This special focus is jointly sponsored by the Center for Discrete Mathematics and Theoretical Computer Science (DIMACS), the Biological, Mathematical, and Physical Sciences Interfaces Institute for Quantitative Biology (BioMaPS), and the Rutgers Center for Molecular Biophysics and Biophysical Chemistry (MB Center). This meeting is also sponsored by The Center for the Development of a Virtual Tumor (CViT), and The National Cancer Institute's Integrated Cancer Biology Program.
Title: Molecular profiles of breast cancer progression
We develop a new robust technique to analyze micro array data which uses a combination of principal components analysis and consensus ensemble k-clustering to find robust clusters and gene markers in the data. We apply our method to a public microarray breast cancer dataset from Ma et al. (2003) which has expression levels of genes in normal samples as well as in three pathological stages of disease; namely, atypical ductal hyperplasia or ADH, ductal carcinoma in situ or DCIS and invasive ductal carcinoma or IDC. Our method averages over clustering techniques and data perturbation to find stable, robust clusters and gene markers. A major result of our analysis is that different sets of patients seem to progress to the same final phenotype along different functional pathways. Our findings are validated on external gene expression microarray datasets.
Title: Improved Breast Cancer Diagnosis and Prognosis by Computational Modeling and Image Analysis
In order to better understand the progression of breast cancer, four models of tumor progression pathways were evaluated by comparing the results of computer simulation and clinical observations. The models described the progression through the grades of ductal carcinoma in situ (DCIS) and though grades of invasive ductal carcinoma (IDC). The best transition rates through grades were sought for each model. Each of the four models could simulate, to various degrees of success, the observed frequencies in which grades of DCIS and grades of IDC co-occurred in biopsies from the same patient. Three of the models were based on the traditional assumption that DCIS is a precursor to IDC. However, the model that produced the best correspondence with observations did not assume that DCIS was a precursor to IDC, but instead described a parallel progression of grades of DCIS and grades of IDC diverging from a common progenitor. A prediction of this parallel progression model was tested by quantitative image analysis of human biopsy specimens. Thirty-nine nuclear image features were extracted from 200 nuclei of each of 80 DCIS patients whose clinical outcome was known. A discriminant function of image features was derived that was prognostic for patient survival. Therefore, computer modeling suggested a new parallel pathway of breast tumor progression that is consistent with clinical observations of biopsy specimens and of patient outcome. (Joint work with J.-A. Chapman, W.A. Christens-Barry, H.L. Lickley, N.A. Miller, J. Qian, L. Sontag, Y.Yuan. Supported by NJCCR 1076-CCR-S0 and NIH U56 CA 113004).
Title: Modeling the migration of glioma cells with a cellular automaton
Malignant glioma cells are able not only to proliferate but also to migrate towards healthy tissues. The Malignant glioma cells are able not only to proliferate but also to migrate towards healthy tissues. The capacity to invade surrounding tissues makes glioblastoma particularly dangerous, because at the moment of diagnosis the tumour has already spread over large distances. The mechanisms of cellular invasion is still poorly understood. Our ambition is to better understand the mechanisms governing the motility of glioma cells. To this aim we studied glioma cell migration in simple experimental conditions: migration in two dimensions over a substrate of collagen. In order to model migration, we developed a cellular automaton which describes migration of tumour cells over time and compared our simulations with the experimental results of the spreading of spheroids in vitro. The cellular automaton is based upon a hexagonal lattice. A given hexagon can be occupied by a single cell. A cell can migrate at each time step to one of six free neighbour hexagons, according to predefined rules. These rules are introduced in order to model two distinct dynamical effects: cell inertia and cell-cell attraction. The notion of inertia is associated to a persistence of the direction of cell motion. The cell-cell attraction can be due to some homotype attraction like chemotaxis (in which a soluble factor attracts cells) or to some physical contact like gap junctions which link two (or more) neighbouring cells. The strength of the attraction between cells can vary substantially, a stronger attraction leading to a reduced migration and thus to a limited spreading of glioma cells. With our model, we find that the best agreement between simulations and experimental results is obtained for a maximum cell-cell attraction while inertia plays only a minor role.
1 IMNC, UniversitŽe Paris VII-Paris XI, CNRS, UMR 8165, B?at. 104, 91406 Orsay, France 2UnitŽe INSERM 492 (IM3), 94010 CrŽeteil, France 3Plate-forme d'Imagerie Cellulaire et Tissulaire de l'IFR10, Institut Mondor de MŽedecine MolŽeculaire
Title: Simulating Tumor Growth Models using Particle Methods
We present a particle-based framework for the simulation of tumor growth based on a sharp-interface formulation of the evolution of the tumor boundary. The present Lagrangian level set formulation relies on the description of the interface using smooth particles which convect with the transport coefficients. The simulation of Cell proliferation and migration is straightforward in the present particle framework. We demonstrate the validity of the method in a simple linear nutrient-cell density reaction-diffusion model within the tumor boundary. We illustrate how this framework enables large-scale simulations of the dynamics of strongly deforming geometries.
