A pharmacologically based multiscale mathematical model of angiogenesis and its use in investigating the efficacy of a new cancer treatment strategy

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Abstract

Tumor angiogenesis is the process by which new blood vessels are formed and enhance the oxygenation and growth of tumors. As angiogenesis is recognized as being a critical event in cancer development, considerable efforts have been made to identify inhibitors of this process. Cytostatic treatments that target the molecular events of the angiogenesis process have been developed, and have met with some success. However, it is usually difficult to preclinically assess the effectiveness of targeted therapies, and apparently promising compounds sometimes fail in clinical trials.

We have developed a multiscale mathematical model of angiogenesis and tumor growth. At the molecular level, the model focuses on molecular competition between pro- and anti-angiogenic substances modeled on the basis of pharmacological laws. At the tissue scale, the model uses partial differential equations to describe the spatio-temporal changes in cancer cells during three stages of the cell cycle, as well as those of the endothelial cells that constitute the blood vessel walls.

This model is used to qualitatively assess how efficient endostatin gene therapy is. Endostatin is an anti-angiogenic endogenous substance. The gene therapy entails overexpressing endostatin in the tumor and in the surrounding tissue. Simulations show that there is a critical treatment dose below which increasing the duration of treatment leads to a loss of efficacy.

This theoretical model may be useful to evaluate the efficacy of therapies targeting angiogenesis, and could therefore contribute to designing prospective clinical trials.

Introduction

Tumor angiogenesis is a process by which new blood vessels are formed from the existing vasculature and carry additional nutrients and oxygen to tumor cells, allowing them to proliferate. The development of the primary tumor mass and the metastatic dissemination of tumor cells require angiogenesis. It is generally accepted that a tumor, which needs nutrients and oxygen to grow, cannot increase beyond a few millimeters cubed without an enhanced blood supply (Folkman, 1990). During tumor growth, a molecular cascade drives the transition from the avascular stage to the vascular stage: new vessels are formed from the surrounding existing vasculature, migrate towards the tumor cells, and penetrate the tumor mass to deliver oxygen and nutrients to the tumor cells. This also means that some tumor cells can escape from this primary tumor, and enter the bloodstream via the newly formed immature and permeable blood vessels to form new tumor masses, also known as metastases, at distant parts of the body (Zetter, 1998, Folkman, 2002). In most cases, the presence of metastases is correlated with the malignancy of the tumor, and indicates a poor prognosis for the patient (Zetter, 1998).

Indeed, angiogenesis, or new vessel formation, results from a complex molecular balance between numerous pro-angiogenic and anti-angiogenic endogenous substances (Hanahan and Folkman, 1996). The complexity of angiogenesis is partly due to the existence of a number of such factors. In breast cancer for instance, up to seven distinct pro-angiogenic factors can be expressed (Relf et al., 1997).

Over the last 25 years, several mathematical models of angiogenesis have been developed (see Mantzaris et al., 2004 for a review). Discrete mathematical models, based on cellular automata, have usually been used to predict the structure of extra- and intra-tumoral vascular networks. Anderson and Chaplain (1998) have published the first discrete model of tumor-induced angiogenesis. This model was derived from a continuous model. Several more recent publications are based on the same approach (Plank et al., 2004, Kevrekidis et al., 2006). Continuous models of tumor-induced angiogenesis are based on ordinary or partial differential equations governing the change in endothelial cell density, and the concentrations of tumor pro-angiogenic factors and of fibronectin (a component of the extracellular matrix). From the physical point of view, these models focus mainly on the endothelial cell diffusion, chemotaxis and haptotaxis (Anderson and Chaplain, 1998, Kevrekidis et al., 2006). It has also been proposed that mathematical models of angiogenesis can be coupled with those of tumor growth. In Zheng et al. (2005), the authors have proposed a vascular tumor growth model in which the tumor growth model proposed by Byrne and Chaplain (1996) is coupled with a continuous-discrete model of angiogenesis, such as that described by Anderson and Chaplain (1998). In Hogea et al. (2006), the authors have coupled a model of tumor growth with a simplistic continuous model of angiogenesis. Several models of vascular tumor growth also include some mechanical constraints, such as wall shear stress, pressure. For instance, Alarcón et al. (2005) have coupled a cellular automaton with ordinary differential equations (ODEs) to describe some of the interactions that occur between the tissue scale, including vascular structural adaptation, the cellular scale and the intracellular scale. In Bartha and Rieger (2006), Lee et al. (2006), and Welter et al. (2008), the authors have used models based on probabilistic cellular automata to investigate the mechanisms leading to abnormal tumor vasculature, and the effects of such vascular heterogeneities on tumor growth and drug delivery. In order to circumvent the numerical cost of such cellular automata, Macklin et al. (2009) have decided to combine a continuous model of tumor growth with a discrete model of tumor-induced angiogenesis, such as that described by McDougall et al. (2006). Thanks to this combination, they were able to take into account the impact of blood flow on changes in the vascular network. Due to their complexity, these models are only qualitative, and even so can only integrate one or two of the molecular factors that drive the angiogenesis process. Moreover, the underlying tumor growth model is often very simplistic and fails to take cell cycle regulation into account.

