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Mathematical Oncology |
1 Department of Pathology and Laboratory Medicine, and Division of Engineering, Brown University, Providence, Rhode Island; 2 Department of Biology, California Institute of Technology, Pasadena, California; Departments of 3 Mathematics and 4 Biomedical Engineering, University of California, Irvine, California; 5 School of Health Information Sciences, 6 Division of Nanomedicine, and 7 Department of Biomedical Engineering, University of Texas Health Science Center; Departments of 8 Experimental Therapeutics and 9 Systems Biology, The University of Texas M. D. Anderson Cancer Center; 10 Department of Bioengineering, Rice University, Houston, Texas; 11 Department of Biomedical Engineering, The University of Texas, Austin, Texas; 12 Department of Mathematics, University of Tennessee, Knoxville, Tennessee; and 13 USC Center for Applied Molecular Medicine, University of Southern California, Los Angeles, California
Requests for reprints: Vittorio Cristini, University of Texas HSC-SHIS, 7000 Fannin no. 600, Houston, TX 77030. Phone: 713-500-3965; Fax: 713-500-3929; E-mail: vittorio.cristini{at}uth.tmc.edu.
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Abstract |
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| Major Findings: We apply a biologically founded, multiscale, mathematical model to identify and quantify tumor biologic and molecular properties relating to clinical and morphological phenotype and to demonstrate that tumor growth and invasion are predictable processes governed by biophysical laws, and regulated by heterogeneity in phenotypic, genotypic, and microenvironmental parameters. This heterogeneity drives migration and proliferation of more aggressive clones up cell substrate gradients within and beyond the central tumor mass, while often also inducing loss of cell adhesion. The model predicts that this process triggers a gross morphologic instability that leads to tumor invasion via individual cells, cell chains, strands, or detached clusters infiltrating into adjacent tissue producing the typical morphologic patterns seen, e.g., in the histopathology of glioblastoma multiforme. The model further predicts that these different morphologies of infiltration correspond to different stages of tumor progression regulated by heterogeneity.
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This partial differential equation is derived from the conservation of mass (11, 12). The three-dimensional in vivo environment is modeled as a mixture of two viable tumor species (each with a volume fraction of In words: temporal rate of change in a species at any tumor location equals the amount transported by the bulk tumor motion and cell adhesion, plus net result of mass creation/loss due to cell proliferation/death.
Tumors is a mixture of cells, interstitial fluid, and ECM. Cell adhesion is modeled through flux J using an approach from continuum thermodynamics (12).
Cell velocity ui of species i is a function of tissue oncotic (solid) pressure and cell mobility due to chemotaxis and haptotaxis. Right side: changes in pressure p create motion counteracted by cell adhesion In words: species movement depends on oncotic pressure from cell proliferation, adhesion forces, and relative strengths of chemotaxis/haptotaxis.
Tumors are viscous, inertialess fluid; interstitial fluid and cell motion through the ECM is similar to fluid flow in a porous medium. Cells move with a mass-averaged velocity arising from a generalized Darcy-type constitutive law for velocity from excess forces due to chemotaxis and haptotaxis. Cells prefer to adhere to one another rather than the host, modeled by the energy as a function of total solid tumor fraction. The tumor/host interface is well-delineated. Although the model is general, here total solid and liquid volume fractions are assumed constant. Therefore, separate fluid hydrostatic pressure and mechanical oncotic pressure due to cell-cell interactions are calculated. Energy E is a function of total tumor volume.
Equations specify net sources of mass for two tumor viable species, directly linking Eq. 1 to the cell phenotype through hypothesized phenomenological functional relationships involving cell substrates (local oxygen or nutrient concentration n) through the tumor interstitium (12). For species S1, right side terms represent, from left to right, volume fraction gained from mitosis (rate In words: the mass of original species increases through cell proliferation and decreases through apoptosis, necrosis, and mutation to the mutated species. The mass of mutated species increases through mitosis and mutation (from the original species), and decreases through apoptosis and necrosis.
Cells are composed entirely of water (12). Mitosis and necrosis are proportional to substrate concentration n (44). As mitosis occurs, an appropriate amount of water from the interstitial fluid is converted into the cell mass (12). Lysis represents a loss of solid mass converted into water that is absorbed into the interstitial fluid. Necrosis occurs only at sufficiently low nutrients.
This partial differential equation describes cell substrate concentration n across tumor tissue. Vessels originate randomly from existing vasculature (data not shown) around the growing tumor in response to vascular endothelial growth factor produced by hypoxic tissue. The first term on the right side models the diffusion of substrates n (with coefficient D) into tumor tissue, the second term represents the source of substrates at the vasculature denoted by In words: steady-state cell substrate concentration across a tumor region equals the amount that diffuses into the region plus the production from the vasculature minus the amount uptaken by tumor cells.