Title: Multiscale modelling of vascular tumour growth
A hybrid cellular automata approach will be used to develop a multiscale model of vascular tumour growth in which subcellular, cellular and tissue scale phenomena are interlinked. Numerical results will then be presented to show that the manner in which the different physical processes are coupled can have a significant impact on the system's dynamic behaviour. Finally the therapeutic implications of these results will be discussed, with attention focussing on treatments that involve combinations of cytotoxic and anti-angiogenic drugs.
Title: Modeling carcinogenesis
Carcinogenesis is a complex multistep process that is often described as "somatic evolution." We have proposed that uccessful adaptation to varying microenvironmental proliferation constraints plays a crucial role during carcinogenesis. We have developed mathematical models to examine the dynamics of carcinogenesis using evolutionary game theory and a hybrid cellular automaton approach. This allows us to specifically investigate the hypothesis that regions of premalignant lesions in the later stages of carcinogenesis develop a substrate limited environment due to growth away from the blood vessels which remain separated from the tumor cells by an intact basement membrane The modified CA approach extends the evolutionary models of carcinogenesis to include spatial parameters that accommodate these boundary conditions. We find that selective forces in tumoral regions furthest from the blood supply act to favor cells whose metabolism is best-suited to respond to local changes in oxygen, glucose and pH levels. The model predicts three phases of somatic evolution in the substrate limited regions of premalignant lesions. Initially, regional cell survival and proliferation is limited due to diminished oxygen levels. This promotes adaptations that drive further cellular evolution that includes constitutive upregulation of glycolysis or resistance to acid-mediated toxicity. The models demonstrate the morphology of the premalignant lesions will initially vary depending on the sequence of adaptations. However, the final cellular phenotype has a significant general proliferative advantage because it will consistently alter the local environment (by acidifying it through constitutively increased glycolysis) in a way that is toxic to its competitors but harmless to itself. Eventually this allows the population to proliferate throughout the premalignant lesion and breech the basement membrane to form an invasive cancer. The evolutionary sequence and morphologic variations have now been observed experimentally in tumor spheroids and pathologic specimens of DCIS. This work illustrates the value of combining mathematical modeling with empirical investigations in understanding the complex, non-linear dynamics of carcinogenesis.
Title: Modeling the effects of vasculature evolution on early brain tumor growth
Mathematical modeling of both tumor growth and angiogenesis have been active areas of research for the past several decades. Such models can be classified into one of two categories: those that analyze the remodeling of the vasculature while ignoring changes in the tumor mass, and those that predict tumor expansion in the presence of a non-evolving vasculature. However, it is well-accepted that vasculature remodeling and tumor growth strongly depend on one another. For this reason, we have developed a two-dimensional hybrid cellular automaton model of early brain tumor growth that couples the remodeling of the microvasculature with the evolution of the tumor mass. A system of reaction-diffusion equations has been developed to track the concentration of VEGF, Ang-1, Ang-2, their receptors and their complexes in space and time. The properties of the vasculature and hence of each cell are determined by the relative concentrations of these key angiogenic factors. The model exhibits an angiogenic switch consistent with experimental observations on the upregulation of angiogenesis. Particularly, we show that if the pathways that produce and respond to VEGF and the angiopoietins are properly functioning, angiogenesis is initiated and a tumor can grow to a macroscopic size. However, if the VEGF pathway is inhibited, angiogenesis does not occur and tumor growth is thwarted beyond 1-2 mm in size. Furthermore, we show that tumor expansion can occur in well-vascularized environments even when angiogenesis is inhibited, suggesting that anti-angiogenic therapies may not be sufficient to eliminate a population of actively dividing malignant cells.
Title: Multiscale Analysis of Genetic Networks in Cancer Cells
Cellular phenotypes are determined by the dynamical activity of large networks of co-regulated genes. Thus dissecting the mechanisms of phenotypic selection requires elucidating the functions of the individual genes in the context of the networks in which they operate. In this talk I will describe ongoing efforts in Columbia University's MAGNet center to develop computational methods for the deconvolution of cellular networks in cancer cells. In the first part of the talk I will describe a novel information-theoretic algorithm, ARACNE, designed to detect direct genetic regulatory interactions using microarray data. I will present analysis of networks inferred from human B lymphocyte and T-cell lymphoblastic leukemia cells, compare ARACNE's performance to that of similar algorithms on a synthetic dataset, and present theoretical proofs of the reconstructed networks. I will then discuss recent extensions to the ARACNE framework that allow the identification of higher order dependencies between gene expression profiles, and demonstrate how this method can be useful in predicting modulators of transcriptional interactions. I will then describe a method designed to identify enriched DNA binding sites in a set of promoters by using data from orthologous genomes. Finally, I will present recent efforts to design a unified framework incorporating these diverse data sources.