Since the angiogenic process was first identified as a key process in tumor development a few years ago, pharmaceutical companies have been looking for inhibitors. Several anti-angiogenic molecules have been identified and tested in clinical trials but, as is all too often the case with targeted therapies, efficacy has been difficult to demonstrate. This makes it rather difficult to assess attempts to optimize treatment. New anti-cancer drugs are designed to target a particular cancer process, unlike standard chemotherapeutic compounds that have a cytotoxic effect on all proliferative cells. Targeted therapies, which are also known as “cytostatic treatments”, act mainly at the molecular level. For instance, some anti-angiogenic drugs, the best known being Bevacizumab (Avastin, Roche), prevent vascular endothelial growth factor (VEGF), a pro-angiogenic endogenous substance, from binding to Flk-1 receptors. These receptors are located on the membrane of endothelial cells, which constitute blood vessels. Such anti-angiogenic drugs inhibit endothelial cell proliferation and, in consequence, prevent new blood vessels from forming, without any direct toxic effect on healthy cells.

To make it possible to analyze the effect of such molecular-targeted treatments by means of mathematical models, we need to include the main molecular entities in multiscale models of tumor growth. In this paper we describe a pharmacologically based continuous mathematical model of angiogenesis and tumor growth. At the molecular level, we were careful to use pharmacological laws to model the activation of angiogenesis as the result of the binding of major angiogenic molecular substances to their respective receptors. This molecular-level model was embedded in the macroscopic model, based on reaction–diffusion partial differential equations, which described the spatio-temporal change in the densities of the unstable and stable endothelial cells that constitute the blood vessel wall. At each of the time steps in the model, sources of oxygen were defined according to the spatial disposition of the endothelial cells. The oxygen concentration was then computed, and introduced as an input signal into the cell cycle model of tumor cells. Indeed, depending on the local concentration of oxygen, we assumed that cancer cells would proliferate, die, or enter the quiescent compartment. In the model, quiescent cells, deprived of oxygen, secrete vascular endothelial growth factor (VEGF). This in turn activates angiogenesis, and this constitutes the feedback loop of the model.

We applied our model to an analysis of the efficacy of a new anti-angiogenesis treatment. This treatment relies on the overproduction of an endogenous anti-angiogenic substance known as endostatin, which is commonly secreted by tumor cells (O’Reilly et al., 1997). Endostatin is a 20-kDa, C-terminal fragment of collagen XVIII, and has been shown to be a potent endogenous inhibitor of angiogenesis (O’Reilly et al., 1997). Endostatin gene therapy consists of infecting tumor cells with adenoviruses encoding a wild-type endostatin gene (Folkman, 2006). When feasible, these adenoviruses are injected directly into the tumor mass, which induces targeted local overexpression of endostatin. This approach is currently being evaluated in clinical trials (Lin et al., 2007, Li et al., 2008). However, as is often the case for targeted anti-cancer therapies, it is hard to identify the best treatment strategy.

Using the multiscale model of tumor growth and angiogenesis to carry out a qualitative analysis of the effect of this treatment, we provide some indications about the best way to optimize this cancer treatment strategy.

Section snippets

Underlying biological hypotheses and an overall view of the mathematical model

Due to the complexity of the problem, we first collected biological information from the literature, and made some simplified hypotheses about the various biological phenomena involved. The diagram shown in Fig. 1 was developed in collaboration with biologists. It provides a schematic description of the relationships between the different entities we will model in this study.

Our model includes two main species of cells. The first is that of the endothelial cells that constitute the wall of

Equations of the mathematical model

Our virtual tumor is described by the densities of tumor cells (number of tumor cells per unit of volume) in the proliferative, quiescent and apoptotic phases.

Newly formed blood vessels are represented by the density of endothelial cells (continuous approach), with a distinction being made between stable and unstable cells. Unstable cells proliferate and migrate towards the source of VEGF, whereas stable cells, which are required for blood to flow, are static.

We modeled the tissue

Numerical results

We simulated the computational model, with the initial conditions mentioned above, on a 100×100 discrete elements grid. The partial differential equations of our model were discretized using a finite volume method. We also used a penalization method to ensure that the concentration of oxygen was fixed in the functional blood vessels. Our time unit is the half-day (12 hours (h)), and our space unit is the millimeter (mm).

Discussion

The process of tumor angiogenesis is recognized as being a key process in tumor development. In consequence, anti-angiogenesis therapies are being investigated throughout the world as a promising way to treat cancer patients. In this paper, we propose a mathematical model of angiogenesis and tumor growth. This model is based on a set of partial differential equations that describe the behavior of endothelial cells, that constitute blood vessel walls, tumor cells, as well as of some major pro-

Acknowledgments

F.B. is funded by Institut National du Cancer (INCa), the French National Cancer Institute. The authors wish to thank the teams Evaluation and Modelization of Therapeutic Effects (of CNRS UMR5558), and Therapeutic targeting in Oncology (EA3738) from the University of Lyon (France) for valuable advice, J.Y. Scoazec (Inserm U865, Lyon) for relevant discussions, and the Camille Jordan Institute (CNRS UMR5208) from the University of Lyon for computation resources. The authors are also grateful to

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