Nonlinear simulation of the effect of microenvironment on tumor growth

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Abstract

In this paper, we present and investigate a model for solid tumor growth that incorporates features of the tumor microenvironment. Using analysis and nonlinear numerical simulations, we explore the effects of the interaction between the genetic characteristics of the tumor and the tumor microenvironment on the resulting tumor progression and morphology. We find that the range of morphological responses can be placed in three categories that depend primarily upon the tumor microenvironment: tissue invasion via fragmentation due to a hypoxic microenvironment; fingering, invasive growth into nutrient rich, biomechanically unresponsive tissue; and compact growth into nutrient rich, biomechanically responsive tissue. We found that the qualitative behavior of the tumor morphologies was similar across a broad range of parameters that govern the tumor genetic characteristics. Our findings demonstrate the importance of the impact of microenvironment on tumor growth and morphology and have important implications for cancer therapy. In particular, if a treatment impairs nutrient transport in the external tissue (e.g., by anti-angiogenic therapy) increased tumor fragmentation may result, and therapy-induced changes to the biomechanical properties of the tumor or the microenvironment (e.g., anti-invasion therapy) may push the tumor in or out of the invasive fingering regime.

Introduction

Cancer is a fundamental scientific and societal problem, and in the past few decades, vast resources have been expended in an effort to understand the root causes of cancer, to elucidate the intricacies of cancer progression, and to develop effective prevention and treatment strategies. In this paper, we present and investigate a model for solid tumor growth that incorporates features of the tumor microenvironment. Using analysis and nonlinear numerical simulations, we explore the effects of the interaction between the genetic characteristics of the tumor and the tumor microenvironment on the resulting tumor progression and morphology. Implications for cancer therapies are discussed.

Cancer is marked by several increasingly aggressive stages of development. The first stage, carcinogenesis, is believed to be characterized by a sequence of genetic mutations that promote growth (i.e., acquisition of oncogenes), circumvent apoptosis (i.e., inactivation or loss of tumor suppressor genes), or hinder DNA repair processes, thereby increasing the probability of acquiring oncogenes or inactivating tumor suppressor genes (e.g., see Hanahan and Weinberg, 2000; Lehmann, 2001). In the second stage of development, avascular growth occurs as the cancer cells proliferate and form an in situ cancer. The local production of matrix-degrading enzymes and subsequent degradation of the extracellular matrix (ECM) may also play a role in providing room for the tumor to expand into the surrounding tissue. (See Hotary et al., 2003 and the discussion throughout Anderson, 2005.) Since the tumor lacks a vasculature, nutrients (e.g., glucose and oxygen) are received only by diffusion through the surrounding tissue. As the tumor grows, less nutrient reaches the center of the tumor. Interior cells become hypoxic, begin to die (necrose), and are broken down by enzymes. As cell death in the tumor interior balances with cell proliferation on the boundary, a spherical tumor may reach a diffusion-limited size, usually on the order of 2–4 mm. However, if the tumor boundary acquires an irregular shape, additional nutrient becomes available to the tumor interior due to the increased surface area to volume ratio, and continued growth may result. Indeed, there are now a number of in vitro studies in which complex growth morphologies have been observed (e.g., see Bredel-Geissler et al., 1992; Mueller-Kleiser, 1997; Hedlund et al., 1999; Enmon et al., 2001; Frieboes et al., 2006b).

The next stage of tumor growth, angiogenesis, is characterized by the development of a tumor-induced neovasculature that grows from the main circulatory system toward the tumor in response to the imbalance of proangiogenic growth factors that are released by hypoxic cells in the tumor (e.g., vascular endothelial cell growth factor or VEGF) relative to anti-angiogenic growth factors (e.g., angiostatin) present in the tumor microenvironment (Carmellet and Jain, 2000). In the final stage of tumor progression, vascular growth, the tumor is supplied with nutrients from the newly developed, although typically inefficient vasculature (Jain, 1990, Haroon et al., 1999, Hashizume et al., 2000). Additional mutations and epigenetic events may occur that lead to increased cellular motility and greater production of matrix-degrading enzymes that degrade the ECM. This can lead to invasion, where either individual or collections of cancerous cells protrude and/or separate from the tumor and migrate through the surrounding tissue, or metastasis, where the invading tumor cells (or cell collections) enter the blood vasculature and/or lymphatic system and travel to distant locations.

