This is the first paper in a two-part series in which we develop, analyze, and simulate a diffuse interface continuum model of multispecies tumor growth and tumor-induced angiogenesis in two and three dimensions. Three-dimensional simulations of nonlinear tumor growth and neovascularization using this diffuse interface model were recently presented in Frieboes et al. [2007. Computer simulation of glioma growth and morphology. NeuroImage S59-S70], but that paper did not describe the details of the model or the numerical algorithm. This is done here. In this diffuse interface approach, sharp interfaces are replaced by narrow transition layers that arise due to differential adhesive forces among the cell species. Accordingly, a continuum model of adhesion is introduced. The model is thermodynamically consistent, is related to recently developed mixture models, and thus is capable of providing a detailed description of tumor progression. The model is well-posed and consists of fourth-order nonlinear advection-reaction-diffusion equations (of Cahn-Hilliard-type) for the cell species coupled with reaction-diffusion equations for the substrate components. We demonstrate analytically and numerically that when the diffuse interface thickness tends to zero, the system reduces to a classical sharp interface model. Using a new fully adaptive and nonlinear multigrid/finite difference method, the system is simulated efficiently. In this first paper, we present simulations of unstable avascular tumor growth in two and three dimensions and demonstrate that our techniques now make large-scale three-dimensional simulations of tumors with complex morphologies computationally feasible. In part II of this study, we will investigate multispecies tumor invasion, tumor-induced angiogenesis, and focus on the morphological instabilities that may underlie invasive phenotypes.

Despite major scientific, medical and technological advances over the last few decades, a cure for cancer remains elusive. The disease initiation is complex, and including initiation and avascular growth, onset of hypoxia and acidosis due to accumulation of cells beyond normal physiological conditions, inducement of angiogenesis from the surrounding vasculature, tumour vascularization and further growth, and invasion of surrounding tissue and metastasis. Although the focus historically has been to study these events through experimental and clinical observations, mathematical modelling and simulation that enable analysis at multiple time and spatial scales have also complemented these efforts. Here, we provide an overview of this multiscale modelling focusing on the growth phase of tumours and bypassing the initial stage of tumourigenesis. While we briefly review discrete modelling, our focus is on the continuum approach. We limit the scope further by considering models of tumour progression that do not distinguish tumour cells by their age. We also do not consider immune system interactions nor do we describe models of therapy. We do discuss hybrid-modelling frameworks, where the tumour tissue is modelled using both discrete (cell-scale) and continuum (tumour-scale) elements, thus connecting the micrometre to the centimetre tumour scale. We review recent examples that incorporate experimental data into model parameters. We show that recent mathematical modelling predicts that transport limitations of cell nutrients, oxygen and growth factors may result in cell death that leads to morphological instability, providing a mechanism for invasion via tumour fingering and fragmentation. These conditions induce selection pressure for cell survivability, and may lead to additional genetic mutations. Mathematical modelling further shows that parameters that control the tumour mass shape also control its ability to invade. Thus, tumour morphology may serve as a predictor of invasiveness and treatment prognosis.

The intracellular and extracellular dynamics that govern tumor growth and invasiveness in vivo remain poorly understood. Cell genotype and phenotype, and nutrient, oxygen, and growth factor concentrations are key variables. In previous work, using a reaction-diffusion mathematical model based on variables that directly describe tumor cell cycle and biology, we formulated the hypothesis that tumor morphology is determined by the competition between heterogeneous cell proliferation caused by spatial diffusion gradients, e.g., of cell nutrients, driving shape instability and invasive tumor morphologies, and stabilizing mechanical forces, e.g., cell-to-cell and cell-to-matrix adhesion. To test this hypothesis, we here obtain variable-based statistics for input to the mathematical model from in vitro human and rat glioblastoma cultures. A linear stability analysis of the model predicts that glioma spheroid morphology is marginally stable. In agreement with this prediction, for a range of variable values, unbounded growth of the tumor mass and invasion of the environment are observed in vitro. The mechanism of invasion is recursive subspheroid component development at the tumor viable rim and separation from the parent spheroid. Results of computer simulations of the mathematical model closely resemble the morphologies and spatial arrangement of tumor cells from the in vitro model. We propose that tumor morphogenesis in vivo may be a function of marginally stable environmental conditions caused by spatial variations in cell nutrients, oxygen, and growth factors, and that controlling these conditions by decreasing spatial gradients could benefit treatment outcomes, whereas current treatment, and especially antiangiogenic therapy, may trigger spatial heterogeneity (e.g., local hypoxia), thus causing invasive instability. (Cancer Res 2006; 66(3): 1597-604)

