Novel Tumor Growth Model

Interdisciplinary Development of a Computational Tumor Model

This work started as a result of a small grant from the National Institutes of Health Grant CA 84509. This was a joint grant between the Harvard Medical School and Princeton University.

Currently, the dynamics of malignant brain tumor growth is a medical mystery. The incidence of primary malignant brain tumors is already 8/100,000 persons per year and is still increasing. The vast majority consists of high-grade malignant tumors such as glioblastoma multiforme (GBM) (see Figure 1). In spite of aggressive conventional and advanced treatments, the prognosis remains uniformly fatal with a median survival time for patients with GBM of 8 months.

The rapid growth and resilience of tumors make it difficult to believe that they behave as random, disorganized and diffuse cell masses and suggest that they are emerging opportunistic systems, that not only adapt to their environment but also change their environment for survival purposes. If this hypothesis holds true, a growing tumor must be investigated and treated as a self-organizing complex dynamical system. This cannot be done with currently available in vitro/in vivo models or common modeling approaches.

MRI of a real tumor A simulated tumor

Figure 1. Top panel: The tumor is located in the upper left portion of the image. The white rim around the tumor is composed of highly active cells, corresponding to the red rim in the right panel. Bottom panel: The outer, red rim corresponds to cells that are dividing rapidly. The middle, yellow region are cells that are alive but are not dividing. The innermost, black region are cells that are necrotic (dead).

The aim of the project is to show that we can model the growth (proliferation and invasion) of brain tumors using concepts from statistical physics, materials science and dynamical systems, as well as data from novel oncological experiments. We expect the work not only to increase our understanding of tumorigenesis, but to provide insight into novel ways to treat malignant brain tumors.

We have developed a novel and versatile three-dimensional cellular automaton model of brain tumor growth [1]. We showed that macroscopic tumor behavior can be realistically modeled using microscopic parameters. Using only four parameters, this model simulates Gompertzian growth for a tumor growing over nearly three orders of magnitude in radius. It also predicts the composition and dynamics of the tumor at selected time points in agreement with medical literature. We also demonstrated the flexibility of the model by showing the emergence, and eventual dominance, of a second tumor clone with a different genotype. The model incorporates several important and novel features, both in the rules governing the model and in the underlying structure of the model. Among these are a new definition of how to model proliferative and non-proliferative cells, an isotropic lattice, and an adaptive grid lattice.

Malignant brain tumors consist of a number of distinct subclonal populations. Each of these subpopulations may be characterized by its own behaviors and properties. These subpopulations arise from the constant genetic and epigenetic alteration of existing cells in the rapidly growing tumor. However, since each single cell mutation only leads to a small number of offspring initially, very few newly arisen subpopulations survive more than a short time. In a subsequent work [2], we quantified ``emergence,'' i.e., the likelihood of an isolated subpopulation surviving for an extended period of time. Only competition between clones was considered; there are no cooperative effects included. The probability that a subpopulation emerges under these conditions was found to be a sigmoidal function of the degree of change in cell division rates. This function has a non-zero value for mutations that confer no advantage in growth-rate, which represents the emergence of a distinct subpopulation with an advantage that has yet to be selected for, such as hypoxia tolerance or treatment resistance. A logarithmic dependence on the size of the mutated population was also observed. A significant probability of emergence was found for subpopulations with any growth advantage that comprise even 0.1% of the proliferative cells in a tumor. The impact of even two clonal populations within a tumor was shown to be sufficient such that a prognosis based on the assumption of a monoclonal tumor can be markedly inaccurate.

We also extended the cellular automaton to study the effects of treatment [3]. By varying three treatment parameters, we simulated tumors that display clinically plausible survival times. Much of our work is dedicated to heterogeneous tumors with both treatment-sensitive and treatment-resistant cells. First, we investigated two-strain systems in which resistant cells are initialized within predominantly sensitive tumors. We found that when resistant cells are not confined to a particular location, they competed more effectively with the sensitive population. Moreover, in this case, the fraction of resistant cells within the tumor was a less important indicator of patient prognosis. In additional simulations, we studied tumors that are initially monoclonal and treatment-sensitive, but that undergo resistance-mutations in response to treatment. Here, the tumors with both very frequent and very infrequent mutations developed with more spherical geometries. Tumors with intermediate mutational responses exhibited multi-lobed geometries, as mutant strains develop at localized points on the tumor's surfaces.