Nutrient diffusion occurs on a shorter time scale (minutes) than cell proliferation (days); hence, there is no time derivative on the left side, indicating a quasi-steady state.
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Introduction |
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Correlations between morphology and cellular dynamics such as mitosis and motility are of fundamental importance here because these dynamics produce the various morphologies that generate the patterns pathologists use for diagnosis with proven diagnostic power. Morphologic analysis based on histopathology, though, is more art than science. Although pathologists describe different features in tumors, the relative effect of each feature on patient outcome and tumor progression remains unknown. To involve the epidemiology of human tumors, either in retrospective studies, which often exclude necessary data, or multi-year prospective studies, is cumbersome and time-consuming. Hence, determining the clinical predictive value from histologic epidemiology remains limited. Nevertheless, the value of histopathologic analysis is that it does not rely on any single feature alone and thus obtains a comprehensive view of the entire morphologic behavior of any particular tumor at the time of biopsy or excision.
Mathematical modeling can provide a rigorous, more precise approach for quantifying correlations between tumor parameters, prognosis, and treatment outcomes. The integration of these elements in a computational model of tumor progression would be an important tool to advance clinical decision-making. Tumors are complex systems dominated by large numbers of processes with highly nonlinear dynamics spanning a wide range of dimensions. Typically, such complex systems can be understood only through complementary experimental investigation and mathematical modeling. Thus, there is a critical need for biologically realistic and predictive multiscale and multivariate models of tumor growth and invasion, and much recent effort has been directed towards this goal (1–7).
We previously described quantitative multiscale models (2, 8–13) to determine the precise functional relationships among quantifiable parameters from analyses of specific phenotypic or genetic alterations in a tumor, and from in vitro experiments (10) and clinical observations (2, 11) of tumor morphology such as cell arrangement patterns at the tumor boundary. These models predict that morphologic instability of a tumor mass, i.e., morphology resulting in "roughness" or harmonic content (14, 15) of the tumor margin, may provide a powerful tissue invasion mechanism because it allows tumor cells to escape the growth limitations imposed by diffusion (even in vitro; refs. 10, 16) and invade the host independently of the extent of angiogenesis (9, 10). Experiments with various glioma models in vivo (17–21) also support these findings. For example, recently published images of rat glioblastoma in vivo (20) showed that whereas the bulk tumor is perfused by blood, infiltrative cell clusters are much less perfused or not at all. These may be universal considerations that apply to tumor invasion across many different tissue types (22, 23).
Changes in tumor cells at the biochemical and genetic levels are also implicated in tumor progression. Mutations in genes that regulate cell cycle and adhesion result in unrestrained proliferation, invasion, and accumulation of further genetic damage characteristic of high-grade disease (24–26). In particular, in glioblastoma, the oncogene EGFR is frequently overexpressed, amplified, or mutated (27), and promotes mitosis (25), tumor progression in vivo (28), and inhibits apoptosis (29). See for example refs. (30–32) for recent mathematical models on the effect of EGFR gene expression on tumor growth patterns, and ref. (33) for phenomenological modeling of multiple mutations. The tumor suppressor genes TP53 and Rb down-regulate cell division (25) and, secondarily affect oxygen/nutrient consumption, whereas PTEN controls angiogenesis, migration, and invasiveness (34). These genes are inactivated in most malignant brain tumors (35).
In the present study, we use a biologically founded, multiscale, mathematical model of tumor progression (8, 11–13) in three-dimensions (Supplemental Fig. S1) to show that molecular phenomena regulating cell proliferation, migration, and adhesion forces generate (including those associated with genetic evolution from lower-grade to higher-grade tumors), in a predictable and quantifiable way, heterogeneous proliferation and oxygen/nutrient demand (and suppression of apoptosis) across the three-dimensional tumor mass, determining its morphology. The model describes physical conservation laws (e.g., of mass and momentum), with conserved variables representing known characteristics of tumor behavior, and hypothesizes phenomenological functional relationships linking genetic and phenotypic effects, the microenvironment, and tissue-scale growth and morphology.