The tumor microenvironment plays a crucial role in these processes (e.g., see Hockel et al., 1996; Enam et al., 1998; Schmeichel et al., 1998; Sanson et al., 2002; Pennacchietti et al., 2003). For example, hypoxic microenvironments lead to the upregulation of HIF-1 target genes in both tumor cells and endothelial cells, including those responsible for the secretion of angiogenic growth factors and matrix-degrading enzymes, metabolic changes such as increased glycolysis, and decreased cell–cell and cell–matrix adhesion (Kaur et al., 2005, Erler et al., 2006, Pouysségur et al., 2006). These conditions are associated with increased tumor invasiveness (Kaur et al., 2005, Erler et al., 2006, Pouysségur et al., 2006) and poor patient outcome (Hockel et al., 1996). However, the effects of the interaction between intra- and extratumoral processes on tumor progression and morphology are not well understood. Mathematical modeling has the potential to provide insight into these interactions through systematic studies of fundamental constituent processes.

Over the past 10 years, the interest in the mathematical modeling and numerical simulation of cancer has increased dramatically. (See the reviews by Adam, 1996; Bellomo et al., 2003; Araujo and McElwain, 2004a; Byrne et al., 2006; Sanga et al., 2006; Quaranta et al., 2005.) A variety of modeling strategies is now available, each of which is well suited to investigate one or more aspect of cancer. Cellular automata and agent-based modeling, where individual cells are simulated and updated based upon a set of biophysical rules, are particularly useful for studying carcinogenesis, natural selection, genetic instability, and interactions of individual cells with each other and the microenvironment. Because these methods are based on a series of rules for each cell, it is straightforward to translate biological processes (e.g., complex mutation pathways) into model rules. On the other hand, these models can be difficult to study analytically, and the computational cost increases rapidly with the number of cells modeled. Because a 1 mm tumor spheroid has over 500,000 cells, these methods can quickly become unwieldy when studying tumors of any significant size. For some examples of cellular automata modeling, see Anderson (2005), Alarcón et al. (2003), and Mallett and de Pillis (2006), and see Mansury et al. (2002) and Abbott et al. (2006) for examples of agent-based modeling.

In larger-scale systems where the cancer cell population is on the order of 1,000,000 or more, continuum methods provide a good modeling alternative. Early work (e.g., Greenspan, 1976; Byrne and Chaplain, 1996a, Byrne and Chaplain, 1996b) used ordinary differential equations (ODEs) to model cancer as a homogeneous population, as well as partial differential equation (PDE) models restricted to spherical geometries. Linear and weakly nonlinear analyses have been performed to assess the stability of spherical tumors to asymmetric perturbations (e.g., Chaplain et al., 2001; Byrne and Matthews, 2002; Cristini et al., 2003; Li et al., 2006, and discussed in the reviews by Araujo and McElwain, 2004a; Byrne et al., 2006) as a means to characterize the degree of aggression. Various interactions of the tumor with the microenvironment, such as stress-induced limitations of tumor growth, have also been studied in this context (e.g., Jones et al., 2000; Ambrosi and Mollica, 2002, Ambrosi and Mollica, 2004; Roose et al., 2003; Araujo and McElwain, 2004b, Araujo and McElwain, 2005; Ambrosi and Guana, 2006). Most of the previous modeling has considered single-phase tumors. More recently, multiphase mixture models have been developed to provide a more detailed account of tumor heterogeneity (e.g., see the work by Ambrosi and Preziosi, 2002; Byrne and Preziosi, 2003; Chaplain et al., 2006).

Very recently, nonlinear modeling has been performed to study the effects of shape instabilities on avascular, angiogenic and vascular solid tumor growth. Cristini et al. (2003) used boundary integral methods and performed the first fully nonlinear simulations of a continuum model of tumor growth in the avascular and vascularized growth stages with arbitrary boundaries. This work investigated the nonlinear regime of shape instabilities and predicted the encapsulation of external, non-cancerous tissue by morphologically unstable tumors. Interestingly, shape instabilities were found to occur only in the diffusion-dominated, avascular regime of growth. The effect of the extratumoral microenvironment was not considered.

Zheng et al. (2005) extended this model to include angiogenesis and an extratumoral microenvironment by developing and coupling a new level set implementation with a hybrid continuous–discrete angiogenesis model originally developed by Anderson and Chaplain (1998). Zheng et al. investigated the nonlinear coupling between growth and angiogenesis. As in Cristini et al. (2003), it was found that low-nutrient (e.g., hypoxic) conditions may lead to instability. Zheng et al. did not fully investigate the interaction between the growth progression and the tumor microenvironment, but their work served as a building block for recent studies of the effect of chemotherapy on tumor growth by Sinek et al. (2004) and for studies of morphological instability and invasion by Cristini et al. (2005) and Frieboes et al. (2006b). Hogea et al. (2006) have also begun investigating tumor growth and angiogenesis using a level set method coupled with a continuous model of angiogenesis. In addition, Frieboes et al. (2006a) and Wise et al. (2006) have recently developed a diffuse interface implementation of solid tumor growth to study the evolution of multiple tumor cell species during progression.

