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Morphologic Instability and Cancer Invasion

Authors' Affiliations: Departments of ¹ Biomedical Engineering and ² Mathematics, University of California, Irvine, California; Departments of ³ Radiology and ⁴ Applied Mathematics, University of Arizona, Tucson, Arizona; ⁵ Department of Chemical Engineering, University of Naples "Federico II," Naples, Italy; ⁶ Davis Heart and Lung Research Institute, Ohio State University, Columbus, Ohio; and ⁷ National Cancer Institute, Bethesda, Maryland

Requests for reprints: Vittorio Cristini, Department of Biomedical Engineering, REC 204, University of California, Irvine, CA 92697-2715. Phone: 949-824-9132; Fax: 949-824-1727; E-mail: cristini{at}math.uci.edu.

	Abstract

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Purpose: A solid tumor embedded in host tissue is a three-dimensional arrangement of cells and extracellular matrix that acts as a sink of oxygen and cell nutrients, thus establishing diffusional gradients. This and variations in vascular density and blood flow typically produce intratumoral regions of hypoxia and acidosis, and may result in spatially heterogeneous cell proliferation and migration. Here, we formulate the hypothesis that through these mechanisms, microenvironmental substrate gradients may drive morphologic instability with separation of cell clusters from the tumor edge and infiltration into surrounding normal tissue.

Experimental Design: We used computer simulations and in vitro experiments.

Results: We provide evidence that morphologic instability could be suppressed in vivo by spatially homogeneous oxygen and nutrient supply because normoxic conditions act both by decreasing gradients and increasing cell adhesion and, therefore, the mechanical forces that maintain a well-defined tumor boundary. A properly working tumor microvasculature can help maintain compact noninfiltrating tumor morphologies by minimizing oxygen and nutrient gradients. In contrast, antiangiogenic therapy, by increasing microenvironmental heterogeneity, may promote morphologic instability, leading to invasive patterns even under conditions in which the overall tumor mass shrinks.

Conclusions: We conclude that therapeutic strategies focused solely on reduction of vascular density may paradoxically increase invasive behavior. This theoretical model accounts for the highly variable outcome of antiangiogenic therapy in multiple clinical trials. We propose that antiangiogenic strategies will be more consistently successful when aimed at "normalizing" the vasculature and when combined with therapies that increase cell adhesion so that morphologic instability is suppressed and compact, noninvasive tumor morphologies are enforced.

The critical role of angiogenesis in promoting tumor growth and invasion has been well demonstrated. However, the results of clinical trials using various drugs to suppress angiogenesis have been mixed. Although some tumor regression can be observed following therapy, length of survival remains unchanged (25–27). Furthermore, it has been observed experimentally that antiangiogenic treatment can exacerbate hypoxic effects (28) and cause tumor mass fragmentation, cancer cell migration, and tissue invasion (29, 30). Systemic treatment with vascular endothelial growth factor receptor-2 antibody inhibited glioblastoma angiogenesis in mice and led to decreased tumor volume, but increased tumor invasiveness along host microvasculature (31). It was shown (32) that hypoxia, instead of inhibiting tumor growth, induced Met tyrosine kinase, which increased sensitivity to hepatocyte growth factor. Other authors (33) noticed a remarkable increase in the number and total area of small satellite tumors clustered around the primary mass in mice treated with vascular endothelial growth factor receptor-2 antibody. These satellites usually contained central vessel cores, and tumor cells often had migrated along preexistent blood vessels over long distances to reach the surface and spread in the subarachnoid space. A modification in the tumor cell invasion pattern at the tumor-normal brain parenchyma interface was reported (34). Tumors from animals belonging to the control group and those under antiangiogenic treatment showed similar invasion patterns consisting of an undefined interface, with trails of invading tumor cells and distant tumor satellites. However, tumors treated against both angiogenesis and invasion showed a well-defined tumor-parenchyma interface, without trails of invading cells or tumor satellites, and showed a clear decrease in the peripheral vessel recruitment. A similar formation of satellite glioblastoma tumors in rats after anti–vascular endothelial growth factor antibody treatment has been observed (35). A correlation of hypoxic status in vivo with increased metastatic spread has also been observed in D-12 melanoma cells and KHT-C fibrosarcoma cells (36, 37).

These highly variable empirical observations illustrate the critical need for biologically realistic theoretical models that integrate tumor proliferation and invasion with vascular density, blood flow, and microenvironmental substrate gradients. Indeed, it is clear from experience in the physical sciences that such complex systems, dominated by highly nonlinear dynamics, can typically be understood only by using appropriate mathematical models and sophisticated computer simulations.