Title: Discrete and continuous modeling of cell migration in the ECM and applications to tumor invasion
Cell migration plays an essential role during both embryonic development (e.g. gastrulation, neural crest migration) and in the normal physiological responses of the adult (e.g. immune response, wound healing). The extracellular matrix (ECM) plays a vital role in regulating movement by both providing a scaffold through which cells can generate traction and imparting specific migratory cues through ECM-bound proteins. The ECM also provides specific guidance to cells through preferential movement by the cells along the matrix fibres, a process known as contact guidance. The acquired ability of tumour cells to break free from the main mass and migrate into the surrounding ECM is a key stage in increased tumour malignancy.
Individual cell migration in the ECM can be classified into two main groups: amoeboid and mesenchymal. In the former, cells move quickly and have negligible effect on the structure of the surrounding ECM. Mesenchymal migration, however, is much slower and extensive matrix degradation takes place through the focussed expression of specific matrix degrading proteins by the cells (pericellular proteolysis).
In this talk, I will describe both discrete and continuous models for amoeboid and mesenchymal cell migration. Numerical investigations will be used to demonstrate a potential role of contact guidance and matrix degradation in directing the macroscopic organisation of cells and the matrix. I will consider applications in the context of models for tumour invasion.
Title: A cell-based model of the development of ductal carcinomas
I would like to present a computational technique that can be applied in 2- and 3-dimensional modelling of various multicellular phenomena, including the avascular tumor growth, the forma- tion of ductal carcinomas and the development of epithelial hollow acini. This model focuses on the biomechanical properties of individual cells and on communications between cells and their mi- croenvironement, but at the same time, it enables all cells to form one complex tissue. The model is based on the immersed boundary method and couples the continuous description of a viscous incompressible cytoplasm and the extracellular matrix, with the dynamics of separate elastic cells, containing their own elastic plasma membrane, uid cytoplasm, and individually regulated cell pro- cesses, such as cell growth, division, apoptosis and polarization. I will show how this technique can be applied in modelling different architectural patterns of ductal carcinomas and the development of normal epithelial tissue in the breast glands.
Title: Modeling chemotherapeutic dose response curves via cell cycle effects
The genetic instability of tumor cells renders them the ability to rapidly become resistant to many chemotherapies. As a result, there is great interest in selecting combination chemotherapeutic regimens that will overcome resistance and exert synergistic therapeutic activity. Traditional analysis of drug combination effects on cells is based on a number of assumptions and idealizations. A more rational mechanistic modeling approach may accelerate the search for effective drug combinations that are tailored to individual responses. The chemotherapeutic drugs, carmustine and etoposide, each nominally induces G2 phase arrest and, secondarily, apoptosis. Despite this similarity in mechanism, we found that the pharmacodynamic responses to these agents is dramatically different on human glioma cell lines. We have developed a cell cycle structured model of chemotherapeutic activity based on the dynamic transitions of cells along the phases of the cell cycle. We show that our mathematical model is able to explain the shapes of the dose response curves of multiple cell lines to these agents. We are able to predict with the model the effects of drug combinations, taking into account variable dose and timing regimens, to determine the most effective strategy. Using this model, we are able to explain several non-intuitive experimental observations involving combinations of chemotherapeutic drugs on glioma-derived cell lines.
Title: Simulated Morphogenesis of Papilloform Ductal Carcinoma
Morphogenesis of cells consists of a number of processes, all of which can be simulated in detail. We consider here results of a simulation in which simple cells are allowed to: (1) reproduce at fixed intervals, (2) migrate either randomly or in response to chemical gradients, (3) interact via cadherins or integrins linked to an extracellular matrix, and (4) differentiate according to prescribed rules. We find that a number of structures appear spontaneously, and we examine in detail the formation of papillae. We find that wavelengths of these structures depend linearly on the ratio of rates of production of progenitors to differentiated daughters, and that papillae grow only approximately exponentially in time, with different early- and late- growth rates. We propose that simulations of this kind may provide a tool for improving the understanding of relations between tumor morphogenesis and staging.