  1. A. R. Kansal, S. Torquato, G. R. Harsh, E. A. Chiocca, and T. S. Deisboeck, Simulated Brain Tumor Growth using a Three-Dimensional Cellular Automaton, Journal of Theoretical Biology, 203, 367 (2000).
  2. A. R. Kansal, S. Torquato, E. A. Chiocca, and T. S. Deisboeck, Emergence of a Subpopulation in a Computational Model of Tumor Growth,Journal of Theoretical Biology, 207, 431 (2000).
  3. J. E. Schmitz, A. R. Kansal, and S. Torquato, Cellular Automation Simulation of Brain Tumor Treatment and Resistance, Journal of Theoretical Medicine, 4, 223 (2002).


Modeling Heterogeneous Tumor Growth in Confined Microenvironments

heterogeneous tumor

Evoluation of a heterogeneous tumor in confined microenvironment.

An in silico tool that can be utilized in the clinic to predict neoplastic progression and propose individualized treatment strategies is the holy grail of computational tumor modeling. Building such a tool requires the development and successful integration of a number of biophysical and mathematical models. Recently, we have generalized the previous cellular automaton model to study the effects of vasculature evolution on early brain tumor growth and proliferative tumor growth in confined microenvironment.

With such progress in modeling individual biophysical mechanisms involved in proliferative tumor growth, we work toward the long-term goal of developing a clinical simulation tool by formulating a cellular automaton model of tumor growth that accounts for several different inter-tumor processes and host-tumor interactions. In particular, the algorithm couples the remodeling of the microvasculature with the evolution of the tumor mass and considers the impact that organ-imposed physical confinement and environmental heterogeneity have on tumor size and shape. Furthermore, the algorithm is able to account for cell-level heterogeneity, allowing us to explore the likelihood that different advantageous and deleterious mutations survive in the tumor cell population. This computational tool we have built has a number of applications in its current form in both predicting tumor growth and predicting response to treatment. Moreover, the latent power of our algorithm is that it also suggests other tumor-related processes that need to be accounted for and calls for the conduction of new experiments to validate the model’s predictions.


  1. J. L. Gevertz and S. Torquato, Modeling the Effects of Vasculature Evolution on Early Brain Tumor Growth. Journal of Theoretical Biology 243, 517 (2006).

  2. J. L. Gevertz, G. Gillies, and S. Torquato, Simulating Tumor Growth in Confined Heterogeneous Environments. Physical Biology 5, 036010 (2008).

  3. J. Gevertz and S. Torquato, Growing Heterogeneous Tumors in Silico. Physical Review E 80, 051910 (2009).


Review: Toward an Ising model of cancer and beyond

Ising Modeling

Left: An Ising model of ferromagnetism. Right: A portion of simulated tumor.

In a recent article, we review the research work that we have done toward the development of an ``Ising model'' of cancer. The Ising model is an idealized statistical-mechanical model of ferromagnetism that is based on simple local-interaction rules, but nonetheless leads to basic insights and features of real magnets, such as phase transitions with a critical point. The review begins with a description of a minimalist four-dimensional (three dimensions in space and one in time) cellular automaton (CA) model of cancer in which cells transition between states (proliferative, hypoxic and necrotic) according to simple local rules and their present states, which can viewed as a stripped-down Ising model of cancer. This model is applied to study the growth of glioblastoma multiforme, the most malignant of brain cancers. This is followed by a discussion of the extension of the model to study the effect on the tumor dynamics and geometry of a mutated subpopulation. A discussion of how tumor growth is affected by chemotherapeutic treatment, including induced resistance, is then described. We then describe how to incorporate angiogenesis as well as the heterogeneous and confined environment in which a tumor grows in the CA model. The characterization of the level of organization of the invasive network around a solid tumor using spanning trees is subsequently discussed. Then, we describe open problems and future promising avenues for future research, including the need to develop better molecular-based models that incorporate the true heterogeneous environment over wide range of length and time scales (via imaging data), cell motility, oncogenes, tumor suppressor genes and cell-cell communication. A discussion about the need to bring to bear the powerful machinery of the theory of heterogeneous media to better understand the behavior of cancer in its microenvironment is presented. Finally, we propose the possibility of using optimization techniques, which have been used profitably to understand physical phenomena, in order to devise therapeutic (chemotherapy/radiation) strategies and to understand tumorigenesis itself.


S. Torquato, Toward an Ising Model of Cancer and Beyond. Physical Biology 8, 015017 (2011).


Emergent Behaviors from A Cellular Automaton Model for Invasive Tumor Growth in Heterogeneous Microenvironments

Invasvie Tumors

Simulated Growth of Invasive Solid Tumors.

