believed to occur when a small avascular tumour exceeds some critical diameter (∼2 mm), above which normal tissue vasculature is no longer able to support its growth (Folkman, 1971). At this stage, the tumour cells lacking nutrients and oxygen become hypoxic. This is assumed to trigger cellular release of tumour angiogenic factors (TAFs) (Folkman and Klagsbrun, 1987), which start to diffuse into the surrounding tissue and approach the ECs of nearby blood vessels. Endothelial cells subsequently respond to the TAF concentration gradient by forming sprouts, dividing and migrating towards the tumour (Ausprunk and Folkman, 1977; Sholley et al., 1984). It takes approximately 10 to 21 days for the growing network to link the tumour to the parent vessel (Gimbrone et al., 1974; Ausprunk and Folkman, 1977; Muthukkaruppan et al., 1982), and this vascular connection subsequently provides all the nutrients and oxygen required for continued tumour growth. An excellent summary of all the key processes involved in angiogenesis can be found in the comprehensive review article of Paweletz and Knierim (1989). More recent individual summaries of selected components of angiogenesis can be found in the papers of Carmeliet (2003), Cleaver and Melton (2003), Ferrara et al. (2003), Jain (2003), Pugh and Ratcliffe (2003), Rafii and Lyden, (2003), Ylä-Herttuala and Alitalo (2003).

At this stage in tumour development, chemotherapy treatments can be administered and tumour cells can be specifically targeted via the newly developed vasculature. However, although most of the drugs used to date have proven to be successful on small animals (e.g. mice), their efficiency in humans remains highly variable from one patient to another. One of the main reasons suggested to explain such variability is the issue of drug delivery to the tumour—it is thought possible that most of the drug can bypass large areas of the target (Jain, 1987, Jain, 1988). In light of this, more theoretical models are being developed in an attempt to better understand vascular architecture and microcirculatory dynamics (Secomb, 1995; Baish et al., 1996; El-Kareh and Secomb, 1997; Quarteroni et al., 2000; Godde and Kurz, 2001; Krenz and Dawson, 2002).

Over the past 10 years or so, there has been a lot of interest in the mathematical modelling of tumour-induced angiogenesis. The modelling has focussed mainly on the key role played by ECs during the formation of the new blood vessels. These models have considered the endothelial cell proliferative and migratory response to different signalling cues, including those associated with the soluble and diffusible angiogenic factors secreted by the cancer cells of the solid tumour itself, as well as those arising from insoluble molecules present in the extracellular matrix (ECM) (e.g. fibronectin). These key interactions of the ECs with angiogenic factors and macromolecules of the matrix have typically led to systems of nonlinear PDEs describing the migration of capillary vessels from a parent vessel through the ECM towards the solid tumour. Some of these models have contained a discrete element, allowing the formation of individual capillary vessels to be examined in detail by tracking the progress of individual ECs. Key papers in this area include the works of Stokes and Lauffenburger (1991), Orme and Chaplain (1997), Olsen et al. (1997), Anderson and Chaplain (1998), Levine et al. (2001), Plank and Sleeman (2004). An excellent comprehensive overview of the modelling done in this area can be found in the review paper of Mantzaris et al. (2004).

By contrast, blood flow modelling in a tumour-induced (micro) capillary network has only been considered relatively recently in papers by McDougall et al. (2002), Alarcon et al. (2003) and Stéphanou et al., 2005, Stéphanou et al., 2006. Blood is a complex fluid, the rheological properties of which lead to interesting feedback mechanisms during perfusion. For example, shear stresses generated within the capillary network by the flowing blood strongly influence vessel adaptation and network remodelling (Lehoux and Tedgui, 1998; Taber, 1998; Quick et al., 2000; Godde and Kurz, 2001; Fisher et al., 2001). These shear stresses are, in turn, affected by blood viscosity, the distribution of which depends upon a non-uniform distribution of haematocrit (volume fraction of red blood cells contained in the blood) within the host vasculature (the Fåhraeus effect). However, the distribution of haematocrit itself depends upon the spatial architecture of the underlying network and so the feedback is established, “the modelling loop is closed”, so to speak. Blood rheology and its influence on the remodelling of microvascular networks have been extensively studied by Pries et al., 1998, Pries et al., 2001a, Pries et al., 2001b both experimentally and theoretically. From these studies Pries and co-workers have formulated a model for vascular adaptation incorporating a number of feedbacks mechanisms. They have demonstrated that the basic requirement for the generation of stable vascular structures involves a combination of both haemodynamic and metabolic stimuli.