Here, we propose the hypothesis that morphologic stability of both normal and tumor tissue in vivo requires a uniform level of substrate concentrations through a robust vascular network. Morphologic instability may occur during solid tumor growth and during response to treatment as the result of oxygen, glucose, acid, and drug concentration gradients driving spatially heterogeneous cell proliferation, migration, and death, and reducing cell adhesion and other mechanical forces in hypoxic and acidotic regions due to disruption of cell-cell and cell-matrix interactions. This results in invasive fingering and branching and even fragmentation and migration of cell clusters into the surrounding tissue because of differential proliferation along the gradient (see recent mathematical and computational analyses; refs. 38–40). In this article, we provide computational and experimental evidence supporting this hypothesis and propose new therapeutic strategies.

	Methods

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Reaction-diffusion in silico model. For full details, please see ref. 40. Briefly, we model the net local rate S of nutrient and oxygen delivery from neovasculature and uptake by tumor cells; nutrient and oxygen diffusion through the tumor interstitium (for simplicity we only describe transport of one chemical species without making distinction between oxygen and cell nutrients; refs. 41, 42); the local specific mass growth rate as the divergence of the tumor cell velocity field u; and the pressure in the tissue by Darcy's law (43). The equations are as follows:

where n is the local nutrient and oxygen concentration; n_v is the concentration in the vasculature (used here as characteristic concentration because it provides an upper bound for the concentration n in the interstitium); p is the pressure in the tumor; p_v is the pressure in the vasculature; ₁ is the transfer coefficient from the vasculature; is the rate of nutrient and oxygen uptake within tumor cells; is the indicator function of vasculature; D is the diffusion coefficient; _M and _D are tumor cell mitosis and death rates (_M = _D = 0 outside the tumor), the latter describing the disintegration of tumor cell mass and the radial effusion of fluid away from the necrotic regions; tumor cells and extracellular matrix are treated as comprising porous media with hydraulic conductivity µ; the notation ()⁺ means max{, 0} and is used for the blood-to-tumor pressure difference term because if it approaches zero, the blood vessel will collapse (44). We assume that transport of molecules across the microvascular wall is primarily due to convection and diffusion (2, 4). Convection is dependent on the difference of vascular and interstitial hydrostatic pressures, whereas diffusion is proportional to the difference of vascular and interstitial concentrations (43). The "tumor" phase encompasses tumor cell matter, interstitial fluid, and extracellular matrix, and therefore no distinction between interstitial fluid hydrostatic pressure and mechanical pressure due to cell-cell interactions is made. These pressures can be different within the tumor tissue (2, 4, 44). Cell adhesion forces are modeled using an equivalent surface tension at the tumor/tissue interface (43). This provides the boundary condition for the jump in pressure [p] = across the tumor/host-tissue interface, where is surface tension and is local curvature of the interface (38). We assume that tumor tissue is saturated with growth factors, and that oxygen and nutrient availability limits cell proliferation, and therefore we model the fraction of cycling cells by n/n_v. We also assume that cell mass density is uniform in the tumor (43) and that regions become instantly necrotic where nutrient and oxygen concentration falls below some specified minimum. Cells undergoing necrosis are hypothesized to produce tumor angiogenic factors that trigger the formation of neovasculature. The model incorporates an angiogenesis component based on the work in ref. 45. This is essentially a reinforced random walk of new blood vessels drawn into the tumor via chemotaxis and other mechanisms. We have previously provided (40) a detailed determination of the microphysical model parameters focusing on the case of glioblastoma multiforme. In particular, a characteristic nutrient and oxygen diffusion length is found to be L = 250 µm. From our in vitro experiments, a typical mitosis rate _M 1 d^–1 was observed. For specific values of cell adhesion and tumor vascularization parameters, please see ref. 40. The above reaction-diffusion model is solved using a novel finite-element/level-set method (40) in two spatial dimensions coupled to an unstructured adaptive mesh technology (46) that allows efficient and accurate solution. This implementation allows for the first time simulation of tumoral lesions through the stages of diffusion-limited dormancy, localized necrosis, vascularization and rapid growth, and tissue invasion.