Variables that characterize the biophysics of tumor growth can be considered in the model, and applied to determine the probabilistic behavior of tumors given their molecular biology and pathologic appearance. By solving the model equations numerically, we predict the combination of variables most likely leading towards tumor invasiveness. At any given time during tumor growth, the model outputs the computed values of all relevant variables at every location within the three-dimensional tumor tissue, e.g., the spatial distributions of cell substrates and tumor cells. Rather than relying on one variable as the sole indicator, the multiparameter computational model enables a more systematic search to be performed for those values of specific variables that reproduce and explain observed tumor behavior. Variables include mutations and phenotypic changes affecting proliferation, apoptosis, nutrient uptake, tumor cell adhesion and motility, and their collective tumor mass effects. By quantitatively linking the invasive phenotype with the observable morphology of the tumor boundary, the model provides a tool for quantitative study of tumor progression as well as diagnostic and prognostic applications.
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Materials and Methods |
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Multiscale tumor model. The model considers genotype, phenotype, and morphologic parameters (Quick Guide, Supplementary Data), and accounts for feedback from the microenvironment, i.e., mutations or phenotypic changes induced by hypoxia (36), as local levels of oxygen/nutrients induce changes in the mutation function. The model also allows for the development of a (hypoxia-induced) migratory phenotype. At the tumor scale, the model is based on first principles describing conservation laws of mass and momentum. Local mass fractions of tumor clones, necrotic, and host tissues are described. Phenomenological parameters describe cell adhesion. Tumor cell migration velocity depends on proliferation-driven mechanical pressure in the tissue, chemotaxis, and haptotaxis due to gradients of chemokines. Cell substrate delivery from the neovasculature (via convection and diffusion; refs. 8, 11, 37) and cellular uptake, and nutrient/oxygen diffusion through tumor tissue (refs. 8, 12) are modeled. The effects of each parameter on outcome can be tested individually or in combination.
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Results |
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3 mm by co-opting the vasculature (which is not shown) while retaining a compact shape with negligible necrosis (Fig. 4A). However, increased nutrient demand generates hypoxic and other substrate gradients pointing radially outwards from the lesion (data not shown). A second, more proliferative clone, is generated by ongoing hypoxia-driven (36) mutations and starts to grow (bottom left corner, shaded area). Its higher cellular uptake introduces perturbations in the spatial gradients of oxygen, further enhancing local hypoxia. These gradients generate spatially heterogeneous cell proliferation and migration. After 4 months, this perturbation triggers a morphologic instability, which noticeably deforms the tumor mass (bottom left). Hypoxia and necrosis are present within the regions in which the more malignant clone grows. Shape instability leads to clusters of clone 2 protruding "finger-like" (darker) regions into the mass of clone 1 first, and the host brain later, growing at the expense of the less proliferative clone and the host tissue. We have also observed that detachment of these clusters may occur in our model (Fig. 2C; ref. 10). These fingers grow away from the bulk tumor and tend to follow substrate gradients.
In 6 months' time, the aggressive, invasive proliferation of clone 2 (darker regions) enables it to infiltrate almost all regions of the tumor, in particular around the boundary, and leads to a higher-grade lesion. A bud-like protrusion emerges on the tumor (bottom left). Hypoxic, necrotic areas continue to expand (Fig. 4B). In 8 months, the glioma aggressively infiltrates the surrounding brain tissue. Clone 1 is being confined by competition with clone 2. Extensive necrosis is present. Additional buds have appeared, and the initial (middle) bud has grown into an invasive finger. Strands and clusters of clone 2 drive the growth of the finger and buds (extent of the darker area). Clone 1 has been mostly eliminated from this region of the tumor, and remains stagnant. In 12 months, the surrounding brain has been severely compromised. Expansion of clone 2, accompanied by continued necrosis, is now the main determinant of tumor morphology. The lesion reaches a size of 4 cm in a little over 1 year of simulated time, consistent with human glioma progression (Fig. 4C shows only a tumor portion at 1 year).
Different tumors are likely to have different genomic instability factors—different types and mutation rates. The idealized tumor in Fig. 4A–C is "programmed" to exhibit the progressive appearance of one highly malignant clone (clone 2). In reality, multiple clones may arise with varying degrees of malignancy.
A histology section of glioblastoma from one patient reveals the tip of a round invading finger (Fig. 4D), consistent with the morphology and size of these infiltrative cell clusters predicted by simulations during the proliferative growth stage (Fig. 4C at 12 months, in which the tip of one protruding cell front is 2 mm in size). The fact that tumor cells rely on vessels beyond the protrusions—and may grow towards blood vessels that they stimulate (11, 38, 39)—also suggests a proliferative growth stage because these vessels increase substrate availability in the microenvironment. Older tumor vessels may have thicker walls that are not as permeable for nutrient/oxygen exchange, and may become occluded due to increased pressure from the tumor mass (Supplementary Data), further promoting substrate gradients.