In Macklin (2003) and Macklin and Lowengrub, 2005, Macklin and Lowengrub, 2006, we also considered a level set-based extension of the tumor growth model that was previously investigated by Cristini et al. (2003) (described above). In these works, we developed new, highly accurate numerical techniques to solve the resulting system of PDEs in a moving domain. These numerical methods are more accurate than those used by Zheng et al. (2005) and Hogea et al. (2006). Using these methods, we modeled tumor growth under a variety of conditions and investigated the role of necrosis in destabilizing the tumor morphology. We demonstrated that non-homogeneous nutrient diffusion inside the tumor leads to heterogeneous growth patterns that, when interacting with cell–cell adhesion, cause sustained morphological instability during tumor growth, as well as the repeated encapsulation of non-cancerous tissue by the growing tumor.

In this paper, we extend the tumor growth models considered by Cristini, Lowengrub, Nie, Macklin, Zheng, and others (for example, Cristini et al., 2003; Macklin and Lowengrub, 2005; Zheng et al., 2005, all of which reformulated several classical models (Greenspan, 1976, McElwain and Morris, 1978, Adam, 1996; Byrne and Chaplain, 1996a, Byrne and Chaplain, 1996b; Chaplain, 2000)) to include more detailed effects of the microenvironment by allowing variability in nutrient availability and the response to proliferation-induced mechanical pressure (which models hydrostatic stress) in the tissue surrounding the tumor. In our model, the region surrounding the tumor aggregates the effects of ECM and non-cancerous cells, which we characterize by two non-dimensional parameters that govern the diffusional and biomechanical properties of the tissue. Fluids are assumed to move freely through the interstitium and ECM, and so such effects are currently neglected. The external nutrient and pressure variations, in turn, affect the evolution of the tumor in our model. Due to the computational cost of 3-D simulations, we shall focus our attention on 2-D tumor growth, although the model we develop applies equally well in three dimensions. In Cristini et al. (2003), it was found that the baseline model predicts similar morphological behavior for 2-D and 3-D tumor growth. This has been borne out by recent 3-D simulations by Li et al. (2006). We note that 2-D tumor growth may be well suited to studying cancers that spread over large areas but are relatively thin, such as melanoma.

Using our model, we shall conduct a systematic investigation of the effect of the microenvironment on tumor growth over a broad range of biophysical parameters. In the process, we shall characterize the behavior predicted by the model and discuss the implications for cancer treatment. We note that by matching the results to known morphologies, one may infer the range of validity of the model and obtain estimates of parameter values; we discuss this at the end of this paper. These simulations are difficult and require the development of accurate numerical techniques, which we present in this paper.

We find that the range of morphological responses can be placed in three categories that depend primarily upon the tumor microenvironment. In nutrient-poor microenvironments, tumors tend to break into small fragments and invade the surrounding tissue, regardless of the mechanical properties of the surrounding tissue. When placed in nutrient-rich tissue, the tumor morphology depends upon the biomechanical characteristics of the tissue. Tumors growing into mechanically unresponsive tissue develop buds that grow into long, invasive fingers. Tumors growing into softer, mechanically responsive tissue develop buds that do not grow, but rather connect with neighboring buds to capture external ECM. The overall morphology remains compact, with a large central abscess containing encapsulated ECM, fluid, and cellular debris similar to a necrotic core. We found that the qualitative behavior of the tumor morphologies was similar across a broad range of parameters that govern the tumor genetic characteristics. Our findings demonstrate the importance of the impact of microenvironment on tumor growth and morphology, and this has implications for cancer therapy: the impact of a therapy on the microenvironment may either positively or negatively impact the outcome of the treatment. A treatment that impairs nutrient delivery in the host tissue (e.g., using anti-angiogenic drugs) may increase tumor fragmentation, whereas a treatment that normalizes nutrient delivery may reduce or prevent tumor fragmentation. Therapies that affect the biomechanical responsiveness of the tumor or surrounding host tissue (e.g., anti-invasion therapy that alters cell–cell or cell–matrix adhesion) may either cause or prevent invasive fingering.