In the work of McDougall et al. (2002), flow simulations through the vascular networks were performed to investigate the efficiency of chemotherapy treatments as they passed from a nearby parent vessel to the tumour surface via a network of capillary vessels. These vessels had been stimulated to grow by chemical factors secreted by the tumour cells themselves (i.e. tumour-induced angiogenesis). The capillary vessels were generated from an angiogenesis model of Anderson and Chaplain (1998)—the growth of the vascular network was described by a discrete formulation of the set of governing partial differential equations and the migration of each individual endothelial cell was traced as it emerged from a parent vessel. Endothelial cells migrated via a biased random walk process with three main components of movement: (i) random motility, (ii) chemotaxis in response to a generic TAF released by the tumour cells, and (iii) haptotaxis in response to fibronectin gradients generated in the ECM as the ECs migrate (a combination of degradation and production). Flow modelling techniques used previously in the context of petroleum engineering to model the flow of water, oil and gas through the interstices of a porous rock (McDougall and Sorbie, 1997) were adapted to model blood and drug flow through these microvascular networks. Although blood was rather crudely considered to be a Newtonian fluid in this early work, results from McDougall et al. (2002) highlighted two important effects that could be responsible for the failure of some therapy regimes. First, it was found that a considerable amount of the drug injected into the parent vessel simply by-passed the tumour by way of the highly interconnected capillary network. The second effect related to the dilution of the drug as it became dispersed throughout the tumour-induced vasculature: the concentration of any drug reaching the tumour became so dilute as to have little effect on the tumour cells. Simulations were then performed to investigate ways of reducing these two detrimental phenomena and thereby optimize the drug uptake by the tumour. Increasing the mean capillary radius of the capillary bed and/or decreasing the blood viscosity both led to a significant increase in the drug uptake. Although these results were interesting from a qualitative perspective, this early model was somewhat naïve, with blood perfusion modelled as the flow of a Newtonian fluid through rigid cylindrical capillaries.

The paper of Stéphanou et al. (2005) extended the work of McDougall et al. (2002) by examining how the removal of certain capillaries affected the distribution of blood flow in the system. Capillary pruning algorithms were designed to reflect how different anti-vascular and anti-angiogenic drugs were thought to operate in vivo. Simulations demonstrated that drug uptake could be increased by up to 130% via the random removal of vessels and this suggested the possibility of developing a new cancer treatment strategy, viz, coupling the administration of an anti-angiogenic drug (to preliminarily optimize the vasculature) prior to chemotherapy treatment (thereby ensuring maximum delivery). In this paper modelling and simulations were also carried out on fully three-dimensional networks.

Vascular adaptation processes have also been recently incorporated into models comprising regular capillary networks (Alarcon et al., 2003) and capillary networks originating from tumour-induced angiogenesis (Stéphanou et al., 2006). The aim of the work of Alarcon et al. (2003) was to model the oxygen distribution within a two-dimensional regular hexagonal network of blood vessels and to determine its influence on the dynamics of a colony of normal and cancerous cells. Results from computational simulations of the model produced inhomogeneous distributions of haematocrit and oxygen tension and highlighted the important role played by hypoxic cells during tumour invasion. The work of Stéphanou et al. (2006) modelled an adaptive vasculature associated with tumour-induced angiogenesis and considered how this adaptive remodelling affected the supply of oxygen and drugs to the tumour cells. However, in both papers, the flow modelling was carried out a posteriori, either after simply generating a hexagonal hollow vessel network a mano (Alarcon et al., 2003) or after an initial cell migration model had produced a hollow vessel network (Stéphanou et al., 2006).

The aim of the current paper is to implement a number of significant improvements in the modelling approach by considering the flow of a non-Newtonian fluid in a dynamic adaptive network, i.e. a network that evolves both spatially and temporally in response to its associated flow distribution. We present results corresponding to a number of different stages in the formulation of what we call our dynamic adaptive tumour-induced angiogenesis (DATIA) model.

The model of DATIA presented in this paper begins with an implementation of a formulation of endothelial cell migration at the capillary sprout tips that explicitly takes into account the important function of matrix degrading enzymes (such as matrix metalloproteinases, MMPs; urokinase plasminogen activators, uPAs) during angiogenesis in the absence of flow (Levine et al., 2002; Lolas, 2003). We explicitly incorporate mediation in vessel growth via ECM proteolysis by specific enzymes produced by ECs. A number of recent publications have demonstrated the importance of enzymes from the MMP family and their involvement in the regulation of the various stages of the angiogenic process (Davis et al., 2000; Yan et al., 2000; Hidalgo and Eckhardt, 2001; Sternlicht and Werb, 2001). These MMPs are involved in the migration of ECs within the ECM, EC proliferation, and the remodelling of the basement membrane of newly formed vessels. Their importance is such that these proteinases and their regulation form new targets for cancer treatment. As our ultimate goal is to propose a global modelling framework within which to further investigate new treatments, it is important to incorporate explicitly the MMP effect into the modelling. Another key aspect of the MMP issue relates to their effect on transmural transport—tumour vascular networks are very leaky and proteolytic activity plays a vital part in the breaching of non-mature capillary membranes by ECs during sprout formation. Although we will not take vessel leakiness into account in the present paper, this aspect is currently under investigation in an attempt to describe drug diffusion through the tumour tissue itself. Matrix degrading enzymes also play a key role in the invasion of tissue by cancer cells, something which is closely tied in with tumour-induced angiogenesis (Anderson et al., 2000; Lolas, 2003; Anderson, 2005).

