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# Predictive oncology: multidisciplinary, multi-scale in-silico modeling linking phenotype, morphology and growth

^{1,}

^{5}Hermann B. Frieboes,

^{2,}

^{5}Xiaoming Zheng,

^{2}Robert Gatenby,

^{3}Elaine L. Bearer,

^{4}and Vittorio Cristini

^{1,}

^{2,}

^{5}

^{1}Department of Biomedical Engineering, University of Texas, Austin

^{2}Department of Mathematics, University of California, Irvine

^{3}Department of Radiology, University of Arizona

^{4}Department of Pathology and Laboratory Medicine, Brown University Medical School, Providence, RI

^{5}School of Health Information Sciences, University of Texas Health Science Center at Houston, Texas

## Abstract

Empirical evidence and theoretical studies suggest that the phenotype, i.e., cellular- and molecular-scale dynamics, including proliferation rate and adhesiveness due to microenvironmental factors and gene expression that govern tumor growth and invasiveness, also determine gross tumor-scale morphology. It has been difficult to quantify the relative effect of these links on disease progression and prognosis using conventional clinical and experimental methods and observables. As a result, successful individualized treatment of highly malignant and invasive cancers, such as glioblastoma, *via* surgical resection and chemotherapy cannot be offered and outcomes are generally poor. What is needed is a deterministic, quantifiable method to enable understanding of the connections between phenotype and tumor morphology. Here, we critically review advantages and disadvantages of recent computational modeling efforts (e.g., continuum, discrete, and cellular automata models) that have pursued this understanding. Based on this assessment, we propose and discuss a multi-scale, i.e., from the molecular to the gross tumor scale, mathematical and computational “first-principle” approach based on mass conservation and other physical laws, such as employed in reaction-diffusion systems. Model variables describe known characteristics of tumor behavior, and parameters and functional relationships across scales are informed from *in vitro*, *in vivo* and *ex vivo* biology. We demonstrate that this methodology, once coupled to tumor imaging and tumor biopsy or cell culture data, should enable prediction of tumor growth and therapy outcome through quantification of the relation between the underlying dynamics and morphological characteristics. In particular, morphologic stability analysis of this mathematical model reveals that tumor cell patterning at the tumor-host interface is regulated by cell proliferation, adhesion and other phenotypic characteristics: histopathology information of tumor boundary can be inputted to the mathematical model and used as phenotype-diagnostic tool and thus to predict collective and individual tumor cell invasion of surrounding host. This approach further provides a means to deterministically test effects of novel and hypothetical therapy strategies on tumor behavior.

## THE ROLE OF PREDICTIVE SCIENTIFIC COMPUTATION AS “IN‐SILICO” CANCER MODELING

### Cancer progression and invasion: current understanding

A wealth of empirical evidence links disease progression with tumor morphology, invasion, and associated molecular phenomena. However, not only is there a lack of quantitative understanding of the underlying physiological processes driving tumor-scale behavior, in particular, morphology at the tumor-host interface, but the qualitative explanations themselves may be indecisive or inconsistent. For example, a positive correlation of cell adhesion molecules (integrins) and cancer cell migration was observed in glioma cells (Tysnes *et al.*, 1996), yet integrins can also serve as negative effectors that impede invasion and progression (Zutter *et al.*, 1995). Similarly, conflicting data on the function of proteases in tumor invasion and metastasis (Friedl & Wolf, 2003) is illustrated by variable results from clinical trials of potential anti-invasive therapies (Lah *et al.*, 2006). While the primary role of angiogenesis in promoting tumor growth and invasion has been well demonstrated, the results of clinical trials using various drugs to suppress neovascularization have yielded mixed results. Despite encouraging signs of tumor regression following anti-angiogenic therapy, in some cases length of survival remains the same (Kuiper *et al.*, 1998, Bernsen *et al.*, 1999, Bloemendal *et al.*, 1999). In addition, experimental observations indicate that anti-angiogenic treatments may exacerbate hypoxia (Steeg, 2003), and paradoxically promote tumor fragmentation, cancer cell migration, and host tissue invasion (Page *et al.*, 1987, Kunkel *et al.*, 2001, Seftor *et al.*, 2002, Lamszus *et al.*, 2003, Bello *et al.*, 2004).

### Links between cellular‐ and tumor‐scale

In spite of abundant experimental and clinical data surrounding molecular and cellular phenomena, it is difficult to quantify their aggregate effect on gross tumor-scale behavior using conventional methods that, for the most part, investigate isolated mechanisms. Prognosis of cancer patients suffering from highly invasive tumors, such as glioblastoma, is grim despite advances in surgical and chemotherapeutic treatment, since not all tumor cells can be removed or treated because of limited delineation between healthy and tumor tissue at the tumor border, which may lead to fatal recurrences (Kansal *et al.*, 2000a). In particular, mechanisms governing glioma invasion likely include intrinsic properties of cell proliferation, migration, and adhesion. Glioma cells have been experimentally shown to infiltrate and scatter throughout the entire central nervous system after a period of only seven days post-implantation (Chicoine & Silbergeld, 1995, Silbergeld & Chicoine, 1997, Swanson *et al.*, 2000). This might be one reason why current treatments that focus on surgery, radiation, and chemotherapy, while perhaps having an effect on primary bulk mass characteristics, may fail to extend survival time.