Additional histology sections from four glioblastomas (Fig. 5 ) reveal protruding fronts of cells pointing away from a necrotic area into an area of the host brain where neovascularization is evident. These invading fronts are also consistent with the tumor boundary morphology predicted by simulation in the proliferative growth regime (Fig. 4A–C). Although infiltrative shapes were consistently observed in histologic sections (Supplementary Data), the model predicts that their size may vary based on the stage of growth. For example, these shapes can be extremely slender in the hypoxic growth regime, down to single rows of palisading cells migrating up substrate concentration gradients (Fig. 3A and B), and thus, away from hypoxic regions, as seen in histology (Fig. 3C and D).
Supplemental Fig. S2 reports additional histologies showing invasive fingering. Such morphologies are predicted by the model to occur in the proliferative growth stage, in which there is increased substrate availability. This is confirmed by the presence of viable vessels in the histopathology acting as sources of substrates around the invasive fingers.
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Discussion |
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The central finding of this work is that tumor growth and invasion are not erratic or unpredictable, or solely explained through genomic and molecular events, but rather are predictable processes obeying biophysical laws, driven by microenvironmental substrate gradients, and regulated by genotypic, phenotypic, and microenvironmental parameters (e.g., cell adhesion, proliferation, and motility). The model enables quantitative study of invading cell clusters as functional units that move as complex systems. Substrate gradients, e.g., of nutrient, oxygen, growth factors, and metabolites, result from diffusion, cellular activity, and heterogeneous delivery and removal. This leads to local hypoxia, nutrient starvation, acidosis, necrosis, and the pleiomorphic appearance of tumors. The underlying physical mechanism of collective cell migration, i.e., a gross tumor morphologic instability (2, 8–10, 12–14, 40), maximizes cell exposure to substrates by evading a compact, nearly spherical morphology in favor of infiltrating shapes moving up gradients of substrates such as oxygen. Phenotypic changes that increase nutrient uptake and augment cell proliferation (and also increase cell motility and reduce adhesion) have a quantifiable effect on morphology at the tumor scale. In particular, they trigger invasive fingering into host tissue while inducing necrosis and angiogenesis for certain parameter conditions.
The model predicts that different morphologies of infiltration are associated with phenotypic and genotypic changes, which depend on cell substrate gradients and microenvironmental parameters, and reflect the growth stage of the tumor. In regions of hypoxia, these changes include the down-regulation of proliferation and up-regulation of motility, which under low cell adhesion results in palisading cells forming "Indian files" (Fig. 3A and B). As cells localize in tissue regions that are richer in substrates, i.e., better vascularized, further phenotypic changes occur in which motility is down-regulated and proliferation is up-regulated. This leads to the formation of wave-like patterns of cell rearrangements at the tumor boundary and round infiltrative fingers that can detach from the tumor bulk as clusters (Fig. 4A–C). Actually, different tumor regions may exhibit different growth regimes at the same time depending on heterogeneity in the local microenvironment and in cell genotypes/phenotypes. In our simulations, proliferation and collective migration of more aggressive clones or phenotypes drives tumor infiltration, as observed in patient biopsies (e.g., refs. 41, 42, and this study).
Model predictions of infiltrative morphologies in the hypoxic and proliferative growth stages compare well with the histopathology of human glioblastoma. Tumor microenvironment conditions were similar to those predicted in the model with respect to vascular distribution, implied cell substrate availability, and palisading of malignant cells between vascular and avascular regions. These findings are further supported by published correlations of invasion with hypoxia and necrosis induced in glioma by characteristically abnormal, inadequate, and hemorrhagic vasculature or by antiangiogenic therapy (17, 18, 36, 43, 44). A formal comparison between observations and model predictions, e.g., measured via a "merit function," could be performed as the next stage to confirm the model hypotheses.
By quantifying the close connection between the observable morphology of the tumor boundary and cellular/molecular dynamics, the work presented here provides a quantitative tool for the study of tumor progression and diagnostic/prognostic applications. This connection is important because the dynamics that give rise to various tumor morphologies also control invasiveness. In particular, by describing morphology as a function of parameters dependent on cellular and environmental phenomena (45, 46), the model quantifies, under the unifying umbrella of morphologic stability analyses, the often seemingly diverse and unrelated morphologies and invasive phenotypes, e.g., palisading cells, round fingers, and clusters. It is generally observed (22) that premetastatic phenotypic transitions (e.g., epithelial-mesenchymal transitions) follow a collective migration stage, and are regulated by the environment (e.g., local hypoxia; refs. 45, 46). Invasive characteristics may strongly influence whether a tumor can be effectively treated by local resection and may suggest specific treatment options (1, 33, 47). Observation of tumor morphology, for example, could indicate the presence of hypoxia, and therefore, the potential to respond to oxygen-dependent treatments such as radiation therapy and certain chemotherapy treatments.