Using our model, we also investigate the internal structure of the tumors, including the volume fractions of the necrotic and viable portions of the tumor. We find that even during growth, the internal structure tends to stabilize due to apparent local equilibration of the tumors as characteristic feature sizes and shapes emerge. We also find that whereas the tumor morphology depends primarily upon the microenvironment, the internal structure is most strongly influenced by the genetic characteristics of the tumor, including resistance to necrosis, the rate at which the necrotic core is degraded, and the apoptosis rate. These results are not at all obvious from the examination of the model and underlying hypotheses alone. By hypothesis, the microenvironment, tumor genetics, and tumor morphology are all nonlinearly coupled. The tumor genetics determine biophysical properties like growth rates, which, in turn, are mediated by microenvironmental factors such as available nutrient supply. One would then expect that the tumor genetics have a greater impact on tumor morphology, and, indeed, Cristini et al. (2003) found that the tumor genetics completely determine the morphological behavior when the microenvironment is not taken into account. While the important role of the microenvironment is consistent with experiments in the literature, the observed dominance of the microenvironment in determining the morphology is intriguing. Likewise, the weak dependence of the internal tumor structure on the microenvironment and morphology is difficult to predict a priori. The model can be analyzed to make this prediction for tumor spheroids, but such an analysis ignores variation in tumor morphology and does not lead to obvious conclusions for the general case, where the morphology (and presumably volume) of the necrotic core depends upon the morphology of the tumor boundary.

We note that while our model captures the basic features of tumor growth, it does not currently incorporate the effects of elastic and residual stress, ECM degradation, signaling by promoters and inhibitors, angiogenesis, and competition between tumor subpopulations. These effects represent model refinements that can readily be added to our current modeling framework. We shall discuss our plans to address these and other refinements in the closing remarks in Section 5.

The contents of this paper are as follows: in Section 2, we describe the tumor growth and microenvironment models, non-dimensionalize the resulting systems, and present an analysis of the internal structure of tumor spheroids that will be helpful in understanding non-spherical growth. In Section 3 and Appendix A, we give the important features of our level set/ghost fluid method and extend our technique to solve the Poisson-like equations on the full domain. In Section 3.2, we present a convergence study to demonstrate the accuracy of our technique. In Section 4, we present the results of a parameter study of tumor growth in a variety of microenvironments, categorize the characteristic tumor morphologies, investigate the causal link between microenvironment and tumor morphology, and analyze the link between the internal tumor structure and the tumor genetic parameters. In Section 5 and throughout the text, we discuss the clinical implications of the behavior predicted by our model. In Section 5, we also summarize our work, address known deficiencies in the model, and discuss ongoing modeling refinements.

Section snippets

Governing equations

We study and extend a model for solid tumor growth that applies equally well in two and three dimensions (Cristini et al., 2003, Macklin, 2003, Macklin and Lowengrub, 2005, Zheng et al., 2005), which is a reformulation of several classical models (Greenspan, 1976, McElwain and Morris, 1978, Adam, 1996; Byrne and Chaplain, 1996a, Byrne and Chaplain, 1996b; Chaplain, 2000). We model an avascular tumor occupying a volume Ω(t) with boundary Ω, which we denote by Σ. The tumor is composed of a

Numerical method

We adapt and apply the numerical techniques we recently described in Macklin (2003) and Macklin and Lowengrub, 2005, Macklin and Lowengrub, 2006. Because we anticipate frequent tumor morphology changes (e.g., the tumor breaks into fragments or tumor fragments merge), we use the level set method: we introduce an auxilliary “level set” signed distance function ϕ satisfying ϕ<0 inside Ω, ϕ>0 outside Ω, and ϕ=0 on the tumor boundary Σ. See Fig. 2. For more information on the level set method and

Numerical results

We now investigate the effects of the tumor microenvironment on the morphology and growth patterns of 2D, avascular tumors growing into piecewise homogeneous tissues. In all simulations, we set the apoptosis parameter A=0 because the tumors are assumed to ignore inhibitory signals for self-destruction (apoptosis). We numerically compute the solutions using a computational mesh with Δx=Δy=0.08. All tumors are simulated to a scaled non-dimensional time of T=Gt=λMt=20, where t is dimensional

Discussion and future work

In this work, we have extended previous models of tumor growth and developed a framework to investigate the interaction between avascular solid tumors and their microenvironments during growth. In particular, we model the perfusion of nutrient through the tumor and the surrounding microenvironment, the build-up of pressure in the tissue from the proliferation of cancerous cells, cell–cell and cell–ECM adhesion, and the loss of tumor volume due to necrosis.

Following previous models of solid

Acknowledgments

We thank the National Science Foundation Division of Mathematical Sciences and the UCI Department of Mathematics for their financial support. We thank Vittorio Cristini and Hermann Frieboes at the UCI Department of Biomedical Engineering for valuable discussions and for their generosity in providing unpublished experimental data from their 2006 study in Frieboes et al. (2006b). We thank Alexander “Sandy” Anderson and Mark Chaplain at the University of Dundee and Steven McDougall at Heriot Watt

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