### A novel, in‐silico approach to cancer modeling

In this review/position paper we describe a multidisciplinary method integrating mathematical models with experimental (*in vitro* and *in vivo*) and clinical data. This methodology reflects an “engineering” approach that views tumor lesions as complex micro-structured materials, where three-dimensional tissue architecture (“morphology”) and dynamics are coupled in intricate, complex ways to cell phenotype, which in turn is influenced by factors in the microenvironment. Cellular and microenvironmental factors act both as tumor morphology regulators and as determinants of invasion potential by controlling the mechanisms of cancer cell proliferation and migration (Friedl & Wolf, 2003, Sierra, 2005, van Kempen *et al.*, 2003). In particular, recent experimental results demonstrate that interactions between cellular proliferation, adhesion, and other phenotypic properties are reflected in both tumor-host interface morphology and invasive characteristics of tumors (Tysnes *et al.*, 1996, Zutter *et al.*, 1995, Lah *et al.*, 2006, Kuiper *et al.*, 1998, Bernsen *et al.*, 1999, Bloemendal *et al.*, 1999, Steeg, 2003, Page *et al.*, 1987, Kunkel *et al.*, 2001, Seftor *et al.*, 2002, Lamszus *et al.*, 2003, Bello *et al.*, 2004, Friedl & Wolf, 2003). The goal is then to create computational (*in silico*) multiscale tools capable of predicting the complexity of cancer at multiple temporal and spatial resolutions, with the aim of supplementing diagnosis and treatment by helping plan more focused and effective therapy *via* surgical resection, standard chemotherapy, novel treatments (e.g., angiogenic, anti-invasive), or some combination of them. The tools would quantitatively examine the effect of tumor morphology regulators, which include tissue rigidity, density, adhesiveness, microenvironment gradients (e.g., oxygen, nutrient, growth factors), and the combinatorial effects of oncogenes (controlling cell proliferation, motility, and nutrient consumption) and tumor suppressor genes (controlling cell apoptosis and motility) on gross morphology. They would also define the degree of diffuse invasion of tumor cells peripheral to the tumor mass that may be beyond the detection of current non-invasive medical imaging techniques (Swanson *et al.*, 2000), or extrapolate tumor invasiveness and metastatic potential from its morphology in fixed tissue. In-silico model development is built upon continuum, discrete, and in particular cellular automata models (e.g., see Hogea *et al.*, 2006, Ayati *et al.*, 2006, Chaplain & Lolas, 2005, Garner *et al.*, 2005, Bru *et al.*, 2003, Jackson, 2004, Castro *et al.*, 2005, Painter, 2000, Khain *et al.*, 2005, Khain & Sander, 2006, Sander & Deisboeck, 2002, Boushaba *et al.*, 2006, Stein *et al.*, 2007; see also reviews Adam, 1996, Bellomo & Preziosi, 2000, Chaplain & Anderson, 2003, Friedman, 2004, Araujo & McElwain, 2003, Byrne *et al.*, 2006, Swanson *et al.*, 2003, Sanga *et al.*, 2006, Quaranta *et al.*, 2005, Nagy, 2005, Hatzikirou *et al.*, 2004, Frieboes et al., 2006b, Sinek *et al.*, 2006, Sanga *et al.*, 2007, Macklin & Lowengrub, 2007).

### Incorporation of patient data: predictive modeling

Multi-scale modeling quantifies the time- and space-dependent physics and chemistry (e.g., diffusion of substrates, mechanical forces exchanged among cells and with the matrix, molecular transport, receptor-ligand interactions, pharmacokinetics determinants) underlying tumor biological behavior. We envision that future simulators, building on the developments described herein, will operate as described in Figure 1. Initial conditions are obtained from patient data, such as MR (magnetic resonance) and CT (computed tomography) images and histopathology, which are used to set phenotypic and other model input parameters (e.g., proliferation rate). For example, CT image information would be translated voxel by voxel (using a computer program) to the coordinate system of the multi-scale model, e.g., a finite-element computational mesh discretizing the space occupied by a tumor and surrounding host tissue (Cristini et al., 2001; Zheng et al., 2005a,b; Anderson et al., 2005; Cristini and Tan, 2004). This input data would also include physical information about viable regions and cell density therein, necrosis, vasculature, blood flow, and other specific details from histopathology (Bearer and Cristini, MS submitted, Frieboes et al., 2007). The computer model then calculates local tumor growth, angiogenesis, and response to treatment under various conditions (Zheng et al., 2005a, Wise et al., MS submitted, Frieboes et al., MS submitted, Frieboes et al., 2007) by solving in time and space conservation (e.g., diffusion equation) and other laws at the tissue scale. These laws are linked to the cell molecular biology by functional relationships and parameters informed by the biopsy data. Computational solutions are obtained using finite elements and other numerical techniques (e.g., Zheng et al., 2005a; Frieboes et al., 2007). Additional patient data obtained from tissue culture, gene arrays, proteomic profiling, and other means would sharpen these parameter estimations (Frieboes et al., 2006a) in order to enable accurate prediction of behavior. Further details on the parameter estimation procedure are described below and in references cited.