Understanding tumor invasion and metastasis is of crucial importance for both fundamental cancer research and clinical practice. \textit{In vitro} experiments have established that the invasive growth of malignant tumors is characterized by the dendritic invasive branches composed of chains of tumor cells emanating from the primary tumor mass. The preponderance of previous tumor simulations focused on non-invasive (or proliferative) growth. The formation of the invasive cell chains and their interactions with the primary tumor mass and host microenvironment are not well understood. Here, we present a novel cellular automaton (CA) model that enables one to efficiently simulate invasive tumor growth in a heterogeneous host microenvironment. By taking into account a variety of microscopic-scale tumor-host interactions, including the short-range mechanical interactions between tumor cells and tumor stroma, degradation of the extracellular matrix by the invasive cells and oxygen/nutrient gradient driven cell motions, our CA model predicts a rich spectrum of growth dynamics and emergent behaviors of invasive tumors. Besides robustly reproducing the salient features of dendritic invasive growth, such as least-resistance paths of cells and intrabranch homotype attraction, we also predict nontrivial coupling between the growth dynamics of the primary tumor mass and the invasive cells. In addition, we show that the properties of the host microenvironment can significantly affect tumor morphology and growth dynamics, emphasizing the importance of understanding the tumor-host interaction. The capability of our CA model suggests that sophisticated \textit{in silico} tools could eventually be utilized in clinical situations to predict neoplastic progression and propose individualized optimal treatment strategies.


Y. Jiao and S. Torquato, Emergent Behaviors from A Cellular Automaton Model for Invasive Tumor Growth in Heterogeneous Microenvironments. PLoS Computational Biology 7, e1002314 (2011).


Diversity of Dynamics and Morphologies of Invasive Solid Tumors


Simulated Growth of Solid Tumors in Confined Heterogeneous Environment.

Complex tumor-host interactions can significantly affect the growth dynamics and morphologies of progressing neoplasms. The growth of a confined solid tumor induces mechanical pressure and deformation of the surrounding microenvironment, which in turn influences tumor growth. In this paper, we generalize a recently developed cellular automaton model for invasive tumor growth in heterogeneous microenvironments [Y. Jiao and S. Torquato, PLoS Comput. Biol. 7, e1002314 (2011)] by incorporating the effects of pressure. Specifically, we explicitly model the pressure exerted on the growing tumor due to the deformation of the microenvironment and its effect on the local tumor-host interface instability. Both noninvasive-proliferative growth and invasive growth with individual cells that detach themselves from the primary tumor and migrate into the surrounding microenvironment are investigated. We find that while it noninvasive tumors growing in ``soft'' homogeneous microenvironments develop almost isotropic shapes, both high pressure and host heterogeneity can strongly enhance malignant behavior, leading to finger-like protrusions of the tumor surface. Moreover, we show that individual invasive cells of an invasive tumor degrade the local extracellular matrix at the tumor-host interface, which diminishes the fingering growth of the primary tumor. The implications of our results for cancer diagnosis, prognosis and therapy are discussed.


Y. Jiao and S. Torquato, Diversity of Dynamics and Morphologies of Invasive Solid Tumors. AIP Advances 2, 011003 (2012).


Spatial Organization and Correlations of Cell Nuclei in Brain Tumors

GBM Cells

Histoglogical images of cells and their assocaited point patterns.

Accepting the hypothesis that cancers are self-organizing, opportunistic systems, it is crucial to understand the collective behavior of cancer cells in their tumorous heterogeneous environment. In the present paper, we ask the following basic question: Is this self-organization of tumor evolution reflected in the manner in which malignant cells are spatially distributed in their heterogeneous environment? We employ a variety of nontrivial statistical microstructural descriptors that arise in the theory of heterogeneous media to characterize the spatial distributions of the nuclei of both benign brain white matter cells and brain glioma cells as obtained from histological images. These descriptors, which include the pair correlation function, structure factor and various nearest neighbor functions, quantify how pairs of cell nuclei are correlated in space in various ways. We map the centroids of the cell nuclei into point distributions to show that while commonly used local spatial statistics (e.g., cell areas and number of neighboring cells) cannot clearly distinguish spatial correlations in distributions of normal and abnormal cell nuclei, their salient structural features are captured very well by the aforementioned microstructural descriptors. We show that the tumorous cells pack more densely than normal cells and exhibit stronger effective repulsions between any pair of cells. Moreover, we demonstrate that brain gliomas are organized in a collective way rather than randomly on intermediate and large length scales. The existence of nontrivial spatial correlations between the abnormal cells strongly supports the view that cancer is not an unorganized collection of malignant cells but rather a complex emergent integrated system.


Y. Jiao, H. Berman, T-R. Kiehl and S. Torquato, Spatial Organization and Correlations of Cell Nuclei in Brain Tumors. PLoS One 6, e27323 (2011).

Questions concerning this work should be directed to Professor Torquato.

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