Title: Towards a collaborative formulation of the Mathematical Principles of Natural Philosophy: Living Matter. The paradigm of In Silico Oncology
The tremendous rate of accumulation of both experimental and observational (clinical) knowledge pertaining to living matter dictates the formulation of a parsimonious system of "laws" in analogy to Newton's Mathematical Principles of Natural Philosophy. This seems to be a necessary step if a rational, coherent and transparent understanding of the biological phenomena is to be sought. Such a system would consist of a finite, yet considerable number of principles and refer to all levels of biocomplexity, as according to D. Noble there is no privileged level of causality. The experimental, observational and theoretical study of cancer, a markedly mutiscale biological phenomenon of obvious clinical importance, may be viewed as an excellent ground for the establishment of a number of such multilevel laws. Their formulation might well be achieved in algorithmic - discrete mathematics terms.
Within this frame the emerging field of in silico (computational) oncology has already provided some quite reliable descriptions of several biological mechanisms characterizing cancer, of both continuous and discrete nature. Obviously cancer is far from being considered a purely deterministic phenomenon. Instead it seems to behave like a mixture of deterministic (e.g. sequence of cell cycle phases) and stochastic (e.g. radiation cell kill probability) processes. Subsequently, stochasticity aspects should always be taken into account. Nevertheless, the more critical knowledge becomes available, the more deterministic the cancer phenomenon appears to become. An illustrative example supporting this hypothesis is that more detailed knowledge of the genetic status of a tumor may lead to a better prediction of its response to therapeutic interventions, thus to an apparently more deterministic tumor behavior.
Based on the previous thoughts, the In Silico Oncology Group, National Technical University of Athens, has developed a number of essentially discrete Monte Carlo simulation models of tumor growth and response to therapeutic modalities ranging from tumor growth and radiotherapy response in vitro to the clinical tumor response to radiotherapeutic and chemotrherapeutic schemes in vivo, based i.a. on actual imaging data. Processed molecular data is used in order to perturb the radiobiological or pharmacodynamic cell kill parameters about their population based mean values. A prototype system of quantizing cell clusters included within each geometrical cell of a discretizing mesh covering the anatomic area of interest lies at the heart of the proposed simulation approach. Cell cycle phase durations and imaging based metabolism distribution define i.a. the quantization equivalence classes considered. Several algorithms have been developed so as to simulate i.a. various macroscopic mechanisms such as tumor expansion/shrinkage and mechanical boundary conditions as well as the effects of particular drugs (e.g. temozolomide) and radiation on the tumor under consideration.
A number of the models developed, mainly referring to imageable glioblastomas, have already been clinically validated to a substantial degree. Long term clinical testing and adaptation procedures are in process. The response of treatment affected normal tissues by radiotherapeutic schemes has also been addressed for certain cases. Currently, a substantial extension of the simulation models to the nephroblastoma (Wilm's tumor) and breast cancer cases is being performed within the frame of the European Commission funded project ACGT (Advancing Clinico-Genomic Trials on cancer), in collaboration with several clinical and technological institutions across Europe. The whole effort profits considerably from the US NIH-NCI supported Center for the Development of a Virtual Tumor (CViT). It is worth noting the remarkably collaborative character of this and other complementary research efforts on a global scale.
The expected practical usefulness of the type of models mentioned would be the possibility of virtually experimenting in silico (on the computer) with the intention of optimizing the cancer treatment strategy based on the specific molecular, histopathologic, imaging and historical data of each individual patient. Deeper understanding of the cancer disease and at the same time of the related natural phenomenon is a further intermediate goal of considerable importance.
Co-authors: Lan Ma(1), John Jeremy Rice(1), Wenwei Hu(2), Arnold Levine(2) and Gustavo Stolovitzky(1)
(1) IBM T.J. Watson Research Center, Yorktown Heights, NY (2) The Cancer Institute of New Jersey, Robert Wood Johnson School of Medicine, New Brunswick, NJ
Title: A mathematical model of the digital response of p53 to DNA damage in single cells
The tumor suppressor p53 is critical to ensure genomic stability. In single cells, the oscillatory p53 response to ionizing radiation (IR), which induces double stranded breaks (DSBs), is "digital," in that the number of oscillations rather than the amplitude shows dependence on the radiation dose. We present a model of single cell p53 dynamics in response to ionizing radiation. In our model, DSB sites interact with a limited pool of DNA repair proteins, forming DSB-protein complexes at DNA damage foci. Both the initial number of DSBs and the DNA repair process are modeled stochastically. The model assumes that the persisting complexes are sensed by ataxia telangiectasia mutated (ATM) kinase, which transduces in an ON/OFF manner the DNA damage signal to the downstream negative feedback oscillator consisting of p53 and its negative regulator Mdm2, a transcriptional target of p53. Our model exhibits coordinated oscillations of p53 and Mdm2 upon IR stimulation, with a stochastic number of oscillations whose mean increases with IR dosage, in agreement with the observed response of p53 to DNA-damage in single-cell experiments.