## MULTI‐SCALE MODELING AND SIMULATION OF TUMOR MORPHOLOGY AND INVASION

### Model development goals and choices

The progression of tumor lesions cannot be completely characterized by studying effects in isolated cells since it is known that the forces and mechanisms regulating the motion of individual cells converge and synchronize into the collective, organized, structural motion of a whole body or cluster of cells (“Functional Collective Cell-Migration Units,” FCCMU) that often precedes the onset of epithelial-mesenchymal and other phenotypic transitions leading to individual cell shedding from a tumor and eventually to metastasis (Friedl & Wolf, 2003). Both individual and collective migration modes are regulated (and complicated) by multifaceted interactions among tumor cells, stroma and tumor microenvironment (Sierra, 2005, van Kempen *et al.*, 2003).

While “discrete” in-silico models (e.g., DiMilla *et al.*, 1991, Dickinson & Tranquillo *et al.*, 1993, Kansal *et al.*, 2000a, 2000b, Patel *et al.*, 2001, Ferreira *et al.*, 2002, Turner & Sherratt, 2002, Leyrat *et al.*, 2003, dos Reis *et al.*, 2003, Anderson, 2005) are able to capture individual cell migration and easily incorporate biological rules, such as cell-cell & cell-medium interactions and motion due to chemotaxis and haptotaxis, they are limited to relatively small numbers of cells due to computational cost, among the other deficiencies and over-simplifications introduced by the discrete approach. In contrast, “continuum” models (e.g., Byrne & Chaplain, 1995a, 1995b, 1996a, 1996b, 1997, Bellomo & Preziosi, 2000, Cristini *et al.*, 2003, 2005, Macklin & Lowengrub, 2005, Frieboes *et al.*, 2006a, Li *et al.*, 2007, Macklin & Lowengrub, 2007), describing tissue matter as a continuum medium rather that discrete individual cells, capture the collective motion of FCCMUs with less computational expense.

The fact that collective migration is often associated with relatively higher degrees of cell differentiation (Friedl & Wolf, 2003) than for the case of single-cell migration suggests that molecular mechanisms are relatively more robust across a tumor cell population. Thus, the multitude of cells can be averaged out and re-described as a single multi-cellular FCCMU unit obeying deterministic dynamics laws, while still employing mathematical models of single-cell migration when needed, e.g., to describe epithelial-mesenchymal transitions (Friedl & Wolf, 2003). Moreover, the domain size of realistic discrete simulations is limited to sub-millimeter-size in-vitro tumor spheroids or in-vivo patches of tumor tissue. We propose that discrete models of cell proliferation and migration should be coupled to continuum models of FCCMU to extend the computational capability to realistic, cm-size three-dimensional tumor lesions as defined and described in the following. A *hybrid*, multi-scale modeling methodology (Cristini *et al.*, 2006) that links continuum (i.e., tissue-scale) with discrete (i.e., cellular-scale) formulations with appropriate functional relationships of cell adhesion and migration due to environmental conditions should provide, over the next decade, a more comprehensive understanding of the molecular basis of diversity and adaptation of cell migration, thus more efficiently and accurately predicting invasion potential from real-time tumor morphology.

This approach has the advantage that well-established engineering methods and analyses of morphology can be applied (e.g., based on continuum methods when possible). Experimental measurements, computer simulations and morphologic stability analyses can be used to study, in detail, microenvironment transport processes (e.g., of oxygen, nutrients, chemokines, growth factors), cell motion and proliferation, signaling pathways and molecular phenomena regulating cell cycling, cell-cell communications and expression of cell adhesion molecules and matrix degrading enzymes. For example, the link between hypoxic gradients and invasion, and between normoxic conditions and compact non-infiltrative tumor morphologies, can thus be explained “by exploiting the ability of mathematics to model physical and biological systems in ways that enable prediction and control” (quoting John Lowengrub, Chair, Mathematics, UC Irvine).

### Significance of in‐silico modeling: a novel hypothesis‐generation tool

The computational models described in this paper represent important steps in generating hypotheses that postulate functional relationships linking the effect of molecular/cellular changes to tumor-scale morphology and invasiveness. By directly solving the mathematical equations describing underlying physical and chemical processes occurring within tumors, the complex biology of tumor behavior and the often hidden mechanisms of growth and invasion automatically are unveiled and can be accurately quantified in virtual, in-silico simulation space. Examples of novel hypotheses generated from simulations studies and tested in experiments will be provided in the following. Although these types of models are not multi-scale *per se*, parameters characterizing cell response to substrate concentration can be interpreted as representing underlying biochemistry and molecular biology driving tumor-scale dynamics, specifically an invasive phenotype. However, modeling of tumor behavior and cell microenvironment remains a challenge. Existing mathematical models are only capable, in general, of recapitulating a *posteriori* the highly variable empirical observations of morphology, once appropriate *phenomenological* parameters that do not incorporate direct molecular-scale description have been “fitted” to the experiments.