The model may be used to study system perturbations by therapeutic intervention and may aid in the design of novel clinical end points in therapeutic trials. By integrating the model with patient data for key tumor phenotypic and microenvironmental parameters (2), model results could be used to enhance clinical outcome prognostication. Initial conditions regarding tumor physical location, structure, and vasculature, e.g., obtained from contrast-enhanced magnetic resonance imaging and possibly coupled with computed tomograpy would be translated using a computer program to the model coordinate system (2), e.g., a finite-element computational mesh discretizing the space occupied by tumor and host tissues (12). Viable region spatial information and microvasculature structure would be obtained from histopathology (11). Vasculature-specific information could be defined from dynamic contrast–enhanced computed tomograpy, yielding blood volume, flow, and microvascular permeability parameters (2). Other input data include cell scale parameters (e.g., proliferation rates). The model then calculates local tumor growth, angiogenesis, and response to treatment under various conditions by solving in time and space the conservation and other equations at the tissue scale.
The model allows predictions of cellular and molecular perturbations that may alter invasiveness and that can be measured through changes in tumor morphology. Therefore, morphologies obtained from the model could be used to both understand the underlying cellular physiology and predict subsequent invasive behavior. For example, novel individualized therapeutic strategies could be designed in which microenvironmental and cellular factors are manipulated to decrease invasiveness and promote well-defined tumor margins—an outcome that would also benefit treatment by improving local tumor control through surgery or radiation. In addition to existing strategies that act on relevant cellular behaviors (e.g., promotion of tumor cell adhesiveness; refs. 10, 14, 22), or that target oncogenes such as EGFR, tumor morphologic stability could be enhanced by improving nutrient supply (9, 40), thus enforcing a more homogeneous microenvironment and normoxic conditions. This could be achieved through "vascular normalization" (9, 48) or uniform nanoparticle delivery (49), e.g., releasing oxygen and antiangiogenic drugs. Furthermore, by maintaining microenvironmental homogeneity, effects of genetic mutations that lead to morphologic instability may be minimized (e.g., ref. 50), without direct intervention at the genotype level.
Applying biologically founded, mathematical modeling to quantify the connections between the microenvironment, tumor morphology, genotype, and phenotype may direct prognosis beyond the limitations of current methodologies, and suggest new directions in the way we think about cancer growth and invasion.
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Disclosure of Potential Conflicts of Interest |
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Acknowledgments |
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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.
We acknowledge Robert Gatenby (Moffitt Cancer Center) for useful discussions, Xiangrong Li (UC-Irvine) for Fig. 2A, Ed Stopa and the Pathology Department (Rhode Island Hospital) for autopsied specimens, Aleksey Novikov and Bryan Kinney (E.L. Bearer's lab) for technical assistance, and Henry Hirschberg (UC-Irvine) for information about recent results (20).
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Footnotes |
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8 Linear stability analysis can be used to determine analytically how infinitesimal perturbations evolve at the tumor surface in time, as illustrated in refs. (9, 13–15). 
Received 10/ 7/08. Revised 3/23/09. Accepted 3/27/09.
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References |
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on T, Owen MR, Webb SW, Maini PK. Modeling aspects of cancer dynamics: a review. Philos Trans R Soc A 2006;364:1563–78.[CrossRef]This article has been cited by other articles:
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  H. Enderling, A. R.A. Anderson, M. A.J. Chaplain, A. Beheshti, L. Hlatky, and P. Hahnfeldt Paradoxical Dependencies of Tumor Dormancy and Progression on Basic Cell Kinetics Cancer Res., November 15, 2009; 69(22): 8814 - 8821. [Abstract] [Full Text] [PDF]
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Top
Abstract
Quick Guide to Equations...
i, i = 1,2), dead tumor (
and assumed to move freely) flowing through the ECM, which is treated as a porous medium. From left to right: change of volume fraction with respect to time; bulk transport by tumor mass with local velocity ui; fluxes Ji that account for mechanical interactions among cell species [based on a generalized Fick's Law (
i mediated through an energy variation 
[for specific forms of this energy E and its variation, see ref. (
n due to soluble gradients of cell substrates and oxygen n 
nV; haptotaxis 
M,1), and lost to apoptosis (
C as a function of distance from the vessels (1 – n/nV) and vascular pressure (1 – p/pV), and the third term represents uptake 
i by tumor species 