Cell culture. Data were obtained as part of a study of an in vitro and in silico model of tumor invasion.⁸ Briefly, the ACBT (grade 4 human glioblastoma multiforme) cell line was cultured to grow tumor spheroids using a liquid-overlay technique (47). Photographs were taken with an Olympus camera mounted on top of a Leitz microscope at x100 magnification with a photography window of 1,130 x 1,430 µm. Spheroids were then fixed in paraformaldehyde and embedded in paraffin. Histologic cross sections were obtained by slicing in 6 µm increments and staining according to H&E. Photographs were taken at x160 magnification.

	Results

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The two-dimensional computer simulation results shown in Fig. 1 are to be interpreted as a "phase diagram" for cancer that summarizes possible morphology states obtained by varying two model parameters: one associated with cell adhesion forces and the other with microvascular density (see Methods). Cell survival and proliferation are dependent on substrate availability and, thus, vascular density. Cellular adhesion is dependent on a variety of membrane proteins, such as E-cadherins and integrins, which maintain the cell position through contact with other cells, the basement membrane, and the extracellular matrix. However, there is also some interrelationship of vascular density with cell adhesion because hypoxia and extracellular acidosis associated with reduction in vascular density diminish cell adhesion through acid-induced extracellular matrix degradation and loss of gap junctions and through the other mechanisms reviewed in Introduction. In other words, reduced vascular density will favor diminished cell adhesion although mutations in intracellular pathways may result in reduced adhesion even in the presence of normal extracellular concentrations of oxygen and acid. Solid thick contours in Fig. 1 denote the calculated interface between tumoral and nontumoral tissues, and black filled regions denote necrotic areas. Newly formed immature capillaries are also shown during angiogenesis. Some capillaries have formed loops through anastomosis and conduct blood, whereas others have not. Dotted contours describe the density of "free" endothelial cells migrating chemotactically after sprouting from preexisting vessels (not shown) in outer tissue towards the source of angiogenic factors in the tumor. This in silico model (39, 40) enables us to describe via computer simulations tumor-induced angiogenesis and perfusion of growing lesions.

View larger version (73K): [in this window] [in a new window]   Fig. 1. Snapshots at dimensionless time t = 1,500 from three simulated tumoral lesion morphologies corresponding to different values of cell adhesion and vascularization parameters. Tumor/tissue interface is depicted by thick solid perimeter. Frame A, high cell adhesion; frame B, low cell adhesion; frame C, low cell adhesion and with higher microvascular density or more efficient vascularization. Arrows, morphologic transitions following different therapy strategies. Photographs a to d show morphologic instabilities observed in vitro in human glioma spheroids. Individual spheroid subcomponents are highlighted in c. Spheroid cross section (d) shows that subcomponents are mainly composed of viable cells at the time of separation. Bar, 130 µm.

View larger version (50K): [in this window] [in a new window]   Fig. 2. Top six frames, entire time evolution (t = 0, 300, 600, 900, 1,200, and 1,500, from left to right) of calculated profiles of local cell nutrient and oxygen levels n/nv in the tissue for the low cell adhesion case (Fig. 1, frame B). Bottom six frames, corresponding lesion morphology during this time evolution. Time is rescaled by the mitosis time M–1 and distance by the diffusion length L (see Methods). Simulated antiangiogenic therapy is started at t = 1,000.

View larger version (22K): [in this window] [in a new window]   Fig. 3. Simulated two-dimensional tumor mass (A) and shape factor (B) defined as (tumor perimeter)2 / (4 tumor area) versus time for simulations shown in Figs. 1 and 2. A, high cell adhesion; B, low cell adhesion and antiangiogenic therapy starting at t = 1,000; C, low cell adhesion and vasculature normalization starting at t = 1,000; D, same as B but without therapy. Tissue density is assumed to be constant and equal to 1 (40), and thus mass is nondimensionalized with L2 (see Methods). The perimeter is the total length of the tumor/tissue interfaces for each frame in Figs. 1 and 2, and the area is the total mass of each lesion.