Here we propose that the next decade of investigation should focus on the task of developing predictive multi-scale models (e.g., see Figure 2) that incorporate new, functional relationships among macro-scale parameters characterizing differences in, and transitions among, cellular patterns, and variations in the molecular repertoire used by tumor cells to regulate proliferation, adhesion and other phenotypic properties.

**...**

This methodology is expected to improve current modeling efforts because a multi-scale approach connects previous work focused on specific scales (e.g., molecular) and processes (e.g., gene transformation), affording the possibility to go beyond the current reductionist picture of tumor invasion and migration (Friedl & Wolf, 2003, Keller *et al.*, 2006, Sierra, 2005, van Kempen *et al.*, 2003, Wolf & Friedl, 2006, Kopfstein & Christofori, 2006, Yamaguchi *et al.*, 2005, Elvin *et al.*, 2005, Sahai, 2005, Friedl *et al.*, 2004, Friedl, 2004, Condeelis *et al.*, 2005, Ridley *et al.*, 2003, Jones *et al.*, 2000) by providing a platform to study cancer as a *system*. Next, we describe biologically founded, in-silico modeling efforts of tumor progression (e.g., Zheng *et al.*, 2005a, Sanga *et al.*, 2006, Bearer and Cristini, MS submitted; Frieboes et al., 2007; Wise et al., MS submitted; Frieboes et al., MS submitted) relying on known characteristics of tumor behavior (Cristini *et al.*, 2003, Zheng *et al.*, 2005a, Anderson *et al.*, 2000, Cristini *et al.*, 2005, Sinek *et al.*, 2004, Frieboes *et al.*, 2006a, Macklin & Lowengrub, 2007) to predict the combination of variables most likely driving progression towards invasiveness.

This effort builds on an approach (e.g., Cristini *et al.*, 2003, Zheng *et al.*, 2005a, Cristini *et al.*, 2005, Frieboes *et al.*, 2006a, Li *et al.*, 2007) that includes reformulations and generalizations of mathematical models (Greenspan, 1976, Byrne & Chaplain, 1996a, 1996b, Adam, 1996, Chaplain, 1996, Lowengrub & Truskinovsky, 1998, Leo et al., 1998, Lee et al., 2002, Macklin & Lowengrub, 2005 & 2007, Garcke et al., 2004, Jacqmin, 1999, Anderson et al., 1998, Bellomo & Preziosi, 2000, Ambrosi & Preziosi, 2002, Byrne & Preziosi, 2003, Chaplain *et al.*, 2006, Ambrosi & Guana, 2006, Chaplain & Anderson, 2003, see also Jackson & Byrne, 2002), solved numerically using state-of-the-art algorithms and techniques (Zheng *et al.*, 2005a and 2005b, Wise *et al.*, 2005, Kim *et al.*, 2004a, Kim *et al.*, 2004b, Wise *et al.*, 2004, Cristini et al., 2001, Berger & Colella, 1989, Brandt, 1977; Wise et al., manuscript submitted). Figure 2 describes the main component modules (Vasculature, Tumor, and Genotype) of this model along with equations that represent mathematically the relevant biological parameters.

### Determination of functional relationships and parameter values

The specific process of multi-scale model “training” relies on conducting experiments (Figure 2) in which molecular factors are measured in the cell and the environment, and outcome of tumor growth (e.g., morphology, shape, extent of vascularization and invasion) is correlated with expression of these factors. This data allows estimation of the mathematical model parameters and functional relationships by perturbing these parameters and comparing the resulting simulation predictions of morphology against direct measurements, thus leading, through an iterative process that reveals deficiencies in modeling choices and triggers refinements in the relationships introduced, to a validated mathematical model with calibrated constitutive parameters. By virtue of its predictive power, this approach (Cristini *et al.*, 2006) can help plan new experiments by identifying parameter regimes of noteworthy behavior–regimes that might otherwise be time-consuming and costly to discover by systematic experimentation.

Theoretical (e.g., Frieboes *et al.*, 2006a) and experimental work (e.g., Gatenby *et al.*, 2006, Frieboes *et al.*, 2006a, and reviews by Chomyak & Sidorenko, 2001, Kim, 2005, Mueller-Klieser, 2000 and references therein) can be used to develop and test functional relationships, and to estimate the microphysical parameter values of a multi-scale *in silico* model. Examples (Cristini *et al.*, 2006) of these functional relationships include those between expression of membrane transport proteins (e.g., glucose transporter-1 and Na/H exchanger) and hypoxia/proliferation; between extracellular matrix macromolecules (e.g. tubulin, actin), haptotaxis and chemotaxis; and between cell-cell adhesion parameters as an increasing function of oxygenation, e.g., from recent measurements by Robert Gatenby (personal communication) showing a gradient of cell-adhesion molecules (E-cadherins) opposed to hypoxia.