We extend the diffuse interface model developed in Wise et al. (2008) to study nonlinear tumor growth in 3D. Extensions include the tracking of multiple viable cell species populations through a continuum diffuse-interface approach, onset and aging of discrete tumor vessels through angiogenesis, and incorporation of individual cell movement using a hybrid continuum-discrete approach. We investigate disease progression as a function of cellular-scale parameters such as proliferation and oxygen/nutrient uptake rates. We find that heterogeneity in the physiologically complex tumor microenvironment, caused by non-uniform distribution of oxygen, cell nutrients, and metabolites, as well as phenotypic changes affecting cellular-scale parameters, can be quantitatively linked to the tumor macro-scale as a mechanism that promotes morphological instability. This instability leads to invasion through tumor infiltration of surrounding healthy tissue. Models that employ a biologically-founded, multiscale approach, as illustrated in this work, could help to quantitatively link the critical effect of heterogeneity in the tumor microenvironment with clinically observed tumor growth and invasion. Using patient tumor-specific parameter values, this approach may provide a predictive tool to characterize the complex in vivo tumor physiological characteristics and clinical response, and thus lead to improved treatment modalities and prognosis.

The flow of interstitial fluid and the associated interstitial fluid pressure (IFP) in solid tumors and surrounding host tissues have been identified as critical elements in cancer growth and vascularization. Both experimental and theoretical studies have shown that tumors may present elevated IFP, which can be a formidable physical barrier for delivery of cell nutrients and small molecules into the tumor. Elevated IFP may also exacerbate gradients of biochemical signals such as angiogenic factors released by tumors into the surrounding tissues. These studies have helped to understand both biochemical signaling and treatment prognosis. Building upon previous work, here we develop a vascular tumor growth model by coupling a continuous growth model with a discrete angiogenesis model. We include fluid/oxygen extravasation as well as a continuous lymphatic field, and study the micro-environmental fluid dynamics and their effect on tumor growth by accounting for blood flow, transcapillary fluid flux, interstitial fluid flow, and lymphatic drainage. We thus elucidate further the non-trivial relationship between the key elements contributing to the effects of interstitial pressure in solid tumors. In particular, we study the effect of IFP on oxygen extravasation and show that small blood/lymphatic vessel resistance and collapse may contribute to lower transcapillary fluid/oxygen flux, thus decreasing the rate of tumor growth. We also investigate the effect of tumor vascular pathologies, including elevated vascular and interstitial hydraulic conductivities inside the tumor as well as diminished osmotic pressure differences, on the fluid flow across the tumor capillary bed, the lymphatic drainage, and the IFP. Our results reveal that elevated interstitial hydraulic conductivity together with poor lymphatic function is the root cause of the development of plateau profiles of the IFP in the tumor, which have been observed in experiments, and contributes to a more uniform distribution of oxygen, solid tumor pressure and a broad-based collapse of the tumor lymphatics. We also find that the rate that IFF is fluxed into the lymphatics and host tissue is largely controlled by an elevated vascular hydraulic conductivity in the tumor. We discuss the implications of these results on microenvironmental transport barriers, and the tumor invasive and metastatic potential. Our results suggest the possibility of developing strategies of targeting tumor cells based on the cues in the interstitial fluid.