In our in vitro experiments (Fig. 1, photographs a-d) on a human glioblastoma multiforme cell line (see Methods), it was observed that spheroids first grow to a diffusion-limited millimeter size. When proliferation rates are low, cell adhesion and possibly other mechanical forces exerted through the extracellular matrix are sufficient to maintain compact spheroid shapes. When proliferation rates are high (at high serum or glucose concentrations), spheroids are observed to become unstable, assuming dimpled shapes as predicted by the simulations. Photographs a and b in Fig. 1 reveal that subspheroids sometimes emanated from the main tumor, separating as a "bubble" of cells or as spheroid fragments. This process could repeat itself on subspheroids, leading to recursive subspheroidal growth and separation as the main mechanism of spheroid morphogenesis and invasion of the surrounding environment. Photograph c shows a colony of successive generations of spheroids from one original "mother" spheroid (dashed white lines are guides to the eye to identify individual subspheroids). See Fig. 2 (bottom frames) at times t = 300 and 600 for comparison. Photograph d is a histologic cross section of a spheroid in the process of splitting off a subspheroid. The main spheroid has a viable rim of cells and a central necrotic area. The subspheroid has mostly proliferating cells and may further grow after separating to develop its own necrotic core. The morphologic instability observed in our simulations and experiments is characterized by growing low-wave-number modes [i.e., a few "bumps" (typically three or four) develop on the original spherical tumor surface and evolve into separate spheroids]. Low-wave-number unstable modes are typical of diffusion-driven morphologic instabilities (38). These instabilities are regulated by a model parameter (38) that is proportional to the ratio _M/ describing the relative strength of cell proliferation driving instability and cell adhesion resisting it. Thus, increasing proliferation or decreasing cell adhesion has an equivalent effect of favoring instability. Note that for high values of this parameter (e.g., very low local cell adhesion), even high wave numbers become unstable (38). This implies that if a single cell mutates and becomes highly aggressive, it can leave the tissue mass migrating and proliferating radially away from the main tumor.

The two frames in Fig. 2 at t = 1,200 and t = 1,500 (top and bottom) correspond to a simulated antiangiogenic therapy on this lesion. Therapy is initiated at dimensionless time t = 1,000 and is simulated by decreasing the rate of transfer of nutrients and oxygen from the neovasculature (see Methods). Note the large necrotic area in black formed at the center of the lesion. Curves labeled "B" in Fig. 3 reveal that at the onset of antiangiogenic therapy, the tumor mass initially shrinks by about half, and then later recovers through renewed proliferation. Hypoxia caused by therapy leads to enhanced tumor fragmentation, as documented in vivo in ref. 34 and by others. Interestingly, the frames in Fig. 2 also show that some tumor cell clusters tend to co-opt the vasculature to maximize nutrient uptake, as documented previously (31, 33, 35). Tumor fragmentation is quantified in our simulation by the large shape-factor increase illustrated in Fig. 3B. In contrast, the curves labeled "D" in Fig. 3 correspond to evolution of this same lesion without the therapy. When comparing "D" (no therapy) with "B" (therapy), it is important to note that without therapy, the tumor mass is higher than after therapy, which is an expected result. However, the shape factor after therapy is ca. 50% higher than without therapy, reflecting increased scattering and invasiveness in response to hypoxia.

Frame C in Fig. 1 corresponds to low cell adhesion (as in B), but with a more uniform and efficient distribution of newly formed vessels. The simulation predicts that this "vascular normalization" leads to reduced oxygen and cell nutrient gradients, and hence to reclustering of cells into bigger fragments with a more compact morphology. This result could be achieved by pruning immature and inefficient blood vessels, leading to a more normal vasculature of vessels reduced in diameter, density, and permeability (50, 51). The effect is quantified in Fig. 3 (curves labeled "C") from a similar simulation where also cell adhesion was increased. The shape factor and thus the invasiveness of the lesion are minimized. Note that the more rapid tumor growth predicted after vascular normalization is an artifact of our oversimplified implementation of vascular normalization therapy in the reaction-diffusion model (see Methods) by increasing the rate of transfer from the vasculature and the local vessel density, which responds by not only producing a more homogeneous nutrient profile but also by producing a higher level of nutrient. In addition to this, in vivo, mass growth with compact, nearly spherical shapes would be mechanically constrained by surrounding tissue.

	Discussion

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Individual cells possess a broad spectrum of migration and invasion mechanisms that can be modified in response to different conditions (49), such as hypoxia. The consequences of this variability are highly multiscalar as morphology, oxygen and nutrient gradients at the tumoral scale, and molecular interactions at the cell scale are strongly linked so that hypoxia affects cell adhesion and motility properties, which in turn are responsible for the stability of the morphology at the tumoral scale. Our simulations predict that the motion of cells in a tumor mass in the presence of heterogeneous distribution of oxygen and nutrients places some cells in more favorable conditions, leading to greater migration and proliferation in certain regions. A morphologic instability of the tumor mass arises if this growth cannot be contained by weak adhesion or other mechanical forces. Higher proliferation occurs near the leading edges, creating fingering growth into the surrounding tissue and possibly separation of cell clusters that migrate up following nutrient and oxygen concentration gradients. This phenomenon may provide a mechanism for tumor cells to overcome diffusion limitations by abandoning compact mass growth. The fragmented, infiltrative resulting morphology would enable tumor cells to access oxygen and nutrients in surrounding tissue, in principle without the need for increased vascular supply. This effect is striking in the in vitro experiments presented where diffusional instability enables tumor cells to effectively overcome diffusion limitations and virtually invade the entire environment, growing to a much larger total mass than the initial single spheroid would have done had it remained compact.