*In vivo* animal models (e.g., dorsal wound chamber by Gatenby *et al.*, 2006) can supply detailed measurements of angiogenesis and blood flow, which provide additional constraints to the *in silico* model to determine parameter values associated with a developing neovasculature. Computational models of angiogenesis (Levine *et al.*, 2002, Plank & Sleeman, 2003, 2004, Sun *et al.*, 2005, Stephanou *et al.*, 2005, McDougall *et al.*, 2006) (Figure 2) can account for endothelial cell chemotactic and haptotactic movement, proliferation, development and remodeling of capillaries and the flow of blood through the local pressure and other constraints. Under *in vivo* conditions, additional measurements can be performed to determine pH and pO_{2} gradients that provide further functional constraints on the parameters relating to proliferation and cellular adaptation to hypoxia (Cristini *et al.*, 2006). Finally, *in vivo* measurements of matrix degradation at the tumor-host boundary due to acidosis and proteases can provide parameter values for the invasion component of the *in silico* model (Cristini *et al.*, 2006).

## COMPUTATIONAL MODELING: A FRAMEWORK FOR LINKING PHENOTYPE, MORPHOLOGY, AND CANCER INVASION

Extensive mathematical modeling has produced preliminary quantifications of the links between invasive, malignant cell phenotypes and tumor-scale morphologies. These involve cell-cell and cell-extracellular matrix (ECM) interactions, cell motility, micro vessel density and acidosis, and local concentration of cell substrates (Mareel & Leroy, 2003). We illustrate representative discrete models in each of these areas. We then review recent mathematical and computational studies of a continuum FCMU model developed in our group. Both discrete and continuum models are based on conservation formulations such as described in Figure 2.

### Effects of cell‐cell and cell‐matrix interactions

The effects of tumor cell and environment heterogeneity on the overall tumor morphology were recently studied (Anderson, 2005) using a model that captured spatial distribution of oxygen, matrix-degrading enzymes, and matrix molecules in the tumor microenvironment, as well as tumor cell properties, e.g., migration and proliferation. Results support the notion that tumor cell-matrix interactions ultimately control tumor shape by driving tumor cell migration *via* haptotaxis and chemotaxis towards fingering, invasive tumor morphologies (Figure 3 A and B).

**...**

However, degradation of ECM, specifically surrounding the tumor boundary, may have a stronger influence on invasion than cell-cell adhesion. Using a derivation of a Potts model (Wu *et al.*, 1982) that incorporates homotypic and heterotypic adhesion as well as secretion of proteolytic enzymes that drive haptotaxis along their gradient, a more quantitative perspective into the role of cell adhesion and proliferation in promoting an invasive phenotype was obtained (Turner & Sherratt, 2002). The model assumes genetic mutations affect cellular adhesiveness, secretion of matrix degrading enzymes, the ability to undergo taxis along gradients, and proliferation rate (Turner and Sherratt, 2002, Stetler-Stevenson *et al.*, 1993). Using the maximum host tissue penetration as an index of invasiveness, simulation results predict that increases in the secretion of matrix degrading enzymes in synergy with increases in cell proliferation and haptotaxis can produce fingering morphologies at the tumor-host interface as cells adhere to the ECM and spread into host tissue. The model hypothesizes a functional relationship between proliferation rates and changes in adhesiveness based on experimental evidence (Huang & Ingber, 1999).

The notion that formation of fingering protuberances at the tumor-host boundary is primarily due to an intrinsic physical property termed rigidity of the host environment to resist tumor growth has also been computationally examined (dos Reis *et al.*, 2003). Low rigidity allows a tumor to expand through the host environment resulting in a well-defined tumor-parenchyma interface, whereas higher rigidity forces a tumor to grow by invading the host tissue resulting in a fingering morphology (Figure 3, C and D), as predicted by simulation results. In addition, cell adhesion changes growth patterns from fractal morphologies at the tumor-host interface to compact shapes.

### Effects of cell motility

Computational investigations of the invasiveness of high, medium, and low-grade gliomas illustrate that the ratio of tumor growth rate and cell motility can quantify the invasive nature of a tumor (Swanson *et al.*, 2000). Specifically, this ratio might be useful in predicting a tumor’s invasive and metastatic potential; high proliferation rates and low motility correspond to lower grade tumors with less invasive potential whereas low proliferation rates and high motility correspond to higher-grade tumors with more invasive potential.

In contrast, a 3D cellular automaton model of glioblastoma capable of predicting tumor growth according to four microscopic parameters (probability of division, necrotic thickness, proliferative thickness, and maximum tumor extent) successfully predicted tumor-scale dynamics of a test case for untreated glioblastoma progression compiled from medical literature; simulations reproduced data such as lesion radius, cell number, growth fraction, necrotic fraction, and volume doubling time at particular time points (Kansal *et al.*, 2000a, 2000b). Human glioblastoma patients have a median survival time of 8 months from diagnosis (Kansal *et al.*, 2000a), which these models (Kansal *et al.*, 2000a, 2000b) accurately predict using primary tumor volume as an indicator for survival.