Despite major advances in the study of glioma, the quantitative links between intra-tumor molecular/cellular properties, clinically observable properties such as morphology, and critical tumor behaviors such as growth and invasiveness remain unclear, hampering more effective coupling of tumor physical characteristics with implications for prognosis and therapy. Although molecular biology, histopathology, and radiological imaging are employed in this endeavor, studies are severely challenged by the multitude of different physical scales involved in tumor growth, i.e., from molecular nanoscale to cell microscale and finally to tissue centimeter scale. Consequently, it is often difficult to determine the underlying dynamics across dimensions. New techniques are needed to tackle these issues. Here, we address this multi-scalar problem by employing a novel predictive three-dimensional mathematical and computational model based on first-principle equations (conservation laws of physics) that describe mathematically the diffusion of cell substrates and other processes determining tumor mass growth and invasion. The model uses conserved variables to represent known determinants of glioma behavior, e.g., cell density and oxygen concentration, as well as biological functional relationships and parameters linking phenomena at different scales whose specific forms and values are hypothesized and calculated based on in vitro and in vivo experiments and from histopathology of tissue specimens from human gliomas. This model enables correlation of glioma morphology to tumor growth by quantifying interdependence of tumor mass on the microenvironment (e.g., hypoxia, tissue disruption) and on the cellular phenotypes (e.g., mitosis and apoptosis rates, cell adhesion strength). Once functional relationships between variables and associated parameter values have been informed, e.g., from histopathology or intra-operative analysis, this model can be used for disease diagnosis/prognosis, hypothesis testing, and to guide surgery and therapy. In particular, this tool identifies and quantifies the effects of vascularization and other cell-scale glioma morphological characteristics as predictors of tumor-scale growth and invasion.

Purpose: A solidt umor embedded in host tissue is a three-dimensional arrangement of cells and extracellular matrix that acts as a sink of oxygen andce ll nutrients, thus establishing diffusional gradients. This and variations in vascular density and blood flow typically produce intratumoral regions of hypoxia andac idosis, and may result in spatially heterogeneous cell proliferation and migration.Here,we formulate the hypothesis that through thesemechanisms,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 usedc omputer simulations and in vitro experiments. Results:We provide evidence that morphologic instability could be suppressed in vivo by spatially homogeneous oxygen andn utrient 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 andn utrient gradients. In contrast, antiangiogenic therapy, by increasing microenvironmental heterogeneity, may promotemorphologic instability, leading to invasive patterns even under conditions inwhich 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 inmultiple clinical trials.We propose that antiangiogenic strategies will be more consistently successful when aimedat ‘‘normalizing’’ the vasculature andw hen combinedw ith therapies that increase cell adhesion so that morphologic instability is suppresseda nd compact, noninvasive tumormorphologies are enforced

Vascular targeting of malignant tissues with systemically injected nanoparticles (NPs) holds promise in molecular imaging and anti-angiogenic therapies. Here, a computational model is presented to predict the development of tumor neovasculature over time and the specific, vascular accumulation of blood-borne NPs. A multidimensional tumor-growth model is integrated with a mesoscale formulation for the NP adhesion to blood vessel walls. The fraction of injected NPs depositing within the diseased vasculature and their spatial distribution is computed as a function of tumor stage, from 0 to day 24 post-tumor inception. As the malignant mass grows in size, average blood flow and shear rates increase within the tumor neovasculature, reaching values comparable with those measured in healthy, pre-existing vessels already at 10 days. The NP vascular affinity, interpreted as the likelihood for a blood-borne NP to firmly adhere to the vessel walls, is a fundamental parameter in this analysis and depends on NP size and ligand density, and vascular receptor expression. For high vascular affinities, NPs tend to accumulate mostly at the inlet tumor vessels leaving the inner and outer vasculature depleted of NPs. For low vascular affinities, NPs distribute quite uniformly intra-tumorally but exhibit low accumulation doses. It is shown that an optimal vascular affinity can be identified providing the proper balance between accumulation dose and uniform spatial distribution of the NPs. This balance depends on the stage of tumor development (vascularity and endothelial receptor expression) and the NP properties (size, ligand density and ligand-receptor molecular affinity). Also, it is demonstrated that for insufficiently developed vascular networks, NPs are transported preferentially through the healthy, pre-existing vessels, thus bypassing the tumor mass. The computational tool described here can effectively select an optimal NP formulation presenting high accumulation doses and uniform spatial intra-tumor distributions as a function of the development stage of the malignancy.