We previously used (38) linear stability analysis and a nonlinear simulation to show for the first time the onset of tumor fragmentation due to a diffusional instability during both undisturbed growth and chemotherapy. We also showed the stabilizing effect of uniform vascularization. Later, via in silico experiments (39), we showed tumor fragmentation under chemotherapy employing adjuvant antiangiogenic therapy. This idea was first introduced by Jain (50, 51) with the aim of homogenizing vascular delivery of chemotherapeutic drugs. In light of these results, antiangiogenesis solely with an end to eradicating a tumor may not be the most prudent course.

Note that these conclusions modify the conventionally view that angiogenesis per se is a key milestone leading to invasion and metastasis. Rather, our results show that local hypoxia or lack of cell nutrients due to imperfect and inefficient neovasculature leads to invasion. In contrast, we find that a well-vascularized tumor may be morphologically stable due to homogeneous cell proliferation. Thus, diffusional instability may represent a fundamental alternative to angiogenesis in triggering invasive growth. In other words, we do not dispute observations that angiogenesis results in adoption of the invasive phenotype. We propose, however, that disordered angiogenesis leads to regions of heterogeneous cell proliferation and migration, which, in turn, leads to an invasive morphology. Ironically, ideal angiogenesis will lead to tumor mass that is compact and spherical and occupying a limited volume of tissue. In this case, further growth requires additional angiogenesis to increase nutrient access. In contrast, disordered vascularity leads to fingering and instability such that the same tumor mass might extend across a larger space and not require angiogenesis for further growth.

We propose diffusional instability as a universal physical mechanism underpinning compact or invasive cancer morphologies during undisturbed lesion progression and during response to therapy. This physically based framework unifies numerous clinical and experimental observations. Current cancer therapy may unwittingly contribute to tumor morphologic instability and consequent tissue invasion. The data presented in this article suggest new therapy strategies based not on killing as many endothelial or cancer cells as therapeutically possible but rather on controlling tumor morphology by suppressing instability, thus leading to compact noninvasive and resectable lesions. Morphologic control would rely on two key parameters: tumor vascularization and cell adhesion. Effective tumor treatment could be achieved by combining antiangiogenic therapy, aimed at vascular normalization (51), with anti-invasive drugs, such as Met inhibitors (52–54) or hepatocyte growth factor antagonists (55, 56). This approach could achieve efficient tumor mass control or shrinkage and prevent tumor fragmentation and infiltration of the healthy tissue by cancer cell clusters. Note that by providing adequate and uniform nutrient and oxygen, there would be the additional benefit that more benign clones would be maintained, helping to keep malignant clones under control by competition for oxygen and cell nutrients. As the arrows in the phase diagram (Fig. 1) indicate, anti-invasive therapy alone would increase cell adhesion, thus leading to more compact morphologies. Vascular normalization would have a similar effect by making oxygen and nutrient supply more uniform. In contrast, antiangiogenic therapy aimed at killing the neovasculature indiscriminately can be harmful by exacerbating oxygen and cell nutrient gradients and, thus, triggering morphologic instability. Verification could be done in future experiments by manipulating tumor morphology in vivo through variations in tumor vascularization and cell adhesion that replicate our computational results. In fact, in the near future, vascular normalization could be effectively achieved using novel, nanoparticle-based delivery strategies (57). Finally, we note that it has been recently shown (58) that inefficient oxygen delivery leads to fingered-type growth using a cellular automaton model.

	Footnotes

 
The costs of publication of this article were defrayed in part by the payment of page charges. This article must therefore be hereby marked advertisement in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.

⁸ H. Frieboes, X. Zheng, C-H. Sun, B. Tromberg, R. Gatenby, and V. Cristini. An in silico and in vitro model of tumor invasion. Cancer Res, in review.

Received 4/18/05; revised 6/ 8/05; accepted 7/ 5/05.

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