### Effects of micro vessel density and acidosis

The potential importance of micro vessel (MV) density and acidosis in promoting tumor growth and invasion has been demonstrated through recent computational models (Patel *et al.*, 2001, Gatenby & Gawlinski, 1996, Gatenby & Gawlinski, 2003, Gatenby *et al.*, 2006). Simulations show that the production rate of H^{+} ions by cancer cells, due to their increased dependence on anaerobic glucose metabolism, is linked to an optimal micro vessel (MV) density such that the microenvironment favors tumor cells over normal cells, hence promoting growth and invasion (Patel *et al.*, 2001). MV density below this optimal value produces an environment too acidic even for cancer cells, while MV density above the optimum reduces or even completely negates the advantage enjoyed by cancer cells over normal cells in an acidic environment, thus inhibiting overall tumor growth and invasion by promoting nutrient competition.

Depending on the metabolic phenotype, various tumor morphologies can be predicted including invasive, fingering protrusions seen in experiments and with other *in silico* models. This and other modeling and experimental work further supports the acid-mediated tumor invasion hypothesis (Patel *et al.*, 2001, Gatenby & Gawlinski, 1996, Gatenby & Gawlinski, 2003, Gatenby *et al.*, 2006), illustrating the potential importance of MV density in driving pH gradients in the microenvironment and associated tumor-scale behavior. Such microenvironmental factors, in addition to cellular dynamics, are thus quantitatively linked to tumor-scale morphology.

### Effects of cell substrate concentration

Competition for nutrient and oxygen amongst normal and cancer cells, in addition to cell proliferation, motility, death, and secretion of matrix degrading enzymes, may be an important factor driving tumor invasion. Using a cellular-automaton model (Ferreira *et al.*, 2002), cell dynamics were described where at each time step, a cell (of type normal, cancer, or necrotic tumor) has equal probability of dividing, migrating, or undergoing necrosis; each action is governed by the local substrate concentration. Cells are modeled to release a series of enzymes that progressively degrade the ECM, thus providing more space for tumor cells to invade. In Figure 3, E and F, fingering morphologies are predicted by the model (and observed) as a result of high proliferation rates demanding large amounts of substrates. Predicted tumor morphology remains compact in situations of high nutrient supply and low cell consumption, while cell clusters expressing a phenotype that increases nutrient consumption exhibit thinner “fingers.”

### Continuum‐based parameter‐sensitivity studies of FCCMU

#### Morphologic instability as a mechanism of tumor invasion

The current conceptual framework of continuum FCCMU models is based on reaction–diffusion formulations (Figure 2). Accordingly, tumor morphologic “stability” is regulated by the competition of pro- and anti-migratory/proliferative factors. When the former prevail, complex, unstable FCCMU patterns can develop (Cristini *et al.*, 2003; Li *et al.* 2007). The power of this approach is that it is based on a physical mechanism that can account for the various invasive morphologies observed, and is thus potentially predictive of tumor growth. This mathematical analysis of morphologic stability has suggested that tumor tissue dynamics is regulated by two dimensionless parameters: the parameter *G* quantifies the competition between local tumor mass growth due to proliferation, and cell adhesion that tends to minimize tumor surface area and thus maintain compact nearly spherical tumors; the parameter *A* quantifies tumor mass shrinkage due to cell death (these parameters are obtained from some of those listed in Figure 2 using dimensional analysis; for the sake of simplicity, the associated cumbersome formulation is not reported here: see Cristini *et al.*, 2003, Li et al. 2007). During glioblastoma tumor growth *in vitro*, cell proliferation is observed in a viable region where nutrients, oxygen, and growth factor levels are adequate, and cell death and necrosis in the inner regions where diffusion limitations prevent these substances from being present in adequate levels (Frieboes *et al.*, 2006a). *In the presence of these substrate gradients*, morphology can be “unstable”, i.e., invasive, when cell adhesion is weak (large *G*). In contrast for small *G*, spheroid morphology is “stabilized” (i.e., spherical or nearly spherical) by cell adhesion (Cristini *et al.*, 2003).

This is illustrated in a morphologic stability diagram (Cristini *et al.*, 2003, Frieboes *et al.*, 2006a), Figure 4, A. The *G*-curves divide the parameter space into stable (left) and unstable (right) regions. The *G*-curves are more shifted to the left as cell adhesion decreases (higher *G*), thus reducing the range of sizes of tumors that will be morphologically stable. The *in silico* model parameters *A* and *G* were calibrated using data from “stable” spheroids (shaded area) until agreement was obtained between the model calculations of morphology and growth and *in vitro* measures of tumor growth curves and thickness of the viable rim of cells. The model was then tested by predicting morphology stability conditions for an independent set of experiments (filled symbols) where the cell medium levels of growth factors and glucose were changed over a wider range to manipulate glioblastoma cell proliferation and adhesion (Frieboes *et al.*, 2006a).