The main aim of this paper is to investigate the impact of blood perfusion during angiogenesis and the paper continues with a discussion of a series of models related to capillary flow and vessel remodelling. Whilst previous approaches by McDougall et al. (2002) and Stéphanou et al. (2005) have made the assumptions of constant blood viscosity and invariable vessel radii, and other work by Alarcon et al. (2003) and Stéphanou et al. (2006) has considered remodelling of hollow capillary networks a posteriori, the modelling formulation in this current paper is extended to account for variable blood viscosity and evolving capillary radii coupled directly to the network as it is growing. The model considers a number of stimuli affecting vessel diameter that account for the influence of the wall shear stress (WSS), the intravascular pressure, and a metabolic mechanism depending on the blood haematocrit. In addition, the angiogenesis model of Anderson and Chaplain (1998), where sprout branching depended only upon local TAF concentration, is extended here to take into account the important feature of shear-stress-induced capillary sprouting and branching. Network architectures are produced that adapt dynamically through adjuvant branching of vessels experiencing high shear stresses during perfusion. The flow induces additional branching, resulting in topological perturbations in the developing architecture that become magnified during successive growth periods. These differences in network structure subsequently alter the distribution of flow (and hence shear stress) throughout the vascular bed and so it is the additional branching mechanism that effectively couples the migration and flow models.

Computational simulation results are presented corresponding to a number of different stages in the formulation of the full DATIA model. The simulation results are presented stage-by-stage in such a way as to show the evolution of the model—this allows the effect of each of the model components to be assessed individually and facilitates direct comparison with previous work. The next section begins with a simulation demonstrating the effect of explicitly incorporating matrix degrading enzyme (MDE) activity into the migration model. At this stage, no flow/remodelling is considered and only sprout branching at the vessel tips is allowed. Subsequent results incorporating flow show the impact of dynamic remodelling and shear-induced vessel branching upon global network architecture. Next, a number of key physical and biochemical parameters are varied in the model to assess their effect upon the network architecture. Finally, in order to quantify the efficiency of these different networks in carrying blood-borne material (e.g. nutrients, drugs) to the tumour, a series of simulations involving the continuous infusion of a “passive tracer-drug” is described. In doing so, some insights are offered into the evolution of chemotherapeutic agents during treatment and a number of new therapeutic targets for tumour management are subsequently identified, thus providing a rational biomechanical basis for an effect which has been known for over 30 years (Salsbury et al., 1970; Le Serve and Hellmann, 1972). The paper concludes with a discussion section summarising all the main results and offering directions for future model development and study.

Section snippets

A modelling framework for DATIA

Although the ultimate aim of the work presented in this paper is to produce a model of angiogenesis that couples cell migration with blood flow, it will first be beneficial to examine each of these components in isolation.

Simulation results

Results are presented in the following section corresponding to a number of different stages in the formulation of the full DATIA model. The section begins with a simulation demonstrating the effect of explicitly incorporating MDE activity into the endothelial migration model. At this stage, no flow/remodelling is considered and only sprout branching at the vessel tips has been allowed (i.e. there is no vessel branching due to WSS). The main point of interest in this first set of computational

Transport through adapted networks: implications for nutrient supply and drug delivery to solid tumours

In the previous section, the effects of varying a wide variety of physical and biochemical parameters upon vascular architecture were investigated in the context of DATIA. These included investigating the network structure sensitivities to shear-induced branching, blood haematocrit, cell–matrix interactions (via the haptotaxis coefficient) and intravascular pressure. The aim of the work presented in this section is to quantify the efficiency of these different networks in carrying blood-borne

Discussion and conclusions

An extensive theoretical investigation of the process of tumour-induced angiogenesis has been presented in this paper and results from computational simulations of the mathematical model have highlighted a number of important new targets for therapeutic intervention. The approach considered has been to integrate and explicitly couple a hybrid model of endothelial cell migration (Anderson and Chaplain, 1998) with a network flow model (McDougall et al., 2002; Stéphanou et al., 2005, Stéphanou et

Acknowledgements

This work was partly supported by the European Community, through the Marie Curie Research Training Network Project HPRN-CT-2004-503661: “Modelling, Mathematical Methods and Computer Simulation of Tumour Growth and Therapy”.

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