*et al.*(2006a) with permission from the American

**...**

A remarkable result of this study was that the *in silico* model was capable of predicting growth and invasion of tumors from experiments that were not used for model training, thus successfully testing, under relatively simple highly controllable *in vitro* conditions, the hypothesized phenomenological relationships of adhesion and proliferation with substrate levels and their effects on tissue-scale growth and morphology (e.g., see Figure 2). Computer simulations and experimental observations of “unstable” spheroids that develop protrusions and detachment of cell clusters are shown in Figure 4, B. As described below, these infiltrative morphologies are also universally observed in tumors in vivo and in data from patients.

#### Clinical relevance

Morphologic instability during tumor growth is predicted to result from genomic changes that produce variations in sub-tumor clonal expansion, rates of mitosis and apoptosis, oxygen consumption, and diffusion gradients. This physical hypothesis is corroborated by in vitro and *in vivo* observations (e.g., Rubenstein *et al.*, 2000, Kunkel *et al.*, 2001, Lamszus *et al.*, 2003, Bello *et al.*, 2004, Frieboes *et al.*, 2006a) and by patient data. In a study of several clinical samples of glioblastoma multiforme from multiple patients (Figure 5), histology reveals protruding fronts of cells in collective motion away from a necrotic area into an area of the host brain where neo-vascularization is present, thus following substrate gradients (Bearer and Cristini, MS submitted; Frieboes et al., 2007). This data strongly resembles the morphology of the tumor boundary predicted by computer simulation (Bearer and Cristini, MS submitted; Frieboes et al., 2007) and by the *in vitro* experiments described above (Frieboes et al., 2006a). These infiltrative shapes were consistently observed in high-grade gliomas, although their size may vary.

*reproduced with permission from Elsevier*) showing tumor (bottom) pushing into more normal

**...**

#### Effects of phenotype on morphology and growth

In Figure 6, A, different morphologies predicted by a continuum FCCMU model (Zheng *et al.*, 2005a, Cristini *et al.*, 2005) are shown, starting from the same initial condition of a spherical tumor. The model predicts that when cell taxis, but no proliferation is present **(a)**, cells tend to align in chains or strands. When cell proliferation is significant **(b)**, more “bulb like” protrusions form and detach into the host. These are also predicted to be more hypoxic. In all cases, these complex morphologic patterns developed because cell adhesion parameters were set very low ("morphologic instability,” Cristini *et al.*, 2003). Corresponding structures observed after inducing hypoxia **(c)** *in vitro* (proliferation was inhibited) (Pennacchietti *et al.*, 2003) and (**d**) *in vivo* (Rubenstein *et al.*, 2000) are reported for comparison. The underlying molecular phenomena (Friedl & Wolf, 2003) responsible for the prevalence of one over another morphology, and for the spatial frequency of finger-like protrusions originating from a primary tumor, are captured by (phenomenological) model parameters describing proliferation and taxis and the associated convective cell fluxes on one side, and cell-adhesion forces on the other (Cristini *et al.*, 2003).

**...**

This model was further used to predict changes in the system dynamics following a range of possible perturbations of parameters related to cell-adhesion forces and to oxygen distribution in the environment, with the goal of providing suggestions for novel treatment protocols aimed at restoring normoxia and thus preventing “unstable” morphologies (Cristini *et al.*, 2005, Frieboes *et al.*, 2006a). Since these phenotypic and environmental parameters are also associated with invasion, this perturbation study provided preliminary quantification of the effects of anti-invasive and “vasculature-normalizing” anti-angiogenic therapeutic strategies that alter the balance of morphology-stabilizing and -destabilizing micro-environmental and molecular processes. In particular, this study helped explain the undesirable effects on morphology following current anti-angiogenic therapy due to exacerbation of micro-environmental hypoxic gradients and enhancement of cell migration (as reviewed above).

In Figure 6, B, case **(a)** corresponds to sufficiently high cell adhesion so that the simulated tumor growth is morphologically “stable”. Due to hypoxic gradients, necrotic regions have formed where concentrations are inadequate, leading to a diffusion-limited tumor size. Angiogenic factors (not shown) emanate from the peri-necrotic regions and diffuse outward, reaching pre-existing vessels and triggering neo-vascularization of the lesion. Even after angiogenesis, the model predicts that lesion **(a)** will maintain a compact shape because of high cell adhesion.

Case **(b)** corresponds to low cell adhesion, in which the tumor experiences morphological instability driven by hypoxic gradients as it progresses (Cristini *et al.*, 2003, Zheng *et al.*, 2005a, Byrne & Chaplain, 1997, Macklin & Lowengrub, 2007). Cell adhesion is insufficient to maintain proliferating cells together. The lesion breaks up into fragments, or detached cell clusters (Friedl & Wolf, 2003), moving away following outward gradients of nutrient and oxygen concentration (Zheng *et al.*, 2005a, Macklin & Lowengrub, 2007). The model predicts that anti-invasive therapy enforces a morphology transition from **(b)** to **(a)** by increasing cell adhesion.

Case **(c)** corresponds to low cell adhesion (as in **(b)**), but with a more spatially uniform distribution of vessels. The simulation predicts that this “vascular normalization” would lead to reduced oxygen gradients, and hence to suppression of instability and to clustering of cells into a more compact tumor 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 (Jain, 1990, 2001, 2005). In contrast, after anti-angiogenic therapy (**(c)** to **(b)**), increased scattering of tumor cell clusters in response to hypoxia is predicted, as documented *in vivo* by Bello *et al.* (2004) and by others. Remarkably, the simulations also predict that in this case some tumor cell clusters tend to co-opt the vasculature to maximize nutrient uptake, as documented previously *in vivo* (Kunkel *et al.*, 2001, Lamszus *et al.*, 2003, Rubenstein *et al.*, 2000).

Figure 7 shows a summary of some of the biology revealed by the predictive model presented here (Zheng et al., 2005, Sinek et al., 2004 and in press, Cristini et al., 2005, Frieboes et al., 2006 and 2007, Sanga et al., 2006) under the categories of Tumor, Microenvironment, Treatment Response, and Vasculature, including gross tumor morphology in 3-D **(A)**, gradients of cell substrates **(B)**, tissue fragmentation in response to chemotherapy involving large nanoparticles and adjuvant anti-angiogenic therapy **(C)**, and tumor vasculature with both conducting and non-conducting vessels (**D**).

## CONCLUSIONS AND FUTURE WORK

The research direction we envision focuses on the development and application to tumor biology of quantitative methods traditionally confined to engineering and the physical sciences. Indeed, it is clear that such complex biological systems dominated by large numbers of processes and highly complex dynamics are very difficult to approach by experimental methods alone. They can typically be understood only by using appropriate mathematical models and sophisticated computer simulations complementary to experimental investigations. In the innovative and powerful multidisciplinary approach reviewed here, mathematical and computational modeling completes the circle of discovery: laboratory experiments provide data that, in turn, informs the construction of a mathematical model that can then predict behavior and guide the design of future experiments to test these predictions.

This multi-scale approach captures tumor progression by taking into account ongoing molecular and cellular scale events (Martinez-Zaguilan *et al.*, 1996, Schlappack *et al.*, 1991, Rofstad *et al.*, 1999, Gatenby *et al.*, 2006, Jensen, 2006). One of the key links established in a more quantitative manner is that mutations drive increased cellular uptake, which introduces perturbations in spatial gradients of oxygen and nutrient; these gradients enhance hypoxia and cause heterogeneous cell proliferation and migration leading towards diffusional shape instabilities. This supports the hypothesis that cellular and extra-cellular properties driving tumor growth and invasiveness also determine tumor morphology (Cristini *et al.*, 2005, Frieboes *et al.*, 2006a) and suggests that morphological characteristics including neo-vasculature and harmonic content of the tumor edge should serve as predictors of tumor growth (Cristini *et al.*, 2006).

Predictive modeling assumes that criteria and critical microphysical conditions for tumor invasion can be formulated in terms of physical laws linking tissue architecture and morphology to cell phenotype. Future multidisciplinary investigations should exploit the power of predictive modeling that allows observable properties of the tumor, such as its morphology in general and specifically the cell spatial arrangements at the tumor boundary, to be used to both understand the underlying cellular physiology and predict subsequent invasive behavior. We envision this research taking steps towards further establishing the dependence of tumor cell motion into surrounding host tissue on the balance between cell proliferation and adhesion, as well as perturbations caused by microenvironmental factors such as oxygen, nutrient, and H^{+} diffusion gradients. This will include the continuing application of mathematical and empirical methods to quantify the competition between gradient-related pro-invasion phenomena and molecular forces that govern proliferation and taxis, and forces opposing invasion through cell adhesion. In addition, a more detailed description of the complex *in vivo* environment, which better recapitulates the conditions of tumors in patients, would be valuable.

Currently, pathologic analysis is often limited to a set of morphological features that are rarely quantitatively assessed (the main quantitative factors are mitotic rates and size of invasive tumor “fingers”), and these measures differ depending on the types of tumor. “Degree of pleiomorphism” (variable phenotypes) is also used as a prognosticator, although this has no absolute quantitative definition and is subjective. Multi-scale modeling of cancer would allow predictions of cellular and molecular perturbations that alter invasiveness and are measured through changes in tumor morphology. This opens the possibility of designing novel individualized therapeutic strategies in which the microenvironment and cellular factors are manipulated with the aim of imposing compact tumor morphology by both decreasing invasiveness and promoting defined tumor margins—an outcome that would benefit cancer therapy by improving local tumor control through surgery or chemotherapy.

## Acknowledgements

We gratefully acknowledge John Lowengrub and Steven M. Wise (Mathematics, U.C. Irvine), and the reviewers for helpful comments and suggestions. We thank Ed Stopa at Rhode Island Hospital and the Columbia University Alzheimer's Brain Bank for archived human glioma specimens. Funding from NIH-NIGMS RO1 GM47368 and RO1-NS046810_(E.L.B.), from the National Science Foundation (V.C.) and National Cancer Institute (V.C.; R.G.).

## Footnotes

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